/
Text
R. W. HAMMING
Bell Telephone Laboratories
NUMERICAL METHODS
FOR SCIENTISTS
AND ENGINEERS
MC GRAW-HILL BOOK COMPANY, INC.
NEW YORK, SAN fRANCISCO, TORONTO, LONDON
1962
<I> H 3 H K 0-
MaTeMaTHQeCKaH
6H6JIHOreKa
l1HIKeHepa
P. B. XEMMI1Hr
4V1CJ1EHHbIE METOllbI
ARH HayqHWX pa60THHKOB H HH eHepOB
rIepeBO!{ C aHrJlMficKOro
B. n. APnA3APOBA, r.c. PA3HHO
iI A. B. YCKOBA
no.n pe.naKlI.l1ett
P. C. fYTEPA
H3Jl,A/iHE BTOPOE. HCnPABJIE/i/iOE
H3.l1.ATEJJbCTBO <HAYKA
f JJABHA5I PE.l1.AKUI-151
ct>1-13HKO-MA TEMA TW-lECK 0 nHTEP A TYPbl
MOCKBA 1972
518
X 37
Y 1J.K 518
'-IHCJleHHble MeTOJ].bl J].JlSI HaYIJHblX pa6oTHHKOB H HHmeHepoB.
X e M M H H r P. B.
KHHra nOCBHllJ,eHa 'lHCJleHHblM MeTO,IJ.aM MaTeMaTH'leCKOrO aHaJlH3a, I1CnOJlb-
3yeMblM Ha COBpeMeHHblx 3J1eKTpOHHbIX BbIqHCJII-ITeJlbHbIX MalllHHax. OHa COCTOHT
H3 '1eTblpex '1aCTeH.
4acTb I, j],l1cKpeTHOe I1CqHCJleHHe KOHe'lHblX pa3HocTeH (rJl. 1-6), 11311araeT
OCHOBHble rrOHHTHR KOHe'lHbIX pa3HocTetl, CYMMHpoBaHHH KOHeqHbIX qHCJlOBblX
PRJ],OB H KOHeqHblX pRJJ,OB <PYPb
4acTb II, rIpH6J1H)/{eHHe MHOrOqJleHaMH (rJl. 7-20), cOJ],ep)/{HT 113J10}1{eHf1e
KJlaCCH'IeCKHX '1HCJleHHbIX MeTOJ],OB HHTeprrOJlRu,HI1, '1HCJleHHOrO HHTerpHpOBaHI1R H
''lHCJleHHOrO pellleHHR J],H4J4JepeHu,HaJlbHbIX ypaBHeHHH, OCHOBaHHblX Ha anrrpoK-
'CHMau,HH 4JYHKU,HH 06bI'IHbIMH aJlre6pal1qeCKHMH MHOrOq. eHaMH. rIpH 3TOM pac-
-cMarpHBJIOTCR npH6J1!IJIJ:eH!IR B CMblCJle TOqHOrO COBnaJ],eHHR B Y3I1JX, B CMblClle
\HaI1MeHbWHX KBaJ],paToB H B CMblC. e HaHMeHbwero OTKJlOHeHHH no 4e6blweBY.
4acTb III, HeMHOrOqJleHHble npH6J1H)/{eHHH (rJl. 21-27), nOCBRllJ,eHa arrrrpoK-
cHMau,HH 4JYHKU,HH C nOMOllJ,blO 3KcrroHeHUHaJlbHblX, a TaKJIJ:e C nOMOllJ,blO pRJJ,OB
H HHTerpaJla <Pypbe.
4acTb IV, AllropHTMbI H 3BpHCTHqeCKHe MeTOJJ,bl (r JI. 28-32), KpoMe HeKOTO-
pblX H3BeCTHblX aJlfOpl1TMOB J],JlH OTblCKaHI1H KopHeH 4JYHKlII1H H J],JlH pR.1IJ 3aJ],a'l
JI!IHeHHOH aJlre6pbl, paccMaTpHB aeT npHMepbI MOJ],eJlHpOBaHHH, llpH leHeHI1R MeTOJJ,a
MOHTe-KapJlo 11 HeKoTopble HrpoBble 3aJ],a'lH. OTJJ,eJlbHaR 3aKJllO'll1TeJlbHaH rJlaBa
nOCBRllJ,eHa BonpocaM opraHH3au,HI1 Bbl'lHCJlHreJlbHOH pa60rbl.
TpeTbH H 'leTBepTaR qaCTH KH!IrH cOJ],epJIJ:ar pHJJ, HOBbIX 3a,IJ.a'l H MeTO,IJ.OB. 1-13J10-
}1{eHHe Bcex 'lH('JleHflbIX MeTO)lOB cOnpOBO)/{)laeTCH pa360poM npHMepoB H3 Bbl'lHC-
ilHTe. bHOH npaKTI1KH JBTopa.
Ta6J1HU 32, pl1CYHKOB 43, oH6J1HOrpa4JH'leCKHX CCblJlOK 44.
Bropoe H3)laHHe neqaTaerCR C MaTpliU npeAbI)J.yw.ero, IICnpaBJIeHbI JIlIlllb 3aMe-
qeHHble One'laTKH.
p, B, XeMMUHe
41lCJIEHHblE METO)J.bl
()\JUJ HaY'!HbIX pa6oTHHKOB H HHJKeHepoB)
M., 1972 r., 400 CTp. C HOlOl.
Pel\aI<Top r. 51. nUpoeo8a
TeXH. pel\aKTOp C. 51. LUKARP
KoppeKTopbl M. q" A AeKCee8a , M. JI. JIuneAuc
Ile'!aTb C MaTpHlI,. IloJ\rlHcaHo K ne'!aTH 5/1 1972 r. 5YMara 60-x90 1 /16' <PH3. ne'!. Ol. 25.
YCJIOBH. ne'!. JI. 25. Y'!.-H3J\. Ol. 23,02. THpaJK 27000 3K3.
UeHaKHHrH! p. 83 K. 3aKa3 N, 18.
113AaTeJIbCTBO ({HaYKa ).
fJIaBHaSI pe,n,aKIJ,H5I cPH3HKO-M3TeM3TH4ecKOlI JIHTepaTypbI.
1 I 7071, MocKBa, B-7 I, JIeHHHcKHII npocneKT, 15.
Op,!\eHa TpYl\oBOrO KpacHoro 3HaMeHH MocKoBcKa>r THnorpacjJH>r N, 7 «HcKpa peBOJIlOlI,HH»
rJIanTIOJIHrpacjmpoMa KOMHTeTa no ne'!aTH npH COBeTe MHHHcTpoB CCCP.
r. MocKBa, TpexnpY,!\HbJH nep., 9,
2-2-4
6 S-7 2
OrJIABJIEHHE
rIpe.lUICJlOBHe pe.u.aKTopa nepeBo.u.a .... . . . . . . . . . . . . . . . . . . . .. 12
113 npe.u.HCJlOBHR aBTopa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 14
'I A C Tb I
}I.HCKPETHOE HCtJHCJIEHHE KOHEtJHblX PA3HOCTEH
r JI a B a 1. HCQHCJJeHHe pa3HOCTefi. . . . . . . . . . . . . . . . . . . .
1.1. BBe.u.eHHe H CI1CTeMa 0603Ha<leHHrt. . .. . . . . . . . .
1.2. Pa3HocTHblH onepaTop . . . . . . . . . . . . . . . . . . . . .
1.3. rIoBTopHbIe pa3HocTH . . . . . . . . . . . . . . . . . . . . . .
1.4. Ta6J1Hl.lbl pa3HoCTeH. . . . . . . . . . . . . . . . . . . . . . .
1.5. <PaKTopHa,1bI . . . . . . . . . . . . . . . . . . . . . .
1.6. j],eJleHfle MHorO'MeHOB . . . . . . . . . . . . . . .
1.7. 411cJla CTHp.1HHra nepBoro porra . . . . . . . . . . . . . .
1.8. 4HcJla CmpJlHHra BToporo pOllJ . . . . . . . . . . . . . .
1.9. IlpHMep . . . . . . . . . . . . . . . .
1.10. AJlbTepHaTHBHble 3aMeqaHHH . .. .................
1.1 I. 06LU,He 3aMeqaHHR H cnpaBKH.. .................
r II a B a 2. norpewHocTH oKpyr JJeHHSI .....................
2.1. BBeJJ,eHHe . . . . . . . . . . . . . . . .
2.2. 06J1aCTb OTBeTa . . . . . . . . . . . . . .
2.3. ,D,BoHHaR TOqHOCTb . . . . . . . , . . . .
2.4. C<leT co 3HaQaLU,HMH pa3pHlIaMfj
2.5. CTaTHCTI1'1eCKHH nOllxo.u.. . . . . . .. . .
2.6. CJlY'IartHoe oKpyrJlcHHe. . . . . . . . . . . . . .
2.7. llepeMcHHaH TOqHOCTb . . . . . . . . . . . . . . . . . . . . .
2.8. Ol.\eHKa II1YMa B Ta6.1Hl.\e . . . . . . . . . . . . . . . . . . . . . .
2.9. TeopHR «MJlallwero 3Haq3LU,CrO pa3pHlIa» . . . . . . . . . . . . . .
2.10. TeopHH CTapwero 3Ha'laLU,ero pa3p>JJ\a» .. . . . . . . . . . . .
2.11. AHaJlH3 pacnpocTpaHeHHR OIIlH6KI1 UpH He60.1LllJOM BbIYJlCiIC-
HHH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.12. 06LU,He 3aMCQaHHR H 6H6nHorpaq)f1R . . . . . . . . . . . . . . . .
r II a B a 3. HCQHCJJelUle CYMM . . . . . . . . .. . . . . . . . . . . . . . . .
3.1. BBe.1leHHe H CI1CTeMa 0603HaQeHI1i'1 . . . . . . . . . . . . . . . . . .
3.2. <POpMy.1b1 CYMMHpoBaHHR . . . . . . . . . . . . . . . . . . . . . . .
3.3. CYMMHpoBaHHe no qaCTHM. . . . . . . . . . . . . . . . . . . . .
3.4. 06LU,He 3aMeqaHHR ........'..................
r JI a B a 4. BbllJHCJJeHHe 6eCKOHelJHbiX PSI.llOB. . . . . . . . . . . . . . . . .
4.1. BBclleHHe . . . . . . . . . . . . . . . . . . . . . . .
4.2. MeTo.ll. KYMMepa. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
17
19
21
23
27
29
32
34
35
36
37
37
37
38
39
39
40
41
41
41
47
49
52
53
53
53
56
58
59
59
59
61
6
OrJlABJJEHVlR
4.3. HeKoTopble cneUHaJlbHble CYMMbl ......
4.4. MeToJl 3i1J1epa. . . . . . . . . . . . . . . . . .
4.5. HeJlHHeHHOe npe06pa30BaHHe. . . . . . . . .
4.6. CTeneHHble pRJlbl ...............
4.7. Pa3J10JKeHHe no cneUHaJlbHblM 4JYHKU,HRM.
4.8. I-1HTerpaJlbl KaK npH6J1HJKeHHR CYMM .
4.9. JJ.HraMMa-4JYHKu,HJJ. . . . . . . . . . . . .
r II a B /I 5. YpaBHeHHH B KOHe'lHbIX pa3HOCTHX
5.1. CHcTeMa 0603Ha'leHHH . . . . . . . . . . . . . .
5.2. rIpHMep pa3HoCTHoro ypaBHeHHR nepBoro nopHllKa .
5.3. rIpHMep ypaBHeHHR BToporo nopRJJ,Ka. . . . . . . . .
5.4. JJHHer'lHble pa3HocTHble ypaBHeHI1H C nOCTORHflbIMH K034J4JH-
UHeHTaMI1 . . . . .
5.5. rIpHMep . . . . . .
r II a B a 6. KOHe'lHble pHJlbl ypbe
6.1. BBeJleHHe . . . . . . . . . .. . .
6.2. OpTOrOHaJlbHOCTb Ha JlHCKpeTHOM Ml!OJKeCTBe To'-/eK. . .
6.3. TO'-/HOCTb pa3J10JKeHHR . . . . . . . . . . . .
6.4. Bbl'lI1CJleHHe K034J4JHu,l1eHToB . . . .
6.5. MeToJl JlBeHa,IJ.uaTI1 OpJlHHaT. . . . .
6.6. MeToJlbl C MHHHMYMOM YMHOJKeHHH
6.7. Pa3J10)/{eHHe no KocHHycaM . .
6.8. JJoKaJlbHble pRJlbl <Pypbe. . . . . . . . . .
62
62
66
67
68
68
69
7\
71
72
74
75
76
78
78
79
81
83
85
87
87
88
LJACTb II
nPHSJUUKEHHE MHOrOlJJIEHAMH - KJIACCHlJECKHH
lJHCJIEHHblH AHAJIH3
r JI a B a 7. BBe.n.eHHe B MHOrO'lJleHHble npH6J1H>KeHHH . 90
7.1. OpHeHTaUHR . . . . . . . . . . . . . . . . . . 90
7.2. AJlbTepHaTHBHble 4Jop MY JlHpOBKH. . . . . . 92
7.3. Y3J10Bble TO'-/KH, HH<p0pMau,I1R. . . . . . . 95
7.4. KJlacc 4JYHKU,I1H . . . . . . . . . . . . . . . . . . . 96
7.5. Cor. acHe . . . . . . . . . . . . . . . . . . . . . . . 97
7.6. TO'-/HOCTb . . . . . . . . . . . . . . . . . . . . . . . 98
r II a B a 8. HHTepnOJlHII.HH MHOrO'lJleHaMH. ,Il,aHHble C np0l13BOJlbHblMH
npOMejj{YTKaMH . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 99
8.1. <P11J10C04JHR . . . . . . . . . . . . . . . . . . .. ... . . 99
8.2. I-1HTepnOJlJJUHOHHble MHOrOqJlellbl. . . . . . . . . . . . . . 99
8.3. MeToJl HIITepnOJlRUHH JJarpallJKa. . . . . . . . . . . . . . 103
8.4. V1l1TepnOJlRUHOHHaR 4JopMYJla HblOToHa . . . . . . . . . . 106
8.5. JJ.pyraR 4JopMa JlJlR Ta6J1Hu,bl pa3Jle. eIlHbIX pa3110cTeH. 109
8.6. rIorpelliHocTb MHOrO'lJleHHOH annpOKCHMau,HH . . . . . . 110
8.7. TpY,IJ.HOCTH npI16J1H)/{eHI1R MHOrO'lJleHOM. . . . . . . . . . 113
8.8. 0 Bb!60pe Y3J10BbIX TOqeK. . . . . . . . . . . . . . . . . . 116
r II a B a 9. HHTepnOJlHII.HH MHOrO'lJleHaMH. PaBHoOTCTOHllI.He Y3J1bl. 117
9.1. <PopMYJla HblOToHa JlJlR HHTepnOJlHpOBaHHH . . . . . . . 117
9.2. I-1HTepnOJlHpOBaHHe B Ta6J1Hu,ax ....... . . . . . . . . . . 118
9.3. POM60BH,IJ.HaR llHarpaMMa ..................... 119
9.4. 3aMeqaHHR K BblBeJleHHblM 4JopMYJlaM. . . . . . . . . . . . . . .. 123
9.5. CMellIaHHble HHTepllOJlRUHOHHble 4JoPMYJlbl . . . . . . . . . . . .. 124
OrJJABJJEHKE 7
r naB a 10. EAHHbllt MeToA HaxmKAeHHIl HHTepnOJIIlII.HOHHbIX 4>OPMYJI 125
10.1. BBe.l\eHl1e ................................ 125
10.2. HeCKOJIbKO THlIH'JHbIX 4JOPMYJI HHTerpl1pOBaHHJJ . . . . . . . .. 127
10.3. <PHKCl1pOBaHHbJe Y3J1bl. . . . . . . . . . . . . . . . . . . . . . . .. 132
10.4. HeKoTopble npHMepbl 4JOPMYJI. . . . . . . . . . . . . . . . . . .. 135
10.5. 3Ha'JeHHJJ 4JYHKU,1111 H npOH3BO.l\HOH B 4JHKcHpoBaHHblx TO'lKaX 137
10.6. CB060.l\Hble Y3J1bl; KBa.l\parypa raycca. . . . . . . . . . . . . .' 139
10.7. CMelllaHHblfi cJlY'Jafi . . . . . . . . . . . . . . . . . . . . . . . . .. 141
10.8. 3aMe'JaHHJJ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 142
10.9. JJHHefiHble orpaHH'JeHHJJ Ha Beca . . . . . . . . . . . . . . . . ., 144
10.10. <PopMYJla rperopl1 . . . . . . . . . . . . . . . . . . . . . . . . .. 147
10.11. BblBo.l\bl. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... 150
r JI a BaIl. 0 HaxmKAeHHH OCTaTO'lHOrO 'IJIeHa IIJOPMYJIbl. . . . . . .. 152
11.1. rIoTpe6HocTb B OCTaTO'JHOM 'JJleHe . . . . . . . . . . . . . . . ," 152
11.2. DOpJJ.I\OK OCTaTO'JHoro '1J1eHa. . . . . . . . . . . . . . . . . . . .. 152
11.3. <PYHKl\HJJ BJlI1JJHI1R . . . . . . . . . . . . . . . . . . . . . . . . . .. 153
11.4. CJlY'JaH, Kor.l\a a (s) I1MeeT nocToRHHblfi 3HaK . . . . . . . . .. 156
11.5. CJlY'Jafi, Kor.l\a 4JYHKU,HJJ BJlHJJHHJJ MeHReT 3HaK . . . . . . . ., 158
11.6. CJla60e MeCTO B MeTO.l\e PR.I\OB TefiJIopa . . . . . . . . . . .. 160
r JI a B a 12. «)OPMYJIbl AJI51 onpeAeJIeHHblX HHTerpaJIOB . . . . . . . . .. 161
12.1. BBe.l\eHHe .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 161
12.2. <POPMYJlbl HbIOToHa-KoTeca. . . . . . . . . . . . . . . . . . . " 164
12.3. VlCnOJlb30BaHHe 4JOPMYJlbl rperopl1. . . . . . . . . . . . . . . .. 166
12.4. OTKpblTble 4JOPMYJlbl . . . . . . . . . . . . . . . . . . . . . . . .. 168
12.5. KBa.l\paTypa raycca . . . . . . . . . . . . . . . . . . . . . . . . .. 169
12.6. <POPMY Jlbl HHTerpHpOBaHHR CMelllaHHoro rayccoBoro THna . .. 170
12.7. CYMMHpOBaHl1e PH.I\OB. . . . . . . . . . . . . . . . . . . . . . . .. 171
12.8. 34J4JeKTbl 3aMeHbl nepeMeHHort .. . . . . . . . . . . . . . . . .. 172
12.9. I-1HTerpaJlbl C napaMeTpoM . . . . . . . . . . . . . . . . . . . . .. 173
r JI a B a 13. HeOnpeAeJIeHHble HHTerpaJIbl . . . . . . . . . . . . . . . . . .. 173
13.1. Onl1CaHHe cOJJ,ep)/{aHHR rJlaBbl 11 CHCTeMa 0603Ha'JeHHfi .... 173
13.2. HeCKOJlbKO npOCTbIX 4JOPMYJl .l\JlJJ Heonpe.n:eJleHHblX I1HTerpa-
JlOB . _ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 175
13.3. 06111I1H MeTO.l\ ............................. 177
13.4. 0lllH6Ka BCJle.l\CTBl1e oT6paCbIBaHHJJ 'JJleHOB ........... 178
13.5. YCTOH'IHBOCTb. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 181
13.6. WYM oKPyrJleHI1R . . . . . . . . . . . . . . . . . . . . . . . . . .. 184
13.7. I-1TOrl1 .................................. 186
13.8. HeKoTopble 06111lfe 3aMe'laHI1H. . . . . . . . . . . . . . . . . . .. 187
13.9. 3KcnepHMeHTaJlbHaH npoBepKa YCTOrt'JlfBOCTH . . . . . . . . .. [89
13.10. rIpHMep HHTerpaJla cBepTKH, HJlJlIOCTPHpYlOlllHH HJJ,elO YCTOH-
'1HBOCTIf. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. l89
r JI a B a 14. BBeAeHne B JJ:n4>lIJepeHII.HaJIbHbIe ypaBHeHHIl ........ 191
14.1. rIpl1po.l\a H CMbICJl JJ,H4J4JepeHl\HaJlbHhIX ypaBHeHHii . . . . . .. 191
14.2. DOJle HanpaBJleHI1H .......................... 192
14.3. 4HCJleHHOe pellleHHe. . . . . . . . . . . . . . . . . . . . . . . . .. \93
14.4. rIpHMep . . . . . . . . . . . . . . . . . . . . . . . . . . . \95
14.5. Y CTOH'! HBOCTb MeTO.n:a npocToro npofHo3a . . . . . . . . . . ., 197
14.6. YCTOH'!HBOCTb KOppeK[U1H . . . . . . . . . . . . . . . . . . . . .. 198
14.7. HeCKOJlbKO 06111HX 3aMe'laHI1H. . . . . . . . . . . . . . . . . . .. 200
14.8. CHCTeMbJ ypaBHeHHii. ......................... 201
8
OfJJABJTEHI1E
r JI a B a 15. OomaSl TeopHSI MeTOll,OB nporHo3a H KoppeKuHH . . . . . .
15.1. BBeJl,eHHe ................................
15.2. OUII16Ka OT oT6pacblBaHHR '1J1eHOB . . . . . . . . . . . . . . . . .
15.3. YCTOH'IHBOCTb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.4. rIoMexH o\{pyrJleHHR. . . . . . . . . . . . . . . .
15.5. rIpOfH03 /]0 TpeM TOqKaM . . . . . . . . . . . .
15.6. npOfHO:Jbl THna MHJ/Ha . . . . . . . . . . . . . .
15.7. rIporH03bI THrra AlI,aMCa-5alll4JopTa.
15.8. 06ll1.He 3aMeqaHHR a BbI60pe MeTOJ],a . . . . .
15.9. BbI60p rrpofH03a . . . . . . . . . . . . . . . . . . . . . . . . . .
15.10: HeKoTopble 4JOPMYJlbI . . . . . . . . . . . . . . . . . . . . .
15.11. BbI60p Illara H oueHKa TO'lHOCTH . . . . . . . . . . . . . . . . .
15.12. 3KcrrepHMeHTaJlbHaR rrpoBepKa. . . . . . . . . . . . . . . . . . .
r JI a B a 16. CneUHaJlbHble MeTOll,bl HHTerpHpoBaHHSI OObll{HOBeHHblx
lI,H<!><!>epeHUHaJlbHblX ypaBHeHHii . . . . . . . .
16.1. BBeJJ.eHHe H 06ll1.ee onHcaHHe . . . . . . . . . . . . . . . . . . . .
. 16.2. MeTOll,bI PYHre-Kyna . . . . . . . . . . . . . . . . . . . . . . . .
16.3. MeToJJ,bI JJ.JlR ypaBHeHI1R BToporo rropRJJ.Ka, Kor.n:a OTCYTCT-
ByeT y'. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.4. JJHHeHHblC ypaBHeHI1H . . . . . . . . . . . . . . . . . . . .
16.5. MeToJl" KOTOpbIH HCllo.1b3yeT 3Ha'leHHR Y, y' H Y" . . . . . . .
16.6. CJlyqai'r, KorJJ.a peWeHl1e TPYJIHC arrrrpOKCHMl1pOBaTb 7II0ro-
'1J1eHOM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.7. KpaeBbIe 3a,IJ.aqH . . . . . . . . . . . . . . . . . . . . . . . . . . . .
r JI a B a 17. MeToA HaHMeHbUlHX KBall,paroB. TeopHSI .,. . . . . . . . .
17.1. BBeLleHHe ....... '. . . . . . . . . . . . . . . . . . . . . . . . .
17.2. MeToJJ, HaHMeHbllll1X KBaJJ,p')ToB . . . . . . . . . . . . .
17.3. j],PYfHe KpHTep1111 . . . . . . . . . . . . . . . . . . . . .
17.4. OWH6KH C HOpMaJlbHblM pacnpcLleJleHHeM. .
17.5. rIpoBeJIeHl1e rrOJl,xoJJ,Hll1.ero MHOrO'lJlefla . . .
17.6. OpTOrOHaJlbHble 4JYHKUHII . . . . . . . . . . . . . . . .
17.7. OOll1.l1e CBoHcTBa opTOrOHaJlbHblX <jJYHKlU111 .....
17.8. HepaBeHcTBo 5ecceJlII H rrOJIHOTa. . . . . . . . . . . .
17.9. MeTo,IJ. HaHMeHbllIHX KBaJ],paToB H K034J4JHllHeHTbl <Pypbe
17.10. OpTOrOHaJlbHble MHOrOqJleHbI. . . . . . . . . . . . . . . . .
17.11. KJlaCCHqeCKHe opTOrOHaJlbHbIe MH rO'lJleHbI . . . . . . . .
17.12. CpaBHeHl1e MeTOJI3 HaHMeHbllIliX KB3JIpaTOB H pa3.JollieHHR B
crerreHHble pHJJ.bI . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.13. MeToJJ, HaHMeHbllIHX KBaJJ.paToB C OrpaH!1'1eHHH IH; rrpOJlo.1-
)/{eHl1e npHMepa 113 1.9.. . . . . . . . . . . . . . . . . . . . . .
17.14. rIoCJleJJ,Hl1e 3aMe'laHI1H 0 MeTOJIe lIaHMCHblllHX KBaJIp3TOB . .
r,JI a B a 18. MeTo,ll; HaHMeHbUlHX KBa,ll;paToB. npaKTHKa. . . . . . . . . .
18.1. 06ll1.He 3aMe'laHHH 0 MHOrOqJleHHOM cJlY'lae . . . .
18.2. Tpexq.1eHHoe pCKyppeHTHoe COOTHOJlleHHe . . . . . . . . . . . .
18.3. rIoCTpoeHHe KB33HopTOrOHaJlbHblX MHOrOQ.1eHOB . .
18.4. HeMHOrOQlleHHblH cJlY'laH. . . . . . . . .
18.5. HeJlHHeHHblc napaMerpbl . . . . . . . . . . .
r JI a B a 19. MHOrOQJIeHbl qeoblUleBa . . . . . . . . . .
19.1. BBeJIeHHe ....................
19.2. HeKoTopble TOlliAeCTBa . . . . . . . . . . . .
19.3. KpHTepHH 4e6bll1IeBa . . . . . . . . . . . . .
19.4. 3KoHoMH3aUHR . . . . . . . . . . . . . . . . . .
202
202
204
205
209
209
210
212
213
214
215
216
219
220
220
221
222
224
225
226
229
232
232
232
234
234
237
240
242
244
245
247
249
250
251
252
252
252
253
255
255
256
257
257
259
260
262
OfJJABJlEHf1E 9
19.5. MeXaHH:3au,HR "pou,ecca 9KOHOMH3all,HI1 . . . . . . . .. 263
19.6. CMelll..eHHble MHOrO'lJlCHbl 4e6wJJCBa. . . . . . . . . . . . . .' 265
! 9.7. 't-npoll,ccc JJaHu,ollla . . . . . . . . . . . . . . . . . . . . . . . . . . 266
19.8. BH,noH3MeHcHHc 't-MeTOJJ:a. . . . . . . . . . . . . . . . . . . . . . . 268
19.9. HecKo. bKO 3aMe'laHHH 0 '1C6b1l1IeBCKOM npH6J1J-f}KCH1H1. . ., 270
19.10. KpHTepHH COBna,IJ.CHHR MOMeHTOB . . . . . . . . . . . . . . .. 270
r JI a B a 20. Pal(HOHaJIbHbIe <PYHKl(HH . . . . . . . . . . . . . . . . . . . ., 272
20.1. BBeJJ,eHHe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 272
20.2. HcnocpcJJ:cTBeHHblH nOJJ:xoJJ:. . . . . . . . . . . . . . . . . . . ., 273
20.3. 4e6blllleBcKoe npH6J1H)/{eHnC pau,HoHaJlbHblMH 4JYHKlI.I1HMH.. 274
20.4. 06paTHble pa3HOCTH (CHMMCTpH'IHble) . . . . . . . . . . . . .. 275
20.5. rIpl1Mep ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
ACTb III
HEMHOrOtJJIEHHbIE nPHEJIH>KEHHSI
r JI a B a 21. nepHOAHlJeCKHe <PYHKl(HH. AnnpOKCHMal(IUI <t>YPbe. 280
21.1. UeJib 9TOH TCOpHH. . . . . . . . . . . . . . . . . . . . . . . . .. . 280
21.2. 3aMeHa ncpeMeHHblx 11 Bbl60p Y3J10B. . . . . . . . . . . . 281
21.3. PHJJ:bl <PYPbC; nepHo,nH'IeCKHC HBJleHI1R .. . . . . . . . . . . . . 282
21.4. VlHTCpnOJlHlUlH ncpl1o,nI1IJCCKI1X ljJYHKUHH . . . . . . . . . . . ., 285
21.5. \ilHTerpHpoBaHHc . . . . . . . . . . . . . . . . . . . . . . . . 288
21.6. MeTo,n 06ll1..ero onepaTopa . . . . . . . . . . . . . . . . . 290
21.7. HeCKOJlhKO 3aMC'IaHHH OTHOCHTCJlbHO 06ll1..cro MCTOlla . . .. 293
r Jl a B a 22. CXO,lJ,HMOCTb pH,lJ,OB cl>ypbe. . . . . . . . . . . . . . . . . . .. 294
22.1. CXOJJ,HMOCTb CTeneHHblX PH,IJ.OB H PR,nOB <Pypbe . . . . . . .. 294
22.2. <PYHKU,HI1 C npoCTbIM pa3pblBoM. . . . . . . . . . . . . . . . 295
22.3. <PYHKU,HR, IIMelOllI..aR HenpepblBHblC npOH3BO,IJ.Hble 60Jlee Bbl-
COKoro nOpHLJ,Ka .......... . . . . _ . . . . . . _ . . . 297
22.4. Y JlY'IWCHHe CXO,lJ,HMOCTH pRLJ,a <PypbC . . . . . . . . . . . . 298
22.5. C"enp MOlll..HOCTH . . . . . . . . . . . . . . . . . . . ::>99
22.6. 5!B.1eHHc rH66ca . . . . . . . . . . . . . . . .. . . . . . . . ;-;00
22.7. CHfMa-MHO)/{I1TCJlH JJaHllowa. . . . . . . . . . . . . . . . . . . 301
22.8. CpaBHeHHe MeTOJJ:OB CXOJJ:HMOCTlI . . . . . . . . . . . . . . . . 303
22.9. TexHHKa JJ:H4J4JepeHu,HpoBaHHR no JIaHuolllY . . . . . . . . . :304
r JI a B a 23. HenepHO,lJ,H'IeCKHe ljJYHKl(HH. HHTerpaJi cl>YPbe ... . . . . 305
23.1. UeJib r JI aBbl. . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
23.2. 0603Ha'lCHHR H KpaTKoe H3J10)/{CHHe pe3YJlhTaTOB ....... 306
23.3. I-1HTerpan <PYPbC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3W
23.4. rIpe06pa30BaHHe <PYPhe HeKoTopblX ljJYHKu,nM. . . . . . . . ., 311
23.5. <PYHKU,IUI C OrpaHI1'1eHHblM cneKTpoM II TeopeMa Bw60pKH . . 313
23,6. TeopeMa cBepTKH ........................... 315
23.7. 34J4JeKT KOHe'lHOfO CYMMHpOBaHI1H. . . . . . . . . . . . . . . . . 316
r Jl a B a 24. JIHHefiHble !J1I1JIbTpbl. CrJIa»CHBaHHe H AH<p<pepeHl(HpOBa-
HHe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
24.1. BBe)J.eHHc . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
24.2. rIpHMep npocToro CfJlamHBalOllI..CrO 4JHJlbTpa. . . . . . 318
24.3. npHMep nOCTpOeHI1R 4JHJlbTpa . . . . . . . . . . . . . . . :H9
24.4. <t>HJlbTpbJ Bo06ll1..c . . . . . . . . . . . . . . . . . . . . . . . . . . . ;{20
24.5. AHa.u13 npOCTblX 4JOPMYJI ll.TH lll14JljJcpCnu,HpoBaHHR . . . . . . 321
24.6. KaK H36c)/{aTb Bbl'll1CJlCHHii np01:l3BO,D,HblX? . . . . . . . . . . .. :322
10
OfJJABJJEHI1E
S 24.7. MeToll <P11J10Ha . . . . . . . . . . . . . . . . . . . . . . . . .. 323
S 24.8. 3aKJllO'IHTeJlbHble 3aMe'laHHJJ. . . . . . . . , . . . . . . . . . 325
r naB a 25. IfHTerpaJIbl H .lI.HlfJlfJepeHII.HaJIbHble ypaBHeHHSI ....... 326
25.1. COllepJKaHHe rJlaBbl . . . . . . . . . . . . . . . . . . . . . . . . . . 326
S 25.2. MeTOiJ. rrepeiJ.aTO'lHOH 4JYHKU,HH ,IJ.JlJI HHTerpHpoBaHHH. . . . . . 327
S 25.3. 06Il\He 4JOPMYJlbl HHTerpHpOBaHHJI . . . . . . . . . . . . . . . 331
S 25.4. j],H4JcpepeHu,HaJlbHble ypaBHeHHR. . . . . . . . . . . . . . . . . . . 332
25.5. rIocrpoeHHe 4JHJlbTpOB rro MeTOiJ.Y 4e6blIlleBa . . . . . . . . . . 334
S 25.6. HeKoTopbIe lleTaJlH MeTOiJ.a 4e6blIlleBa . . . . . . . . . . . . . . 336
r naB a 26. 3KCrrOHeHII.HaJIbHaSi arrrrpOKCHMaII.HSI .. . . . . . . . . . . . . 340
S 26.1. BBeJJ,eHHe ................................ 340
S 26.2. 0 HaXOJKiJ.eHHH 4JOPMYJl, HcnOJlb3YlOIl\HX 9KCrrOHeHTbI, KOriJ.a
rrOKa3aTe.1H 9KcnOHeHT 113BeCTHbl . . . . . . . . . . . . . . . . . . 340
S 26.3. HeH3BecTHble rrOKa3aTeJlH. . . . . . . . . . . . . . . . . . . . . . . 342
S 26.4. npeiJ.yrrpeJKJJ,eHHR ........................... 343
S 26.5. 3KcrroHcHTbl H MHOrO'lJleHbl . . . . . . . . . . . . . . . . . . . . . 344
S 26.6. OCTaTO'lHble 'IJleHbl .............. . . . . . . . . . . . . 344
r naB a 27. OCo6eHHOCTH................. . . . . . . . . . . . . . 344
S 27.1. BBeiJ.eHl1e ....................... . . . . . . . . . 344
27.2. rIpHMep HHTerpaJla C oc06eHHocTbl0 B 6eCKOHe'lHOCTH . . .. . 345
27.3. Oc06eHHocTb B JlI1HeHHOM iJ.H4J4JepeHUHaJlbHOM ypaBHeHl1H . . 346
27.4. 06Il\He 3aMe'laHHJI. . . . . . . . . . . ; . . . . . . . . . . . . . .. 349
tj ACT b IV
AJ1rOPIfTMbI H 3BPHCTHQECKHE METO.IT.bI
r II a B a 28. Haxo}J{)I.eHHe HYJIeR . . . . . . . . . . . . . . . . . . . . . . . . . 350
28.1. AJirOpHTMbI H 9BpHCTl1'1eCKHe MeTO,IJ.bl. . . . . . , . . 350
28.2. MeToiJ. lleJleHl1R rrOnOJIaM iJ.JlJI HaXOJK)leHHR KOpHR 4JYHKUHIi.. 351
28.3. JJHHeHHaR HHTepnOJlJIu,l1JI . . . . . . . . . . . . . . . . . . . . . . . 352
28.4. rIapa60JlH'IeCKaR HHTeprrOJIIIUHR. . . . . . . . . . . . . . . . . . . 352
28.5. HeKoTopble 06Il\He 3aMe'laHHR. . . . . . . . . . . . . . . . . . . . 353
28.6. MeTa.!! EepCToy .!!IIJI HaXO)/{.!!cHH.R KOMrrJlCKCHbJX KopHeH MHO-
rO'lJleHa ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
r II a B a 29. CHcTeMbI JIHHeRHblx aJIre6paH ' lecKHx ypaBHeHHR. . . . . . 359
29.1. BBe,IJ.eHHe ................................ 359
29.2. MeTo,IJ. l1CKJllO'ICHHR faycca .. . . . . . . . . . . . . . . . . . . . 360
29.3. BapHaHTbl MeTOlla faycca . . . . . . . . . . . . . . . . . . . . . . 362
S 29.4. MeToJJ, faycca-3aHiJ.e.1R . . . . . . . . . . . . . . . . . . . . . . . 363
29.5. nOBblIlleHHaH TO'lHOCTb . . . . . . . . . . . . . . . . . . . . . . . . 364
29.6. 06Il\He 3aMe'laHHR. . . . . . . . . . . . . . . . . . . . . . . . . . . 364
r naB a 30. 06paUl,eHHe MaTpHII. H co6CTBeHHbie 3HaQeHHSI .... . . . 365
S 30.1. BBe,IJ.eHl1e ................................ 365
S 30.2. 06paIl\eHHe MaTpHu,b! MeTO,IJ.OM liCKJllO'IeHHR no fayccy . . .. 365
30.3. 3aiJ.a'la HaXO)/{iJ.eHliR c06cTBeHHblx 3HaQCHHH . . . . . . . . . . . 366
30.4. HaHMeHbwHe c06cTBeHHble 3HaQeHHR. . . . . . . 368
30.5. HCCKOJlbKO 3aMeQaHHH ....... . . . . . . . . 368
r naB a 31. HeKoTopble rrpHMepbl MO.ll.eJIHpOBaHHSI . . . . 369
31.1. BBe,IJ.CHHC ...... . . . . . . . . . . . . . . . . . . . . . . . . 369
31.2. IlpocTort rrpl1MCp JJ.HcKpeTHOrO MO.ll.eJlHpOBaHHH . . . . . . . . . 370
OrJlABJlEHVlE 11
31.3. rIpHMep MOJIeJlHpOBaHHH CKJla.QCKHX onepauHt!. . . . . . . . . . 374
31.4. TpexMepHble KpecTHKH - HOJlHKH . . . . . . . . . . . . . . . . . . 375
31.5. 06llJ,He 3aMe'laHI1H 0 .QHcKpeTHoM MO.l1eJlHpOBaHHH . . . . . . . 379
31.6. HenpepblBHoe MO.QeJlHpOBaHHe. . . . . . . . . . . . . . . . . . . . 380
r naB a 32. CJlY'IafiHbIe 'IHCJla H MeTOJl,bI MOHTe-l\apJlo. . . . . . . . . . 381
32.1. rIoHiITHe CJlY'IaHHOrO 'II1CiIa . . . . . . . . . . . . . . . . . . . . . 381
32.2. reHepHpoBaHHe CJlYtJaHHblx 'IHCeJl B MaIllHHe, pa60TalOllJ,eH B
,QBOH1JHOH CHCTeMe. . . . . . . . . . . . . . . . . . . . . . . . . . . 382
32.3. reHepHpoBaHHe CJlY'IaHHblX 'Il1ceJl Ha .QecHTH'IHOH MaIllI1He . . 386
32.4. j],PYfHe pacnpe.QeJleHHH . . . . . . . . . . . . . . . . . . . . . . . . 386
32.5. MeTO,Q MOHTe-Kap,10 . . . . . . . . . . . . . . . . . . . . . . . . . 388
32.6. EllJ,e O,QHa HJlJllOCTpau,HR MeTO,Qa MOHTe-[{apJlo .... . . . . . 389
32.7. MeTo,Q )/{YJlHKa . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 390
r naB a N+ I. HCKYCCTBO BbI'IHCJlIITb P.JlII HHmeHepOB H Y'IeHbIX. .. 391
N t 1.1. BalliHocTb Borrpoca . . . . . . . . . . . . . . . . . . . . . . . . 391
N 1.2. 4TO Mbl c06HpaeMcH .QeJlaTb C OTBeToM? . . . . . . . . . .. 392
N 1.3. 4TO Mbl 3HaeM? . . . . . . . . . . . . . . . . . . . . . . . . . . 393
N 1.4. 06,IJ.YMbIBaHHe Bbl'lHCJleHHH . . . . . . . . . . . . . . . . . . . 394
N t 1.5. f1oBTopeHHe npe,IJ.blJJ,YllJ,HX IIlarOB . . . . . . . . . . . . . .. 395
N 1.6. Ou,eHKa YCHJlHH, He06xo,IJ.HMb1X ,QJlH pellleHHH 3a,IJ.a'lH . . . 395
N 1.7. 113MeHeHHR nepBOHa'laJlbHOrO nJlaHa. . . . . . . . . . . . . . 396
N + 1.8. <PHJlOC04JHR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
N + 1.9. 3aKJIlO'IHTeJIbHble 3aMe'laHI1JJ . . . . . . . . . . . . . . . . . . 398
J1HTepaTypa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 399
nPEilHcnOBHE PEilAKTOPA nEPEBOilA
JilM5I P. B. XeMMnHra - H3BeCTHOrO aMepHKaHCKOrO YtJeHOrO, 6h1Blllero
npe3H.n:eHTa aCCOll.Hall.HH no BbJ411CJIHTeJIbHbIM MaIllI1HaM, pyKoBo.n:HreJJ5J
MareMaTl1tJeCIwtl CJIY)j{6bl «Bell Telephone Laboratories» - Hero pa60Tbl
B 06JJacT\J BbJtlHCJJHTeJJbHot! MaTeMaTHKH H TeopHH HHcpOpM3UHH .lI.OCT3-
rOtJHO XOp0ll10 H3BeCTHbl H He HY)j{.n:alOrCH B OC06bIX peKOMeH.n:aI1HHX.
Tpy.n:HO, o.n:HaKO, y.n:ep)j{aTbCH OT I1CnOJIb30B3HI1H npe.ll.OCTaBHBllletJCH B03-
MO)j{HocrH peKOMeH.lI.OBaTb tJHT3TeJIIO 3aMetJ3reJIbHYlO KHHry.
J{HHra «4HCJJeHHbIe MeTo.n:bI .n:JIH H3YtJHbIX pa60rHHKOB 11 HH)j{eHepoB»
6eccnopHO HBJJ5IerCH Bbl.n:alOmHMC5I 5!BJJeHl1eM B MaTeMaTHtJeCKoti JIHTepa-
Type. OHa Y.ll.HBHTeJIbHblM 06p330M COlJeT3 T ll1Hpory OXBaTblBaeMoro
MaTepHaJJa, rJJy6HHY no.n:xo.n:a K HeMY H npaKTHIJHOCTb B JIY4weM CMbJCJIe
9Toro CJIOBa, Hl1r.n:e He nepexo.n:muylO B Y3KHtI npaKTHUH3M.
Cpe.n:H Y)j{e .n:OBOJIbHO MHOrCIJHCJIeHHblX KHHr no Bbl4HCJIHTeJIbHotJ M3-
TeMaTlIKe KHHra P. B. XeMMI1Hr3 Bbl.QeJISlerCH 11 no co.n:ep)!{aHI1IO H no
cpopMe.
Dpe)j{.n:e Bcero, B HelJ Hall1JJH IllHpOKOe 11 nOJIHOe OTpa)j{eHHe H.n:eli
D. J1. 4e6blweBa. He TOJJbKO B 3apy6e)j{Hotl, HO H B Hall1elJ PYCCKOtJ JII1-
TepaType MHorotJHCJIeHHble acneKTbl 4e6blweBcKHx H.lI.etl H MerO.QOB 4HC-
JIeHHOrO aHaJIH3a He nOJJY4aJIH eme CTOJIb nOJIHOrO H llIHpOKoro OCBe-
meHI1S1. .ilpyrotl oc06eHHocrblO KHHrH, OTHOCHmetJCH K ee cO.Qep)j{aHI1IO,
SlBJI5IeTCH 60JIbWOe BHHMaHHe, y.n:eJIHeMOe pa31IH4HOro po.n:a HeMHOrOIJ. eH-
HblM npH6JI1J)j{eHHSlM. B KHHre LlOCTaT04HO no.n:p06HO paCCMOTpeHa an-
npOKCHMaUHH cpYHKUHH paUI1OHaJJbHblMI1 H 9KcnOHeHUl1aJJbHbIMH, a TaK)j{e
cpYHKUI1HMH C orpaHHlJeHHblM cneKTpOM. DOCJIe.n:Hee oc06eHHo HHTepeCHO
H HMeeT 60JIhll10e 3HaIJeHHe .lI.JIII npaKTHKll npllMeHeHHH IJIlCJIeHHbIX MeTOLlOB.
YKa3aHHbIe oc06eHHOCTil cO.ll.ep}KaHI1H JJerKO 06"bHCH5!IOTCH 3aMerHblM
OTnelJaTKOM, H3JIO)j{eHHbIM H3 Hero JIHIJHblMH HaYIJHbIMH HHTepeCaMI1 aB-
Topa, II 5!BJIHIOTCH CJJe.QCTBl1eM e.QllHoro II HOBorO no.n:xo.n:a K BbI4HCJJH-
TeJJbHoti MaTeMaTIlKe - C T04KH 3peHllSl TeopHH HHcpopMaUHI1. Eha T04K3
3peHH5! npOBO.ll.HTC5I CHCTeManJtleCKH, H ee npeHMymeCTBa 6y.n:YT JJerKO
3aMeIJeHbl IJIlTaTeJJHMH.
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k=l
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KOHelJHblX p5lJJ:ax, lJHCJIO 3HalJeHI1ii cjJYHKlI.HH KOHelJHO. IT03TOMY BOnpOChJ
cYIlI,eCTBOBaHH5I H OrpaHHlJeHHOCTH 3JJ:eCb 06bllJHO He npe.ll.CTaBJI5IIOT TPYJJ:-
HocTetl; JIerKO BH.lI.eTb, BbInOJIHeHbl HJIH He IIbl!10JIHeHbI Te HJIH HHble YC-
JIOBHH. OrpaHHlJHBWHCb TaKHMI1 npOCTblMI1 C,lYlJaHMH, MO)J{HO paCC fOTpeTb
60JIbWYIO lJaCTb MeTOJJ:OB lJHCJIeHHOrO aHaJlH3a TOJlbKO B KOHelJHOpa3HOCT-
HblX COOTHOWeHH5IX, He KaCa5lCb 60. ee CJIO)J{HbIX BonpOCOB.
ITOCKOJIbKY BbIlJHCJJHTeJlbHa51 MaW11Ha MO)J{eT BbInOJIHHTb JIHWb KOHelJHOe
lJl1CJIO Onepall.H!!, 51CHO, lJTO npeLleJIbHbl!! nepexoJJ: HenocpeJJ:CTBeHHO Ma-
WHHO!! BbInOJIHeH 6bITb He MO)J{eT. B CBH3H C 3THM II lJHCJIeHHOM aHaJJlme
npHXOJJ:HTCH 3aHHMaTbC5I IIonpocaMH nOJIYlJeHH5I Oll.eHOK pe3YJIbTaTOB npe-
l(e. bHhIX nepeXOJJ:OB npH nOMOIll,H KOHel/HOro 4HCJIa JJ:e!!CTBI1!!.
CTOHT BcnOMHHTb, lJTO LlHcjJcjJepeHlI.HaJIbHhle ypaBHeHH5I, BCTpelJalOIlI,HeCH
Ha npaKTI1Ke, 06bJlIH0 B03HHKalOT H3 cjJH311lJeCKI1X 3a.ll.alJ lJepe3 nocpeJJ:cTBo
KOHelJHOpa3HocTHblX cxeM. ECJIH 06paTHbl!! nepexoJJ: OT HenpepblBHoro
CJIYlJaH K JJ:HcKpeTHoMY HeB03MO)J{eH, TO HMelOTCH OCHOBaHH5I COMHeBaTbCH
B c!J113HlJeCKOM CMblCJIe COOTBeTCTBYIOIlI,HX ypaIlHeHH!!.
(r= I, 2, ..., n).
18
I1C411CJJEHI1E p A3HOCTEI1
IT J1. !
Bo BTOpOtl lIaCTH rrpeJJ:nOJIaraeTCH, 4TO B rrpOMe)!(YTKe Me)!(JJ:Y 33)J:aH-
HbIMH T04KaMH (Y3JIaMH) HaWa cpYHKlLHH rrpeJJ:CTaBJIHeTCH UJIU anrrpOKCH-
MHPyeTcH MHOr04JIeHOM HJIH palLUOHaJIbHotl cpYHKlLuetl; B TpeTbetl 4aCTH
paCCMOTpeHO rrpeJJ:CTaBJIeHHe cpYHKlLUHMU C OrpaHI14eHHbIM CrreKTpOM *),
CYMMOtl rrOKa3aTeJIbHbIX cpYHKlLUtl HJIH cpYHKlLU'HMH KaKOrO-JIH60 JJ:Pyroro
BUJJ:a, orrpeJJ:eJIHeMOrO XapaKTepOM OC06eHHOc-retl. Bo BCeX CJIY4aHX, Ol(Ha-
KO, MbI 6YJJ:eM OrrepHpOBaTb CO 3Ha4eHUHMI1 cpYHKlLUtl Ha l(aHHOM JJ:UCKpeT-
HOM MHO)!(eCTBe T04eK.
I1HOrJJ:a 6bIBaeT BbIrOIJ:HO UCrrOJIb30BaTb MHO)!(eCTBO HepaBHOOTCTOHIIJ.HX
Y3JIOB. Ho JJ:a)!(e H B aTOM CJIY4ae HJJ:eH H cpopMaJIbHbIe I rrpe06pa30BaHHH
4aCTO CBH3aHbI C HJJ:eHMH H rrpe06pa30BaHHHMU JJ:JlH paBHOOTCTOHW,HX Y3JIOB.
TaKHM 06pa30M, MeTOJJ:bI, HCrrOJIb3yeMbIe B KOHe4Hopa3HOCTHblX cxeMax
C paBHOOTCTOHIIJ.HMU T04KaMH HBJIHIOTCH OCHOBOtl 60JIbwetl 4aCTH 4HCJIeH-
Horo aHaJIH3a.
nepBaH 4aCTb KHUTH cyw,eCTBeHHO UCI10JIb3yeT napaJIJIeJIH Me)!(JJ,y UC-
lIHCJJeHUeM pa3HOCTetl H .lmcpcpepeHlLHaJIbHbIM HC4HCJIeHHeM. npeJJ:I10JIara-
eTCH, 4TO 4HTaTeJIb 3HaKOM C JJ:HcpcpepeHlLHaJIbHblM I1C4UCJIel-meM. Bce TPYJJ:-
HOCTH, KOTopbIe MorYT B03HUKHYTb y 4UTaTeJIH B rJIaBaX 1, 3 H 5, 6YJJ:YT,
BepoHTHo, 06"bHCHHTbCH HeJJ:OCTaT04HbIM rrOHHMaHueM cooTBeTcTBYIOw,ero
MaTepUaJIa B JJ:HcpcpepeHlLUaJIbHOM HC4HCJIeHHH.
EOJJblllHHCTBO cOBpeMeHHblx BbI4UCJIHTeJIbHbIX MalllUH pa60TaeT B pe-
)!(HMe <mJIaBalOIIJ.etl 3arrHTot!». IlJJH JJ:ecHTu4HQtl CU.CTeMbI C4HCJleHHH aTO
03Ha4aeT, 4TO 4HCJIa rrulllYTc5I B BUJJ:e
= 0,31415927. 101, 1 O = 0,31415927 .10 2 ,
"'/10 = 0,31415927 .10°,
H T. JJ:. Eho - 06IIJ.enpUHHToe 0603Ha4eHl1e**), H MbI BcerJJ:a 6YJJ:eM npeJJ:-
rrOJlaraTb, qTO Bbl4UCJIeHUH BeJJ:YTCH C IlJIaBaIOIliet! 3aIlHTot!, eCJIU TOJJbKO
He orOBapHBaeTC5I rrpOHIEHoe.
B aTot! KHl1re rrpeJJ:nOJIaraeTCH, 4TO Bce BbI4HCJIeHI-IH npOH3BOJJ:HTCH Hel
YHHBepCaJIbHot! BbI4HCJIHTeJIbHot! MalllHHe 11 '1-;:0 Bbl4HCJIHTeJIbHblM lLeHTpOM
06eCrreQeHbI: YJJ:OBJIeTBOpHTeJIbHaH CHCTeMa rrporpaMMHpOBaHHH, YJJ:OBJIeT-
BOpHTeJIbHaH CHCTeMa KOHTpOJIH H pa3YMHaH 6H6JII10TeKa rroJJ:rrporpaMM.
EJJ:UHCTBeHHbIM PyqHbIM BbI'mCJJeHUeM rrpeJJ:nOJIaraeTCH paC4eT, Heo6-
XOJJ:UMbIt!, 4T06bI y6eJJ:UTbCH, 4TO JJ:3HHaH rrporpaMMa pa60neT HaJJ:Jle)Ka-
IliUM 06pa30M, a TaK)!(e C4eT, Tpe6yeMbIt! JJ:JIH oueHKH BpeMeHH peWeHUI1
Ha BbI4HCJIHTeJIbHotl MaWUHe.
*) TepMHH 4JYHKU.HII e Orp3HH'IeHHblM cneKTpoM» eeTb TeXHH'IeCKOe BbIpa-
)/{eHHe, npH6J1H3HTeJlbHO 03HatJaIOr.u.ee, qTO y 4JYHKU,HI1 eyr.u.ecTBYIOr qaCTOTbl TOJlb-
KO B .n:aHHofi rroJloce. TaK, t/JYHKU.HR, HMeIOr.u.alI qaCTOTbI B rrO.10ce aT I JJ,O 10
nepHo.n:oB B ceKYH.n:y, orrHCblBaeT onpe.n:eJleHHblH KJlaCC 4JYHKU,HH e OrpaHH'IeHHbIM
eneKTpoM. TotJHoe onpe.n:eJleHHe 6Yller llaHO B cooTBeTcTBYIOr.u.eM MeCTe KHHfH
(eM. rJl. 23).
**) 80 MHOfHX MaIllI1HaX nOKa3aTeJlb CTerreHH (nopRJJ,oK) 3arrHCb1BaeTC5I pa3-
Jll1qHbIMH YCJlOBHblMI1 cnoc06aMH; B qaeTHOCTH, HarrpHMep, K rrOKa3aTeJlIO CTeneHIi
npl16aBJlHIOT Te I1J1H f1Hble KOHeTaHTbI. .LJ.JliI HaWHX u.eJlefi ,)TO He HMeeT HI1KaKOrO
3HaQeHlHI.
,1.2]
PA3HOCTHblM OnEPATOP
19
3aMeHa nepeMeHHOrO
xk=a+kh
B nepBOM npHMepe SToro naparpa a HOBOtt nepeMeHHott
Xk - a k (k 0 I
t k = ---,z-= = , 1, ..., n- ) (1.1-1)
npHBO.ll.HT paccMoTpeHHbltt CJIyt-tatt K Y.ll.06HOtt CTaH.lI.apTHott opMe. 6JIaro-
.lI.ap51 9TOMY B nepBott IJaCTH, eCJII1 He C.lI.eJIaHO m:1KaKHX orOBOpOK, BbIpa-
>KeHHe f (x) 6Y.ll.eT paCCMaTpHBaTbC5I .lI.JI5I 3HaIJeHHtt x = 0, 1, "', n - 1.
HHor.ll.a Y.ll.06HO nOJII.30BaTI.C5I 0603HaIJeHH5IMH
[(0)=/ 0 ,
[(1)=/1>
C TOIJKH 3peHl151 pe3YJJI.Tan BI.IIJHCJIeHHtt 06I.IIJHO 6bIBaeT 6e3pa3JJHIJHO,
113MeHI1M JIH MI.I .lI.aHHYIO 3a.ll.aIJY B COOTBeTCTBl-I1-I C e.ll.I-IHI-IlJ.et! H3MepeHI-I5I,
HJJI-I >Ke npeo6pa3yeM p!l3HocTHoe BbIpa>KeHl-Ie B COOTBeTCTBI-IH C lllarOM
B 3a.n:aIJe. Sro aHaJIOrH4HO,npeo6p a30BaHI11O OKPY>KHOCTI1
(x-a) +(v- b) =r
B aHaJII-ITI1IJeCKott reOMeTpI111 3aMeHott nepeMeHHWX
x-a=x',
y-b=y'
B OKPY>KHOCTb
X'2 + y' = r .
STO npeo6pa30BaHI1e MO>KHO paCClVlaTpI1Ban C .lI.BYX TOIJeK 3peHI151. MO>KHO
CIJI1TaTb, IJTO. I13MeH5IeTC5I KOOp.ll.I1HarHa51 CI1CTeMa, IJTO npHBe.ll.eT K HOBI.IM
KOOp.ll.l-IHaTaM .lI.JI5I Tex >Ke caMbIX TOIJeK OKPY>KHOCTI1. B .lI.pyroM CJJYIJae
MO>KHO npI-IH5ITI., IJTO OKPY>KHOCTI. nepeXO.ll.HT B HOBoe nOJJO>KeHI1e, TOr.ll.a
KaK KOOp.ll.I1HaTHa51 CI1CTeMa OCTaerC5I HenO.ll.BI1>KHott.
YnpamHeHH5I
1.1-1. IlycTb'x npHHHMaeT 3Ha'leHI1R 11, 9,7, ..., - 11. HaHTI1 npeo6pa30-
BaHHe, KOTopoe npHBO.ll.Hr era K CTaH.lI.apTHOH 4JopMe 0, I, ..., II.
Or BeT: t=(-x+II):2.
1.1-2. rIpHBeCTH K CraH!lapTHOfi 4JopMe x = 3; 3,5; 4,0; 4,5; ...; 10.
OTBer: t=2(x-3).
1.2. Pa3HocTHbIM onepaTOp
OCHOBHbIM O:1epaTOpoM B I1CIJHCJIeHl-IH KOHeIJHbIX pa3HocTett 5IBJI5IeTC5I
pa3HocTHbltt onepaTop 11, onpe.ll.eJIeHHbltt paBeHcTBoM
I1f (x) = f (x + h) - f (x).
(1.2-1)
STOT onepaTOp 3HaKOM IJHTaTeJJlO H3 allaJII13a, r.ll.e OH nCnOJIb3yeTcfI
npn onpe.ll.eJJem-II-I np0I13BO.ll.Hott. TeCHa51 CB5I3b Me>K.lI.Y npOH3BO.ll.HOi! II
20
HCYHCJJEHHE PA3HOCTEA
[fJJ. I
p33HOCTblO 51BJJ5IeTC5I OCHOBOt1 60JIblllHHCTBa npHMeHeHHIt KOHe'IHOpa3HOCT-
HbIX BbIp3)J{eHHtI n:JI5I npH6.1H)J{eHH5I BbIpa)J{eHHt! H3 n:HcjJcpepeHllHaJIbHOrO
HCIJHCJIeHlUl.
OnepaTOp /),. MO)J{HO npen:CTaBJI5ITb K3K cYIlI.eCTBYIOIlI.Ht! OTn:eJJbHO If
n:et!CTBYIOIlI.Ht! Ha cjJYHKll.HIO I(x) TOIJHO TaK )J{e, KaK MbI IJaCTO paCCM3T'
pHBaeM npOH3Bon:HYIO dfd ) KaK onepaTOp :X ' n:et1CTBYIOIlI.Ht! Ha I (x),
H HHTerp aJJ ) I (x) dx - KaK HHTerpaJJbHbIt! onepaTOp )... dx, n:et!CTBYIO-
IlI.Ht! Ha cjJYHKll.HIO I (x).
Pa3HocmHbiU onepamop AllHeeH, KaK n:HcjJcjJepeHll.l1aJIbHbIIt H miTe-
rpaJIbHbIt!, T. e. eCJJH a H b nOCT05lHHbI, TO
/),. [al(x) + bg(x)] = a /),.I(x) + b /),.g(x).
(1.2-2)
JIHHet!Hoe CBOt!CTBO onepaTopa n:eJIaeT onepaTOp /),. oco6eHHO npOCTbIM
n:JI5I npl1MeHeHI151 BO MHorHX CJJYIJa5lx.
B KaIJeCTBe npHMepa paCCilWTpHM cpYHKll.HIO
y=ax +bx+c.
I-1cnOJIb3Y5I (1.2-2), nOJJYIJHM
/),.y=a/),.x 2 +b/),.x+c /),.1 =a[(x+h)2_x2] +b [(x+h)-x] +
+ c [1 - 1] = 2ahx + ah 2 + bh. (1.2-3)
Pa3HOCTHbIt! onepaTop, n:et!cTBY5I Ha npoH3Ben:eHHe, n:aeT
/),. [f(x) g(x)] = I(x + h) g(x + h) - I(x) g(x) =
= I(x + h) g(x + h) - I-{x + h) g(x) + I(x + h) g(x) - I(x) g(x) =
= I(x+ h) /),.g (x) + g(x) /),.I(x)
UJIH
/),. [I (x) g(x)] = I(x) /),.g(x) + g(x + h) /),.I(x).
(1.2-4)
Ecm! nCKJIIOIJHTb TO, IJTO aprYMeHT On:HOt! H3 cpYHKll.Ht! eCTb x + h,
cjJopMYJIa (1.2-4) cooTBeTcTByer cpopMYJJe n:HcjJcjJepeHll.HaJIbHOrO HCIJI1C-
JIeHH5I
d
dx [I (x) g(x)] = I (x) g' (x) + g(x) j' (x).
AHaJIOrnIJHO .D:JI5! OTHOmeHH5! cjJYHKll.Ht! HMeeM
/:" f(x) _ f(x+1z) f(x) =
g(x) -g(x+h) g(x)
_ f(x+ h) g (x) -g (X + h)f(x) + [f(x)g (x) - f(x) g (x) 1
- g (x)g !x+h) - -
_ g (x) /::'f (xi - J (X) !:!.g {X) 11.2-5 )
---g(x)g(x+h) · \
1;\ 1.31
nOBTOPHbIE P A3HOCTH
21
4TO Om!Tb HarrOMlmaeT COOTBeTCTBYIOIlJ,YIO CPOPMYJIY D:ml D:H4>4>epeHlJ,HaJlb-
HOfO HC4HCJJeHHI1, a HMeHHO:
d f (x) g (x) f' (x) - f (x) g' (x)
dx g (x) = g2 (x)
B KHHfax no aHam13Y 06bI4HO D:aeTCH D:JlIIHHaH Ta6JJHlJ,a <POPMYJI D:Jl51
npOIl3BOD:HWX. AHaJJOfl14HaH Ta6J1l1ua IICnOJJb3yeTCI1 11 B HC4HCJleHHII pa3-
HOCTet!. DpHBeD:eM CJleD:YIOllI.YIO KOPOTKYIO Ta6J1l1uy:
Asin(ax+b)=2sin a; cos[a(x+ )+b].
A cos (ax+b)= - 2sin a; sin [a (x+ ) +b],
A tg (ax + b)= sin ah sec (ax + b)sec la(x+ h)+ b],
Aa x = aX (a h - 1),
A2 x = 2 x (2 h - 1),
A In'; = In (1 + ).
(1.2-6)
(1.2-7)
(1.2-8)
(1.2-9)
(1.2-10)
(1.2-11)
PORb, KOTOPYIO HfpaeT' 411CJlO e B D:1I<P4>epeHUllaJIbHOM IIC4IICJleHIIII.
B I1C4HCJleHHI1 KOHe4HbIX p a3HOCTet! B HeKoTopoM OTHOIlIeHI111 I1fp aeT
'IHCJJO 2. l1MeHHo, eCJlI1 a h = 2, TO Aa x = aX. B 'IaCTHOCTH, eCJlII a = 2
II h = 1, TO A2 X = 2 x .
Ynpa>KHeHHe 1.2-1. rIpoBepHTb paBeHcTBa c (1.2-6) no (1.2-11).
1.3. nOBTOpHbIe pa3HOCTH
TaK KaK !:if(x) eCTb <PYHKlJ,HH OT x, TO, rrpllMeHI1B K Hett CHOBa one-
paTop A, MO}\{HO nOJIY'II1Tb
A [Af (x)] = A2f (x). (1.3-1)
cho 0603Ha'leHIIe cooTBeTcTByeT 0603Ha'leHHIO D:JI5I BTOpOtt npOH3BOD:Hott
B D:11<pcpepeHlJ,HaJlbHOM HC'IHCJleHI1I1
d df (x) _ d 2 f (x) _ d 2 f (x)
dx - (dX)2 - -rfX2 .
Boo6llI.e
B npl1Mepe
Arf(x) = A [Ar-l f(x)].
(1.3-2)
y(x)=ax 2 +bx+c
Mbl HMeJlH (yp aBHeHne (1.2-3»
Ay (x) = 2ahx + ah 2 + bh;
TOfD:a
A2y (x) = 2ah 2 .
22
HC411CJJEHHE PA3HOCTEA
[Tn. t
OpO.ll.OJDKaSl reM >Ke cnoc060M, nOJIY4aeM
ABy (x) = O.
TOT <jJaKT, 4TO A By (x) = 0, - He CJIY4at!HOCTI" 8 CJIeLICTBI1e Ba >KHOt!
TeopeMI,I:
o C HOB H a 51 T e 0 p e M a H C 4 H C JI e H H 51 P a 3 HOC T e t!. JI./lJl MHO-
20tf,/lefta c mene HII n
y(x)=ao+ alx + a2x2 +...+ anx n (an =F 0)
n-Jl pa3H.OC mb noc mOJlH.H.a II paBH.aann!h", (n + 1 )-Jl pa3H.OC mb paBH.a
H.Y/llO. J10I<a3aTeJIbCTBO JJer4e BcerO nOH5ITI" paccMoTpeB npe.ll.BapI-ITeJIbHO
13I<YIO JIeMMY:
JJ e M M 3. EC/lll Y (x) - .MH.020tf,/leH. CmeneH./l n, mo Ay (x) ecmb ..flHO-
20tf,lleH. CmeneH.ll n - 1 (T. e. CTeneHI, Ay (x) pOBHO H3 eLII1HI1IJ:Y MeHI,lile,
4eM CTeneHI, y (x».
J1 0 I< a 3 aTe JI I, C T B 0 JJ e M M hI. J1JI5I <jJYHI<U,HH Y (x) = x n , HCnOJIb-
3Y5I p33JJO>KeHHe no 6HHOMY *), H3XO.ll.HM
"
y(x)=(x+h)"-x"= C(n, k)h"-kXk_ X "=
k=O
h n-I + n (n-l) h 2
=n x 2
"-2 + + h n
x '" .
T31<HM 06pa30M, npH npHMeHeHHH onepaTop3 A 4JIeH x n CTaHOBHTC5I MHO-
r04JIeHOM CTeneHH n - 1 co CT3plill1M 1<09cpcpHu,HeHToM p3BHhlM nh.
Y1cnOJIb3Y5I CBOt!CTBO mlHet!HOCTH (ypaBHeHHe (1.2-2», H3XO.ll.I1M, 4TO one-
p3TOp A YMeHbUIaeT CTerreHI, 1<3>K.lI.OrO 4JJeHa B MHOr04JIeHe Ha e.ll.HHHUY.
l.{JIeH nxn-1ha n He MO>KeT YHH'lTO>KHTbC5I, CJIe.ll.OB3TeJIbHO, JIeMM3 .lI.O-
K33aHa.
J1 0 I< a 3 aTe JJ b C T BOT e 0 p eM bl. TIpHMeHHB I< MHOr04JIeHY n-tt
CTeneHH .lI.Ol<a3aHHY!O JIeMMY n pa3, y6eJ),\'IMC5I, 4TO ero n-51 pa3HOCTb no-
CT05lHH3 H 1<09cpcjJHUHeHT npH an eeTb n!h n . CJIe.ll.YlOllI.He p33HOeTH 06pa-
llI.alOTC5I B HYJIb. 3THM TeopeM3 .lI.Ol<a3aHa.
TIpHBe.n:eHHa51 TeopeMa HMeeT 60JIbUIOe 3Ha4eHHe B I<JIaeCI14eCKOt!
qaCTH 4HCJJeHHOrO aH3JIH3a.
Ynpa»meHHe 1.3-1. OOJlarail h = I, BblqHCJIHTb BTOpylO H '1eTBepTYro pa3-
HOCTI1 MHOrO'lJleHa y = Xi - 4x 8 + 6x 2 - 4x + I.
o T BeT: ,12y = 12x2 + 2; Ll4y = 24.
*) MbI HCrrOJlb3yeM )lJlR 611HOMI1aJlbHblX K094J4JHIJ.HeHTOB 60JIee cTapoe 060-
3HaqeHHe C (n, k) BMeCTO nonYJIRpHOrO B HaCTORLQee BpeMR ( ) (HJlM JIro60ro
.JIPyroro C BepXHHM 11J111 HH)/{HHM, 11J1f1 TeM M .JIPYrHM BMeCTe HH.JIeKCaMH), TaK
KaK rrpe)/{Hee o603Ha4eHHe Jlerqe Ha6parb, OHO MO)/{eT 6bITb HarreqaTaHO Bbl-
qHCJUueJlbHoi1 MaIllHHofi 11 JlyqIlle Bblr JlRJJ,HT B cepe,IJ.HHe rrpe.aJlO)/{eHI1R H 4Jop-
MYJlbI.
1.41
T ABJJJ1l\bl P A3HOCTEfi
23
1.4. TaOJJHlI.bJ pa3HocTeH
ECJlH nplixo.n:I1TC51 HcrrOJlb30BaTb BblCOKHe pa3HocTII, nOJle3HO npe.n:-
CTaBJl51Tb HX paCnOJlQ)KeHHbIMI1 B BI1.n:e Ta6J1Hu,bl (CM. Ta6J1I1u,y 1.4-1, r.n:e
rrpHHHMaeTC51 h = 1), XOT51, B03MO)j{HO, pa3HOCTH He pacrrOJlO)j{eHbl B Ta-
KOM nOp51.n:Ke B MaIlIHHe.
T a 6 II H U, a 1.4-1
Ta6JJHu.a pa3HocTeA
x Y (XI y (XI 2y (XI 3y (xl
° y (0)
/ly (0)
1 Y (I) /l2y (OJ,'
/ly (1) /l3y (0)
2 Y (2) /l2y (I)
/ly (2) /l3y (1)
3 Y (3) /l2y (2)
/ly (3) /l3y (2)
4 y (4) /l2y (3)
/ly (4)
5 y (5)
T a 6 JI H U, a 1.4-2-
Ta6JJHu.a pa3Hocrefi .n:JJH
X
Si (x) = Si t dt
o
X I 51 (X) I I 2 I 3
0,0 0,0000
999
0,1 0,0999 -2
997 -6
0,2 0,1996 -8
989 -1
0,3 0,2985 -9
980 -5
0,4 0,3965 -14
966 -2
0,5 0,4931 -16
950 -3
0,6 0,5881 -19
931 -3
0,7 0,6812 -22
909 -3
0,8 0,7721 -25
884 -3
0,9 0,8605 -28
856
1,0 0,9461
B Ka4eCTBe npHMepa Ta6J1Hu,w pa3HocTefi paCCMOTpl1M 3Ha4eHH
HHTerpaJlbHOrO cHHyca (Ta6J1Haa 1.4-2). TIo o6merrpHH51TblM o603Ha4e-
HI151M pa3HoCTH nHIlIYTC51 TaK, KaK eCJlH 6bl 3arr51Ta51 B .n:eC51TH4HOM 4HCJle
CJle.n:OBaJia 3a nOCJJe.n:HHMH pa3p51.n:aMH, OTBe.n:eHHbIMH lJ,Jl51 3Ha4eHH51 <PYHK-
aHH. 4T06w rrpoBepHTb apH<pMeTHKY, 3aMeTHM, 4TO B Ta6mme 1.4-2
cYMMa 4HCeJl B JIJ060M cToJl6ae pa3HocTefi, rrpl16aBJleHHa51 K BepxHeMY
'II1CJlY COCelJ,HerO CToJl6u,a CJleBa, paBHa HI1)j{HeMY 411CJlY B TOM )j{e CTOJI6-
ae CJleBa.
Vl3 TOJlbKO 4TO lJ,OKa3aHHofi TeopeMbl CJIe.n:yeT, 4TO eCJlI1 y (x)
eCTb MHOr04J1eH CTeneHI1 n OTHOCI1TeJlbHO x, TO n-)! pa3HOCTb 6y.n:eT
24
MC4McnEHHE PA3HOCTEA
(fJJ. I
KOHCT3HTOti. ECJJI1 6bI MbI XOTenH npoT36ymlpOBaTb MHOr04neH CTenerl!1 n
B HeCKonbKHX paBHOOTCT05lW,HX T04Kax, TO, B npHHlume, MOIKHO BbI4HC-
J1HTb BepXHlle 4Hcn3 B KaIK.n:OM CTon6l1.e H nOCTpOHTb BCIO OCTaJIbHYIO
Ta6JJHlI.Y, nonb3Y5ICb TOJIbKO CJ10IKeHlleM.
.ITJI5I HJIJIIOCTp3l1.11H B03bMeM KBa.n:p3THblt! TpeX4JIeH
y (X) = 3x 2 - 6x + 9;
y (0) = 9
1'1 BbI4HCJJHM ero 3Ha4eHI'I5I B T04K3X OT X = 0 .n:o X = 10 C Ill3rOM h = 1.
MbI I'IMeeM
Ay(x)=6x-3,
A2y (x) =6,
Ay(O)=- 3,
A2 Y (0) = 6.
DOCTpOllM aTY Ta6JIHlI.Y (Ta6nHlI.a 1.4-3, ompaBHbIe 'IHcna Ha6paHbJ IKHp-
HblM rupHcpTOM). C .n:pyrot! CTOpOHbJ, MOIKHO 6bJJIO 6bI BbI4HCJIHTb nep-
Bble TPH 3H34eHH5I H BOCnOJIb30B3TbC5I HMH
.n:JI5I OTblCK3HH5I p33I-1OCTei1.
B 3TOM npllMepe .n:JI5\ BbI4HCJI'eHH5I O'le-
pe.n:Horo 3H3'leHHH KBan:paTHoro TpeX4JJeHa
nOTpe6oBaJIOCb .n:Ba CJIOIKeHH5I; Bo06w,e 'lT06bI
Bbl4llCJII1Tb O'lepe.n:Hoe 3H34eHHe MHOr04JIeHa
n-t! CTeneHH, Tpe6yeTC5I n cnOIKeHHtt.
60JIbll1HHCTBO BbI4HCJJeHHtI cet!'lac D,e-
JIaeTC5I B CHCTeMe nJIaBaIOw,etl 3a115lTotl, HO
nepBOHa4aJIbHbie HCxo.n:Hble .n:aHHble, KOTO-
pble 4aCTO CHHMa!OTC5I Henocpe.n:CTBeHHo
C aBTOMaTI14eCKH perHCTpl1pYIOW,HX npH6o-
pOB, 6blBaIOT TOr.n:a B CHCTeMe cpHKCHpOBaH-
HOt! 3an5lTotl. DpH 3THX 06CT05lTenbCTB3X
YKa3aHHbJtI MeTO.n: BbJ4HCneHHH MHOrO'IneHa
oco6eHHO IIone3eH. HeKOTopoe BHHM3HHe
CJle.n:yeT, Brrp04eM, y.n:enHTb pacrrpOCTpaHe-
HHIO OWH6KH BCneD,CTBHe HCnOJIb30B3Hll51
np1l6nHIKeHHoro 3Ha4eHI'151 pa3HOCTH B Bbl-
411CJIeHH5IX.
XOTH T36JIHlI.bl 1.4-2 H 1.4-3 Ha4HH3IOT-
C5I C HYJIH, 43CTO rrOJIe3HO npe.n:CTaBJI5ITb
T36mn.lY pacrrpocTpaH5IIOw,etlC5I HeorpaHH-
4eHHO B 060llX HarrpaBJIeHH5IX H, B 4aCTHO-
CTH, npe.n:cfaBJl5lTb HHTepeCYIOw,ee Hac reKyw,ee MeCTO HaXOD,51w,I1MC5I B
HYJIe. Ta6JIHlI.a 1.4-4 nOKa3bIBaeT, KaK e.n:HHCTBeHHa51 oll1H6Ka Ha OD,HY
eD,I1HHlI.Y B cpYHKlI.llH Y (x) = 0 pacnpoCTpaHHeTcH B ra6JlHu.e pa3HoCTet!.
.llonYCTHM, 4TO oWI16Ka 6blJIa apll x = O. Tor.n:a y (0) = 1, a Bee Apyrue
3Ha4eHHjj Y (x) = O.
T a 6 JI 11 U a 1.4-3
y (x) =3x 2 -6x+9
x !J .:1!J .:1}
[/ :S- >6
1
2 9 6
9
J 18 {j
15
4 33 0
21
5 54 ""' {j
27
6 81 0
JJ
7 1/4 6
39
8 15.3 5
45
g 1M 6
5J
10 249
IljltJoC'jl/(tJ!J-(IO}=,J(}(}-5(}r]= 24]
S 1.4J
T ABJlIiUbl P A3HOCTEfi
25
T a 6 JI H U a 1.4-4
T aOJnIll.a pacnpoCTpaHeHHSI OllIHOKH
x I y t.y t. 2 y t. 3 y t. 4 y t.5y t. 6 y
-3 6
6 :
-2 0 6 .....................
0 ................... . i
-I 0 ...................... I -4
.................... 1 -3 (0
0 1 -2 6 -20
................... -I 3 -10
1 0 ....H....'........... I -4 15
0 -I 5
2 0 ...... 'PO I
-6
0 :.""""0"'''''''' -I
3 0 0 .... 0 ....... 1
0 0 :.........O".......
4 0 0 0 :
0 0 :
5 0 0 :
0 :
6 0 :
:
: :
1..JHCJIa B k-M cToJI6ue pa3HocTet! paBHbI (- 1 Y C (k, r). YTo6bI .lJ:OK3-
33Tb 9TO, BBe.lJ:eM onepaTop C.lJ:BHra E, KOTOpbIt! orrpelLemIeTc5I rrp3BHJIOM
(npHHI1MaeM h = 1)
E[f(x)]=/(x+ 1).
(1.4-1)
5ICHO, QTO orrepaTop JIImeeH, TaK KaK
E [aJ(x) + bg(x)] =aE [f(x)] + bE [g(x)).
IloBTopHoe ero rrpHMeHeHI1e .lJ:aeT
e [f(x)] = f!EE.. .f! [f(x)] =J(x+n).
n paa
VlCXO.lJ:51 li3 9Toro, MbI orrpe.lJ:eJJ5IeM .lJ:JI5I n = 0
EO [f(x)] = I (x),
a .lJ:JI5I OTpHuaTeJIbHOrO rrOKa33TeJJH
En [f(x)] = I(x - n).
26
I1C"II1CJJEHI1E P A3HOCTEH
[TJJ. t
MbI Y)I(e 33Me'l3JIH, '1TO 11H 0 r.n;a y.n;06HO npe.n;CT3BJI5ITb
OT.n;eJIbHO OT CPYHKlI.I-IH, Ha KOTOpylO OH .n;ei!CTByeT. ECJm B
>oT6pOCI-IM H3CT05lmyIO CPYHKlI.HIO, TO nOJIY'II-IM OnepaTOpHOe
Dpl-IMep TaKOra paBeHCTB3:
onepaTOp
TO)l(.n;eCTBe
paBeHCTBO.
fl=£-1,
(1.4-2)
r.n;e I - TO)l(.n;ecTBeHHbIi! onepaTop 1. I(x) = I (x). DpaBHJIbHOCTb STora
'onepaTopHoro paBeHCTBa CJIe.n;yeT H3 npOCToro 3aMe'laHI-I5I
fly (x) = JI (x + 1) - Y (x) = £ fy (x)] -1. Y (x) = (£ - l)y (x).
TaKI-IM )l(e 06pa30M CJIe.n;yeT, '1TO
k
flk=(£_l)k= L:(-1i- r C(k,r)F.
r=O
(1.4-3)
ITJJ5I JI106oi! CPYJlKlI.l-I1-I Y (x) onepaTopHoe paBeHcTBo .n;JI5I flk .n;a T
,q>OPMYJIY
k
flk y (n) = (- ll- r c (k,r) y (n + r),
r=O
(1.4-4 )
'KOTOpylO '1aCTO Ha3bIBalOT «pa3HOCTHOi! CPOPMYJJOi! narpaH)I(a» (OCHOBa-
'HHe J],JI5ITaKOrO Ha3BaHH5I 6y.n;eT npHBe.n;eHO n03)1(e). vb ypaBHeHH5I (1.4-4)
MO)l(HO Hai!TI1 OT.a:eJIbHble pa3HOCTI-I, He BbI'II-ICJI5I5I Bcei! Ta6JII1l1.bI pa3HoCTei!.
DpHMeHI-IM Tenepb 9TO pa3JIO)l(eHHe K CPYHKlI.l-IH y ueJIO'll-ICJIeHHOrO
>aprYMeHT3 x, onpe.n;eJIeHHoi! .n;BYM5I JIeBbIMH cToJI611.aMH Ta6JII-IlI.bI 1.4-4:
{ 0,
y(x)=
1,
x =;f: 0,
x=O.
Bce '1JIeHbI CYMMbI B npaBoi! '1aCTI1 CPOPMYJIbI (1.4-4) 06paT5ITC5I B HYJIb,
,KpoMe o.n;Horo, KOTOpbIi! H .n;aCT cooTBeTcTBYlOmHi! 6I-IHOMI1aJIbHbIi! KOScp-
<pHlI.HeHT.
<POPMYJIY (1.4-4) MO)l(HO HCnOJIb30BaTb B 06paTHYIO CTOPOHY, KorlIa
H3BeCTHO 3Ha'-leHHe pa3HocTH, a o.n;Horo 3Ha4eHH5I CPYHKlI.HH HeT. B Ka-
'1eCTBe xap'aKTepHoro npHMepa npe.n;nOJIO)l{HM H3BeCTHbIM, '-ITO rneCTbIe
pa3HocTH HJIH B TO'lHOCTI1 paBHbI HYJIIO, HJIH HaCTOJIbKO MaJIbI, '1TO HX
MO)l(HO C'IHTaTb npU6JJH)I(eHHO paBHbIMH HYJIIO, a HaM He XBa-raeT OD:HorO
3Ha'leHH5I CPYHKlI.I1H. 0603Ha4HM ero Yo' a CMe)l{HbIe 3Ha'leHH5I 6y.n;YT Y -3,
y-9' Y-l' Yl' Y9' Y3' Tor.n;a, I1CilOJIb3Y5I (1.4-4) C k=6 H n=-3, I-IMeeM
Y-3 - 6Y_9+ 15Y_l - 20yo+ 15Yl- 6Y9 + Y3= 0
,J1JJH
I
Yo= 20 [CY-3 + Y3) - 6CY_2+ Y2)+ 15CY_l + Yl)].
(1.4-5)
1.51
<l>AKTOPI1AJJbJ
27
Bn cpOpMYJJa 04eHb nOJJe3Ha, KOrJJ:a OJJ:HO 3Ha4eHHe B rJJaJJ:KOt! Ta6-
JJHUe OTCYTCTByeT 113-3a HeBHHMaTeJJbHOCTH. OJJ:HaKO 1JOJJ:06HOt! CPOPMYJJOt!
HaJJ:O 1JOJJb30BaTbC5I C OCTOpmKHOCTblO, TaK KaK OTCYTCTBYIOw.ee 3Ha4eHHe
MO)J{eT 51BHTbC5I CJJeJJ:CTBileM oco6eHHOCTil B CPYHKUilil (HarrpilMep, JJ:eJJeHil51
Ha HYJJb). 3aMeTilM, 4TO pa3HOCTil 4eTHoro nOp5lJJ:Ka ilC1JOJJb30BaTb JJ:JJSI
STOt! ueJJH MHoro JJer4e, 4eM pa3HOCTH He4.eTHOrO nOp5lJJ:Ka, TaK KaK JJ:JJSI
4eTHbIX pa3HOCTet! HMeeTC5I eJJ:HHCTBeHHbIit MaKCHMaJJbHbIit 4JJeH.
Ynpa)l{HeHHSI
1.4-1. BbI'IHCJlHTb f(n)=n 2 -79n+1601 J1.JIH nor 0 IlO 10. (3aMe4aHHe:
f (n) npocToe '1HCJlO IlJlR n = I, 2, ..., 79.)
1.4-2. BbI'JHCJlHTb Ta6J1Hu,y Ky60B OT 10 JJ,O 20, I1cnoJlb3YR Ta6J11IUY pa3HocTefi.
1.4-3. rI peIlnOJlo)J{I1M, '1TO 3Ha4eHHe npH x = 0,6 B Ta6JII1ue 1.4-2 OTCYT-
CTjjyeT; C'IHTaH, 'ITa '1eTBepTble pa3HOCTH paBHbl HYJlIO, BbI4HCJIHTb Si (0,6).
1.4-4. Bbl'lHCJlHTb npH6J1mKeHHO Si (1,1), HCnOJIb3YH (1.4-4).
1.5. aKTOpHaJIbI
B aHaJJI13e Ba)J{HYIO pOJJb HrpaeT CPYHKlI.H5I x n . B p5l.n:e Tet!JJopa
co
f(x)= anx"',
n=O
HanpHMep, npOH3BOJJbHa51 CPYHKI1.I151 pa3JJo)J{eHa no CTeneH5IM x. BTa Be-
JJ:yw.a51 pOJJb CPYHKlI.HH x n B 60JJbWOt! CTeneHll onpe.n:eJJ5IeTC5I TeM, 4TO
OHa y.n:OBJJeTBOp5leT COOTHoweHHIO
d _
dx x n = nx n 1
(n;:;?; 1).
ECTeCTBeHHa51 aHaJJOrH5I Tpe6yeT nOCTpOeHH5I B HC4HCJJeHHH pa3HocTet!
CllCTeMbl CPYHKlI.Ht! gn (x), KOTopbIe y.n:oBJJeTBOp5lIOT aHaJJOrH4HbiM COOT-
HOWeHH5IM
I1gn (x) = ngn_l (x)
(n ;:;?; 1).
(1.5-1 )
COOTHoweHlllO (1.5-1) YJJ:oBJJeTBOp5lIOT rjJatCmOpZlallbHble
onpe.n:eJJeHHWe paBeHCTBaMI1
gn(x)=x(n) =x(x - l)(x - 2)... (x -n+ 1)
go (x) =x o = 1.
lleficTBllTeJJbHO,
.MH020t.{lleHbl *),
(n;:;?; 1),
(1.5-2)
I1g n (x)= gn(x+ 1) - gn (x) =
=[(x+ 1)-(x- n+ 1)] x(x-l) ... (x -n+2)=
=nx(x-l) ... [x-(n-l)+ I] =nx(n-1) =ngn_l(X).
*) He nYTaHTe C nl = n (n - 1) (n - 2) ... 2. 1, KOTopoe RBlllleTCJI 3Ha'le-
HHeM x lnJ lIP" x = n.
'28
HC4MCJJEHME p A3HOCTEi%
[rn. t
3aMeTi-IM, {!TO 113 X O = 1 CJle.LI.yeT 0(0) = 1, cooTBeTcTBYIOUJ.ee CTaH.LI.apT-
HOMY O! = 1, H '1TO X(n) HMeeT pOBHO n COMHOlKHTeJlei1.
I1YCTb .LI.3HH351 tjJYHKllJHI f (x) MO)l{eT 6bITb p33J10)l{eHa no tjJ3KTO-
pHaJlbHbIM MHoro'weH3M, 3HaJlOfH4HO pa3J10)j{eHHIO Tei1J1opa no CTene-
H5IM x, T. e.
co
f(x) = b o + bix + b 2 x(2) + b 3 x(3) +... = 2: bkx(k).
k=O
I1pe.LI.nOJlO)j{HM T3K)j{e, '1TO p33HOCTHbIi1 onep3TOp JlHHeeH .LI.Jl5I 6eCKOHe'l-
HbIX CYMM H '1TO MO)l{HO 6paTb pa3HOCTH nO'lJleHHO. B 3TOM pa3J10)l{eHI-!H
npH X = 0 nOJlY'IaeM
f(O) = boo
f1pI1MeH5I5I onep3Top A K o6eHM QaCT5IM paBeHcTBa, nOJlY'lHM
co
Af(x)= 2: bkkx(lH).
k=1
CHOBa nOJlara51 X = 0, HMeeM Af(O) = b i . f1po.LI.oJl)j{a51 n:efiCTBOBaTb Ta.
.KHM o6p a30M, nOJlY'IHM
00
f:lnf(x) = 2: bkk (k - 1)... (k - n + 1) .x\ -n)
k=n
'H, IIOJlara51 x = 0,
Anf(O)=n!b n .
},\bI nOJlY'IaeM, CJle.LI.OBaTeJlbHO, tjJopMaJlbHbIt! aHaJlOr p5lJJ:3 TetlJlopa
co co
[(X)=! Ak{/ l X(k) = ! C(x, k) f:lk [(0),
k=O k=O
( 1.5-3)
'f'.LI.e C (x, k) - 6HHOMHaJlbHbIe K03tjJtjJHlJ.HeHTbI, KOTOpbIe MorYT 6hITb 3a-
.Jerk,
llHCaHhI KaK k!'
ECJlH OrpaHH'lHBaTbC5I MHOrO'lJleHaMH, TO p5l.LI. Tei1J1opa 6Y.LI.eT C01(ep-
)j{aTh KOHe'lHOe '!HCJIO '1JIeHOB H BOnpOC 0 CXO.LI.HMOCTI-I He B03HHKHeT.
AHaJlOfH'IHO pa3J1o)j{eHHe (1.5-3) 1(Jl5I MHor01JJleHOB CTeneHH n 6Y.LI.eT KO-
:HeCIHbIM
n
f(x) = ! Ak iO) X(k),
k=O
(1.5-4)
TaK KaK no OCHOBHOtl TeopeMe ( 1.3) f:ln+1 f(x) = 0 H Bce pa3HocTH
'DOJlee BhlCOKoro nOp5l1(Ka 06paT5ITC5I B HYJlb, Bhlpa)j{eHHe (1.5-4) H3BeCTHO
J<aK llH./llepnO/lfll1110ftftafl cfiopMY/la HblO/llOH.a.
1.61
.lJ:EREHHE MHor04REHoa
29
Onpe.n.eJIeHlle <jJ3KTOpI13J13 MO)l{eT 6b/Tl. 0606meHO TeM )l{e nYTeM,
K3KI1M nOJJb3YIOTC5I .n.JI5I 0606IlJ,eHl15l onpe.n.eJJeHI1H CTeneHH B 3J1re6pe.
VlCnOJIb3Y5I onpe.n.eJJeHl1e (1.5-2), I1MeeM
x(n) = x(m) (x _ m)ln-m).
ECJIH Tenepb <jJOpM3JIbHO nOJIO)l{HTb n = 0, TO rrOJIY'II-IM
X(O) = 1 =x(m) (x - m)(-m)
'11J\1-I
( ) (-m) 1
x - m = ---.m;;-.
x
33MeHHH x - m H3 y, npl1XO.ll.l1M K cpopMYJ\e
(-m) _ ( _ I
Y -(y+m)(ml-(y+m)(y+m-I)... (y+1) ,
(1.5-5)
'KOTOp351 CJJY)I{HT onpe.n.eJIeHHeM cp3KTOpH3J10B .n.J1H OTpl1u.3TeJJbHblX u.eJIO-
'IHCJIeHHbIX nOK333TeJIetL
33MeTHM, 'ITO eCJIl1 m -=F- 0, TO
X(m)X(-m) -=F- X(O).
Ynpa}JmeHHH
1.5-1. \ilCIIOJlb3YR 4JOpMy.1Y HbIOToHa, HaHTH MHOrO'lJleH, !WTOPblfi "PFlHM.
Mae! CJleJJ,YIOIll,He 3Ha'leHHR (Bce f(n)-"pOC!ble '1F1CJla "PH n<41):
n
o
2
3
4
5
f(n)
41
43
47
5:1
61
71
o T BeT: P (n) =n 2 +n+41.
1.5-2. rIoKa3aTb, 'ITa
3(-3' -
-120'
ll-m, = I
(m+ 1)1 .
1.6. lleJleHl-le MHOrOQJleHOB
lleJ\eJ:IHe MHOr04JJeHOB \1Hor.n.a naeTC5I B aJ\eMeHTapHOM Kypce 3JIre-
6phl. OHO 6yneT 43CTO HCnOJIb30B3TbC5l, a 1I0TOMY 3aCJ\Y)l{\1aaeT aHUMa-
TeJlbHoro p3CCMOTpeHl1H. MbI rrOKa)!{eM, K3K HcrrOJIb30BaTb n.eJIeHl1e MHoro-
'1JJeHOB nJJ5I Toro, I '1T06bl no KoacpcpHUHeHTaM MHor04JIeHa H311TI1 Koact>-
CPHUl1eHThl ero pa3JIO)j{eHI1R no <jJ3KTOpI13JIbHbIM.
ECJIH MHOrO'lJIeH
f(x)=x"+6x 2 -7x+8
30
I1C'-I!1CJlEH!1E P A3HOCTEfi
[TJI. t
.n:eJII1Tb H3 X - 2 n:o Tex nop, nOKa OCTaHeTCH JIl1ll1b KOHCTaHTa, MbJ no-
nY'-ll1M
x 4 +O.x.+6x 2 -7x+8 \ x-2=j],eJlHTeJlh
- x 4 _ 2x. X.+2X2+ 10x+ 13 = 4acTHoe
2x. + 6x 2
- 2x. - 4x 2
lOx 2 - 7x
- 10x 2 - 20x
_ 13x + 8
13x - 26
34 = OCTaToK,
T. e.
I Jx) = x 4 +6x 2 -7x+8 - x3+2x2+10X+13+
x-2 x-2 x-2
I1JII1
MHOrO'lMH )1emnenb lI3CTHoe
j -----r- j
f (x) = x 4 + 6x 2 - 7 x + 8 = (x - 2) (x 3 + 2x 2 + lOx + 13) +
OCT3TOK
-y-
34
DpH X = 2 nOJIy'-IHM
(2)= 34,
TaK 'ITO OCTaTOK OT .n:eJIeH11H Ha x - 2 eCTb B TOt:fHOCTH (2). TaKHM
o6p a30M, Mbl HMeeM cJIe.n:ylOllJ,YIO TeopeMy.
T e 0 p eM a 06 0 C TaT K e "'). Ee/lu .Mli020'l/leH. f (x) paaOe/lUmb Ha
x - a, mo nOAY'lUmCfl
(x)=(x- a)Q (x) + R,
zoe Q (x) ecmb 'taemH.oe, a R - ocmamOK U
f(a)=R.
ECAU R=O, mo a eemb KopeHb (x)=O.
<popMa 3anHCI1 n:eJIeHI1H MO)J{eT 6bITb ynpOllJ,eH3, eCJlH 3aMeTI1Tb, '-ITO
CTeneHI1 x MO)J{HO 6b1JJO 6bl He nHcaTb (npl1 YCJIOBI1H, '-ITO OTCYTCTBY-
IOIlJ,He CTeneHl-l cHa6)J{aIOTCH HYJIeBblMU K09CPCPUll.UeHTaMU). KpOMe TOro,
K09cpcpl111.l1eHTbl qaCTHOro Ha BepxHeti /JUHUU COBnan:aIOT co CTapllll1MI1
K09cpcpHll.l1eHTaMH Ha COOTBeTCTBYlOllJ,HX JIHHHHX cxeMbl 11 He HY)J{n:aIOTCI1
B 3anHCH. HaKOHell., BMeCTO x - a MO)J{HO npOCTO nHcaTb a H CKJJa.n:bl-
BaTb, BMeCTO Toro 'lT06bl Bblt:fHTaTb BHYTpl1 Bblpa)J{eHHI1. Tor.n:a 3anl1Cb
npHHHMaeT BH.l1
LlemneJIb 2 I 1 0 6
2 4
4acTHoe = 1 2 10
-7
20
13
8
26 (1.6-1)
34 = R = OCTaTOK
*) Y Hac ee npUHaTO HaablBarb reopeMofi Seay. (flpultl. ped.)
t 1.6]
EnEHME MHOrO nEHOB
31
TaK KaK R = I (a), TO npHBe}J.eHHbltl aJIrOpHTM MOJKHO paCCMaTpliBaTb
I,aK cnoco6 Bbl HCJIemIH 3HaT.JeHfIH MHOrOT.JJIeHa. B 9TOM CJIYT.Jae CxeMa
npHHI1MaeT BH}J.
Y (X) = {[(X + O)x + 6] X - 7} X + 8. (1.6-2)
.
3aMeTI1M, C O}J.HOti CTOpOHbI, T.JTO }J.JIH n = 6 9Ta cxeMa *) He IIBJIlIeTCH
HafI60JJee 9KOHOMHOti IIO T.JfICJIY }J.etlcTBfItt, eCJIfI MbI XOTfIM MHorOKpaTHO
lJbIT.JfICJIIITb 3HaT.JeHi-lII O}J.Horo fI TOro JKe MHOrOT.JJIeHa wecToti CTeneHfI **)
(HalIpfIMep, B 6H6JIHOTel\Hot! nporpaMMe). TaK, cxeMa
Pl=x(x+a 1 ),
P 2 = (PI + X + a2)(P I + as),
P =(P2+a4)(Pl+a5)+a6
HCIIOJJb3yeT Ha O}J.HY OI:epaU:Hlo MeHbwe. K09<p<pfIU:fIeHTbI ai MorYT 6bITb
H a tJ}J.eHbl T.Jepe3 K09<P<PI1U:HeHTbl Ai MHOrO'lJIeHa
P (x) = X6 + A 1 x 5 + A 2 x 4 + Asxs + A4X2 + A5X + A6'
BblT.JHCJIeHl1e K09<P<PI1ll.l1eHTOB ai OCTaTO'lHO CJIOJKHO, HO ero HYJKHO c}J.e-
JIaTb TOJIbKO O}J.I1H P a3.
I1pI1Be}J.eM eIll,e O}J.I1H npHMep HCIIOJIb30BaHHH }J.eJIeHI1H MHOrO'lJIeHOB.
ECJlH HYJKHO BblT.JI1CJlHTb 3HaT.JeHfIe MHOrOT.JJIeHa P (X) C }J.etlcTBHTeJIbHbIMH
K09<P<PHU:HeHTaMH B KOMIIJJeKcHotl TOT.JKe
Xl = a + ib
(i = -V - 1 ),
TO MOJKHO 06pa30BaTb KBa}J.paTHblti TpeXT.JJIeH
[x 2 _ 2ax + (a 2 + b 2 )]
H, pa3}J.eJIHB MHorOT.JJIeH Ha 9TOT }J.et!CTBHTeJIbHblti KBa}J.paTHbltl TpeXT.JJJeH;
nOJIYT.JfITb
P (X) = [x 2 - 2ax + (a 2 + b 2 )] Q (X) + rlx + r2'
(COKparn.eHHYIO <POPMYJIY, nO}J.o6HYIO (1.6-1), MOJKHO HanHcaTb
paTHoro TpeXT.JJleHa.) I10JlOJKfIB X = xl> Hatl}J.eM
P (Xl) = r 1 x 1 + r2'
H }J.JIIi KB a}J.-
B TaKOM CJIY'-lae
P(X 1 )+P (X 2 ) = rl (Xl +X2)+ 2r2= 2 (rIa + r 2 );
l\aCTO 6blBaeT HYJKHO Bbl'-lHCJIHTb He P (Xl), a HMeHHO STY BeJIHT.JHHY.
*) Y Hac ee npHHRTO Ha3blBaTb cxeMOH ropHepa. (npUM. pea.)
* ) H 6J11o)(eHHe npHHa)(Jle)/{IIT Mou,KHHY. CM. J. To d d, Motivation for
WorkIng 1IJ Nume\-ical Analysis, Commun. Pure and Appl. Math., vol. 8,
pp, 97-116 (1955). [PYCCKHH nepeBoJJ,: To)J, JJ" MOTIIBbl )(JlR pa60Tbl B 0611acTI1
IJHCJleHHoro aHaJlH3a. MaT. npOCBelIl.eHHe, BbIIJ. I (1957), 76-86. CM. T8K)/{e
B. 5L rI a H, HeKoropble CXeMbJ )(JlR Bbl1JHCJleHHH MHOrOIJJleHOB c BelIl.eCTBeHHblMH
K03<!J<!J I1I .tHeHTaMI1, 1J.AH CCCP 127, .N2 2 (1959), 266-269. (npUM. pea.)]
32
I1C411CJIEHME P A3HOCTEI1'
[Tl!. 1
BepHeMC5I Tenepb K 3a,u:a'le pa3JIO}j(eHI1H paHHOrO MHOrO'lJIeHa B CYMMY
no x(n). YT06bl c.aeJIaTb &TO, Ha.ao pa3peJIHTb nOCJlen.OBaTenbHO lHoro-
tJJleH Ha x, tJaCTHoe Ha (x - 1), cne.aY!OllJ.ee tJaCTHoe H3 (x - 2) H T. .a.,
c.aeJJaB n warOB. TaK KaK peJJeHHe Ha x TpHBHaJlbHO, TO JI.Jl5I npHBeJl.eH-
Horo Bblwe MHorOtJJIeHa
P(x)=x 4 + 6x 2 -7x+8,
6epH no OlJepeJl.H J1.eJlHTeJlH 0, I, 2, 3, nonYlJaeM
o
1
1 7 /
2 6
31 13
3
11 6
HJm, B COOTBeTCTB!1!1 C (1.6-2),
P(x)=({[l (x -- 3) + 6](x - 2)+ 13} (x - 1) + 0) x+ 8,
P (x) = X(4) + 6X(3) + 13x(2) + 0 . x(i) + 8.
2
6 -7 I
1 7
o
+8
3
(1.6-3)
(1.6-4)
PaBeHCTBO (1.6-3) .aaeT y.n06HY!O <j:>OPMY .II.JIH BbItJHCJlHTeJlbHbIX pa6oT, a
(1.6-4) 60Jlee OTtJeTJlHBO BbIAeJI5IeT K09<j:><j:>HuHeHTbI.
Ynpa:dmeHIHI
1.6-1. IlpepcTaBHTb x 3 B BIIJIe CYMMbl 4JaKTOpHaJlbHbIX MHorO'liIeHOB.
o T BeT: X'81 +3x(21 +X(1I.
1.6-2. IlycTb P (x) = x" - 5x 4 +3x 2 + I; Bbl'JHCJlI1Tb P (1 + i) +P (f -I).
o T BeT: 34.
1.7 *. £IHeJIa CTHpJIHHra nepBoro pona*)
TaK KaK QJYHKUHH x(n) Ba}j(Hbl B I1C4HCJIeHHH pa3HoCTet!, a x"-
B AH<j:><j:>epeHUHaJIbHOM HClJHCJIeHHH, HCHO, lJTO Mor JIH 6bJ 6blTb nOJIe3HbI
COOTHoweHHH Me}j(.ay 9THMH <j:>YHKUHHMH. YTo6bl BbIpa3HTb x(n) lJepea
CTeneHH x, HamJWeM
n
x(n) = S(n, k)x k (1.7-1)
k=O
H Bbl4HCJIHM K09<j:><j:>HuHeHTbl S (n, k). 3TH K09<j:>QJHUHeHTbI Ha3bIBa!OTC!I
If.UCAaAf1I CmUpAUli2a nepB020 poiJa. llJl5! n = 1 HaXOll.HM
x(1)=x=S(l, O)+S(l, 1) x,
oTRypa S(l, 0)= 0; S(I, 1)= 1.
*) Ilaparpa4Jbl !.7 H 1.8, IJOMC'IeHHble 3SC3110'lKOn, He 6YllYT HCUOJIbSo-
BaTbCfI B reKCTe HHrJ\C, KpOMe 1.9.
1.7]
'lI-1CJ1A CTI-1PJ1I-1HfA f1EPBOfO POJJ,A
33
IlJlH n = 2
x(x 1)=x 2 -x=S(2, 0)+S(2, 1)x+S(2, 2)x'1,
OTKY.II. a
S(2, 0)= 0;
S(2, 1)=-1;
S(2, 2)= 1.
BMeCTO TOro '1To6bJ .n:eticTBoBaTb TaKHM )j{e o6pa30M .II.aJJbWe, nOJJY'HlM
o61l1.ee peKyppeHTHoe COOTHoweHHe, Hanl1caB
x(n+1) = (x - n) x(n).
OTCIO.rra, HCrrOJlb30BaB .II.Ba)j{.II.bl (1.7-1), nOJlY'IaeM
n+l '.'1> n
S(n+l, k)xR=(x-n) S(n, k)x R =
k=! R=\
n
= [S(n, k-1)-nS(n, k)]xR+S(n, n)x n + 1 -nS(n, 0). (1.7-2)
R=l
O'leBHJ!,HO, 'ITa J!,JJH Bcex n cnpaBeJ!,JlHBO S(n, n)= 1, TOfJ!,a KaK .lI.JJH
n>O HMeeM S(n, 0)=0. f1pl1paBH5IB K09<P<PHll.HeHTbI B ypaBHeHHI1 (1.7-2),
nOJlY'IHM peKyppeHTHoe COOTHOllleHHe, KOTopoe J!,aeT B03MO)j{HOCTb Ha-
XO.n:I1Tb 'IHCJla CTHpJIHHra nOCJIe.n:OBaTeJJbHO:
S(n+l, k)=S(n, k-l)-nS(n, k) (k=1, 2, ..., n).
(1.7 -3)
IlJI51 'IuceJl CTl1pJI!1Hra HeT npoCTotl <POPMYJIbl, KaK J!,JI5I 6HHOMHaJlbHblX
KOs<P<PHIJ,HeHToB. Ta6JIHIJ,a 1.7.1 .n:aeT HeKoTopbIe U3 STHX 'IHCeR.
T a 6 II H U, a 1.7-1
qHCJIa CTHpJlHHra nepBoro pOAa S (n, Il)
I 2 3 4 D
1 I
2 -1 I
3 2 -3 1
4 -6 11 -6 1
5 24 -50 35 -10 1
Ynpa>KHeHHe 1.7-1. PacnpOCTpaHI1Tb Ta6J111I.J,y 1.7-1 eme Ha o!l.HYCT(!' oIL lt'
!l.JlR n=6.
2 P. B. XCMMli!lr
34
11 C4I1CJ!EHI1E P A3HOCTEi%
(rn. I
, 1.8 *. IJJfcJla CTJfpJlJfHra BToporo pOJJ:a
lJllCila CmllpllUH.Za 8mopozo poiJa Bblpa)\(atOT x" qepe3 cpaKTOpHaJIb-
HbIe MHOr04JIeHbI
n
x n = 2: cp (n, k) X(k).
k=1
(1.8- 1)
KaK Ii B npe.n:bI.lJ:yrn,eM naparpacpe, BbI4HCJIHM CHa4aJIa HeCKOJIbKO 3Ha4e-
HHt!, a 3aTeM Hat!JJ:eM o6rn,ee peKyppeHTHoe COOTHorneHl-Je.
D.JISJ n = 1 I1MeeT MeCTO x = 'f (1, 0) + cp (1, 1) x, oTKY.lI.a cp (1, 0) = OJ
cp(l,l)=1. JJJIR n=2 HanHrneM x 2 =cp(2,0)+cp(2,1)x+
+cp(2,2)x(x-I), oTKYJJ:a cp(2, 0)=0; 9(2,1)=1; cp(2, 2)=1.
PeKyppeHTHoe COOTHorneHne CJJe.lI.yeT n3 paBeHcTB3 Xn+l = x . x n :
n+1 n n
'f (n + 1, k)X(k) =x 2: 'f (n, k) X(k) = 2: 'f (n, k)(x - k)X(k) +
k"=1 k=1 k=1
n n n
+ kcp (n, k) X(k) = cp (n, k) X(k+1) + 2: kr.p (n, k) x(P) =
k.=1 k=1 k=1
n
= 2: [9 (n, k - 1) + kcp (n, k)] X(k) + 'f (n, n) x(n+l) - SO (n, 0).
k=J
s:ICHO, tlTO SO (n, n) = 1 .lI.JIIJ JIlo6oro n, Tor.lI.a KaK J:J:JJSJ n> 0 HMeeM
'f (n, 0) = O. TIpnpaBHSJB K09rpqml1.neHTbI npn nO,lJ,06HbIX 'weHax 8
lPaKTOpHaJJaX (TaK KaK OHH JIHHet!HO He3aBHCHMbI), rrOJIY4llM peKYP-
peHTHoe cOOTHorneHlle
'f (n + 1, k) = SO (n, k - 1) + kp (n, k). (1.8-2)
1--13 SToro COOTHorneHnSJ MO)\(HO rrOCTpOHTb Ta6JJHUY 4llCeJI CTllpJIHHra
aIoporo pO.lI.a (CM. Ia6mm:y 1.8-1).
Ta6J1I1u.a 1.8-1
4HCJIa CTHpJIHHra BToporo pOJJ:a
I 2 3 4 f>
1 1
2 I 1
3 1 3 I
4 I 7 6 1
5 I 15 25 10 I
,.
YnpamHeHue 1.8-1. PacnpocTpaHHlb la6J1uu,)' I. -l eUJ,e Ha O.D.HY CTPOKY,
AMI Ii = O.
. 1.9]
OPI1MEP
35
1.9. fipMMep
TIpHBe.l1eM npHMep HCnOJJb30BaHHSI cpOpMYJIbl HblOToHa II 4HCeJI CTHp-
mlHra. 3a.l134a COCTOHT B TOM, 4T06bl Bbl4HCJIHTb
Y
g(y) = - d d- \ ; dx,
y y-x
O y I,
eCJII1 3Ha'IeHHSI lex) 3a.l13Hbi B TO'IK3X X = 0; 0,1; 0,2; ...; 1, O.
O'IeBH.l1HO, 'ITO npl1 BepxHeM npe.n.eJIe, Kor.n.3 X = y, nO.l1blHTerpaJIb-
Hoe BblpmKeHl1e CT3HOBHTCSI Heonpe.n.eJIeHHbIM. 33MeTHM T3K)J{e, 4TO pe-
8yJIbT3T HHTerpHpOB3HHSI .l10JJ)J{eH 6bITb npo.n.l1cpcpepeHlJ.HpOB3H no y; HO
eCJII1 .n.HcpcpepeHJJ.Hp0I33n no y no.l1 8H3KOM HHTerp3JI3, TO nOJIY'IeHHbltJ
B pe3YJlbT3Te HHTerp.1J1 6y.n.eT P3CXO.n.I1TbCSI. TaKI1M 06p330M, npe.l1J1O)J{eH-
HaSl 33J1.a'I3 Tpe6yeT HeKOToporo BHI1MaHI1S1.
H34HeM C 33Me'I3HI1S1, 4TO eCJlI1 6bl f (X) 6blJIO paBHO x1l, TO HY)J{HO
6blJIO 6bJ Bbl4l1CJI11Tb npOCToe Bblpa)J{eHl1e
y
d \ xndx
g(y) = dy V;=X .
3.n.eCb eCTeCTBeHHO nOJlO)J{I1Tb
x=y sin 2 B,
dx = 2 Y sin B cas B dB,
Ii Mbl nOJlY4I1M, 4TO n:JISI f (x) = x1l HCKOMaSl cpYHKJJ.I1H p3BHa
,,/2
d 2"n+l \ ( 2 + 1 ) 11
g(y) = dy }fy sin 2M! e de = n Vy Y W 2n +1>
rone
" 2
2M! 2 . 4 '" (2n)
W 2n + 1 = sin B dB = I . 3 . 5 ... (2n + 1)
o
2n
2n + 1 2n-1>
W 1 =L
n.aJIee, aaMenlM, 4TO eCJl\1
10
l(x)= a n x 1J ,
11,=0
TO
10
g(y)= v 1I 0 (2n + 1) W 2n + 1 anY n .
ECJlH Tenepb .n.onYCTHTb, 4TO I1CTI1HH3S1 cpYHKJJ.I1S1 f (x) MO)J{eT 6MTb
aurlpOKCI1MHpOB3H3 .a.OCTaT04HO T04HO MHOr04JIeHOM .a.eCSlTotJ CTeUeHI1 no
2. ,
36
I1C4I1C.flEHV1E p A3HOCTEH
[fJJ. I
,a:3HHbIM Y3J/OBbIM r04K3M, TO OCr3erCH TOJ/bKO H3ttTH an' O,a:HH H3 cno-
COOOB H3XQ)K.a:eHI151 an - HCnOJlb30BaHHe 410PMYJlbl HblOTOHa (1.5-4)
C IllarOM no x = 0,1:
((x) = 1(0) + 10xb.j(0) + lOx (l -I) 2/(0) + ... + (10X)(10) 10 reO)
(B pa3HocTHX npl1HHMaeTcH h = 0,1), 4T06bl 3aTeM 4epe3 4HCJra CTHp-
JlHHra nepBoro po.a:a (ypaBHeHHe (1.7-1» nOJlY4HTb K034141 H uHeHTbJ an
npH CTeneHHX x. f{aK oy.a:eT nOKa3aHO n03)1{e, HCnOJlb30BaHI1e ,a:eCHTbIX
pa3HocTett MO)l{eT OKa3aTbC5I onaCHblM; HO B 3a.a:a4e, H3 KOTOpOtt pO.a:HJlC5I
9TOT npHMep, r.a:e .a:aHHble OblJlI1 rpyoblMI1 11 O)l{l1.a:aeMbltt pe3YJlbTaT He
HaMHoro JJY4Ille, OH BblfJl5l.QeJl Y.QoBJleTBOpI1TeJlbHbIM. 06 9TOM MO)l{HO
Cy.a:HTb no cJle.a:YIOU(I1M 41aKTaM: (1) rp a41HK almpoKcHMHpYIOill.ero MHoro-
4J1eH3 Bblr JlH.a:eJJ P33YMHO; (2) pe3YJlbTaTbl OblJlH COBMeCTHMblMH C .Qpy-
rHMH 43CT5IMI1 41H3114ecKott TeOpHI1, KOTopaH 113Y4aJl3Cb, H (3) pe3YJlbTaTbl
CTHMym1pOBaJIH .a:aJJbHettIIIYIO pa60TY,
1.10. AJIbTepHaTHBHbIe 3aMeqaHHH
Bbl60p onpe.a:eJIeHHH pa3HocTHOfO onepaTopa (paBeHcTBo (1.2-1»
/(x)=/(x+h)- I (x)
OblJI npOH3BOJIbHbIM. MO)l{HO OWfO Obi BMeCTO 9TOfO BblopaTb 06paTHble
pasHocTH
Vf(x) = I (x) - I(x - h).
4TO npOU30llIJIO ObI, eCJIU ObI MbI 9TO c.a:eJIaJIu? B 'laCTHOCTH, B03paCTa-
1Oill.l1e 41aKTOpl'IaJlbHbIe MHOr04JIeHbI *)
(n)x=x(x+h)(x+2h) ... [x+(n-l)h]
.lI.aJIH ObI, aHaJIOrH4HO (1.5-1),
v(n)x = n [(n-1) xJ.
OTCIO.a:a MO)l{HO nOJJytJHTb oopaTHYlo 410PMYJIY HbIOTOHa, COOTBeTCTBYIO-
!1 YIO (1.5-4). AHaJlorl1'mo B03HI1KJIH 6bl HOBbIe 4HCJIa CTHpJlI1Hra nepBoro
H BToporo po.a:a (n, k) H 0 (n, k). OHI1 CB5I3aHbI co CTapbJMH 411CJJaMH
CTHpJIHHra paBeHCTBaMH
(n, k)=(_1)n+kS(n, k)=IS(n, k)\, o(n, k)=(_I)'Z+kr.p(n, k).
TaKHM 06pa30M, TeOpI1H, COBepmeHHO aHaJIOrH4HaH TeOpHH, KOTOpa51 6bIJIa
pa3BHTa BbIme, MorJIa ObI OCHOBbIBaTbC5I Ha OOpaTHblX pa3HOCT5IX. <Pop-
MYJIbI, KOTOpbJe nOJIY'JaIOTC5I H3 TaKotl TeOpHH, nOJIe3HbJ C!1eUHaJIHCTY,
1-10, Bep05lTHO, 51BJl5IIOTC5I pOCKOillblO .lI.JJ5I nOTpeOHTeJI5I,
*) B OTJJH'JI1C OT BBC)\CHHblX B 1.5 OHI1 0603HaQaJOTCii In 'x. (/JpU.M. pea.)
2.11
BBEI\EHI1E
37
UeHTpaJIbHble pa3HOCTH H Cpe.n:HHe pa3HOCTH npHBO.n:HT eUJ,e K .n:BYM
CI10C06aM 3arrHCVI, KOTOpble VlHOr.n:a npVlMeHHIOTCH, HO OHVI TaK)I{e HBJI5IIOTC5I
npOCTO cnOC06aMH 3anHCI1 H .n:alOT He60JIhllIOe y.n:06CTBO JIHWb B HeKO-
TOpblX CJlY4aHX; CJIe.n:OBaTeJIbHO, OHH, nO-BH.n:HMOMY, He CTOHT 5eCllOKot!-
CTBa rroTpe6HTe JI5I .
1.11. OOllI.lIe 3aMeQaHOH 0 C"paBKO
MbJ .n:aJJH JIHllIb KpaTKoe BBe.n:eHHe B HC'IHCJJeHHe KOHe4HblX pa3Ho-
CTet!, XOTH Ha ::ny TeMY Hanl1CaHbI lJ.eJIhle KHHrH. OlJ:HOtJ H3 JIY4llIHX
IJBJIlJeTCIJ KH!1ra )Kop.n:aHa [19] *).
VlC4!1CJIeH!1e KOHe4HhlX pa3HocTet! MO)l{eT 6hITb np!1MeHeHO .n:JlSJ no-
JlY4eHI1IJ MHorHX !1HT peCHhlx pe3YJIhTaTOB, TaK!1X KaK
,
n
S(n, k)<p(k, m)=a(n, m),
k=O
r.n:e a (n, m) - «Cll.tf80/l Kpo , HeT<:epm> paBHblt! HYJIIO, eCJJl-l m -=F n, H
e.n:HImll.e, eCJIH m = n. CnpaBe.n:JlHBOCTh cpOPMYJIbl cJIe.n:yeT H3 npoCToro
3aMe4aHHH, 4TO eCJJI-J pa3JIO)l{!1Th cpaKTOp!1aJl no CTeneHHM x, a 3aTeM,
Ha06opOT, CTeneHH x no cpaKTOpHaJlaM, TO Bce K09cpcpHUl1eHThI, KpOMe
K09cpqmlJ.I1eHTa npH Ha'laJlbHOM cpaKTOpHaJIe, 05paTIJTCH B HYJIb. TaKHe
pe3Y.'lbTaTbl pelllw npl1MeHlJlOTCIJ B npaKT!1'1eCKI1X BhI4HCJleHI1SJX, HO' HHTe-
peCHbI C r04K!1 3peH!151 MaTeMaTHK!1.
MHorne KH!1rH HCnOJIb3YIOr onepaTopHble MeTO.n:bl. Mbl HCnOJIb3yeM
JlHWh oneparopbl II !1 E B HeCKOJlbKI1X npOCTblX KOHe4HblX CJlY4a5lX H
rlU.areJIbHO H36eraeM npHMeHeHHH llO.n:03pHTeJIbHbIX 5eCKOHe4HhiX npOlJ,ec-
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Ta6JII1na 4HCeJI H Ka)f{.n:oe 4.HCJJO HMeeT 01lll16KY oKpyrJIeHH5I, He 3aBHC5I-
w.y OT OCTaJIbHblx. 3TO - Ta 4aCTal! B rrpaKTH4.eCKotl pa60Te cHTyaUH5I,
KOr,lJ,a HY)f{HO COC4HTaTb MHoro BapHaHToB 3a,lJ,a4.11 ,lJ,JI5I paBHOOTCTOI!-
w'HX Y3JIOB He3aBHCHMOro nepeMeHHoro. TaKHM 06pa30M, MbI rrbITaeMCH
42
norPEU!HOCTI1 OKPYrJJEHI1
I r JJ.
OTBenITb H3 CJlelI.yIOIllI1t1 BOnpOC: nYCTb l1.aHa Ta6mma OTBeTOD BbI'UlCJIeHfUJ
f (x) 11)1\1 rpynnbJ paBHOOTCTOHIlUIX X; liTO HBJI5!eTCH npel1.nOJlar3eMhIM
«WYMOM oKpyr JJeHH5!» B "TI-IX OTBeTax?
3aMeTHM, '-ITO Mhl Hl1KOHM 06pa30M He IIhITaeMC5J Hat1TH HHKaKUX
OJCTeMaTI14eCKHX 01ll1160K. MhI XOTHM J1HIllb, nOJIY4HB pe3YJJbTaT H .ll0-
63B11B ewe HeMHoro BhI4I1CJIeHHI1, OIleHHTb mYM B OTBeTax.
Ta6Jl11Il3 1.4-1 nOK33hJBaeT, KaK e.llHHCTBeHHa5! oIlIl16Ka p3cnpOCTpaHH-
eTCH B Ta6J1HUe pa3HOCTeU. MCIIOJIh3Y5! J1HHel1Hoe CBOI1CTBO onepaTopa ,
MO)!{HO npel1.CTaBI1Th Ta6JJl1U,y pa3HoCTel1 3aTa6y JIIJpOBaHHoti CPYHKIlHI1
KaK CYMMY Ta6JIHUhl pa3HOCTeti HCTHHHOt! lj'JYHKUI1H H n6Jl1-1Uh! pa3HocTeti
WYMa. 9Ta nOCJIeLI:H5!51 B CBOIO 04epeLI:b MO)!{eT 6hITh npeLI:CTaBJleHa KaK
CYMMa Ta6J1HLl pa3HoCTeti 3JJeMeHTapHbiX WYMOB, npH4eM '-51 Ta6JJl1ua
cooTBeTcTByeT WYMY e, Ha roM MeCTe B nepBOM cToJI6u.e Ta6Jl1llJ,hl IllYMa.
CYMMapHhlfi WYM B k-fi pa3HOCTI1, B COOTBeTCTBHI1 C (1.4-1), eCTb
k
k en = h (_l)k-r C(k, r) $n+" (2.8-1)
,=0
Ero rpaHHIlhl MorYT 6b!Tb JJemo Hat1LI:eHhl 113 orpaHH4eHH51 1Ia OTJI:eJJhHbll1
WYM B Ta6JIHUe. ECJlI1 $, OrpaHHlIeHO BeJIH4HHOH e, T .e. I $, I e, TO
k
I ken l hC(k,r)e=2ke. (2.8-2)
r=O
llJI!l k = 10, Hanpl1Mep, 3TO npe.llCTaBJIHeT TblC5JlJeKpaTHoe YBeJll14eHHe IllYMa.
TIOCKOJIhKY rpaHHUhJ IllYMa B k-11 pa3HocTH TaK neCCHMHCTI14Hhl,
HMeeT CMblCJI B03BpaTHThC51 K CTaTHCTH4eCKOMY nOl1.XOLI:Y ECTeCTBeIlHO
npel1.nOJIO)!{I1Th, 4TO OTpHuaTeJIbHa5! 11 nOJIOllWTeJIbHaH olliu6KH en palmo-
BepO!lTHhI, H n03TOMY CpeLI:Hee 3HalleHI1e M (e) MHO)!{eCTBa en eCTb Hy.1b.
Hac HHTepecyeT l1.HCnepCHH 0 ( k an) pacnpel1.eJIeHI1S1 k-it pa3HOCTll. f1YCTb
II:HCnepCHSI 0 (en) WYMa oKpyr JIeHHH an eCTb ,,2, TaK 'ITa
,,2 = 0 (an) = M.(e ) - M2 (an) = M (e ),
rLI:e Cpe.l!,Hee 3Ha4eHHe 6epeTCSI, KOHe4HO, no MHO)!{eCTBY. llJISI .l1HCnepCHH
k-tl pa3HOCTI1, HCnOJIb3Y5I (2.8-1) H TOT <paKT, 'iTO Cpe.l1Hee 3Ha4e-
HHe Sn eCTb HYJlb, nOJIYllHM
k
o ( k Sn) = M [LJ (- ll-r C(k, r) en+rr.
O _
BbInOJIHHB B03Be.l!,eHHe B KB8l1.p8T, HaXOl1.HM
M [ (- 1)'( - II C(k, r}C(k, s) e n +, $n+s] =
t,_
= M l h (k, r) e .w+ 2 h (- 1)'+5 C(k, r) C(k, s) Sn.w £n+8l.
, ,<s
!I 2.8'
OI..\EHKA illYMA B TABJ1HI..\E
43
LI:JJ5I ynpoUleHlHI 9TorO BblpmKeHH5I HcnoJlb3yeM CJle.n:YIOUlHe 06CTO-
51TeJlbCTBa. no npe.n:nOJImKeHHIO UlYMbl oKpyrJleHHII IOn He3aBHCHMbi H
M(6 n )=0. OTCIO.n:a CJle.n:yeT, 4TO M(6 n + r 'Sn+s)=O rrpH r#8. KpoMe
TorO, 113BeCTHO, 4TO
k
L: C2(k, r)= C(2k, k)
r=O
(ynpa)l{HeHl1e 2.8-2). TaKHM 06pa30M, nOJlyqaeM
M [ C2(k, r) 6 ] =
r
= L:C 2 (k, r)M(E ) =a 2 C(2k, k)= 2k a2. (2.8-3)
r'
Ta6J1HlI.a 2.8-1 .n:aeT npe.n:CTaBJl eHl1e 0 p OCTe K09cpcpHlI.HeHTa YCHJleHH5I
wyMa, KOTOpbltl eCTb npOCTO -Vc (2k, k).
T a 6 JI H U. a 2.8-1
I{09<f!qJHIJ,HeHTbl YCHJIeHHSI oKpyrJleHHSI
nOpH- CpeJ\HeKBaJ\- nOp"- CpeJ\He-
KBaLl.paTHQ.
J\OK LJ,HcnepCH" paTHI.IHoe MaKCHM)TM J\OK LJ,HcIIepcHH Hoe 3naqe- MaKCHMYM
pa3- C 121,. k1 3Haqem1e mYMa wYMa 2 k pa3- C (2k. II) HHe wYMa wYMa 2 k
nOCTH VC( 2k. k) HOCTH V C (2k. Ii)
I 2 1,414 2 I 4 70 8,367 16
2 6 2,449 4 5 252 15,875 32
3 20 4,472 8 6 924 30,397 64
TeopeTH4eCKHtI pe3YJlbTaT, KOTOpbltl 3.n:eCb nOJlyqeH, Tpe6yeT ycpe.n:-
HeHI-HI BeJlHqIJHbl lI3 MHO)KeCTB3 IOn- Ha np3KTnKe 9TO C.n:eJl3Tb HeB03-
MO)KHO 11 Mbl npn6eraeM K 06bl4HOMY B TaKOt! CI1TyaUHn npHeMY: 3aMe-
HSJeM Cpe.n:Hee 3H3qeHne MHO)KeCTBa IOn Cpe.n:HnM 3H3qeHneM CPYHKUUI1
(CTaHLJ,apTHbltt npIJeM, HCnOJlb3YIOUlI1t!CII B 9prO.1l.I1qeCKOtl TeOplm).
MO)KHO 6blJJO 6bI .n:YM3Tb, 4TO Cpe.n:Hee 3Ha4eHI1e k-x pa3HOCTet!
B Ta6J1HIJ.e 1.4-1 .n:aeT HenJlOXYIO oueHKY cpe.n:Hero 3Ha4eHI1SJ MHO)KeCTBa
(B npe.n:nOJlO)KeHHH, qTO npUMeH5IeTC5I 9pro.n:U4eCKut! npUHunn), HO, K co-
)KaJleHHIO, k-e pa3HOCTn CHJJbHO KOppeJll1pOBaHbI.
LI:et!CTBIJTeJlbHO, K09cpqmll.I1eHT KOppeJl5ll1.HH .lI.Jl5I CMe)KHbIX 3Ha4eHutl
paBeH
"" (- 1)' C (k, r) (_1)'-1 C (k, r-I) __ C (k, r) C (k, r-I)
.1 C (2k, k) V C (2k, k) - "':' C (2k, k) _
C (2k, k + I) k
=- C(2k, k) =- k+1
(2.8-4)
44
nOTPEUHlOCTM OKPYTJJEH!15!
ITJ!.
(CM. ynpall<HeHI-!e 2.8-3). KorJI.a k - 00, K09cpcpluJ,HeHT KOppeJUIu,tUl CTpe-
MHTCSI K -- I H rIOKa3bIBaeT, 'ITO 'IHCJIa B k-M cToJl6u,e C JI Y 'I a.lt H bl,
H 0 He He 3 a B H C H M bl, KaK 'IaCTO rIOJlaralOT; OHI1 HMelOT YCTOtt'IHBYIO
TeHJI.eHu,HIO 'Iepe.n:OBaTb 3HaKH +, -, +, -, ...
CHJJbHa51 OTpHu,aTeJlbHa51 KOppeJISlUH51 y rrOCJleJI.OBaTeJlbHbIX k-x pa3-
HOcTett JI.aer B03MQ)KHOCrb 06HapY}j{HBaTb OTJI.eJIbHble OIlIH6KI1 B T36JIHue.
SrOT MeTOll JlY'Iwe Bcero rrOHHMaeTCH Ha rIp HMepe. PaCCMOTp I1M 1'36-
mIUY 2.8-2, r.n:e, KaK 06bl'IHO, I1HWYTCH JlHWb 110CJJe.n:Hl-le 3HaKI1 pa3Ho-
CTett. IlOBelleHI1e '1eTBepTblx pa3HocTett, HaH60JJbWHe 3Ha'leHHSI KOTOpbIX
crpYl1rrI1pOBaHbI OKOJIO 350., 3aCTaBJIHeT I1peJI.rIOJIaraTb oWH6KY B CPYHK-
UHH JI.JJ51 350. Ilo9ToMY Mbl rrpe1J.rIOJIaraeM, '1TO qeTBepTbIe p a3HOCTH
rrOCTOSlHHbJ I1 '1TO rIPI1 350. eCTb o1lHl6Ka. TaK KaK CTpYKTypa 'IeTBepTblx
pa3HocTet! H3BeCTHa (ypaBHeHHe (1.4-3»,
1, - 4, 6, - 4, 1,
TO MO}j{HO nblTarbCH npH6J1H}j{eHHO yn:OBJlerBOpnTb CJlen:YIOw.UM ypaB-
HeHHHM:
41 = -48 + C, - 59 = 68 + c, 37 = -48 + C.
PeruaSl I1X, rrOJlY'laeM rrpH6J1H3I1TeJlbHO 8 = - 10, TaK '1ro rrpaBI1J1bHOe
3Ha'IeHHe CPYHKUI1H B 350. Jl:OJl}j{HO 6bITb 6,421. IloJI.craBI1B 6,421 BMecro
6,411 ,nOJIY'IHM Ta6J!I1UY P a3HocTett:
-4
8
-50
-468 9
6,421 -41
-509 10
-31 -3
7
0
Ilpon:eMoHCTpnpoBaB CHJlbHYIO KOppeJlHu,HIO, KOTOpMI cyw.ecrByeT
Me)l()l,y nOCJlen:OBaTeJlbHbIMH 'IHCJlaMH B CToJI6IJ,e k-x pa3Hocret!, H rrpo-
HJJJJIOCTpHpoBaB, KaK aTHM MO}j{HO BOCrrOJIb30BaTbC51, BepHeMcH K Hawett
rJlaBHott 3aJI.a'le ou,eHKH IllYMa B Ta6JII1ue. IlpeJI.BapHTeJJbHo rIpen:nOJlo}j{HM,
'1ro JI.HWepCHH fJ IllYM3 B ra6JII1u,e 3Ha'lelmtt CPYHKU,HH H3BeCTHa; MbJ
xorHM ou.eHlHb Jl:HWepCHIO IllYM3 B k-M CToJl6u,e. ECJJ!-! HCTHHHble 3Ha'le-
HHH CPYHKU,HH B HaIllet! ra6JJHu,e 6bJJJH 3H3'JeHHSlMH MHOrO'IJJeHa, ero pa3-
HOCTH, Ha'JHHa51 C HeKoToporo nop51n:Ka, JI.OJJ}j{HbI 6bIJJH 6bl o6parHTbcH
B HYJlb, a TO, '1TO OCraJJOCb 6bI B Ta6JJHu,e pa3Hocrei1, 6bJJlO 6bJ cJJen:-
CTBHeM., wYMa. 3aTpYJl:HeHHH COCT051T B TOM, 'JTO CPYHKUH51 He 06513a-
TeJJbHO 51BJJ51eTC51 MHorO'JJJeHOM; 60JJee TOro, JI,a}j{e eCJJH .9Ta CPYHKUHSI-
MHOrO'lJJeH, TO, He 3HaH ero CTeneHH, MbJ He 3HaeM, KaKoro fJOp51JI.K3
pa3HOCTH HY)I{HO B351Tb.
.2.8J
OUEHKA lliYMA B TABJJHUB
45
T a 6 JI 11 U, a 2.8-2
TeopeTH-
II eCKH e
fpa- c n (z) 6 6 2 68 U QCTBepTLle
JlYCbJ pa3HocTH
BCJle.ltcTBHe
OIIH:t6KH E
0 8,346 [
- 44
5 8,302 -87
-131 4
10 8,171 -83 3
-214 7
15 7,957 -76 -I
7;' 67 -290 6
20 -70 6
-360 12
25 7,307 -58 -14 +6
-418 -2
30 6,889 -60 41 -46
-478 39
35 6,411 -21 -59 +66
-499 -20
40 5,912 "-41 37 -46
-540 17
45 5,372 -24 -10 +6
I -564 7
50 4,808 -17 -2
-581 5
55 4,227 -12 °
-593 5
60 3,634 -7 -2
-600 3
65 3,034 -4 -2
-604 I
70 2,430 -3 2
-607 3
75 1,823 -0 -4
-607 - 1
80 1,216 -I 2
-608 1
85 0,608 0
-608
90
I
Ha npaKTHKe B 6JJarOnpH5ITHhIX CJJYlfa5lX pa3HOCTH HMelOT TeH.ll.eHlI.HIO
CHalfaJIa YMeHplliaThC5I no BeJlI1lfHHe, 3 3aTeM YBeJIHlfHB3ThC5I, O.ll.HOBpe-
MeHHO HCnhlThlB351 CI1JIhHhle KOJIe63HI151 113-3a OTp!1lJ.aTeJIhHot! KOppeJI5IlI.I1H.
B aTOM CJJYlfae 06hl4HO C411TalOT nepBhlt! cToJI6eu pa3HOCTet!, CJIe.ll.YIOw.HlI
8a MI1HI1MaJIhHhIM, cyw.eCTBYIOw.HM BCJIe.ll.CTBHe WYMa, XOTH, H3CKOJIhKO
1J3BeCTHO 3BTOPY, COOTBeTCTBYIOw.He HCCJIe.ll.OB3HH5I no aTOMY Bonpocy He
46
nOTPEUlHOCTI'! OKPYTJlEHI'!51
!Tn. ,
npOBOIlHJJHCb. TaKHM 06pa30M, CpellHee 3HaqeHHe KBallpaTOB pa3HOCTeil
B 9TOM cTOJJ6u.e Mbl npl1HHMaeM 3a Ol1.eHKY KBa.apaTOB pa3HoCTetl, ycpe.n:-
HeHHbIX no MHO)l{eCTBY. Pa3.l1.eJll1B 9TO cpellHee 3HaqeHI1e Ha C (2k, k).
nOJIyqaeM Ol1.eHKY KBa.apaTa ypOBHSJ WYMa B HCXO.aHOti Ta6J1Hl1.e.
T a 6 J] fI U a 2.8-3
raMMa-<t>YHKIJ.HSI r (x)
x I T(x/ I A 1 A2 I A3 I A4
1,0 1,000
-49
I, I 0,951 , +16
-33 -4
1,2 0,918 +12 3
-21 -1
1,3 0,897 +11 -1
-10 -2
1,4 0,887 +9 +2
-I °
1,5 0,886 +9 -2
+8 -2
1,6 0,894 +7 +2
+15 0
1,7 0,909 +7 +2
+22 +2
1,8 0,931 +9
+31
1,9 0,962
B KaqeCTBe npHMepa paccMoTPHM Ta6J1I1I1.Y oKpyrJleHHblX 3HaqeImA
r (x), npl1Be.ll.eHHblX B Ta6JJI1u.e 2.8-3. Cpe.a:Hee 3HaqeHHe [Air (X)]2 =
13
=(9 + 1 + 4 + 4 + 4 + 4): 6 ="3' Pa3.a:eJIHM ero Ha C(8,4) = 70:
3 o = 0,062 = ai - oIJ.eHl<3 WYMa B Ta6J1Hl1.e.
TeOpeTHqeCKaH Ol1.eHKa OCHOBaHa Ha paBHOMepHOM pacnpe.a:eJleHHlI
cnyqatiHoro oKpyrJleHlISJ
1/ 2
a = Xi dx = 1 1 2 = 0,0833.
_1/2
(2.8-5)
BBH.Il.Y Toro, qTO Ta6J/Hl1,a, B3RTaH B grOM npHMepe, HeBemiKa, 6mI-
BOCTh nOJlyqeHHblX Ou.eHOK
a1 = 0,25,
a2 = 0,29
MO)l{HO CqHraTb XOPOUIH,M nO.lJ.TBep)I{.a:eHHeM TeopHH. TOqHOe BblqHCJIeHHe
OUJI16KH oKpyr JIeHHH .aaeT
aa= 0,32.
,2.91
TEOPIUI .MJJA.D.WEfO 3HA4AW,EIO PA3PSI.D.A.
47
Ynpa)lmeHIHI
2.8-1. PaCKpbJB 06e '1aCTH paBeHcTBa (1+t)a+b=(l+t)a(l+t)b lIupa-
paBHHB K094J4Jf1lJ,HeHTbJ npH O.l\llHaKOBblX CTeneHIlX t, nOJlY'IHTb
a
C(a+b, r)= C(a, s)C(b, r-s).
5=0
2.8-2. B ynpa)/{HeHHH 2.8-1, nOJlO)/{IiB a = b = r, nOJlY'lHTb
r
C (2r, r) = C2 (r, s).
5=0
2.8-3. B ynpa)/{HeHHll 2.8-1, nOJlO)/{I1B a = b = n, r = n + 1, nOJlYQIlTb
n
C (2n, n+ 1)= C (n, s) C (n, 8-1).
I 5=0
2.9. TeopliH «MJIaJJ.mero 3HaQam:ero pa3pHJJ.a»
PaCCMOTpl1M OJJ:HY 113 .n:BYX HOBbIX Teopl1fi, ueJJb KOTOpblX I10Ka3aTb,
KaK pa3BHBaIOTC5I 3 cjJCpeKTbI oKpyr JIeHH5I. OHI1 MorYT 6bITb Ha3BaHbi
«TeopHI1 OKpyr JIeHH5I B MaJIOM». OTKa)l{eMC5I 3JJ:eCb OT CTaTI1CT114ecKoit
MO.n:eJI11 2.5 11 nocTpoHM Hawe MHO)l{eCTBO, nOJIar3<}jj 'I I1CJIa, HaJJ: KOTO-
pbIMI1 npOH3B01UITC5I apH<pMeTHlJeCKl1e onepaUHH, sapbl1pYJOUJ.I1MI1C5I. TaKI1M
06pa30M MO)l{HO 6Y1J.eT H3YlJaTb, KaK pacnpOCTpaHHIOTC5I OWI16K11 OKpyr-
JIeHI151 B Tl1nl1lJHOM He60JIbWOM KYCKe Bb]lJl1-
CJJeHHt!.
Kor.n:a BbI'IHCJJeHHb]e lJOCJIa oKpyr JI5IIOTC5I,
K3JKeTC5I pa3YMHbIM npeJJ:110JIOJKHTb, lJTO COBep-
WaeMa51 oWH6Ka (B .n:efiCTBI1TeJIbHOCTH 113MeHe-
HHe BCJIe.n:CTBue oKpyrJJeHI1H) paBHOMepHO pac-
npe.n:eJIeHa MeJK.n:y - 1/ 2 11 1/ 2 110CJIe.n:Hero 3Ha-
4aUJ,ero p33pll.n:a. LI.pyrnMI1 CJJOBaMI1, JJJ060e
aHalJeHl1e oW116KH CTOJJb )/{e B03MOJKHO, KaK
JII060e .n:pyroe B nOM I1HTepBaJIe. Ha BepOHTHOCT-
HOM H3bJKe BepOHTHOCTI> P {Xl ,s; c ,s; X2} =
= X2 - XI (XJ,s; X2 B (- J h, 1/ 2 » (pI1C. 2.9-1).
B 3TOtl Mo.n:eJIH .n:onycKaeTcH HenpepbIBHoe pacnpe.n:eJJeHl1e 11 I1fHopl1pyeT-
CR TOT 04eBH.n:HbIt! <paKT, lJTO .n:eticH!11TeJIbHOe pacnpe.n:eJ]eHHe MalllHHHoro
OKpyr JIeHHII .n:OJJJKHO 6bITb .n:I1CKpeTHbIM, TaK KaK MorYT BCTpelJaThC5I
JII1111b HeKOTopbIe oKpyrJJeHI1H (HanpHMep, 110TOMY 4TO MalllHHa onepl1pyeT
C 4HlJIaMH KOHe4Hofi .n:JIHHbJ
DOCMOTpHM Tenepl>, lJTO npOHcxo.n:HT, Kor.n.a CKJIa.n:blBaeTCH HeCKOJII>KO
'-lHeeJI. l3y.n.eM: npeJJ:nOJIaraTb, 4TO Bee lJHeJIa B nJIaBaI01ll.eM BHJJ:e HMeIOT
O.n:I1H H TOT )/{e nop5l1J.OK 11 4TO OH He MeH5IeTC5I HH B O.n.HOM 113 CJIO-
)/{eHHtI II BhI4HTaHHt!.
D.m HalJaJIa paCCMOTpl1M cJIOJKer-me Jl,BYX lJHCeJI. OlJeBH.n:HO, Oll1H6Ka
CYMMbJ JleJKHT B HliTepBaJJe OT - 1 .n:o 1 nOCJIe1J.HerO aHaLIaw.ero paapH.a.a.
P(x)
J
-JjZ
o
1/2 x
PHC. 2.9-1. nJlOTHOCTb pac.
npeJ!eJleHHH OIl1l16KH oKpyr-
JleHHR .l\JlR OLLHoro '1HCJla.
48
nOfPEUJHOCTVI OKPYfJJEHVlSI
Im.2
nYCTb $1 - OIllH6Ka nepBOrO CJraraeMOrO, e - OIllH6Ka BTOpOrO,
e - 01lH16Ka CYMMbI. P3CCMOTpHM Bep05lTHOCTb TOrO, qTO e = - a (a 0).
BbI60p e l Orp3HH'JeH I1HTepB3JIOM OT _I/ ,n:o I/ - a, 3 e onpe,n:e-
JI51eTC5I HCKJIIO'-fI1TeJIbHO Bbl6opOM el' TaKI1M 06pa30M, I1JJOTHOCTb BepOHT-
HOCTH eCTb
AHaJIorH'-fHO 110CTyrraeM rrpUMeHHTeJIbHO K nOJIO)l{HTeJIbHbIM OIlIUOKaM.
TaKUM 06pa30M, Mbl nOJJyqaeM TpeyrOJIbHOe pacnpe,n:eJIeHHe, H306pa)l{eH-
Hoe Ha pHC. 2.9-2.
BMecTo Taro '-IT06bI paCCMaTpl1BaTb CYMMbI Tpex '-fHCeJJ, 33TeM '-fenI-
pex, n5lTH 11 T. ,n:., 06paTHMC5I K lI.eHTpaJIbHot1 npe,n:eJIbHot1 TeopeMe,
KOTOpaf!, no Kpat1Het1 Mepe HHTYHTHBHO, WH-
pOKO U3BeCTH3. B HallieM CJIY4ae, Kor,n:a cpe,n:-
Hee 3Ha4eHHe OIlll16KI1 paBHO HYJIIO, 3Ta Teo-
pei'lIa r JlaCI1T, 4TO eCJJI1 '-fHCJIO OKpYfJleHHbIX
4UCeJI, KOTopbIe CKJIa,n:bIB3IOTC5I, YBeJIH4HBaeTC5I,
TO CPYHKlI.l151 pacnpe,n:eJIeHHH ,n:JI5I OIllH6KH CYM-
Mbl CTpeMHTCf! K HOpMaJIbHOMY pacrrpe,n:eJleHHIO
p {x e x+dx} = ,;_ e-1/2 (x/a)2 dx
a r 27t
p(X)
7
-7
PHC. 2.9-2. f1J10THOCTb pac-
npeJJ,eJlCHHR 01llH6KH OK-
pyrJleHHR JJ,JlR CYMMbI
'-IHCeJl.
1/2- a
P' (a) =p (a) = del = 1 - a.
-1/2
C HeKOTopbIM COOTBeTCTBYIOIlI.HM 0, KOTopoe
6y,n:eT onpe,n:eJIeHO n03)j{e. <paKHI4eCKH 3Ta CXO-
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BbI6paHHbIX 113 paBHOMepHoro pacnpe,n:eJIeHH5I, HcnOJIb3yeTcSI KaK xopowee
npH6JIH)j{eHue K e,n:HHCTBeHHOMY 4UCJIY, BbI6paHHOMY U3 HOpM3JJbHoro.
paCl1pe.n:eJlemHI (CM. 32.4). MbI Y)j{e BbI411CJIHJlH
1/9
D(e)= r e de= 112 '
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(2.9-1 )
.lI.JISl CYMMbI He3aBHCHMblX OIllH6oK oKpyr JIeHH5I e = $1 + e + ... + en
HMeeM
D(e)=D(.sei)= ...H2>iYdeld'i ... den'
i i
l1cnOJIb3Y51 paBeHCTBa
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i,} i
fi 2.101
TEOPM:5I «CTAPlIlEro 3HA'-IAIJJ:Ero PA3PjI,lI.A»
49
TaKI1M 06pa30M, npl1paBHHBaR .n:I1CnepCHIO .n:BYX pacnpe.n:eJIeHHtt, nOJIY-
qaeM O = n/12. ECJII1 n .n:OCTaTOqHO BeJIHKO, HOpMaJIbHOe pacnpe.n:ene-
HHe 51BJJ5IeTC5I xopolIletl annpOKCHMaIJ.Hett. 11ml HOpManbHOrO pacnpe.n:eJJe-
HH5I CYilleCTByeT CHMMeTpll'-!Hbltl I1HTepB3n, B KOTOpOM C BepoflTHoCTblO
50% JIe)!{HT olIlH6Ka:
OT - 0,6745,'1 .n:o + 0,67450.
'if
TaKI1M o6pa30M, Mbl BH.n:I1M, '-ITO .n:JJHHa nonOBHHbl HHTepBaJ!a paCTeT KaK
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chy Mo.n:eJlb He CJle.n:yeT npUHUMaTb Bcepbe3, HeCMOTp51 Ha cJlo)!(l-lOe
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He npoucxo.n:UT c.n:BurOB npH BblpaBHUBaHUU IIOp5!.n:KOB H HOpMaJlH3aIJ.HH,
CJlHlllKOM HepeaJlHCTHqHa. Ka}K.lI.blt! pa3, Kor.n:a IIpOlJCXO.n:HT c.n:Bur, lJMeeTC5!
TeBJJ.el-lIJ.H5! B03Bp3TUTbC5I K paBHOMepHOMY paClIpe.n:eJleHUIO. TaKUM o6pa-
30M, npH CJJO)!{eHHI1 U Bblq!1TaHHU Ha nocJle.n:l-lue 31-1aqaillUe pa3p5l.n:bI
p,etlCTBYIOT KaK 6bl JJ.Be npOTHBOnOJIO)!{Hble CUJJbl: OJJ.Ha TOJlKaeT B Ha-
npaBJlel-ll1H HOpMaJJbHOrO pacnpe.ll.eJIeHH5I, Tor.n:a K3K p,pyra51 - K paBHO-
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'1epe3 .lI.eJleHl1e.
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6bJJIH Bbl6p3Hbl cJlyqafiHblM 06pa30M (He AonycKaJll1cb 6e3pa3MepHbie
50
nOTPEiliHOCTI1 OKPyrJlEHI151
(tJT. 2
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(15]. Ha6moJl.aeMOe pacnpe.ZJ:eJIeHlle cTapwero pa3p5l.ll.a nOKa3aHO B Ta6-
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rpopMYJIe 100 [In (n + ]) -In n], onllCblBalOw.etl JIOrapHCpMHlIecKoe pac-
npe.ZJ:eJIeHHe. MaJIeHbKlle pa3HocTll nOKa3blBalOT, liTO B Teopnn, YTBep-
)I(QaIOIl!,efl, '-ITO CTaplime paSp5lJI.bl pacnpeJI.e. eHbl JIOrapllrpMHlIeCKll, eCTb
KaKotl-To CMbICJI.
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HYIO BblllllCJlllTeJlbHYIO MaWllHY C
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pa3pHJI.OB paBHOMepHO. ECJJH llCCJleJJ,OBaTb Bce CYMMbl llJJH npOH3Be-
JI.eHlHI, KOTopble MorYT 6bITb nOJIYlIeHbI, TO 06HaPYiKI1TCH CHJIbHOe CMe-
w.eHHe B CTOPOHY '-Il1ceJl C MeHbWHMll CTapWHMll pa3pHJI.aMn.
3TO P3CCYiKlI.eHI1e MO)l{eT 6bITb npOBeJI.eHO 60Jl e CTporo. 6YJI.eM
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OT 1 JI.O 10, I1MeHHO
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1011110,- 91nz
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TEOPM}'I .CTAPUlEfO 3HA4AillEro PA3PRLlA.
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2. t.JacTHoe HMeeT nJIOTHOCTb pacnpeJJ.eJIeHH5I
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cTpeMHTC5I K npeJJ,eJJbHotl
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lI.YIOUJ,eM: "
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B lIeTBepTO CJlyqaeB eCTb C.n:BHr Ha OJJ.llH pa3p5lJJ.;
B OJJ,HOt! BOCbMOll CJIY4aeB CJJ,BOf Ha JJ,Ba pa3p5lJJ.a;
B OJJ,Holl weCTH3JJ.aaTot! CJIY4aeB CJJ.Bor Ha Tpn pa3p5lJJ.a fI T. JJ,.
52
norPEUIHOCT!1 OKPYfJ1EHl15/
I fJJ. 2
ECJIIJ pBa 'lIJCna B3HTb! 1J3 paBHOMepHOrO pacrrpepenemlH 1";; x < 10,
TO B rrepBOM CJIYlJ3e rrJIOTHOCTb CYMMI.J TpeyrOJlI.H3H, n.a:ymaH OT HYJlH
rrplJ Z = 2 K rrHKY nplJ Z = 11 IJ na.ll.alOmaH orrHTb .lI.O UYJIH rrplJ Z = 20
(pHC. 2.10-2). TaK KaK rrJlOma.ll.b .lI.OJlJKHa 6I.ITb 1, TO MI.! HMeeM rrHK,
I
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pa)j{aeTcS! B IIJlOma.ll.H
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2.11. AHaJIH3 pacnpocTpaHeHHH OillHOKH npH HeOOJIbillOM
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pl1S OCHOBaHa Ha npHMeHeH1111 04eBI1.lJ,HOrO np116JIil)j{eHHH
df
at = dx ax,
11 MI.! ilHor.a:a 6Y.ll.eM ero HCnOJlb30BaTI..
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53
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Kor.aa HY)I{HO C,IJ,eJlaTI. pa3YMHYIO ou:eHKY T04HOCTH BI.J4HCJleHHtJ ,lJ,O Toro,
KaK BI.J4I1CJleHHe C,IJ,eJIaHO, HCrrOJJI.3Y5J JIHlllI. 06IlJ,YIO CTPYKTYPY II pa3-
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APyr ,lJ,Pyra.
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OKpYfJJeI-lIl51 CJIIIIlIKOM 4aCTO KOppeJIIIpOBaHbJ, ,lJ,orrYIlJ,eHile HX l-Ie3aBilCII-
MOCTI1 HeCrrpaBe,lJ,JII-IBO, a 3Ha'1IIT, orpaHH4HTI.C5I rrpO)J,BH)I{eHileM JIIIlllI. B I-Ia-
npaBJIeHHHX, HaMe"leHHI.JX paCCMOTpeHHI.JMII BI.JllIe TeOpH5IMH, He Y)J,aCTCfi.
fJ1ABA 3
HCqHCJJEHHE CYMM
3.1. BBe.ll.eHRe R CRCTeMa 0003HaQeHRn
Vlc4HCJIeHHe pa3HocTet!, BBe,lJ,eHHOe B r JI. 1, 6blJI0 3aTeM B fn. 2
I1CnOJlI.30BaHO rrpII paccMoTpeHIIH Ba)l{HOrO Borrpoca 06 olllH6Kax oKpyr-
JIeHH5J. B not! r JIaBe MI.J B03BpaIlJ,aeMC5J K r JJaBHoti TeMe nepBoti '1aCTH
H 6epeMc5I. 3a HC'mCJleHHe CYMM. VlC'1HCJIeI-lIIe CYMM CB5I3aHO C HC'1IICJIeHHeM
54
I1C411CJIEHI1E CYMM
I rJJ. 3
p23HOCTet!, KaK nHTerpaJJbHOe nC'JI1CJJeHHe C .1J,l1rprpepeHUnaJJbHblM.
3.1J,eCb, KaK H B I1HTerpaJJhHOM nC'JHCneHHH, HaXQ}K.1J,eHl1e 06paTHOrO one-
paTopa, B eyru.HOCTH, oeHOBaHO Ha .lLora.lLKe.
CaMblM y.1J,06HhIM 0603Ha'JeHHeM .1J,JJ5I HC'1HCJJeHH5I CYMM HBJJ5IeTCH ollpe-
b
Jl.eJ1eHI1e 2: I(x) KaK CYMMhl
x=a
I (a) + I(a + 1)+...+ I(b - 1).
31'0 0603Ha'JeHHe (3aMeTHM, '11'0 BO Beet! !;H0t! r JJaBe Mhl CtJllTaeM h = 1)
npuMeHHeTcH 6yneM [3], )!{opJl.aHOM [19] 11 MHOrl1MH .lLPYfI1MH *); OJl.HaKO
OHO He HBJ15!eTC5! ynoTpe6wreJlhHhlM B .lLpyrnx 06JJaCT5!X MaTeMaTHKH, H
I1CnOJlb30BaHI1e ero MOrJlO 6hl nplIBeCTI1 K nYTaHl1ue. ITO-BJ1.lLI1MOMY,
JlY'1ille Bce me I1MeTb .1J,eJJO C 3aTpy.1J,HeHHHMI1, KOTopble B03HJ1KaIOT OT
npl1MeHeHI15! HeY.lL06HOrO, HO o6ru.enp11HHTOrO 0603HaQeHI15!
b
2:/(x)=/(a)+ {(a+ 1)+. ..+ {(b).
x=a
MeTO.lLbl CYMMHpOBaHI15! 6Y.lLYT ueJII1KOM OCHOBblBaThCH Ha I1CnOJlb30-
BaHlJH npHMoro pa3HocTHoro onepaTOpa
A{(X) = I(x + 1) - I (x),
a He 06paTHoro pa3HOCTHoro onepaTopa \I H He ueHTpaJJhHOrO pa3-
HOCTHoro Ol1epaTopa. ITpOCYMMupOBaB nOCJleJl.Hee paBeHCTBO aT x = a
Jl.0 x = b - 1, nOJlY'JI1M
b-l
2: Af(x) = I(b)- I (a). (3.1-1)
X=a
31'0 COOTBeTcTByer paBeHCTBY
b
d: ) dx = {(b) - I (a)
a
B I1HTerpaJlhHOM I1C'JUCJleHI1
OCHOBHaH TeOpeMaI1C.II1CJleHU5! CYMM COCTOHT B TOM, 'JTO eCJlll OBe
ifJyrllCl1,llll, onpeOeJleHHbte Ha OllClCpemHO.M .MH0Jl!ecmBe mO'lelC, ll.Me/Om
OUHU U me :JICe nepBbte pa3Hocmu, mo OHU pa3/tU'ta/OmCfl He 6o/tee
'te.M nocmOflHHbt,,f{ c/ta2ae.Mbt.M. 31'0 HaBO,!lJ1T Ha MbleJlh 0 HeOnpe.lLe-
JleHHot! eYMMe, eooTBeTCTBYIOru.el1 HeOnpe.lLeJleHHoMY l1HTerpaJlY, 11 a)IJJ.l1-
TI1BHOI1 KOHCTaHTe B Ta6JJl1ue HeOnpe.lLe. eHHhIX CYMM.
<b
*) TbJOKI1 npe.n.JlaraJi 0603HalJeHI1e KaK 60. ee y.n.o6Hoe.,
x=a
3.\1
BBED.EHIo\E 11 CIo\CTEMA OB03HA'tEHIo\H
55
B HC4I1CJleHI1H 6eCKOHe4HO MaJIblX 1'36JII1JJ.a HeOrrpe.n;eJleHHblX mne-
. rpaJIOB OCHOBblBaeTCH Ha COOTBeTCTBYJOllI.eti Ta6JIHJJ.e IlpOv!3BO.n;HbIX;
TaKHM }I{e 06pa30M Ta6JII1ua HeOnpe.n;eJIeHHblX CYMM OCHOBblBaeTCH Ha
Ta6JIHUe pa3HOCTeti. 3
X(n) = nx(n-1),
npl1MeHHH (3.1-1), rrOJIY411M
b- I
L: b,n+1) - a,n+lJ
x(n) --
- n+1
JC=a
(n =p - 1).
(3. i -2)
lLJm npnMepa rrOJlOiKl1M n = 0, ror.n;a
b-l b-l
( ) b(1) ....:: aU)
X o - 1 -
--0 - I
a a
b-a.
CnOJlb3Y5I 06IHYlO CPOPMYJIY (3.1-2) 11 04eBH.n;HYJO JlI1HetiHocrb one-
paTopa 2:, Mbl 1II0iKeM Haxo.n;I1Tb CYMMbl MHOr04JJeHOB rrYTeM npocroro
npeBpmneHI151 crerreHeti X B cpaKTOpHaJlbl I1JJI1 rrpl1 rrOMOllU1 4HCeJl CTl1p-
JlHHra BToporo po.n;a ( 1.8), HJII1 rroBTopHH .n;eJleHHe MHOr04JJeHOB ( 1.6).
23THM MeTo.n;OM MOiKHO I10Ka3aTb, 4TO
n n
'\1 _ (1) _ (n+ I) n
£..X-""X - 2 '
x=\ \
n n
!x 2 = ![x(2)+x(1)1= (n+l)n 6 (2n+l) ,
x=1 1
n n
X3 = 1: [X(3) + 3X(2) + x(1)l = [ (n1" I) n r,
);=1 I
n
,, (n+l)n(2n+l) 3n 2 +3n-l
k. X = 6 . 5 '
);=1
(3.1-3)
n
'\'15_ [ (n+l)n ] 2 2n2+2n-l
X - 2 3'
);=1 /
n
.X 6 __ (n+ I) n (2n+ I) . 3n 4 +6n 8 -3n+I
""- 6 7'
);=1
n
'"' 7 _ [ (n+ I) n ] 2 3n4+6n8-n2-4n+2
i..JX - 2 6'
x=1
j
56
HC411CJlEHVlE CYMM
[TI1. .
YnpamHeHJfSl
3.1-1. npOBepl1Tb 4JOPMYJlbl JIJlJI x 3 H x., rrpHMeHJlH MeTOJI JIeJleHI1H MHoro-
'1J1eHOB.
3.1-2. I-1cnoJlb3YJl paBeHcTBo I::.C (n, k) = C (n + I, k) - C (n, k) = C (n, k-I),
-noKa3aTb, 'ITO
m
.L;C (x, k) =C (m+ [, k+ I).
x=k
n
3.1-3. rIoKa3aTb, 'ITa (2x-I)=n»,
x=1
.t (2x-l). = n (4n; -I) ;
x=1
n
(2x - 1)8 = n 2 (2n 2 - I).
x=J
. 3.2. <l>0PMYJIbI CYMMHpoBaHHSI
<popMYJIa CYMMI1pOBaHI151 .LJ:JI5I x(n) BepHa raKlKe .D:JISI orpmJ:areJIbHbIX
nOKa3areJIeli (n -=I=- - 1); HarrpuMep,
m m
1 '\'1 ( 'm(-!) _0(-1) 11m
...... x(x+I)- (X-l)-2)= . -I - - m+I +T= m+I '
;>:=1 x=1
(3.2-1)
I10.n:06HblM lKe o6pa30M,
m m
1 1 ! ( -3 ) m(-2) - 0(-9)
- x-I - -
' x(x+I)(x+2) - ( ) - -2
1 1
[ [ [ 1 ]
=2 1.2 -(m+l) (m+ 2 ) .
Pa3HOCTHaSl cpopMYJJa
f1a x = (a - 1) aX)
npllBo.n:llr K CYMMupOBaHllIO reoMerpw.Jeclwi1 rrporpeCCllli
m
am+l_!
x_
a - a-I .
a
<I> OPMYJIbI .D:JI5I pa3Hocreli cUHyca H KOCI1Hyca
f1sin (ax+b)=2 sin -;- cos [a (x+ )+bJ,
A cos (ax + b) = - 2 sin . sin [a (x + ) + h]
. 3.2]
4>OPMYJJbI CYMMI1POBAHI15I
npHBo.n:5IT I< nOJIeSHbIM CPOPMY JIaM
m . cos[a(m +{)+bJ-COS[-%+b]
Slll (ax+b)=- . ( a ) =
=o 2slIJ"2
. [ a (m+ I) ] . [ am +b]
SIIJ 2 SIIJT
= Sin( ) ;
m + ) + - ; +
cos (ax + b)=
=o 2 sin ;
. [ a (m + I) ] [ am + b]
SIIJ 2 COS T
. a
SIIJ'2
61'
(3.2-2),
(3.2-3)
MO)!{HO BBeCTH eru.e MHoro .n:pyrHx aHaJIOfH4HbIX CPOPMYJI, HO MhI He:-
6y.n:eM 06peMeH5ITb HMH 4HTaTeJJ5I.
YnpamHeHHSI
3.2-1. Bbl'lHCJlHTb 1 +2+4+8 + 16+ ...+264.
3.2-2. rIoKa3aTb, 'ITa
2N-l
'"' 'ltX
COS N = 0,
x=o
3.2-3. rIOKa3aTb, '1TO
2N-l
'"' 'ltkx 'ltmx
cos7rcosN=N8(m, k),
x=o
2N-I
'"' . 'ltX '
i.. smf[=O.
x=o
(0 < m+k <2N);
2N-l
'"' 'ltkx 'ltmx
cosH sinN=O;
x=o
2N--I /
. 'ltkx . 'ltmx
L SIIJ H slIJ7r=N8 (m, k),
x o
3.2-4. BbI'IHCJIHTb
(0<m+k<2N).
N
1
"" x(x+I)(x+2)(x+3)'
x=l
58
I1C4:I1CJJEHI1E CYMM
[TJJ.3
3.2-5. Bbl1.JI1CiIHTb
N-I
sin m 6 (6 * 0, .:!:1t, .:!:21t, .. .).
m=O
o T BeT: (I - sinN 6)
(I-sine) .
3.3 CYMMl.JpOBaHl.Je no '1aCTSlM
DpH HHTerpHpOBaHHH, KpOMe Ta6J1HQbl I.JHTerpanoB, rrpHMeHS!IOTCS! .a:Ba
MeTo.rra:
1) gaMeHa rrepeMeHHoro,
2) HrrrerpnpoBam1e no lfaCTHM.
DepBblti n3 9Tr1X n:BYx MeTo.rrOB HerrpHro.a:eH n:JJH HClfnCJJeHrIH CYMM, 3a.
BHC5!!lUJX OT pasHooTcTOHU.tHX aprYMeHToB. Eho .n:enaeT Bbl4HCJJeHHe CYMM
B aHaJJ!1THlfeCKOM KOHelfHOM B!1.a:e, Boo6me rOBOpS!, 60JJee TPyn:HbIM, lfeM
BbIlfHcneHHe mlTerpaJIOB.
C n:pyroti CTOpOHbl, B HC4HcneHHH CYMM eCTb OlfeHb C!MbHhlfi MeTO.n:,
anaJIorl14Hhlt1 !1HTerpl1pOBaHH!O no '-!aCTf!M. VlHTerpnpoBaHI'le no <JaCTHM
OCHOBhlBaeTCf! Ha <jJopMyne .rrJJf! npol13Bo.a:HOfi OT np0I13Be.n:eHI1S!
d (u'V) = ud'V + 'V duo
Vl3 9Toro paBeHcTBa <jJopMYJla n:JJH HHTerpHpoBaHHH no <JaCTHM
U d'V = U'V - 'V du
Haxo.rrHTC5I HHTerpHpOBaHHeM. AHaJlOrHlfHO H3 <jJOPMYJlbI .LJ.JlSI pa3HoCTll
np0I13Be.LJ.eHHSI
(u'V) = u6.'V + 'V (x + 1) 6.u
yMMHpoBaHHeM rroJlYlfaeM
m-I m-l
.L: u6.'V = u (m) 'V (m) - u (0) 'V (0) - .L: 'V (x - I) 6.u (x).
x=O x=O
MO)j{HO BhJ6paTb 'V TaK, 4:T06bl 'V (0) = a (HJlH JlI060MY n:pyrOMY 3MaH-
HOMY 3HalfeHH!O).
B KaIJeCTBe rrpHMepa rrpUMeHeHHSI CYMMHpOBaHHS! rro lfaCT5IM paCCMOT-
pHM Bblpa)j{eHHe
m-I m-I
! xa)!. j m
xa x =- -
a-l0 '-
.x=O 0
aX+! xa x aX+! 1 m
a-l ']= a-l - (a-l)2 0 =
= - [( m - 1 ) am - ma m - I + I ]
(l - a)2 .
4.11
BBE!!EHI1E
59
Bo06me npOMeHeHOe CYMMl1pOB3HH51 no QaCT51M CI1JJbHO HanOMHHaer
npHMeHeH!1e I1HTerp!1pOBaHI151 no lIaCT51M. HanpHMep, CYMMlipOB3H!1e
2 1x sin Ox
k.J
BbInOJJH51eTC51 lLBYKpaTHbIM npHMeHeHHeM CYMM!1pOBaHI151 no QaCT51M' H npH-
BelLeJ-lI1eM nOlL06HbIX 'lJJeHOB. B 9THX npeo6pa30BaHflSIx HeT HOBbIX fllLetI.
3 TOJlbKO CKY4Ha51 anre6pa.
YnpamHeHI1S1
3.3-1. nOKa3aTb, 'ITa
n
'\1 \ . 6 _ 2sin6-2- n [2sinO(n+l)-sinOn]
"'" 2X sm x - 6'
x=1 1+8 sin 2 2"'
3.3-2. BbI'IHc.mTb
n
2:: x 2 cos x.
x=1
3.4. 06mIJe 3aMeqaHIJH
JIHllIb HeMHorHe KOHe4HbIe p51lLbI MO)l{HO npocYMMHpoBaTb fI npe.n:CTa-
BHTb KOMnaKTHotI CPOPMYJJOtI. C lLPyrotI CropOHbI, HeO)!{HlL3HHO 43CTO,
CYMMHPylOTCH HeKOTopble Cneu,l1aJ1bHble PIJAbl, HaI1pI1Mep, COlLep)l{aml1e 6H-
HOMHaJJbHbIe K03cpcpHIJ,HeHTbI. COBeTyeM lJ1UaTeJ1lO, npe)l{lLe lJeM o6pa-
manCH K Bbl411CneHI11O pSIlLa C I10MOmblO MallIHHbl, nOnbITaTbCSI npOCYM-
Ml1pOBaTb ero PYKaMH. YlLaliHoe CYMMHpOBaHHe lIaCTO I1pflBOJJ.HT K 06"bHC--
HeHl110 I1epBOHa4aJIbHotI 3alLalJl1. CI1eu,HaJIbHbIe MeTOlLbl CYMMHpOBamUl pHJJ.OR
cnHllIKOM MHOr04HCJIeHHbl, lJT06bl p aCCMaTp HBaTb HX 3lLeCb. Bep05lTHO,
JIY'IlUHM cnpaBo4HHKOM 110 CYMMHpOBaHHIO p51AOB 51BJI51eTCIJ )f{OJIJIH [18]._
CM. TaK)Ke [1].
I'J1ABA 4
BbllJWCJ1EHI1E 6ECKOHEIlHbIX P5I1I.OB
4.1. BBeJleHHe
B 60J1blUHHCTBe KHHr 0 6ecKoHe4HbIx p51lLax Ha MHorHX CTpaHHu.ax
paccMaTpHBalOTCIJ CXO.lJ.l1MOCTb, paCXOJJ.l1MOCTb H «CYMMHPyeMocn» p51lLOB,
HO I104TH Bce OHH cOBeplUeHI-!O I1peHe6peralOT JJ.eticTBHTeJJbHbJM BbI4HCJIe--
Hl1eM (CYMMHpoBaHHeM) p51lLOB. OJJ.Ha 113 I1pl1l1HH 3Toro COCTOHT B TOM,
'11'0 CYUleCTByeT OlJeHb MaJIO MeTOJIOB .lI.JI51 CYMMl1pOBaHH51 p51.lJ:OB B KOHe'l-
HOM 1111JJ.e. KOHe4HO, eCJ1!1 B03MO)KHO B KOHeliHOM BI111.e nposeCHI Heonpe-
.n.eJleHHoe CYMMHpOBaHHeH p51JI CXOJIHTC51, TO 6eCKoHellHblH p51lL raK)J{e.
60
BbI411CJJEHI1E BECKOHE4HbJX P5TD.OB
I TJJ. 4
MO)!{HO rrpOCYMMUpOBaTb, YCTpeMUB BepXHHI1 rrpe.n:eJI K 6eCKOHe4HOCTU.
B Ka'leCTBe npUMepa MbI HMeJIH (paBeHcTBo (3.2-1»
n
! x (}+ 1) = 1 - n 1 .
x=1
OTc!O.n:a
co
"\'I I - 1
x (x + 1) .
x=1
Boo6rn:e
co
"\'I 1 I 1
"""x (x+ 1)... (x+ k-l) = k-l ' (k- I)I
x=1
(k;:::2). (4.1-1)
3a.n:a4Y aHaJIH3a Hepe.n:Ko MO)!{HO CBeCTH K BbI4HCJIeHH!O 6ecKoHe4Horo
p51.n:a. IlpH STOM 4aCTO Tpe6yeTc51 60JIbIlIP, pa60TbI, 4To6b1 BbI4HClIHTb
p51.n:, 4eM 'IT06bI peIlIHTb rrepBOHa4aJIbHY!O 3a.n:a4Y; HO HHor.n:a npe.n:CTaB-
JIeHHe pH.n:OM HBJIHeTCH rrpeuMyrn:eCTBOM. ILeJIo B TOM, 4TO, CYMMUPYH KO-
He4Hoe 4HCJIO 4JIeHOB p51.n:a, JIer4e CJIe.n:HTb 3a TO'lHOCTb!O BbI4HCJIeHI1Ii,
qeM .n:eIiCTBY51 HHbIM crroco60M. TaK, .n:JIfi BbI'lHCJIeHH5I
x
e- /2 dt
o
.n:JIfi 3HaqeHI1ti X, MeHbIlII1X 1, MO)!{HO pa3JIO)!{I1Tb sKCnOHeHTY B 6ecKo-
He4Hbltt pH.n:. Y1HTerpupY51 n04JIeHHO, nOJIY4uM
.x xoo 00
-/2 _ \ "\'I (_I )k t 2k _ "\'I (_I)k x2k+1
e dt - j ! dt - "'" (2k + 1) kl
o 11=0 11=0
ECJJH MbJ nHTepecyeMcH 3HaQeHUHMH X, MeHbIlII1MH 1, H XOTHM HMeTb
BoceMb BepHblx 3HaKOB, TO .lWCTaT04HO B3HTb 11 4JIeHOB pH.n:a. ILetlcTlH1-
Te.1IbHO, pH.n: 3HaKorrepeMeHHbIIi, ero 4JJeHbI MOHOTOHHbI, OH CXO.n:I1TCH H
rrepBbItl OT6poIlIeHHbIIi 4JIeH HMeeT 3HaMeHaTeJIb npH6JIH3HTeJIbHO 9,2. 10 8 .
EC.1I11 6bI MbI nbITa.1II1Cb BbI4I1C.1II1Tb 3Ha4eHI1e I1HTerpaJJa KaKI1M-HI16YD.b
npn6.1I!!)!{eHHbJM MeTo.n:OM HHTerpHpOBaHWI, TO 3a.n:a4a ou.eHKU OIlIH6KH
6bIJIa 6bI 6o.1Iee TPYD.HOt1.
Ynpa}\{HeHHSI
4.1-1. j],oKa3aTb (jJOPMYJlY (4.1-1).
4.1-2. Hanl1CaTb pHJJ: JJ:JHI
, x
51 (x) = , sin t dt.
u t
CKOJlbKO '1J1eHOB HY)/{HO B3HTb 1JJIR BbllJl1l:II\:JilIH Si (21t) C BoceMbIO JJ:eCRTH'IHblMI
3HaKaMuf
4.2J
METOJl, KYMMEPA
61
9 4.2. MeTon KYMMepa
ECJJI1 Jl,aHHbItJ p5lJI. CXOJI.MTC5I 6bICTpO, TO BbI60p cTIoc06a BbI4llCJIeHI151
ero CYMMbI He rrpeJI.CTaBJI5IeT 3aTPYJI.HeHMtJ. ECJJI1 p5lJI. CXOJI.MTC5I MeJI.JIeHHO,
TO MbI MIlI.eM p5lJI. C M3BeCTHotl CYMMOtJ, KOTOpbItl CXOJI.MTC5I rrpM6JIM3M-
TeJJbHO C TOtl me CKOpOCTbIO, qTO M JI.aHHbItJ p5lJI.. CJIOBa «rrpM6JJM3MTeJJbHO
C TOtJ me CKOpOCTbIO» B JI.etJCTBMTeJIbHOCTM 03HaqaIOT, 4TO 06IlI.MtJ 4JJeH
pa3HOCTH 3TMX JI.BYX p5lJI.OB CTpeMMTC5I K HYJIIO 6bICTpee, qeM 06IlI.I1H qJIeH I1C-
XOJI.HOrO p5lJI.a. HatJJI.51 rrOJI.xOJI.51IlI.HtJ p5lJI., MbI CBOJI.MM 3aJI.aqy K BbIql1CJIeHMIO
CYMMbI p5lJI.a, rrpeJI.CTaBJHIIOIlI.erO pa3HOCTb JI.BYX p5lJI.OB; rrOCJIeJI.HMtJ rro orrpe-
Jl,eJIeHHIO CXOJI.HTC5I 60JIee 6bICTpO. B 3TOM H COCTOHT HJI.e51 MeTOJI.a KYMMepa.
. DyCTb JI.aHHbIt! p5lJI. eCTb
ro
8= 2: ax
x=r
H npe.l1rrOJIOmHM, 4TO MbI 3HaeM CYMMY
ro
8'= 2:cxax,
x=r
r.a.e cx_c rrpH x_co. Tor.l1a
00
8= S; + ! (1- C; )a x .
x=r
B Ka4eCTBe rrpHMepa paccMoTpHM p5lJI.
00
KOTOpbItJ CXOJI.MTC5I
8= ! (x2 1)2 '
x=l
1
KaK Ii' B351B
X
00
8'- I.
- (x-l)x(x+l)
x=2
(c = 1)
B KaqeCTBe p5lJI.a JI.JJ5I cpaBHeHI151, npH YCJIOBHM, 4TO 4JIeH C x = 1 B351T
OTIIeJIbHO, nOJIyqHM
00
S=S' +( S-8')=S' + + [ x
4 (x 2 + 1)2
2
1 J .
(x-l)x(x+l) =
00
_ x 4 _X'-(x 4 +2x 2 +1)
- 4 + 4 + '- x (x' -I) (x' + 1)2
HORMIt p5lJI. CXOJI.WTC5I
ro
1 3x 2 +!
= 2 - '- X lX' - I) (x' + 1)2
:.I
K.H I/x".
0,39711677.. .
62
BbIlj\1CJJEHI1E 6ECKOHE4HblX P5!.[I,OB
(rJI. .
Ynpa>KHeHHSI
O'J
4.2-1. VlCnOJlb30BaTb ! x (X 1 + 1) JI.JJR
1
4.2-2. rIpH6J1H3HTb pe3YJlbTaT ynpa)/{HeHHiJ 4.2-1
00
npn6J1I1JKeHHiJ '\'1_
ka ;..
i
c rrOMOllJ.blO
co
'\'1 1
k.x (x+ 1) (x+2)' !
1
rI p0JI.OJl)/{aTb 110 k-ro wara.
4.3. HeKoTopbIe cnelJ:\iaJIbHbIe CYMMbl
MeToD. KYMMepa Tpe6yeT p5J.1I.0B .lI.JI5I cpaBHemHI fI 3HaHlUI fiX CYMM.
O.ll.Ha 113 caMblX rrOJIe3HbIX rrOCJIe.ll.OBaTeJIbHOCTet\ p5!.lI.QB .lI.JIH cpaBHeHl1S1,
KpOMe pH.lI.OB (4.1-1), eCTb rrOCJIe.ll.OBaTeJIbHOCTb CYMM
O'J
,., I
8k = xk (k=2, 3, ...),
x=1
51BJJ5!!Oill.I1XC5! 3Ha4eHI151MI1 .lI.3eTa-<jJYHKI1.1111 PI1MaHa
(4.3-1)
co
C(z)= ! z
n=1
(CM. Ta6m:my 8.7-1 .nJJ5I z=2, 3, ...,17). 3HaqeHfl5I ,n1IJ! qeTHblX u.eJIblX Z
H3BeCTHb) B KOHeqHOM BI1.l1.e,
71: 2
82= (2)= In
71:'
8 4 = 90 ;
71: 6
8 6 =94.'5;
71: 8
8 8 = 9450 '
HO .lI.JI5! HeqeTHblX qIKeJI KOHe4HbIt! BI1.l1. HeH3BeCTeH.
B MHOI'O'1HCJIeHHbIX KHl1raX II Ta6JIllI1.aX *) llMeeTC5I MHoro D.PYrHX
p5!.lI.OB, llMe!OUJ,HX 113BeCTHbIe CYMMbI, H 'IHTaTeJIb, HeCOMHeHHO, 3HaKOM C
TeMI1, KOTopbIe B03HHKa!OT ll3 pa3JIO)\{eHll51 &JIeMeHTapHblX <jJYHKUHti B p5l,n
M:JKJIOpeHa.
4.4. MeTol{ 3iiJIepa
LlPYrHM MeTo,nOM '-1HCJIeHHOrO cYMMnpoBaHllH pH.non IIBJIlIeTCII MeTOll
3J1Jlepa, J{ HeMY MO)\{HO rrpHt1Tll CJIe.uy!Oill.llM 06'pa30M.
PaccMoTpllM KOHe4Hblt! p5J.n
n-I
\.' t k
L...J ak .
k=O
(4.4-1)
*) CM. [18]H [1].
4.41
METOJI 9i1JJEP A
63
DpHMeHHM CPOPMYJIY CYMMHpOBaHHSI no '-IaCTSlM
0-1 0-1
a (k) A v (k)= [a (k)v(k)] - v (k + 1) tw (k).
a
a
ITOJJQ)J{HM u (k) = all' Av (k) = t h . TaK KaK MbI MOiKeM HCIIOJIb30BaTb
mo6yIO a.llJl.HTHBHYIO nOCTOSlHHYIO, TO B03bMeM
k-I
1 - t"
v(k)= '- t X = I- t .
x=O
TorJIa
n-I n-I
II [ 1 - t" ] n I - t"+!
t. aht = all 1-t 0 - '- --r=T Aak=
k=O k=O
n.-\ n-I
( I - t n ) 1 A t k
=a n I-t - 1-t k. all + I- t 1.. t Aa ll ;
o 0
(4.4-2)
HO
n-I
Aa ll = (aj - a o ) + (a2 - aj) + . . . + (an - an_I) = an - ao'
k=O -
TaK 4TO (4.4-2) npHHHMaeT BH.l1
n-I n-I
k ( I - tn ) an - a o + t '\:1 II A
i... a"t = an \ 1 _ t - 1=T 1 _ t 1.. t <.1 ak =
11=0 0
a-I
a o ant n + t "" t k ,\
= 1 - t - I _ t 1- t":" <.1 ak'
k=D
(4.4-3)
Mbl npYlMeHl'leM CYMMHpOBaHHe no 'l3CTSlM K TpeTbeMY 'lJIeHY, 3aMe-
THB, 4TO OH HMeeT TOT iKe BH.l1, 4TO H HCXO.QHblI! p51.l1, TOJlbKO ak 3aMe.
HeHO Ha A a k B (4.4-3):
n-I n-I
t t" A t ( !:! a o !:!ant n + t "" t" A 2 )
1- t "'- all = 1 - t 1- t - 1- t 1- t '- all .
k=O k=O
T3KHM o6pa30M, B pe3YJIbTaTe npHMeHeHHSl CYMMHpoBaHHR no 'laCTHM
.QBa)K.l1bJ nOJlY4HM
n-.J n-I
'1 k, a o t 11 a o t' k ant" , ant n + 1
,-allt = 1-t + (I-t)i + (1-t) 2...t A ak- l_t - i_ t I-t .
k=O 11=0
64
Bbl4Y1CJIEHYlE BECKOHE4HbIX P5IJ.!.OB
! JJl. 4
fIoCJre r T8KHX npe06p830B8HHtl Bblp8meHHe (4.4-1) npHMeT BH.lI.
n-I r-I n-I
aktk-= J 11 !( J 1 I Y Aiao+( J 1 I Y tkArak-
k=O i=O k=O
r-I
t n ( t ) 1 '
- 1-1 .:... J - t A'an'
i=O
TaK KaK nepBOHa'laJIbHbrt1 pSI.lI. CXOJ1.HTCSI, TO J1.JISI .lI.aHHOrO e> 0 cyrn.e-
cTByeT TaKoe no, 'ITO npH n> no CnpaBeJ1.JIHBO I an I < 2 ' CJJe.ll.OBaTeJIb-
HO, no (1.4-3), I A r an I 0;;;:; e. fIOCJJe.ll.HHtl 'IJIeH B nOJIY'leHHCiM Bblpa)I{eHI1H
CTpeMHTCSI n03TOMY K HYJIIO npH n 00, H MbI HMeeM
00 r-I 00
"\1 k J "\1 ( 1 ) i Ai +( 1 ) '''\1 kAr
akt= I_I l-t /.lao l- t t /.lak'
k=O i=O k=O
(4.4-4 )
fIpe.ll.nOJIO)I{HM, 'ITO 'IJJeHbl pSI.lI.a (4.4-1) H3MeHSIlOTCSI .lI.OCTaTOlJHO r J1a.ll.KO,
TaK lJTO BTopoe CJ1araeMOe cnpaBa CTpeMI1TC5! K HYJIIO npl1 r 00. Tor.n:a
OCTaeTC5I p5l.ll.
00 00
"\1 k I "\1 ( t ) i A i
ak t .1- t l- t /.l ao.
k=O i=O
(4.4'5)
PaCCMOTpl1M npHMep npl1MeHeHHSI YKa3aHHoro MeTO.ll.a. ECJII1 .lI.aH pSIJJ.
ro
" Uk+l t
.LJ Uk' Y KOToporo CTpeMI1TCSI K ,TO MO)I{HO Hanl1CaTb
k=O Uk
00 00
Uk= aktk,
o 0
r.ll.e ak+! CTpeMHTCSI K 1, H npl1MeHHTb MeTO.ll. 9tlJIepa.
ak
HaH60JIee lJaCTO 3TOT MeTO.ll. npHMeHSIeTC5I .lI.JI5! t = -1. fIo (4.4-5)
HaXOJ1.HM
00 00
"\1 ) k 1"\1 (-I)'Ai
(-1 ak=2"'-"2l/.l ao.
11=0 i=O
( 4.4-6)
HHOr.ll.3 npe06p330B3HHe 9tlJIepa .lI.eJIaeT pSI.lI. CXO.ll.51rn.HMC5I 6bICTpee, HO
IiHOr.ll.a II HeT, PaCCMOTpHM CJIe.ll.YIO\HHe npHMepbI.
TlpHMep 1.
00
V i l)k
"'- 2 k .
k=O
II 4.4J
METOJ1 3HnEP A
65
Mbl IIMeeM
1
an = 2 n '
I I -1
A an = 2n+J - 2'i = 2nH ,
i (_1)'
A ao=
H, TaKHM o6pa30M, no (4.4-6) nOJlY4aeM pH.ZI.
00 00 00
(_1)11 _ '\"1 (-Ir . (-I)' _ '\"1 1
-;pi - 2"';;;' 2i 2; - 2 .v,
11=0 ;=0 i=O
KOTOpbl1l CXO.Zl.I1TC}1 6blcTpee nepBOH3lJaJlbHoro.
IlpHMep 2.
00
! (3t .
11=0
J1erKo BI1.Z1.eTb, lJTO
A i a _ (- 2)'
u 0- ..
3'
T3KHM o6pa30M, npeo6pa30BaHI1e 9tlJlepa .n.aeT pH.ZI.
00 00 00
'\"1 (_1)11 = '\"1 (_l)i . (-2)' = '\"1
-'" 2 -'" 2i 3; 2 f.. 3 i ,
11=0 i=O ;=0
KOTOpbltl CX01UITCH HeCKOJlbKO Me.n.JleHHee, 4eM 3a1J.aHHbJlt.
Il p 11 M e p 3. AHaJJ0fl14HO
00 00
'\"1 (_1)11 _ '\1 ( ) i.
..::.. -----:v< - 2 f.. 8 '
k=O ;=0
nOCJleJJ,HHfi pH.ZI. CXOJJ,HTCH 60Jlee MeJJ,JleHHO.
«TOlJKa nepeJlOMa» I ai I 1J.JlH npJ1MeHeHHH MeT01J.a 91lJlepa K TaKoMY
BHaKonepeMeHHoMY pH1J.Y, nO-BI1JJ,I1MOMY, Jle}f{I1T Me}f{1J.Y 1/2 I1 1/3.
Ha npaKTJ1Ke 06bllJHO CYMMHPYIOT nepBble HeCKOJlbKO (cKa}f{eM, 10)
lJJleHOB HenOCpe1J.CTBeHHo H npl1MeHHIOT npeo6paBoBaHHe 9tlJlepa K OCTaB-
IIJI1MCIJ *)
YnpamHeHHII
4.4-1. rIpl1MeHHTb Me'fO)l 9fiJlepa K PRllY
co 00
, '\"1 (-1)11-1 '\"1 1
In2= f.. = f.. k2 k .
11=\ 1i 1
*) CM. [4], a TaK)/{e J. B. R 0 sse r, Trans\ormations to Speed the Conver-
gence of Series, J. Research. N at. Bur. Standards, vol. 46, 1951.
3 P. B. XeMMHHr
66
Bbl411CJJEHI1E 6ECKOHE4HbIX P5l110B
[fJI. 4
4.4-2. rIoKa3aTb, 'ITa
ro ro
(_l)k 1 k!
arctg 1 = i.J 2k + 1 = 2 i.J 1 . 3 . 5 ... (2k + I) .
k=O k=O
4.4-3. CJlO)/{I1Tb nepBble BoceMb '1J1eHOB pa.aa .aJla In 2 li npHMeH!!Tb MeTO.a
3fiJIepa K OCTaBlllHMCR.
. 0 T BeT: 1-1/2 + 1/3-...- IJ8 =0,63452381
1
OCTaBWaRCIJ '1aCTb paBHa:2' 0, 11724074.
4.5. HeJIHHeiiHOe npeoopa30BaHHe
60JIblllHHCTBO rrpeo6pa30BaHHI1, rrpHMeH5JeMbIX B MaTeMaTHKe, mmel1HhI,
HO HeJIHHel1Hoe rrpe06pa30BaHHe '1aCTH'IHWX CYMM p5Jlla
T (Sn) = Sn+1 S n_l - S
Sn+1- 2S n+ S n_l
4aCTO OtleHb nOJIe3HO. ECJIH 6bI T 6bIJIO JIHHeI1HbIM, MW HMeJIH 6w
( 4.5-1)
fI
T(CSn)=CT(Sn)
T(Sn + Un) = T(Sn) + T(U n ).
BTopoe YCJJOBHe, Boo6w.e rOBOp5J, HeBepHO. OIlHaKO crrpaBellJIl-IBO 60JIee
,cJIa6oe paBeHCTBO
T(Sn+C)= T(Sn)+C
(C - const).
Eho rrpeo6pa30BaHI1e rrOJIe3HO B Tex CJIY4a5JX, KorAa p5Jll BelleT ce6f1,
rpy60 rOROp5l, KaK reoMeTpWleCKa5J rrporpeCCH5J. B Ka'leCTBe liJIJJIOCTpa-
QHH 9cpcpeKTHBHoCTH npeo6pa30BaHI151 paccMoTpHM p5l.1l: JJel16HHQa
=4 - 4/3+4/5 - 4/7 + ...
CXOllHMOCTb ero TaK MeAJIeHHa, 4TO npaKTH'IeCKH He3aMeTHa, HO O.1l:HO
npHMeHeHHe npeo6pa30BaHIUJ AaeT cJIe.1l:YIOw.ee:
n I sn \ T(Sn l n I sn I T(Sn l
0 4,00000 5 2,97605 3,14271
1 2,66667 3,16667 6 3,28:37 4 3,14088
2 3,46667 3, \3333 7 3,01707 3,14207
3 2,89524 3,14524 8 3,25237 3,14125
4 3,33968 3.13968 9 3,04184
4.6]
CTEnEHHblE PS!J1bJ
67
TeopIHl SToro npe06p330B3HIHJ H MHorl1X p0JJ,CTBeHHbIX eMY, KOTopbIe
43CTO JJ,3JKe 60JJee CI1J1bHbI, XopOWO H3Y4eH3 11 He 6YJJ,eT paCCM3TpH-
B3ncSI 3JJ,eCb.
3TO npeo6p330B3HHe rrOJle3HO He TOJlbKO JJ,JlSI 06P360TKH MeJJ,JleHHO
CXOJJ,SlW,HXCSI H P3CXOJJ,SlW,HXCSI pSlJJ,OB, HO T3KJKe H JJ,JlSI MHornx I1Tep311.HOH-
HbIX rrOCJJeJJ,OB3TeJlbHOCTet!, T3K JKe K3K MeTOJJ, HblOTOH3 JJ,JlSI H3XOJKJJ,eHHSI
HYJJet! CPYHKlI.HH. ECJlH CXOJJ,I1MOCTb I1MeeT B K3KOM-TO CMbICJJe CKOpOCTb
reOMeTpH4eCKot! rrporpeCCHH, TO rrpeo6p330B3HHe 43CTO YJlY4W3eT JJ,eJlo.
Ynpa}l{HeHHe 4.5-1. rIpHMeHRR npe06pa30BaHHe (4.5-1), BbJ1lHCJlHrb
" [ (- k l )k-l ] .no
4eTBepToro .neCRTH'IHOrO 3HaKa.
k=I
4.6. CTeneHHbJe pnlI:b1
CTerreHHbJe pSlJJ,bl Wl1pOKO rrpHMeHSlIOTCSI B M3TeMaTHKe, oc06eHHo
B JJlmet!HblX 33JJ,a'-l3X. Ho JJ,3JKe B HeJlI1Het!HbIX 33JJ,3'-l3X CTeI1eHHbIe pSlJJ,bI -
I10Jle3Hblt! HHCTPYMeHT. T3K K3K B HeJlHHetlHblx 33JJ,3'-laX BbI4HCJleHHe no-
CJleJJ,OB3TeJlbHbIX K09cpcpI1IJ.l1eHToB 4aCTO O'-leHb TpYJJ,oeMKo, Bbl4I1CJlI1TeJlbH3S!
MaIllHHa JJ,OJlJKHa HaXOJJ,HTb I1X caM3. llJlS! JJ,aHHoro 'lHClia K09cpcpl1I.l.HeHTOB
JlerKO HanHcaTb I1pOrpaMMbl CJlOJKeHH5I, Bbl'lI1TaHH5I, YMHOJKeHH5I, JJ,eJleHHS!,
nOJ1.CT3HOBKH OJJ,Horo CTeI1eHHOrO p5IJ1.3 B J1.Pyrot! H o6paw'eHH5I p5IJJ,a.
TIpeHMY1lI.eCTBO HCI10Jlb30BaHH5I CTeI1eHHbIX PSlJJ,OB I1J1H MHOr04J1eHOB,
KaK B 3aJJ,34e 1.9, B TOM, 4TO 60JlbWYIO 4aCTb .aet!CTBHt! HaJJ, K09cpcpH-
Il.l1eHT3MH MOJKHO BbIl10JlHHTb OJJ,I1H pa3 rrepeJJ, Ha4aJJOM BbI4HCJIeHHt!.
A6CTpaKTHoe MaTeM3TH4eCKoe OI1HCaHHe Toro, 4TO 6bIJIO CJJ,eJlaHO
B 33JJ,a4e 1.9, BbIrJl5IJJ,I1T TaK: 11 T04eK MorYT pacCMaTpHBaTbCSI KaK
OJJ,Ha TO'lKa B 11-MepHoM I1pOCTpaHCTBe. TIepB3S! 0I1ep3UHSI nOJIY'leHHSI
cp3KTOpH3J1bHOrO MHOr04J1eH3 HblOToHa rrpeo6p330B3JIa T04KY HCXOJJ,Horo
npOCTp3HCTB3 B cooTBeTcTBYIOW,YIO TO'lKY Il-MepHoro npoCTp3HCTB3 KOScp-
cpI1UHeHToB. l1cnOJlb30B3H11e 4HCeJI CTHpJll1Hr3 JJ,JISI Bbl411CJleHHSI HeI10CpeJJ,-
CTBeHHoro MHOr04J1eHHOrO npeJJ,cT3BJleHI1S! 6bIJIO p3BHOCHJIbHO H3MeHeHHIO
6a3Hca rrpoCTpaHcTBa K09CPCPHuHeHToB. TIoclIeJJ,YIOw,He 0I1ep3Il.HH 6b1JlI1
npe06pa30BaHH5IMI1 B npocrpaHcTBe K09cpcpHlI.l1eHTOB, TorJJ,a KaK OKOH4aTeJlb-
Hoe BbI4HClIeHl1e 6WlO rrpe06pa30BaHI1eM 06paTHO B HCXOJJ,Hoe npOCTpaH-
CTBO. TaKHM 06pa30M, 60JlbIllHHCTBO JJ,et!CTBHt! npOH3B0J1HJlOCb B I1pe-
06pa30BaHHOM rrpOCTpaHCTBe, a He B I1CXOJ1HOM npOCTp3HCTBe .aaHHbIX
I1 OIBCTa. 8TO x3p3KTepHo JJ,JJ5I MeT0J13 CTeneHHblX pSlJ10B; MbI onepHpyeM
C K03cpcpHlI.HeHT3MI1 p33J10JKeHHSI H B03Bp3m.3eMcSI K JJ,3HHOMY npocTp3H-
CTHy TOJlbKO B KOHlI.e. TIOJJ,06HbIe 33Me43HHSI OTHOCSlTCSI K3K K 3CHMI1TO-
TH'IeCKHM p5IJJ,aM, T3K H K MeToJJ,3M, 06CYJKJJ,3101lI.HMCSI B 3TOt! rJIaBe.
Ynpa»meHH5I
4.6-1. COCTaSI1Tb 6J1oK-cxeMY )J.JlI! nporpaMMbI YMHOJKeHfll! )J.syx creneHHbIX
pll)J.OS c k 4J1eHaMH, tIT06b1 nOJlY'IHTb k 4J1eHOB B pe3YJlbrare.
4.6-2. To )/{e caMoe .nJlR .neJleHUJI.
a.
68
Bbl411CJIEHKE 5ECKOHE4HbiX P5InOB
I [JI. 4
4.6-3. To )/{e caMO!: /lJIH fJO/1.CTaHOBKI! O/1.HOrO pR/1.a /J /1.pyrofi.
4.6-4. K JJ,aHHoMY nHTerpa. y
co
1' \ f (x{f r eXIT
g(T)= ( ) 2 dx
eXIT - I
.
o
npl!MeHWTb MeTOIJ. 1.9 H npHHRTb f (x) = x k . rIoKa3aTb, 'ITO g (T) =
= (n + 2)! Trtt1C (n + 2). BblsecTH OTCIO/1.a, 4TO eCJlH g (T) MO)/{er 6bITb anlJpOKCI1-
MHpOBaHa
N
g (T) = 2: anTntl,
n=O
TO
N
,.., a x n
f(x) = (n+2) C(n+2) '
n=O
4.7, Pa3JIOmeHHe no CnelJ,HaJIbHbIM <PYHKIJ,HHM
f(pOMe pa3JIO)!{eHl-lSJ B CTeneHHble pRabl, B aHaJJU3e 4aCTO npHMe-
HSJeTCSJ pa3J1Q)KeHHe no cneUUaJIbHblM cpYHKUI1SJM, TaKI1M K3K nOJIHHOMbl
J1e)!{3HJJ,pa P n (x), nOJlI1HOMbl J1areppa Ln (x), nOJII1HOMbI 3pMHTa Hn (x),
cpYHKIJ,UH EecceJlSJ in (x) U T. JJ,. TIpH nOBepXHOCTHOM paCCMOTpeHl111 MO-
)!{eT nOKa3aTbCH, 4TO 9TI1 p33JIO)]{eHl-lSJ 6eCIIOJIe3Hbi 113-33 TPYJJ,tiOCTH
BbICJHCJleHHSJ 3Ha4eHHtJ C3MHX Cnell.H3JIbHbiX cpYHKUHtI. OJJ:HaKO 113BeCTHO,
YTO 60JIblllHHCTBO CeMetJCTB CneIUIaJIbHbIX cpYHKUl1tJ YJJ:OBJIeTBOpSJeT TpeX-
YJleHHOMY peKyppeHTHOMY COOTHOWeHHID BHJJ:a
fn+l (X) = A (n, x) In (x) + B (n) In-l (x).
B Tex CJJY4aSJX, KorJJ:a CneUH3JJbHbie cpYHKUHH cyn MHOr04J1eHbl, MHoro-
YJIeHbJ HYJIeBOrO U nepBoro nopSJJJ:KOB oc06eHHO JIerKO BbI4HCJISJTb JJ:JI5J
Ka)]{JJ:oro 3Ha4eHHSJ x. T3KHM 06pa30M, pa60Ta C pa3JIO)]{eHHeM B pSJJJ:
no CneIl,H3JIbHbIM cpYHKUI-I5JM He 60JIbllle, 4eM pa60T3 no BblCJI-ICJJeHHfO
CTeneHHoro pSJJJ:a.
CJJeJJ:yeT 06paTHTb BHHM3Hue Ha H3KOnJIeHl-le 01lI1-I6KH npH I-ICnO/lb-
30BaHHH peKyppeHTHoro COOTHOllleHHSJ JJ:JISJ Bbl4HCJJeHHSJ nOCJIeJJ:OBaTeJlb-
HblX cpYHI{l.(I-It!; HO 06bl4HO OllIH6K3 He CJIHllIKOM 6blCTpO p3CTeT JJ:JI5J
YMepeHHblX (CKa)]{eM, nOp5IJJ:K3 15) 3Ha4eHHtJ I1HJJ:eKca n.
4.8. HHTerpaJIbI KaK npH6JIHmeHHH CYMM
OnpeJJ:eJIeHHblti HHTerpaJI onpeJJ:eJI5IeTCSJ KaK npeJJ:eJI CYMMbl. fl09TOMY
HHTerpaJIbl MorYT 6blTb HCnOJIb30BaHbl JJ,Jl5I npH6J1H)!{eHHOrO BbI4HC;;eHH5I
CYMM. O.aHa 1-13 cpOPMYJI TaKora THlla YKa3bIBaeTC5I B 12.3 Ii MO)]{eT
4.91
.llIHAMMA-<PYHKUI151
69
6WTb SanMCaHa TaK:
n
(' 1 1
10+11+ ... +In= J l(x)dx+ 2 fo+ 2 fn-
o
- /2 (I:),.fo - b.f _I) + 2 1 4 (b. 2 10 + b. 2 fn_2)-
- ;;0 (b. 3 10 - b. 3 fn_a) + i O (b. fo+ b.4fn_ ) +
(4.R-l)
PO,LI;cTBeHHot! CPOPMYJIOt! SlBJI5Ie-rcSl cpopMYJIa 3t1JIepa - MaKJIOpeH3 (l431,
CTp 127-128)
n
. fo+11 +...+ In= I(x)dx+ /o + +In- I ff -f J+
o
+ 7 0 ff " - f "] - 30 40 [fJ" - lit] + (4.8-2)
4.9. lblraMMa-!f>YHKu.IUI
B 3aKJIlO4eHHe EHOt! r JIaBbl BcnOMHHM CPOPMYJIY
(' x m +1
J xmdx= m+l +C,
KOTopaSl HerrpHMeHHMa, KOr.lla m = - I. LletlcTBI1TeJIbHO, rrpM m = - 1
STOT HHTerpaJI orrpe.D.eJI5IeT HOBYlO CPYHKll.MlO In x. AHaJIOrn4HO CPOPMY JJa
CYMMHpOBaHHSl
x-I
x(m+11
., t(m)=_+c
.... . m+1
1=1
HerrpnMeHnMa npn m = - I 11 COOTBeTCTBeHHO orrpe.D.eJI5IeT HOBYlO CPYHK-
llHlO, H33BaHHYlO aU2aMMa-IjJYHlCl1,ueii H 0003Ha4eHHYlO F (x)
x
F(x)= ! -j,
r=1
(4.9-1 )
r.D.e j = 0,5772156649 ... (KoHcTaHTa 3t!JIepa) n, CJIe.D.OBaTeJJbHO,
F(O)= -r-
3Ta cpopMYJIa rrpHMeHSleTcSl, KOr.D.a x - ueJIoe 4HCJIO. Llpyrot! BH.D.
Q:)
F(x)= ! r(r x) -j, F(O)=-j
r=1
( 4.9-2)
AaeT B03MO)\{HOCTb pacrrpoCTpaHHTb orrpe.D.eJIeHHe F (x) Ha HeueJIbJe 3Ha-
4eHI1i1. Ha.D.o nOKa3aTb, CJIe.D.OBareJIbHO, 4ro STO HORoe onpe.D.eJIeHne .D.JISl
70
Bbl411CJJEHI1E EECKOHE'lHblX PSUtoB
rrJJ. 4
UeJlblX X COBI13.lJ.3eT CO CT3pbIM. JJ.JlSI ,noro .QOCT3TO'iHO yCT3HOBI1Tb,
'1TO F (X) Y.lI.OBJIeTBOpSieT ypaBHeHlHo
1
6.F(x)= x+I '
(AHaJlOrH'-!HO I1pn lI:OK33aTeJlbCTBe Toro, '1TO <PYHKLUIII YlI:OBJleTBOpHeT
cpopMYJle HHTerpnpOBaHHH, npOBep5l!OT, 4TO ee npoH3BO.QHaSl eCTb nO.ll.blH-
TerpaJIbHa51 CPYHKUHSI.) IlpI1MeH5I5I (4.9-2), HaXOll:HM
""
A x _ '\1 [ x+ I _ x ] _
F( )- k. r(r+x+l) r(r+x) -
r=1
ro ro
= [ (r+x)(:+x+1) ]= [ r x - r+;+l ]= X l '
r=1 r=1
Me)l(lI:Y .lI.HraMMa-cpYHKu,nett n HaTyp aJIbHblM JIorap ncpMoM cYllI.eCTByeT
cJJe.ll.Y!OllI.ee COOTHoweHHe:
F(x)=; mro (In (x + n + 1) - x I - X 2 -... - x n )'
ECTb .lI.pyra51 CB5I3b Me)l(.lI.Y 113BeCTHblMH cpYHKu,n5lMH
d
dx In [1' (1 + x)] =F (x).
LlaJlbHettn1HM lI:ncpcpepeHuHpoBaHHeM Mbl nOJIY'IHM TaK Ha3blBaeMYlO TpH-
raMMa-cpYHKuH!O
co
d 2 '\1 I ,
dx2 1n [1' (1 + x)] = k. (r+ X)2 = F (x),
1
TeTp araMMa-cpYHKu,H!O
""
d 3 '\1 '1
dx3 1n [1' (1 + x)] = - 2 k. (r+x)3 =-p (x),
1
neHT araMM a-<PYHK UH 10
ro
d 4 '\1 1
tiX4 Jn [1'(1 + x») =6 (r+x)4
1
F''' (x),
H T. lI:.
eTH cpYHKUIHI npoTa6YJInpOBaHbl 135] H MorYT npHMemITbC51 .lIJJ5I
CYMMHpoBaHwl pRlI.OB, 06ll1.Htl '1JIeH KOTOpblX eCTb paunOHaJIbH351 cpYHKU,HSI.
lLJISI rrpoMepa rrpe.ll.I10JIO)l(HM, 4TO Mbl HMeeM p5l.ll.
00
s- pn-. (x)
- '"" Pn (x) I
x=l
II 5.11
CI1CTEMA 0603HA 4EHI1i1
71
r.n:e P n (x) - MHOrO'lJIeH CTeneHI1 n 11 P n 2 (x) no KpatlHet! Mepe Ha .n:Be
CTerreHH MeHhllle, 'IeM P n (x). !1crrOJIh3YH 9JleMeHTapHbIe .n:P06H, MbI MO-
)i{eM Hanl1CaTb (C'II1TaH, 'ITO ai eCTb HYJII1 P n (x»
ex>
s- '" [ + + + + Bt + . B2 +
- x - at x - a 2 . . . x - ak (x - a t )2 (X - a 2 )2 . . .
x=1
+ Bk + + Mt + M2 + + Mk ]
... (X-ak)2 ... (x-a 1 )m (x-a 2 )m ... (X-ak)m'
3.n:eCb HeJJb3H OT.n:eJII1Tb nepBYIO rpynny 'weHOB, TaK KaK OT.n:eJIbHbJe
k
p5lJl:bJ pacxo.n:5ITC5I. OJl:HaKO JIerKO BI1J1:eTb, 4TO Ai = 0, CJIeJl:OBaTeJlbHO,
i=1
MOIKHO Hanl1CaTb
[( _ A1 ) + ( _ 'A2 ) +... + ( _ Ak )]
x-a 1 x x-a 2 x x-ak x t
He MeH5I5I 3Ha4eHI151 CYMMbJ rpynnbJ. Tenepb MbJ I1MeeM pH.n: B BHJl:e,
B KOTOpOM OH MO)i{eT 6bJTb JIerKO npOCYMMHpOBaH. B Ka4eCTBe npHMepa
paCCMOTpHM (CM. [35])
ex>
s= ! (4x+2) (4X I) (4x+3)2 =
\
co
"' [ t 1 t 3 I 1 I 1
= - 4x+2 +4' 4x+l +4' 4x+3 + '2 ' (4x+3J2 =
1 '
1 ( I ) 1 ( t ) 3 ( 3 ) t ' ( 3 )
=4 F 2" - 16 F 4 - 16 F 4 + 32 F 4'
fJ1ABA 5
YPABHEHHSI B KOHEqHbIX PA3HOCTSIX
5.1. CRcTeMa 0003HaQeHR"
Pa3HOCTHbJe ypaBHeHH5I B HC4HCJIeHI1H pa3HocTeff cooTBeTcTBYIOT
AI1<j:J<j:JepeHlJ.HaJIbHbIM ypaBHeHH5IM B AI1<j:J<j:JepeHu,HaJIbHOM HC'-IHCJleHHH, TaK,
HanpHMep, Jl:JISJ HeH3BeCTHot! <j:JYHKU,1111 Y MO)i{HO HMeTb .lI.11<j:J<j:JepeHlJ.l1aJlb-
Hoe ypaBHeHHe
y' +2y=x
HnH pasHocTHoe ypaBHeHHe
Ily -1- 2y =x.
(5.1-1 )
72
YPABHEHVl5I B KOHE4HblX PA3HOCT5IX
I fJI. Ii
JII060I1bITHO, '-ITO B 3TOM BI-ID:e aHaJ/Orl-!51 Me)f(D:Y LUlcpcpepeHUHaJIbHblMI1 11
p a3HOCTHblMH yp aBHeHI-I5IMH He 110J/Ha5l. Ho eCJIH Mbl 110D:CTaBHM
t1y=y(x+1)-y(x)
B (5.1-1), TO 110JJY'lI-IM ypaBHeHHe
y(x+ 1)+ y (x)=x.
(5,1-2)
51cHo, '1TO Bcer.n:a MO)f(HO nepefiTH OT o.n:Horo BH.n:a pa3HOCTHoro
ypaRHeHH5I K .n:pyrOMY. Mbl 6y.n:eM Bcer.n:a nOJ/b30BaTbC5I BTOpofi cpopMofi
3anHCH. B HetJ ypaBHeHl1e MO)f(HO peUIHTb B OCHOBHOM TaK )f(e, KaK Mbl
pernaeM ypaBHeHHe
y'+y=x.
D.eJIO B TOM, '1TO B 3Tofi cpopMe 60JJee 51CHO paCKpblBaeTC5I npl1po.n:a
aa.n:a'lH. TaK, eCJJYI .n:aHO ypaBHeHHe
t1 2 y+2t1y+ y=f(x),
TO MO)f(HO C'IHTaTb, '1TO Mbl J1MeeM pa3HOCTHoe ypaBHeHl1e BToporo no-
p5l.n:Ka. Ho OHO npJ1Bo.n:I1T K ypaBHeHHIO
f(x)=y(x+2)- 2y(x+ 1)+ y(xY+2 [y(x+ 1) - y(x)] + y(x)=
=y(x+ 2),
KOTopoe TpHBHaJJbHo. MaKCHMaJIbHblfi nOp5l.n:oK pa3HocTH aprYMeHToB
HeI-I3BeCTHoil CPYHKUIHI ol1pe.n:eJI5IeT nOp5l.n:oK pa3HocTHoro ypaBHeHI-I5I,
TaK KaK Me)f(.n:y pa3HocTHblMH 11 .n:l-IcpcpepeHUHaJIbHblMl1 ypaBHeHl151Ml1
eCTb npHMa51 aHaJIOrl-!51, CTOHT nOBTopHTb HeKoTop bIe .n:eTa. l1 Toro, KaK
pernalOTC5I .n:HcpcpepeHUl1aJIbHbIe ypaBHeHH5I. KaK H paHbrne, Bee npHeMbI,
KpOMe 3aMeHbi He3aBHCHMoro nepeMeHHoro, npHMeH5IIOTC5I B 060I-IX cJJyya5Jx.
S 5.2. IIpltMep pa3HocTHoro ypaBHeHltH nepBoro nOpH.1I,Ka
4T06bI pernHTb .n:HcpcpepeHUHaJIbHOe ypaBHeHHe
y'+y=x,
(5.2-1 )
CHa4aJJa paCCMaTpllBaeTC5I O.n:Hopo.n:Hoe ypaBHeHHe
y'+y=O
(5.2-2)
H HllJ.eTC5I o6mee perneHHe O.n:Hopo.n:Horo ypaBHeHH5I. Mbl npeD:110JIaraeM,
QTO I1pl1 D:OJI)f(HOM Bbl60pe m perneHl-IeM OD:HOpOD:HOrO ypaBHeHI151 6y-
/l.eT y = e mx . TIO/l.CTaBJI5I5I B ypaBHeHHe, nOJIY'laeM
me mx + e mx = 0,
e mx (m + 1)=0,
m + ! =0 (e mX =I=- 0),
m=-l,
5.21
npl1MEP P A3HOCTHO ro YP ABHEHI-HI nEPBO ro nOPSll!.KA
73
oTKy.n:a
y = Ce- x ,
(5.2-3)
r.n:e C - npOH3BOJJbHa51 nOCT05lHH351.
3aTeM HweTC5I '-IaCTHOe peweHHe nOJJHOrO ypaBHeHHSI. TIpHHHMaH 80
BHI1MaHl1e np3SYlO '-IaCTb, nOJJOlKI1M,
y=ax+b.
D.HrpcpepeHl.ll1aJJbHOe yp aSHeHl1e (5.2-1) 1I.aeT
a+ax+ b=x,
TaK '-lTO
a=1,
b=-1
n
y=x-I
(5.2-4)
eCTb '-laCTHOe perneHl1e (5.2-1).
VITaK, 06LUee perneHl1e eCTb cYMMa (5.2-3) 11 (5.2-4):
y = Ce- x + (x - I),
T. e. cYMMa 06LUero perneHI1H 01l.HOp01l.HOro ypasHeHI151 11 '-IaCTHOro
perneHI1H. KOHe'-lHO, 9TO,T MeTOl!. np I1MeHHM TOJJbKO K JJI1HetiHbIM yp as-
HeHH5IM.
D.JJ5I pa3HocTHoro ypaBHeHHH
y(x+ 1)+ y (x)=x
n.eJJaeTCR TO )Ke caMoe. B 0.lLHOp01l.HOe ypasHeHl1e
y(x+ 1)+ y(x)= 0
MbI n01l.CTasmIeM SMeCTO e mx
y=px,
'-ITO HSJJHeTC5I npocTbIM 113MeHeHI1eM 0603Ha'leHI151, TaK KaK MO)KHO 1I0JJO-
){(HTb em =p. TIOJJYlJaeM
pX (p + 1) = 0,
oTKy.n:a
p = - I 11 Y = C (- 1 t.
D.Jl5I nOJJHOro ypaBHeHI151 MbI onHTb npe1l.nOJIaraeM, 4TO
y=ax+b,
Pa3HOCIHoe ypaBHeHl1e 1I.aeT
a(x+ 1)+b+ax+b=x, 2a= 1, a= 1/2,
74
YPABHEHYl5I B KOHE'-IHbIX PA3HOCT5IX
[W. I)
TaK qTO a + 2b = 0, b = - 1/4 H Y = x/2 - 1/4. Tor,1J.a o6Il1.ee peme-
Hl1e eCTb
Y=C(-lY+ - .
TaKHM o6pa30M, MbI BHJU1M TecHYIO CBH3b JJHHetlHhIX ypaBHeHldi nepBoro
nOpH.llKa paSJIH'lHbIX TunOB.
5.3. I1pllMep ypaBHeHHH BToporo nOpHl(I{a
lIJIH JIUHetlHbIx ypaBHeHutl 60JIee BbICOKoro nopH,QKa npHMeHHeTC5I
ra )l(e TeXHlJKa. PaccMoTpllM, HanpllMep, 'lHCJIa <Pu60Ha'l'lU, KOTopbIe on-
pe.lleJJeHbI ypaBHeHueM
Yn+l = Yn + Yn-l
c
Yo = 0, Yi = 1.
npe)\{,1J.e Bcero HIlI.eM o6Il1.ee pemeHue. Dpo6ySl
y=pn,
nOJIyqaeM
p2_p_1 =0
UJIH
1::!:Y5
p= 2 .
taKHM o6pa30M,
-c ( 1+ V5 ) n +C ( 1- V5 ) 1J
Yn - 1 2 2 2 .
Y CJIOBHe Yo = 0 ,QaeT
0= C i + C 2 ,
a YCJJOBUe Yl = 1:
1'1 = 1 = C i ( I -1; V5 ) + C 2 ( I -2 V5 ) .
H3 9THX ypaBHeHHtl nOJIY'-IaeM
C t = J5 =-C 2 H Yn= :5 [ C-1; V5 r - e-;VS rl.
SrOT npHMep BbI6paH npe)\{.lle Bcero nOTOMY, qTO caMO pa3IIOCTHOe
ypaBHeHue .QaeT B03MO)\{HOCTb JIerKO BbJ'lHCJIHTb no O'lepe.lI.H nOCJIe.lI.OBa.
TeJIbHhIe 'IHCJIa. T aK
Yo=O, Ys=2, Y6=8, Y9 = 34,
Yi=l, y =3, Y7 = 13, YiD = 55,
Y2=1, Y3=5, YB = 21, Yu = 89.
II 5.41
JJVlHEflHbJE YPABHEHI1S'1 C nOCTO\1HHbIMI1 K03<J><J>VlUVlEH1'AMI1
75
HHKaKoro COMHeHH5I B cymecTBoBaHI1H perneHI1S1 He B03HHKa€T, HO o6mee
peweHHe, O'leBH.n:HO, CJJHWKOM Tpy.n:HO npUMeHHTb Ha npaKTI1Ke, XOTSI
OHO, MO)l{eT 6bITb, flOJ/e3HO .n:JI5I BbI'IHCJIeHH5I HeKOTopblX OT.lI.eJIbHblX
'1l1ceJI.
S 5.4. JhmeHHble pa3HocTHble ypaBHeHRH
C nOCTOHHHbIMII Koart>rt>HII.ReHTaMH
DpO.n:OJl)l{HM aHaJlOrmO C JIl1HeItHbIMI1 .lI.HcpcpepeHUHaJJbHbIMH ypaBHe-
HH5IMH. B CJIY'lae .lI.BofiHOro KOpH5I xapaKTepl1CTI-J4ecKoro ypaBHeHH5I MbI
HmeM peweHH5I KaK B Bl1.l1.e e mx , TaK U xe mx . PaccMoTpHM, HanpHMep,
ypaBHeHf-!e
yfl-2y'+y=0.
DOJIaraSl y = e mx , nOJIY'laeM
m2-2m+I =0
HJIH
mt=I,
m2 = 1.
Tor.ll.a eX 1-1 xe x y.n:oBJIeTBOpmoT ypaBHeHl-l1O H oomee perneHl-le eCTb
y = C 1 e x + C 2 xe x .
PaCCMOTpl-lM Tenepb pa3HocTHoe ypaBHeHHe
y(x+ 2) - 2y(x+ 1)+ y(x)=O.
Do.n:CTaBJHm y = pX, nOJlY'laeM
p2_ 2p+ 1 =0,
Pl=I,
P2 = 1.
Tor.n:a Ka)l{.D:Oe 1-13 perneHHtl (It = 1, x (It = x y.n:oBJIeTBOpSleT ypaB-
HeH!11O H oomee perneHHe eCTb
y=C t +C 2 x.
BcnOMHHM, '1TO aHaJlOrH'IHbIM 06pa30M Mbl .n:eficTByeM, Kor.n:a npaBaS!
'1aCTb Heo.n:Hopo.n:Horo ypaBHeHI1S! co.n:ep)l{HT '1JIeH, KOTOpblfi eCTb B pe-
rnemm O.ll.HOpO.LJ:HOrO ypaBHeHI151. B cJlY'Iae
y' + y=e- x
O.n:Hopo.n:Hoe ypaBHeHHe
y'+y=O
HMeeT peweHHe y = Ce- x . D09TOMY 4aCTHoe perneHl-le Bcera ypaBHeHHSI
Mbl nblTaeMCSI HaltTH B BH.n:e
y = kxe-X, ke- x - kxe- x + kxe- x = e- x , k = 1.
76
YPABHEHr15J B KOHE'fHbIX PA3HOCT5JX
[TIT. I)
OKOH43TeJJbHO nOJJyqaeM
y=Ce-X+xe- x =(C+x)e- X
TO'-IHo T3K /Ke, paCCM3TpI1Ba51 pa3HOCTHOe ypaBHE;HUe
y(x+ 1)- y(x)= 1,
MW COCT3BJJ5IeM O Hopo Hoe ypaBHeHl1e
y(x+ 1)- y(x)=O.
OHO I1MeeT perneHl1e y = C, r e C - np0I13BOJlbH351 nOCT05lHHa5l. TI09TOMY,
Kor.n:a MbI yra blBaeM 4aCTHoe perneHl1e, TO np06yeM
y=kx.
TIo.n:CTaBJl5I5I ero, BI1 I1M, 4TO k = 1, CJJe.n:oBaTeJJbHO, o6rn:ee pernelJHe
nepBOHa4aJlbHOrO ypaBHeHI151 eCTb
y = C + X.
MI1J1H- TOMCOH [27] paCCMaTpl1BaeT perneHl'le pa3HocTHbIX ypaBHeHl1fi
60Jlee no.n:po6HO. OH IlOKa3blBaeT, B 4aCTHOCTI1, KaK 3HaHl1e O Horo
perneHI151 O Hopo Horo JJI1HetiHoro pa3HoCTHoro ypaBHeHH5I aeT B03-
MO)!(HOCTb nOHI13l1Th nOp5l.n:oK ypaBHeHI151 Ha e I1HI1UY.
5.5. TIpuMep
UeJlh aToro naparpaq,a - A3Th np!1Mep np!1MeHeHI1S1 pa3HocTHblX
YP3BHeHl1fi I1 nOK33aTh, K3K HeKoTopbIe !1 el-l 11 MeTo.n:bl 1-13 06JJaCTl1
06hlKHOBeHHblx 11q,cjJepeHUl1aJJhHhIX ypaBHeHHI! MO/KHO eCTeCTBeHHWM 06-
p330M npHMeH5ITh K pa3HOCTHhIM yp3BHeHI151M.
3a.n:aQa CTaBHTC5I TaK: HafiTI1 I1HTerpaJJbI
00
l(n)= xne- x sin xdx
(J
(n 0),
00
K (n) = xne- x cos x dx
o
H, B QaCTHOCTI1, fIoKa3aTb, 4TO I (4k + 3) = O.
VIHTerpl1pOBaHHe no QaCT5IM aeT cncTeMY pa3HocTHblX ypaBHeHHfi
n . )
l(n)="2[I(n-1)+K(n-I)], .
K(n) = ; [-I(n-1)+K(n-I)].
(5.5-1 )
I
VI3 cTaH.n:apTHbIx TaOJlI-IU HHTerpaJJOB HaXO HM 1(0) = K (0) ="2'
5.5J
np l MEP
77
3TU pa3HOCTHhie ypaBl eHI151 HMelOT nepeMeHHbIe K09tjJtPUIlUeHTbi ,
n09TOMY I (n) HMeeT MH01KHTeJlb n, I (n + 1) I1MeeT MH01KI1TeJlb (n + 1)
11 T. .11:., TaK '1TO I (n) BeJ.\eT ce651, KaK n!. TaKI1M 06pa30M, MbJ npl1xo-
J.\I1M K nOJ.\CTaHOBKe
I (n) = n! j (n),
K (n) = n! k (n).
1
I1pl1 9TOM Mbl He Yl!JlI1 MH01KI1TeJlb 2 B K03<pqJI1UHeHTax pa3HOCTHbiX
ypaBHeHl1iJ. I1pI1H5IB BO BHI1MaHl1e 11 ero, nOJlY'IaeM npeo6pa30BaHI1e
I (n) = j (n),
n!
K (n) = 2 n k (n),
(5.5-2)
KOTopoe npl1Bo.ll:uT K CI1CTeMe
j (n) = j (n - 1) + k (n -- I),
k(n)=-j(n-I)+k(n-I),
1(0) ! ' J
k (0) - 2'
(5.5-3)
ECJIH 6bl 3TO 6wIa CHCTeM3 06blKHoBeHHbIx J.\11<p<pepeHUHa.7IbHbIX
ypaBHcmltt, MbI HJlI1 nbIT3JII1Cb 6bl BcnOMHHTb, KaK pernalOTc51 CHCTCMhI,
H.7IIl CBCJlI1 33J.\a'lY K O.ll:HOMY ypalmCHl110 BToporo nOp5lJ.\Ka, npO.ll:I1<plpe-
peHUl1pO!3aB OJ.\HO ypa!3Helmc H I1CKJlIO'II1B OJ.\HY ncpeMcHHYIO. I1ocJlc.II:-
Hce, rrO-BI1J.\I1MOMY, JIer'le rrpHcrroc06HIb K pa3HocTHbIM ypaBHeHI151M.
Hal1l1rnCM BepxHee ypaBHeHHe (5.5-3) B Bl1jJe
;(n+ l)=j(n)+k(n)
(5.5-4)
I1CKJIIOl!I1M k (n). 3a-
I1CKJIIO'IeHI1>1 k (n - 1),
11, I1Crr0J1b3Y5I ero BO BTOpOM ypaBHcm:11 (5.5-3),
TeM, I1Cl10JIb3YH nepBoe 113 ypaBHeHl1iJ (5.5-3) JI}I5I
110JlYl!aeM
j (n+ 1) - 2j (n) + 2j (n - 1)= O.
(5.5-5)
3TO - JII1HeiJHoe pa3HocTHoe YP3BHcHue BToporo rrOp51.11:Ka C rrOCT051H-
HblMH K03<P<PI1Ul1eHTaMI1. Hal!aJIbHble YCJlOBI151 HaXO.ll:l1M 113 (5.5-3) f1 (5.5-4):
j(O)={, j(I)= 1. (5.5-6)
06mec pCrneHl1e (5.5-5) CCIb (3.11:eCb i = V -I)
; (n) = C 1 (1 + On + C 2 (1 - i)n,
a Haq3J1bHbIe yc.nOBH51 (5.5-6) .ll:alOT
1
C 1 +C 2 =2' C , (1+i)+C 2 (1-i)=1,
lIJIli
l-i I+i
C 1 - -;r-, C 2 =--;r'
78
KOHE'IHblE PSUlbl <l>YPbB
[fll. 6
TaKI-IM 06pa30M,
j (n) = [(1 + i)'H + (1 - on-I]
H
J (n) = 2 ll [(1 + l)n- 1 + (1 _l)n-l]
(Ha caMOM .n:eJIe J (n) 1-1 j (n), KaK BI1.n:HO 113 rrOJIYtJeHHhlX Bblpa)l(eHl-ltt,-
.u.eticTBHTeJlbHhle tJilCJIa).
4T06bl peUIHTb BTOpyJO 4aCTb 3a.n:a4H, 3aMeTI-IM, 4TO
(1 + l) = 1 + 41 + 61 2 + 4t 3 + l = - 4 = (1 - t)4.
CJle.n:OB8TeJlbHO,
j (4k+ 3) = (_4)k j (3)= ( )k [(1 + 2i +t 2 ) + 1 - 2t +t 2 ] =0.
T8KI1M o6pa30M,
1(4k+3)=0.
YnpamHeHHe 5.5.1. f1YCTb
"
'k(<P)= \ coskO-cosk<P dO
cos 0 - cos <P .
()
f1oKa3aTb, 1JTO JJ:JlR k, paBHoro l1.eJloMY 1JI-ICJlY,
/k+2 (<p)-(2cos<P)lk+1 (<P)+'k (<P) =0, '0=0, It='f!:o
H, CiJe)toBaTeJlbHO,
, (<P) = 1t sin k<P
k sin <P
r.nABA 6
KOHEQHblE PSI1I.bl <l>YPbE
6.1. BBeJ(eHlIe
P!!Abl <t:>ypbe
<XI
[(x) = °2 + (ak cos kx + b k sin kx),
k=1
(6.1-1)
r.n:e
2"
ak= I(x) cos kxdx,
o
l.n
bk= I(x) sin kxdx, (6.1-2)
6.21
OPTOrOHAJJbHOCTb HA LU1CKPETHOM MHmKECTBE TOllEK
79
Chlrp3J111 B3)1{HYlO pOJlb B pa3BI1THI1 M3TeMaTI1KI1 *). HapH.l!Y C 06b1'IHotJ
TeOpI1etJ, B KOTOpOtJ cjJYHKUI1H npe.l!nOJlaraeTCH I13BeCTHotJ Ha BceM HH-
TepBane .l!JJI1HbI 2", cYlUecTByeT TeOpI1H, paccM3TpHBalOlUaH cjJYHKUI1I1,
3aJJ,aHHble Ha .l!HCKpeTHoM MHO)l{eCTBe paBHooTcTOHlUHX T04eK. MbI 6YJJ,eM
nOJlaraTb, 4TO HMeeTCH 2N P3BHOOTCTOHIUl1X T04eK, 11 OCTaBHM CJJY4at!
He4eTHoro 4HCJJa T04eK 2N + 1 B Ka4eCTBe ynp3)1{HeHI1H .l!JJH qnTaTem.
P33JIO)l{eHHe B pH.l! <t>YPbe OCHOBaHO Ha opTOrOHaJJbHOCTH cjJYHKIU1tJ 1,
cos kx, sin kx no OTHOIlleHl11O K I1HTerpnpOB3HI11O no OTpe3KY [0, 27t]
2" I 0,
cos kx cos mx dx = ",
o 2
2" { °
sin kx sin mx dx = '
o
2"
sin kx cos mxdx=O.
o
k::;!=.m,
k=m::;!=.O,
k = m = 0,
k::;!=.m,
k=m::;!=.O,
(6.1-3)
6.2. OpTOrOHaJIbHOCTb Ha JUICKpeTHoM MHOmeCTBe TOqeK
fI pnMeqaTeJlbHO, qTO ecnl1 BMeCTO HHTerpnpOB3Hl1H MbI HcnoJlb3yeM
CYMMl1pOBaHne, TO cjJYHKUl1n
...,
cas (N - 1) x, cos Nx,
sin (N-l)x
1, cas x, cos 2x,
sin x, sin 2x,
...,
OKa3bIB31OTCH opTorOHaJlbHbIMH H3 .l!I1CKpeTHoM MHO)l{eCTBe TOqeK
(6.2-1 )
0,
'It 2'1t
N ' N'
(2N - I) 'It
.., , N
flpeo6pa30B3HHe
'It
x-+ N x
npPlBO.l!HT Hac 1< CI1CTeMe cjJYHI<UI1t!:
1, 'It 2'1t (N-l)'It 'It
COS iIi x, cos iIi x, ..., cas N x, cos N Nx,
'It 2'1t . (N-I)'It
sin ilix, sin ilix, ..., Sill N X.
(6.2-2)
*) 3J1eMeHrapHoe H3J10)/{eHHe pRJJ,OB <PYPbe eM. [17].
80
KOHE4HbIE P5ILlbl <l>YPbE
[fJJ. 6
COOTHOWeHl-HI OpTOfOH3JJbHOCHI Me)/{.lI.Y HHMH (k,,;:;; N,
2 I 1t 1t { O, eCJJH
7 sin N kx sin N mx =
.-. N, eCJJH
)(=0
2N-\
,.., 1t 1t
k. sin N kx cos N mx = 0 ,
)(=0
2N - I ( 0, eCJJI1
1t k 1t N
.t... cos N x cas N mx = , eCJJH
x=O 2N, eCJJH
m N)
k;t=m,
k=m::j:;O,
(6.2-3)
k =1= m,
k = m =1= 0, N,
k=m=O, N,
... ,
... ,
6bIJ/11 J1.0K333Hhl B ynp3)!(HeHHI1 (3.2-3). OHH MorYT p3CCM3TpHBaTbCH
KiJ K 3H3J10rH YCJloBtm nepneHJU1KYJl5IpHOCTH (opTorOH3JJbHOCTH) B 061,14-
HOM TpexMepHOM npOCTp3HCTBe. (CM. 17.6.)
ECJJH cpYHKUH5I MO)/{eT 6bITb 33rrHC3H3 B BI1J1.e
N-I
(x)=;o+ 2: (akcos;Vkx+bk sin ;Vkx)+ a; COS7tX,
k=1
(6.2-4)
TO MO)J{HO HCnOJlb30B3Tb COOUIOWeHH5I opTOrOH3JlbHOCTI1 J1.JJ5I onpe.lI.eJle-
HH5I K09cpcpl1UYJeHTOB p33JJQ}1{eHH5I. l!To6hl nOJIY4HTb a k (k = I, . . ., N - 1),
1t
YMHO)J{HM o6e 43CTH p3ueHCTB3 Ha cas N mx H npocYMMl1pyeM no BceM x.
fIOJlY4HM
2N-I
! I(x) cos mx = Na m
x=O
(l,,;;m,,;:;;N-I),
(6.2-5)
TaK K3K no (6.2-3) Bee OeTaJJbHble lIJJeHbI paBHbI HYJlIO. 3aMeH5I5I
1t . 1t
COS N mx H3 SIn N mx, rrOJlY43eM
2N-1
! (x) sin ; mx = Nb m
x=o
(I m N -- I).
(6.2-6)
OKOHlIaTeJJbHO HMeeM
2N - \ 2N - I
! !(x)= 2N( ;O ) = Na o , ! !(X)COS7tX=2N( a:) =Na N .(6.2-7)
x= 0 x=O
fIOCne.lI.HHe .lI.Be cpOPMYJlbJ HMelOT TOT /1{e BI1.lI., 4TO (6.2-5) !1 (IOK331>1-
BaIOT, n04eMY BblrO.lI.Hee nl1CaTh pa3JIO/1{eHl1e c ao/2 H a N /2, a He c a o
11 aN'
6.31
T04HOCTb PA3JJO}!{EHI151
81
YnpalliHeHHe 6.2-1. PacCMOTpCTb pa3J10)/{eHHe .n. R HetJeTHoro tJHCJla 2N -t- 1
Y3.10i3bIX TOtJeK,
o T B e r:
N
a o ( 2nk . 2nk \
f(x)=2+ -'- akcos 2N+1 x+bksIll 2N+1 X).
k=l
6.3. TOQHOCTb pa3J10llieHHSI
B paCCMaTpHBaeMOM pa3JIO}!{eHHH 2N 3Ha'-leHHt! f(x) (x = 0, ],...
. . ., 2N - I) onpe11,eJISIIOT 2N K03cjJcjJI1lJ,I-JeHTOB ak, b k 11 eCTeCTBeHHO
O}!{HJJ:3Tb, '-ITO CYMMa p5I.na B .n3HHbIX TO'-lKaX 6Y11,eT B TO'-lHOCm paBHa
3HaqeHI:I5IM HaqaJIhHot! cjJYHKUHH (T. e. '-ITO CHCTeMa cjJYHKUI1t! 5IBJ!5IeTCH
nOJIHot! no OHIOWeHl11O K rpynne I1CnOJJb3yeMbIx TO'-leK). 4T06bI 11,OKa-
3aTb 3TO npe11,nOJIO}!{eHHe, 3acjJI1KcHpyeM x = X H neperpynnHpyeM CYM-
MY, npHMeH5I5I paBeHCTBa (6.2-5)-(6.2-7), qT06bI HCKJIIO'-IHTb a k H b k :
N l
f(x)= ,!t(X) [-}+ 2: (cas kx cos kx + sin kx sin kX) +
x k=l
+ + cos 7tX cas 1tx] =
N-I
1 '\1 [ 1 + n - 1 ' - J
= N ./u,!(x) 2 ..::.. cos N k(x-x)+2cOS7tXCOS1tx.
x ',=1
Tenepb HanHilleM
N-I N-l N-l
1 cosifk(x-x)= ! COS k(X-X)++! cosifk(X-x)
k l k=\ k=1
H BO BTOpOt! CYMMe HCnOJIb3yeM TOT cpaKT. '-ITO
cos if k (x - x) = cas (2N - k) (x - x),
'-ITo6bJ I10JIY'{HTb
IV - i N - 1 2N - I
I cos k (x - x) = ! cos k (x-x) + ! cos k (x-x).
k j k=l k=N+l
l1cno,1b3Y5I STO, HMeeM
2N--l
f(x)= 2 !t(X) [ ! cos k (x - x)].
x k=O
82
KOHE'-!HblE P5UJ,bl <l>YPbE
[m.6
Ho Bblpa)l{eHl1e B KBa,lJ,p aTHblX cKo6KaX
2N-l { 2N
! cos :v k(x-x)= 0'
k=O '
paBHO
eCJII1
x=x,
eCJIH
x =F X,
TaK 'ITO OT CYMMbl OCTaeTCH JIl1ll1b 'IJIeH X = X 11
I(x) = 2 /(x) 2N= I(x).
TaKHM 06pa30M, o6e 'IaCTI1 ypaBHeHI1H ,lJ,alOT O,lJ,HO 11 TO )l{e 'II1CJJO.
CYllI.eCTByeT XOpOIliO 113BeCTHoe COOTHOllleHl1e Me)l{,lJ,Y CYMMOI1 KB a,lJ,-
paTOB K09rpcpl1lll1eHTOB 11 CYMMOit KBMpaTOB 3Ha'IeHnl1 rpYHKllHI1
N-l
! [f(x)j2 = N[ ! l- + ! (aJi + bJi) + a 2 }v ]. (6.3-1)
x k=l
YTo6bl ,lJ,OKa3aTb ero, o6p a3yeM CYMMY KBMP aTOB 3Ha'IeHI1I1 cpYHKlll111
N-l
! [I (x)] = ! [ at + ! (ak cos 7v kx + b k sin kX) + N cos 7':X T
x x k=J,
H Bbll10JIHHM YMHO)l{eHl1e. CooTHOllleHI1IJ opTOrOHaJIbHOCTI1 (6.2-5)-(6.2-7)
,lJ,alOT
![f(X)] =( a 2 Y.2N+ !(aJi.N+bt. N)+ ( N Y.2M
x k
'ITO COBrra,lJ,aeT C (6.3-1).
STOT pe3YJIbTaT l\W)I{eT 6hITb HCrrOJJb30BaH ,lJ,JI5I Bbl'II1CJIeHH5I CYMMbI
KBa,lJ,paTOB Olll1160K, KOr,lJ,a I1CnOJIb3YlOTC5I 'IJIeHbl JIl1ll1b ,lJ,O k = M <N-l.
TIYCTb 1 M (x) - CYMMa nepBbIX M rapMOHI1K, TaK 'ITO
M
f M (x) = a 2 + ! (a k COS kx + b k sin kX) .
k=]
Tor,lJ,a
N-1
!(/-fM?=![ ! (akcosHkX+bk sin kX)+ a2" cOS7':XT=
x x k=M-\-1
N-I 1\1
=N[ ! (a%+bk)+ a ]= }>2_N[ }+ !(aJi+bO]. (6.3-2)
k=M-\-1 x k=1
LLpYrHMH CJIOBaMH, I1CrrOJIb30BaHHe 'IJIeHOB JIl1ll1b ,lJ,O k = M B pa3JIO)/(e-
HI1H cpYHKlll111 (6.2-4) ,lJ,aeT HaM rrpH6JJH)I{eHI1e K I (x), OU1I16Ka KOToporo
B CMbJCJIe Hal1MeHblllHX KBa,lJ,paTOB orrpe,lJ,eJI5IeTCH paBeHcTBoM (6.3-2).
\!lccJle,lJ,Y5I, K3K YMeHblllaeTcH (6.3-2) rrpH YBeJIH'IeHHH M, Mbl MO)KeM
ou,eHHTb 3Ha'IeHHe B3HTbiX ,lJ,OrrOJIHHTeJIbHO 4JIeHOB B pa3JIO)l{eHI111 <t>YPbe.
6.4]
r Bbl411CJJEHI1E K03<1><I>I1I..{I1EHTOB
83
6.4. BblQHCJleHHe K09<1HIJHlI,HeHTOB
K03tPcj.JI1Ul1eHThI pa3JIO}KeHI1H (6.2-4) .aaIOTCH tPopMYJIaMH
2N-l
1 '\1 1t
ak= N I(x) cos N kx
x=o
2N-l
b k = ! I(x) sin kX
x=o
(k = 0, 1, 2, ..., N), 1
}
(k= 1,2, ..., N-1). I
}
(6.4-1 )
nOKa3aHO, '1TO 3TH K03cj.Jcj.JI11J.l1eHTbI MO}KHO JIen{Q BbI'IHCJIHTb, 3HaH JII111Ib
cos k H sin k, 6ea HerrOCpe.lI.CTBeHHoro BbI'IHCJIeHHH .lI.pyrHx aHa 'Ie-
Hl1fi CI1Hyca H KocHHyca. cho .lI.eJraeTCH CJIe)J,YIOUJ.HM o6pa30M: rrycTb
Uo=O, U 1 =/(2N-1),
U m =(2 cos -"N k) U m _ l - U m _ 2 +/(2N-m)
(m=2, 3, .." 2N-1).
(6.4-2)
Tor.lI.a
2N-1 )
Nak = I (x) cos kx = (COS k) U 2N _ 1 - U 2N _ 2 + 1(0), I
x=o t
2N-l
Nb k = ! I(x) sin kx = ( sin k) U 2N _ 1 .
x=O J
npmvlOe cj.JopMaJJbHOe )J,OKaaaTeJIbCTBO He nOKa3hIBaeT MexaHH3M3 pa-
60ThI 3Toro MeTO)J,3, TaK 4TO MbI )J,a)J,I1M HeCKOJIbKO 60JIee )J,JII1HHOe )J,OKa-
aaTeJIbCTBO, KOTopoe K TOMY }Ke .lI.aCT eru.e H .lI.onOJIHHTeJIbHhlfi pe3YJIbTaT.
Ha'lHeM C onpe.lI.eJJeHI1H
(6.4-3)
Vo=O, VI = 1,
V m = (2 cos t) V m-l - V m-2
(m = 2, 3, ...) }
(6.4-4)
H nOKa}KeM CHa'laJIa, '1TO
V _ sin mt
m- slnt '
(6.4-5)
TaK KaK (6.4-5) BepHo )J,JIH m = 0; I, TO )J,OCTaTO'lHO npOBepHTb, '1TO
( 2 cos t ) V m _ l - V = 2costsin(m-l)t _ sin(m-2)t_
m-2 sin t sin t
_ sin mt + sin (m -2) t- sin (m- 2) t _ sin mt _ V
- sint - sint - m'
84
KOHE4HbIE P5UlbI <PYPbE
(TJJ. 6
Mbl 3aMe4aeM TaK)/{e, 4TO
t ( V ) _ v _cas t sin mt - sin (m - I) t
cos m m-l - sin t
[sin (m + 1) t + sin (m - 1) t I - sin (m - I) t
sin t
sin (m + I) t - sin (m - J) t t
= 2 sin t = cas m .
(6.4-6)
TaKI1M 06pa30M, 3HaH 3Ha4eHI1H sin t 11 cas t, MbI MO)J{eM BbILJI1CJIHTb,
npl1MeH5!H (6.4-4)-(6.4-6), 3Ha4eHHH sin 2t, cos 2t, sin 3t, cas 3t, '"
..., sin kt, cos kt. TIpH 9TOM MbI .lI.eJlaeM He 6oJlee, '1eM
{ 3k YMHO)/{eHHtt }
HJIH
2k BbI4HTaHHtt
{ 2 3 k YMHOIKeHHtt } .
k BbI4HTamJtt
TIpH BbI'1I1CJIeHI1H 3Ha4eHHtt 9TI1M MeTOJJ:OM npOI1CxoJJ:IJT, KOHermo, nOCTe-
l1eHHaH nOTepH T04HOCTH, HO 06bJ'-!H0 OHa He3H3'1!neJIbHa.
TIoKa)j{eM Tenepb, KaK npHMeHHTb paBeHcrBa (6.4-4)-(6.4-6) K JJ:OKa-
3areJIbCTBY (6.4-3), I1cnOJJb3Y5! (6.4-2).
Bo-nepBbIx, paccMoTpHM cJIY4afi, KorJJ:a Bce I (x) = 0; KpOMe I (j),
It
H nOJIO)KHM t = N k. vb (6.4-2) BHJJ:HO cpa3Y, 4TO
U o = U 1 =. ..= U 9N - J - I =0,
U 9N - i = IU).
TaKHM o6pa30M, U 9N - i COOTBeTcTByer VI' 3a HCKJII04eH!1eM Toro, 4TO
Bce U, KOTopI.Ie BI.I411CJI5!lOTCH, 6YJJ:YT B I (j) pa3 60JJI.lI1e, 4eM COOT BeT-
CTBYJOUJ.11e V. Kor JJ:a MI.I JJ:ottJJ:eM J1,0 U 9N _l> MI.I CJ1,eJIaeM pOBHO i lI1arOI
1-1 eMY 6yJJ:eT COOTBeTCTBOBaTI. Vi' TIo (6.4-5) 11 (6.4-6) pe3}TJJI.T<lTbI,
KoropbIe MbI J-laII1JJH B (6.4-3), paBHbI
f(j) sin j k,
f U) cas j H k.
BepHeMC5! Tenepb K npOH3BOJJbH0t1 CPYHKIJ.HI-I f (x). OHa MO)KeT 6I.ITb
npeJJ:CTaBJIeHa B BHJJ:e CYMMbI CPYHKIJ.Ht1 TOJJbKO 4TO paCC lOTpeHHoro Tlma.
B CI-IJIY JII1Het1HocT11 (6.4-2) H (6.4-3) OTJJ:eJIbHbIe pe3YJIbTI!TbI JJ:JIH Ka)/{-
JJ:ott 1-13 KOMnOJ-1eHT MO)/{HO CJIQ)KHTb, 4TO H JJ:OKa3bIBaeT (6.4-3).
LLJI5! HaXO)/{JJ:eHHR Ka)/{JJ:ofi napI.I ak, b k Tpe6yercH OKOJ10 6N ap!1cpMe-
TI-I4eCKHX onepaIJ.l-ltt, TaK LJTO JJ:JI5! Bcex 2N K09cpcpI1I HeHTOB HY)j{HO cJJ:e-
JIaTb OKOJIO 6N 2 JJ:ettcTBHtt ECJJ!1 6I.I MI!TpHIJ.1! CHHyCOB H KOCHHYCOB 6bwa
B nI!MHT11 MI!II111HbI 11 J1,JJH nOJIY4eH!1H K09cpcp!lIJ.HeHTOB MI.I npoCTo YMHO-
)KaJJH 6I.I MaTpHIJ.Y HI! BeKTOp, TO npH 9TOM npHII1JIOCb 6bI 3aTpl!THTI. 8JV'A
.u.ettcTBHtt. TaKHM 06pa30M, MO)KHO CKa3an, LJTO npeJJ:JIO)J{eHHI.Itt MeTO.lI.
9KOHOM!1T Malli!lHHOe BpeMH JJ:a)l{e no cpaBHeH!lJO C 04eHb 6J1aronp!15!T-
Hott CI1Tyau.!1ett, KOf JJ:,a JJ:aHa MaTp Hl a CHHYCOB !1 KOClillYCOB.
6.51
METon llBEHAfl1LATH OPllHHAT
8f)
IA " . 11 11 V 6
n3 1I.BYX 3Ha4eHHl'! sm N ' cos N ' HCrrOJIb3YII MeTOll -0 03Ha4eH\1ti,
MO)/(HO 110JIY4HTb
11
sin N k,
11
cas N k
(k = 2, 3, ..., N - 1).
OCHOBHOfi npouecc B V-0603HaCIeHHH Tenepb llaeT Tpe6yeMble K09(p<pH-
lJ,HeHTbI. DpH cooTBeTcTBYKJUleM neperrJIeTeHlIl1 V- H U-npoueccoB B J<a)j{-
Ablt1 MOMeHT Heo6xoLLHMO JIlUllb 2N 114eeK LLJUI TeKYillHx llaHHbIX H He-
CKOJlbKO CCIeT4HKOB DporpaMMbI LI:JlII BbI4HCJIeHHII V H U TaK)J{e LI:OBOJlbHO
KopoTKHe, a CJIeLI:OBaTeJJbHO, He 3aHHMalOT MHoro MeCTa B 11aM\lTH.
, 6.5. MeTO)l )lBeHaJI.u.an 0pJI.HHaT
OC06blfi CJIYCI, 3fi KOrLI:3 4HCJIO TOT.JeK pam'Io 12 (N = 6), rrpe.n.cT3B-
JIlleT HHTepec, TaK KaK CIaCTO BCTpe4aeTcII H JIerKO BbIIlOJJHlleTC\I BPYlJ-
HYIO. MbI HMeeM
II
6ak = ! (x) cos i kx,
>::=0
II
6b k = (x) sin ikX.
>::=0
Pa36lma51 (3)K.n.ylO CYMMY 110 x Ha D:Be: OT 0 LI:O 6 H OT 7 Jl.O II,
H nOJlO)!{HB BO BTOpOtl qaCTH x = 12 - x', nOJIYl.fHM
6 5
6ak = !(x) cos i kx + ! (12 - x') cos - - kx',
x=O x'=1
6
6b k = !f(x) sin i kx - ! ((l2 -x') sin i kx'.
x=O x'=]
CYMMa
Pa3HocTb
{(OJ f(l) f(2) {(3) ((4) f(5) ((5)
1(11) 1(10) 1(9) [(8) 1(7)
s (0) s (I) s (2) s (3) s (4) s (5) s (6)
t 9" t (2) t (3) t (4) t (5)
TOfJl.a HMeeM
6
6ak= !S(X) cos i kx,
x=O
5
6b k = ! t (x) sin i kx.
x=J
86
KOHE4HbIE P51llbJ <l>YPbE
r r J1, 6
Pa306beM OmlTb OTpe30K Ha .lJ.Be 4aCTH, Ha 3TOT pa3 OT 0 .lJ.O 3 1'1 OT 4
/1.0 6, nOJIO)\(HB x = 6 - x' BO BTOpOM oTpe3Ke. TOfJJ:a
3 2
6a k = S (X) cas kx + (- l)k S (6 - x') cos i kx'.
x=o x'=o
3 2
6b k = !t(x) sin kx - (- It t(6 - x') sin i kx'.
x=l x'=l
Tenepb MO)\(HO Hal1HCaTb
s (0) s (I) s (2) s (3) t (I) t (2) t (3)
s (6) s (5) s (4) t (5) t (4)
CYMMa u(O) u(l) u(2) u(3) pel) p(2) p(3)
Pa3HocTb v(O) v(l) v(2) q(l) q(2)
PeSYJIbTaT, BbIl111Camlblt! nOJIHOCTbIO, Bblf JIHJJ:I1T TaK:
'6ao = II (0) + II (1) + II (2) + II (3) = [ll (0) + u (3)] + [u (1) + tt (2)],
6al=V(O)+ r;3 '0(1)+ '0(2)=['0(0)+ V(2)]+ 3 '0(1).
I
6a2 =u(0)+2 [ll (1) -ll (2)] -ll (3) =[ll (0) -ll (3)] +
l
+ 2 [ll (1) -ll (2)],
6a3 = v (0) - v (2),
l I 1
6a4 =u(O) - 2u (1) - 2 11 (2) + u(3) = [u(O) +u(3)] - '2 [u(1)+ll(2)J.
6a5=V(0)- r;3 v(1)++V(2)=[V (0) ++V(2)]- r;3 v(I),
6a6 = II (0) - [u (1) -ll (2)] -ll (3) = [11 (0) -/[ (3)] - [u (1) - 11 (2)].
6b j = p(I)+ 3 p(2)+P(3)=[+P(1)+p(3)]+ 3 p(2),
1/3
tib 2 =----y- [q (1) + q (2)],
6b 3 =p (1) - P (3),
6b 4 = r;3 [q(I)-q(2)],
6b 5 =ip(1)- 3 P(2)+P(3)= U p(1)+p(3)]- 3 p(2).
6.7J
PA3JJO}!{EHHE no KOCI1HYCAM
87
3.u:eCb rrpHxo.u:HTC5I .u:eJJaTb MeHbIlJe 60 apH<pMeTi-lIJeCKHX orrepau,Ht! H
60JIbIlJHHCTBO H3 HHX - rrpOCTbIe CJIO}!{eHH5I. 3TO MO}!{HO cpaBHHTb C
6N 2 .= 6 3 = 216 onepalJ,lJSlMlJ, Heo6xo.u:lJMbIMI1 B MeTo.u:e 6.4. fIporpaMMa,
KOHeIJHO, HeCKOJIbKO .u:JJHHHee, HO ImcaTb ee 04eHb J1efKO, TaK KaK HeT
CJIO}!{HOt! JIOfUKI1 H Tpe6yeTcSI JIUIlJb O.u:HO «HerrpocToe» 4HCJJO
V3
= 0,866 025 407 5.
06IUue MeTo.u:bI MorYT H .u:OJI}!{HbI rrpHMeHSlTbcSI B CJIY4a5lX, OTJII1IJHbIX
OT N=6,
Ynpa}!{HeHlle 6.5-1. HaihH BpY'JHYlO pa3J10)/{eHUe <t>YPbe J],JIR f(x) -x (12-x)
(x=O, I, ...,11).
6.6. MeTO]];},1 C MllHllMYMOM YMHomeHllii
Boo6IlJ,e B 3a.u:aIJlJ 3TOt! KHHflJ He BXO.u:HT B.u:aBaTbC5I B .u:eTaJIH TOro.
KaK opraHH30BbIBaTb BbIIJlJCJJeHHe, IJTo6bI 3KOHOMUTb MaWUHHoe BpeM5I.
O.u:HaKO nOKa3aTb, KaK MO)!{HO 3<pcpeKTHBHO BbIIJHCJJ5ITb P5l.u:bI <t>ypbe,
Heo6xo.u:HMO, TaK KaK 06bIIJHO npe.u:rrOIJHTaroT MHOrOljJIeH!1bIe pa3JIO)!{eHH5I,
a TaK)!{e rrOTOMY, IJTO npH UCrrOJIb30BaHHH CYMM <PYPbe 06bIIJHO TpaTHTCS!'
MHoro JI!'JWHero ManlHHHoro BpeMeHH.
4!'JCJ10 YMHO)!{eHUt! MO}!{HO CBeCTH K MHHlJMYMY, BbIIlOJJH5I51 CHa4aJ1a
Bce CJIO)!{eHI151 H MaKCI1MaJIbHO HCnOJIb3Y51 npUBe.u:eHHe TpUfOHOMeTpn4e-
CKHX <PYHKlJ,Ht! K 3HaIJeHI151M B rrepBot! 'IeTBepTu. BOT rr04eMY pa60TaeT
MeTo.u: 6.5. BC5IKHt! pa3, Kor.u:a rrop5l.u:oK rapMOHI1K HBJI5IeTCH .u:eJJIITeJIeM
IJUCJIa 2N, HeKoTopbIe H3 B03MO)!{HbIX 3HaIJeHut! TpHroHoMeTpu4ecKot!
<PYHKUHH He Tpe6yroTCH. fI03TOMY Ha rrpaKTUKe IJaCTO HCnOJIb3yroTCSI 3Ha-
4eHUS! 2N, paBHbIe 12, 24 H 60.
6.7. Pa3JIOmeHMe no KocllHycaM
ECJIH .1l.0nyCTUTb, IJTO <PYHKlJ,U5I {(x) rrepno.u:V4eCKa51 C nepno.u:oM 2N,
TO npH HCrrOJIb30BaHHH 3Ha4eHHt! x = 0, 1, "', 2N - 1 I1JIU - (N - 1),
- (N - 2), ..., 0, 1, ..., N pe3YJIby-3T 6y.u:eT O.u:HH H TOT }!{e. Terrepb
npe.u:rrOJJO}!{HM, IJTO 3a.u:aHa cpYHKUI151, CHIpe.u:eJIeHHaSl .u:JI5I x = 0, 1, ..., N,
H MbI XOTHM pa3JIO)!{HTb ee B p5l.u: TOJIbKO no KocuHycaM. Onpe.u:eJIHM
{( - x) = {(x) .u:JI5I x = 1, 2, ..., N - 1. Tenepb {(x) - IJeTHaH <PYHK-
UIHI. Kor JJ.a Mbl HaXO.lI.HM pa3JIO}!{eH!'Je <pYPbe, TO nOJIY'laeM, '-ITO Bee b k = 0,
TaK KaK
N
bk= ! {(x) sin x,
X=-N+J
"88
KOHE'!HhlE P5lJJbI 4>YPbR
[m. 6
(j CI1HYC - HeqeTH351 CPYHK1UIH. TaKI1M 06pa30M iIOJIyqaeM pa3JIO)KeHI1e lIO
KOCI-IHYC3M. <l>OPMYJIbl AJIH ak MO)KHO yrrpOCTI-ITb
N-I
'\'1 rck
a k = N """ f(x) cos N x =
x=-N+I
N-I
2 [ I '\'1 rck I k 1
= j\/ :2 fo + "'" f (x) cos N x + 2 (- 1) flV .
x=1
OnpeAeJI5IH f (x) KaK ne4eTHYlO rpYHKlI.I-IIO, MO)t{HO 3HaJIOfl14HhIM 06-
pa30M nOJlY4I-1Tb pa3J10)t{eHYIe no cI-IHycaM.
3aMeTi-lM TaK)t{e, 4TO eCJm Mbl HMeeM o6J1acTh 0 x R, TO
3aMeHa
x= (1 + cos t)
AeJlaeT rpYHKlI.I-IIO qeTHot! nepf!OAH4ecKot! rpYHKUf!et!' t. Eha 3aMeHa J):aeT
p5JlJ, npef!Myw.eCTB B Bonpocax CXOAI1MOCTI1, KOTopble 6YAYT o6cy)t{AaTbcH
B fJl. 22.
Ynpa}J{HeHHSI
6.7-1. <PYHKllHR f (x) onpe,QeJleHa Ha HHTepBaJle O,s:; x,s:; 'ltj2. rIoKa3aTb, lJ.TO
MO)/{HO npO.ll.OmKHTb 4JYHKU,HlO Ha - rc < x,s:; rc TaK, lJ.T06bl BCTpelJ. aill1Cb JlHlllb
He'leTHble (lJ.eTHbIe) rapMOHI1KH KOCHHYCOB.
6.7-2. Paccr,lOTpeTb B JJ,eTail5IX 3a,IJ.a4Y HaXO)l{.lI.eHH5I pR.lI.OB <PYPbe TOJlbKO 110
cHHycaM. no lJ.eTHblM (HJlH He'leTlIbIM) rapMoHHKaM CHHYCOB.
6.8. JIoKaJlbHble pHJJ;bl \f>YPbe
Bo MHOff!X 3aAa4ax 5JBJleHf!e MemleTC5I MeAJleHHO, H f!Ae51 «JlOK3J1bHbIX
pHAOB <l>YPbe», K09rpcpf!UI1eHTbl KOTOpblX MeAJleHHO MeHHIOTCH C Te4eHHeM
BpeMeHI1, COBeprneHHO eCTeCTBeHHa. TaK)t{e 60Jlee eCTeCTBeHHO lI.eHTpI1-
pOBaTh o6J1acTh OKOJlO Ha4aJla 11 f!CnOJJb30B3Tb 3Ha4eHI1H OT - N + I
11.0 N, 4eM OT 0 AO 2N - 1. TaKI1M 06pa30M, H3 p3BeHCTB (6.2-5)-(6.2-7)
f!MeeM
t
Om (t) =j\/
N I
I V rc
am(t)= N """ f(x+t) cos j\/ mx,
x=- IV+l
m=O, 1, ..., N, } I
N
J '
xo",-N+I
m= 1, 2, ..., N- 1.
(6.8-1 )
f(x + t) sin N- mx,
6.81
JJOKAJJbHbIE P5ILlbl CPYPbE
89
JlerKo nOJlYlfl1Tb a (t + 1) H b (t + 1) 113 a (t) 11 b (t) H,1n!.HJMep.
N
am(t+ 1)= ! f(x+t+ 1) cas Iv iltx.
x=-N+I
TIOJ!Q)KI1M x + I = x', TOfLla
N+I
am\lll)= ! f(x'+t)cos m(x'-l)=
x'=-N
N
= { ! f (x' + t) cos m (x' - 1) +
x'=-N+I
+(_I)m [f(N+ 1 +t)- f(- N+t)]}.
PaCKphlBa51 KOCHHYC, nOJJY'IaeM
am (t + 1) = [ am (t) cos N m + b m (t) sin ; m] +
. + ( )m [f(N+ 1 +t)- f(- N+ tn,
3aMeTHM, '1TO MbI nOJlb3yeMcII cpl1KCl1pOBaHHbIMI1 (LlJlII LlaHHoti lfaCTOTbI)
1tm . 1tm
MHO)j{HTeJJIIMH cos N 11 sm N H CKJla.D.bIBaeM HJlI1 BbI'II1TaeM .D.Ba HOBbIX
3Ha4eHI111 CPYHKLl.I1I1. AHaJlOfl1'1HO .D.JlII CI1HYCOB:
bm(t+ l)=[-am(t) sin m+bm(t)COSNm].
t.JACTb II
npI1EJII1)f{EHHE MHOr04JIEHAMI1- KJIACCl14ECKl1H
4I1CJJEHHbIH: AHAJIH3
'fJJABA 7
BBElI.EHHE B MHOrOIJJIEHHbIE npI16JI1UKEHHSI
7.1. OpMeHTa HII
TIepBbIe pa60TbI JJlo6oro BbI4!1CJIHTeJHI 06bI'IHO CB5I3aHbI C BbI'I!1CJIeH!1eM
3Ha4eH!11l CPYHKU!1!1. I'JIaBHbIM!1 !1HCTPYMeHTaMH 3JJ,eCb 51BJI5IIOTC5I «o6w.ee
4YTbe» H «MaJIeHbKHe XflTpOCT!1». IJ,eJIb :noll KHHrH - nOKa3an r.naBHbIe
HJJ,eH B 06JIaCTH BbI4HCJIeH!1Il; MbI 6YJJ,eM JIHillb H3peJJ,Ka !1JIJlIOCTpHpOBaTb
acneKT «MaJleHhK!1X xHTpoCTell», XOTH OHH JJ,O C!1X rrop 51BJIH!OTC5I B3}!{HOll
''1aCTh!O HCKyccTBa BhI4!1CJIeHH5I.
B-ropoe, C 4eM 06hl4HO CTaJlKHBaIOTC5I, - :HO HHTeprrOJJ5IUHH HeJJ,OCTaIO-
w.!1X B Ta6JIHue 3Ha'leHHIl, Hanp!1Mep B JIOrapHcpMH4eCKoll HJIH Tpl1rOHO-
MeTp!1'1eCKOIl Ta6mme. Boo6w.e rOBopH, rrpH !1HTepnOJIHUHH HaM JJ,aHO
HeCKOJIhKO Y3JIOB CPYHKUH!1!1 HY}!{HO BhI4!1CJI!1Th rrpH6JIH}!{eHHO HeKOTophIe
3Ha4eH!1H, KOTOphIX HeT B T36JIHue. B 60JJhIllHHCTBe Ta6JI!1U CJ].eJJaHO npeJJ,-
nOJJO}!{eH!1e, '1TO CPYHKUHH BeJJ,eT ce651 Me}!{JJ:Y nOCJJeJJ:OBaTeJIhHO B3HTbIMH
T04KaMH, KaK np5lMaH, XOTH !1HorJJ:a rrpeJJ,nOJIaraeTC5I, '1TO OHa BeJ].eT
ce651, KaK KBaJJ:paTHhlll TpeX4JIeH H JJ:a}!{e MHorO'lJIeH 60JIee BhICOl\01l
'CTeneHH.
t.JacTo JJ:YM a lOT, '1TO rJIaBHbIe rrp06JIeMbJ '1HCJIeHHOrO aHaJJ!13a cocpe-
J].OT04eHbI BOKpyr HHTeprrOJI5IUI-IH; HO 3TO He TaK. K HHM OTHOCHTCH CKopee
TaKl1e onepaU!1!1, KaK !1HTerp!1pOBaHHe, .lI.HcpcpepeHlJ,HpOBaHHe, HaXO}!{JJ,eHHe
HYJIell, MaKCHM!133UI151 H T. JJ:. B CJIY4aHX, KorJJ:a Bce, '1TO Mhl HMeeM liJIH
MO}!{eM BbI4I1CJII1Th, - 3TO HeKOTopbIe Y3JJhI CPYHKUlil1, np!14eM !1 OH!1 06hI4HO
H3BeCTHhI He T04HO, a rrpH6JIHJKeHHO, TaK KaK 6bIBaIOT Hcnop4eHbI no-
rpeillHOCTb!O oKpyrJIeHHH.
KJIaCCH4eCK!11l 4liCJIeHHO-aH8JJI1T!14ecKHtI rroJJ:xoJ]. 3aKJJ!048eTCH B TOM,
4T06hI !1CnOJlh30BaTb HeKOTopbIe Y3JIbI CPYHKU!1!1 J].JJH nOJJY4eH!151 np!16JI!1-
}!{31OIIJ.erO MHOrO'lJIeHa 11 3aTeM BblIlOJIHHTh aHaJI!1T!1'1eCKYIO one.paUH!O Hat!;
9TI1M MHOrO'lJIeHOM. STOT npouecc MO}!{eT 6bITh Ha3BaH «aHaJIHT!14eCKOIl
saMeHotl» ([38], CTp. 51), TaK KaK CPYHKU!1H, KOTOPY!O HeB03MO}!{HO 06.
pa6oTaTh, 3aMeHHeTCH JJ:pyrotl cpYHKU!1ell, HaJJ: KOTOpoll y}!{e MO}!{HO BhIIlOJI-
HlHb aHaJIHTli4eCKY!O orrepa H!O. Harrpl1Mep. B cnoco6e HhlOTOH3 AJIH
7.11
OPI1EHTAlU151
9T
HaXO)l(.11eHI1H HYJlH <j:>YHKIlI1I1 Y = f (X) .D,aeTCH npI16J1f1)1(eHHOe 3Ha'leHl1e Xl
11 BMeCTO KpI1BO I1CnOJlb3yeTcH npHMaH
Y - YI = Y; (X - Xl),
KOTopaH KaCaeTCH rpa<j:>I1Ka <j:>YHKIlI1H B TO'1Ke (Xl> Yl)' TIO.11CTaBJJHH Y = 0,
nOJIYlJaeM 3Ha'lemle X, HBJIHlOlll.eeCH KOpHeM STO HOBO <j:>YHKIlI1I1,
X=XI- Y '
YI
Eho HOBOe 3Ha'leHl1e X ncnOJIb3yeTcH KaK CJle.11Y1OIll.ee npI16JJI1)1(eHHO
3Ha'leHl1e KOpHH.
TIOCKOJIbKY C MHOrO'lJIeHaMI1 JIerKO 06palll.aTbCH, 60JIbllIaH 'laCTb KJJaC-
cn'leCKOrO 'lHCJleHHOrO aHaJIH3a OCHOBblBaeTCH Ha npl16JII1)1(eHI111 MHOrO'lJJe-
HaMI1. O.11HaKO .11J1H MHorl1X lleJJet! npe.D,nOlJl1TalOTCH .D,Pyme KJIaCCbl <j:>YHKIlI1t!
OHI1 6Y.11YT 113YlJaTbCH nOCJIe TOrO,- KaK 6Y.11eT paCCMOTpeHa annpOKCI1Ma-
UHH MHOrO'lJleHaMI1.
Bbl6p aB Y3JIbl 11 KJlaCC npl16JJI1)1(alOIll.HX <j:>YHKIlI1t!, MbI .110 JI)I(Hbl ellJ,
Bbl6paTb O.D,HY Onpe.11eJIeHHYIO <j:>YHKllHIO H3 SToro KJlaCCa nOCpe.D,CTBOM
HeKOToporo KpHTepHSI - HeKOTopot) Mepbl npH6JJH)I(eHHH I1JII1 «cor JIaCI1H»,.
CaMblt) IlIHpOKO npHMeH5JeMbltJ KpHTepHtJ COCTOI1T B Tpe60BaHHH TOro,
lJT06bl npH6JJlj)l(alOlll.a5J <j:>YHKllH5J COBna.11aJIa C 3a.n:aHHbIMH 3HalJeHH5JMH
B Y3JJOBhlX TOlJKax. LLpyrot! 60JIee 06IlLHtJ KpflTepHtJ - «HaHMeHb-
IlIHe KBa.11paTbl» - 03Ha'laeT, 'lTO «CYMMa KBa.11paTOB OTKJIOHeHHt! Me)l(.D,Y
.llaHHbIMH Y3JIaMH H npH6. 1 ]H)I(alOlll.etJ <j:>YHKllHe B Y3J10BbIX TO'lKaX
.D,OJl)l(Ha 6hITb MHHHMaJlbHOH». O.D,HaKO HHOr.D,a npHMeHHIOTC5J H .11pyme'
KpHTepHH.
TIpe)l(JJ:e lJeM Ha'laTb BhllJHCJIeHH5J, MhI .110JI)I(Hbl pellIl1Tb TaK)I(e, KaKYIO
TOljHOCTb Mhl XOTI1M HMeTb B OTBeTe 11 KaKot! KpHTepHH Mbl H36epeM .D,JHJ
H3MepeHI15J STOtt TOlJHOCTH. Bce H3JIO)l(eHHOe MO)l(HO C<j:>OPMY JIHpOBaTh
B BH.11e lJeThlpex BonpOCOB:
1. Kmcue Y3.1lbt .Mbt 6yaeM ucno.llb308amb?
2. KaKou K.IlaCC npu6.1lU:JICalOt4UX rftYHKqUi1 .M.bt 6yae.M. UCnO.llb30-
8amb?
3. KaKou Kpumepuu COZ.IlaCUfl .Mbt npU.MeHU.M.?
4. KaKYlO mot{Hocmb .M.bt xomu.M.?
Ynpa>iiHeHHe 7.1-}' B351B TPH IJJleHa p5l.lla TeHJlopa
(X-X I )2 y" (XI)
Y=Y (XI) + (x-xl)y' (XI) + 2
npH6J111311Tb 4JYHKU,HIO B TOIJKe XI! 0606111.HB 4JOPMYJlbl HblOTOHa ll.JlR HaXO)/{}le.
HHR KopHeH 4JYHKlI.HH.
o T Be r; X =.x _ Y (XI) [ I + Y (XI) yO (Xd ]
I y' (XI) 2 y 2 (xd '
92
BBEII.EHI1E B MHO ro'WEHHbIE l1PI1BJJI1)!{EHI15!
1m. 7
7.2. AJIbTepHanlBHble qJOpMYJIHpOBKH
B OfmcaHHOM TOJlbKO '1TO npOll.eCce, HatlLUJ npH6J1H)!{a!OliLYIO CPYHKlI.l1lO,
MbI 3aTeM npl1MeH5!eM K Hetl aHaJlI1TI1'1eCKYIO Ollepalll1lO 11 Bbl'lI1CJI5!eM
pe3YJIbTaT B HeKOTopotl TO'lKe 11J111 TO'lKax. 06bl'lHO OKOH'IaTeJlbHbItl pe-
3YJIbTaT OKa3bIBaeTC5! JII1HetlHotl KOM611Hall.I1etl 3Ha'leHHtl B nepBOHa'laJlbHblX
Y3JIax. i:ho nOIl.CKa3bIBaeT, '1TO MO)!{HO IIpl1tlTH HenOCpe.ll.CTBeHHO OT Y3JIO-
BblX TO'leK K OTBeTY, He nOJlY'la5! no 1I,0pore annpOKCHMI1PYlOliLetl
CPYHKlI.I1H.
4r06bl C.lI.eJlaTb 3TH 3aMe'laHH5! 60Jlee onpe.ll.eJleHHbIMI1, npe.ll.nOJlO)!{BM,
'1TO I1MeeTC5! HeKoTopbItl JIHHetlHbItl oneparop L, 1I,eficrBYlOliLI1tl B HeKO-
TOpOM KJJaCCe CPYHKlI.l1tl, TaK '1TO
L [al(x) + bg(x)] = aL [I (x)] + bL [g(x)].
(7.2-1)
YlHTerpl1pOBaHl1e, 1I,I1cpcpepeHlll1pOBaHI1e H HHTepnOJl5!lI.l15! - TI111I1'1Hble
npl1Mepbl JIImetlHbIx Ol1eparopOB. HaXO)!{1I,eHl1e HYJl5! - He JJHHenHbItl
DnepaTOp.
Pe3YJlbTar L [lex)] 6Y1l,eM I1CKaTb, HCX01l,5! H3 1I,aHHbiX B Y3J1aX 3Ha-
qeHHtl CPYHKlI.HI1 H ee I1pOH3BO.ll.HbIX. TaKI1M 06pa30M, Mbl HliLeM CPOPMYJlY
BH.lI.a
L !I (x)] = ad(xt)+ ad (X2) +...+ ani (x n ) +
+bd' (Xt) + b 2 f'(X2)+"'+ bnl' (x n )+...+ kd(k-J) (Xt) +
+ kcj(k-t) (X2) +. .. + knf(k- 1 ) (x n ), (7.2-2)
r 1I,e MHOrl1e K03cpcpl111.I1eHTbI MorYT 6bITb paBHbl HY JlIO. D.Jl5! I1poC'rorbl
npe1l,CraBHM, qTO Y Hac eCTb JlHlllb 3HalJeHI15! CPYHKIJ.I1I1, a Bce K03cpcjm-
1I.I1eHTbi b, c, ..., k paBHbl HYJlIO.
fl pe1l,110JlO)!{I1M, '1ro I1CnOJlb3yeTC5I npI16J1H2KeHHe MHOrO'lJleHOM H rpe-
-6yeTC5! TOLJHOe COBna.ll.eHHe ero C .lI.aHHbIMH 3HalJeHH5IMI1 B Y3J10BblX rOqKax.
B03MO)!{Hbl Tpl1 3KBI1BaJleHTHbIX clloc06a onpe1l,eJIeHI15! K03cpcp1111.I1eHrOB ai'
flepBbItl cnoc06, Koropbltl y)!{e paCCMaTpI1BaJIC5!, COCrOI1T B HaXO)!(1I,e-
HI1I1 npI16J1I1)!(aIOliLerO MHOrO'lJleHa 11 npl1MeHeHl111 K HeMY 011epaTopa L;
eTO - TaK Ha3bIBaeMbitl MerO.ll. aHaJlHTI1'1eCKot! 3aMeHbI.
BTOpotl cnoc06 COCTOI1T B TOM, '1To6bI pa3J10)!(HTb I (x) B p5!1I, Ten-
)lopa, nO.ll.CTaBHTb pa3J1O)KeIme B JleBYIO 11 npaBYIO '1aCTH 11 npl1paBIiHTb
K03cpcpl111HeHTbI npl1 O.ll.HHaKOBblX npOH3BO.ll.HbIX. flO5!CHI1M ero nocTpoe-
HHeM, HanpllMep, CPOPMYJlbI CI1MIICOHa
1
I(x)dx= a_d(- 1) + aof(O) + adO).
-I
(7.2-3)
TaK KaK
x 2 8 x'
/(x) = f (0) + xl' (0) + 2f I" (0) + 1'" (0) + 4T pv (0)+...,
7.21
AJJbTEPHATHBHYE$OPMYJJHPOBKH
93
TO B neBofi 4aCTH paSeHCTBa 6YAeM HMeTb
I
f(x) dx = 2/(0) + t" (0) + IIV (0)+...,
-I
TOrAa KaK cnpasa
a_d (- I) = a_d (0) - a_d' (0) + a it f" (0) -
- a it I'" (0) + j fIV (0) + . . .,
ao! (0) = ao! (0),
ad(I)= ad(O) + ad' (0) + ;t t" (0) + t ['" (0) + flY (0)+ ...
Dpl1paSHI1SaH, rrOKa 9TG S03MO)/{HO, K09cpcpHll.l1eHTbI fIpl1 1(0), f' (0), ...,
nOJlY4aeM
2 = a_I + a o + ai'
o =-a_1 +al'
2
a=a_ 1 + ai'
o = - a_I
2
5=a_- 1
+al'
+al+E,
rAe E - OlIJHOKa. PeweHHe 9THX ypaBHeHl1fi AaeT 31-Ia4eHHH a_I = al = 1/3,
ao = 4/3, KOTopble rrpl1S0AHT K 06bI4HOti cpopMYJle
1
I (x) dx = 1/3 t _I + 4/3 10 + 1/3 fl (7.2-4)
-I
TpeTHti cfIoc06 Tpe6yeT, 4T06bl cpOPMY na 6bwa T04HOt! AJJH fIocne-
AOBaTenbHOCTI1 cpYHKll.Hfi I (x) = I, x, x 2 , .,. AO B03MO)/{HO 60nee BbICO-
KOti CTerreHH. OH fIone3eH AnH fIonY4eHI1H CfIell.HanbHbIX <popMyn, 11 MbI
6YAeM I1CfIOJJb30BaTb ero AnH 9TOfi ll.eJII1 B 60JIbWHHCTBe CJJY4aes. OAH3KO
rrpl1 peweHHH KOHKpeTHot! 3aAa4H MeTOA aHaJIHTI14eCKot! 3aMeHbI, npH
KOTOpOM B nepsYlo o4epeAb CTpOHTCH npH6JJI1)/{aIOIUaH cpYHKll.I1H AJIH
IKXOAHbIX AaHHbIX, 4aCTO 6bIs3eT JIY4we, TaK KaK OH AaeT S03MO)/{HOCTb
1l0, Y'1aIOIlJ,eMY pe3YJIbTaTbI BHAen CBOHMH rJIa3aMH, KaKaH I1MeHHO afIrrpoK-
CI1Mall.HH AeJJaeTCH. TaK, B fIpl1Mepe 1.9, KorAa HaXOAI1JICH afInpoKcrr-
MI1PYIOIUHtI MHOrOl\JIeH, MbI Bbll\I1CJI5!JIH ero S TotJKax fIocpeAHHe Me)/{AY
y3JIOBblMH H CTpOI1JJI1 fIOJIY411BwHecH 3Ha4emm, 4T06bI sl1Aen, KaKofi
rpacpl1l{ OHI1 AaIOT.
DOKa)/{eM, 'JTO 9TH TPH cfIoc06a 9KBHBaJJeHTHbI H AalOT OAHH H TOT)/{e
OTser. TpeTHti cnoco6, KOTOpbltI AeJlaeT cpOPMYJIY T04HOt! AJIH pa3JII14HbIX
c-refIeHet! x AO HeKoTopot! CTeneHI1, CKa)/{eM n, AeJIaeT <POPMYJIY T04HOfi
94
BBEJlEHI1E B MHOro4JJEHHhIE f1PI1BJJVDI(EHI15!
[fJJ. 7
TaK)I(e t-I .IJ}ISI Jlio6oro MHOr04JIeHa CTerreHH, He npeBOCXO.ll:Slmet! n. TaKHM
06pa30M, eCJIH CPYHKlI.H5! ecn MHOr04J1eH CTerreHH n (HJlI1 MeHbmet!), TO
cpopMYJla TO'lHaH. Ho eCJI!1 CPYHKlI.HSI eCTb MHorO'JJleH, TO ee pSlJJ, Tet!Jlopa
OKaH4HBaeTCSI IJJleHOM n-ro rropH.lI:Ka H BTOpOt! crroco6 JJ,aCT TOT )l(e pe3YJlb-
TaT. 06paTHoe TaK)I(e BepHO, CJJe.ll:OBaTeJlbHO, BTOpOtl crroco6 9KBI1BaJleH-
TeH TpeneMY.
YTo6b1 nOKa3aTb, IJTO rrepBbIt! cnoco6, KOTOpbItI 3aKJlIOIJaeTCSI B Ha-
XO)l(.lI:eHHU HHTepnOJJSllI.HOHHOro MHOrOIJJJeHa H orrepHpOBaHHH C HHM, 9KBI1-
BaJleHTeH .lI:BYM .n:pyrHM, Heo6xo.n:HMO UMen B paCrrOpSl)l(eHI1U HeKOTOpylO
TeopHIO UHTeprrOJlSlll.HOHHbIX MHorOIJJleHOB, a ee MbI 6y.n:eM pa3BI1BaTb JJHmb
B CJIe.n:YlOllI.et! r JJaBe. I10Ka rrpe.ll:nOJlO)l(HM, IJTO eCJJH .lI:aHbI 3HaIJeHHSI Ii
(i = 1, 2, ..., n) CPYHKlI.UH, TO MO)l(HO nOCTpOHTb MHOrO'lJJeH *)
p(x)=L 1 (x) 11 + L 2 (x) 12 +.. .+Ln(x)ln'
(7.2-5)
KOTOpbIti .lI:JlSI l(x)=l, x, x 2 , ..., x n - J COBrra.ll:aeT C I (x). I1pHMeHHB
orrepaTOp L [/(x)1 K JleBot! 11 rrpaBoti IJaCTSlM (7.2-5), I10JlYIJHM CPOPMYJlY
TI1rra (7.2-2), KOTopaSl HB/meTCH T04HOti .lI:JlH CPYHKlI.utI 1, x, x 2 , ..., x n - 1 .
06paTHO, TaK KaK L [I (x)] JIHHeeH, TO cpOpMYJla, TOlJHaH .n:JlH CPYHK-
Uut! 1, x, X 2 , ..., xn-l, TOlJHa H .1(JJ5I JJI060tl HX JJHHetiHoti KOM6ImaUHH,
CJJe.n:OBaTeJlbHO, .n:JJ5I Jl!1HetiHotl KOM6HHall.HH (7.2-5). TaKHM 06pa30M, Tpe-
THt! crroc06 9KBUBaJJeHTeH nepBOMY.
TpeTI1t1 crroco6, KOTOpbIM MbI 6Y.lI:eM nOJlb30BaTbC5I .1(JlH HaXO)l(.D.eHHH
CPOPMYJJ, SlBJJHeTC5I HaH60JJee ynoTpe6HTeJlbHblM. O.D.HaKO, 'iTo6bI HatlTH
CPOPMYJlY 9THM 06ll1.HM rrpHeMOM, IJaCTO Tpe6yeTcSI 60Jlbme pa60TbI, 'JeM
eCJlH rrpHMeHHTb pa3J1HlJHbIe TpIOKH, KOTopbIe rrpHBO.D.5ITC5I B YlJe6HHKax.
B 3alll.HTY Hamero MeTO.D.a MbI CKa)l(eM: «TIpHHHMaSl BO BHHMaHHe orpOM-
Hoe KOJJHIJeCTBO HMelOllI.HXC5I Terrepb 3HaHHtI, JlYlJme rrpHMeH5ITb O.D.HH 06ll1.HtI
MeTO.D., .lJ,0 HeKOTopoti CTerreHH Herrp0I13BO.D.HTeJlbHbIt!, 4eM u3YlJaTb MHO-
)l(eCTBO crrell.HaJlbHbIX TpIOKOB».
OKOHlJaTeJlbHbIe CPOPMYJlbI TaK)I(e 6e3pa3J1HlJHbl K MeTO.D.Y HX nOJJY-
IJeHH5I. MeTO.D., 6y.n:YlJI1 06ll1.HM, JlerKO .D.aeT HOBbIe CPOPMY JIbI H o6ecrre-
IJUBaeT arrrrapaT .lJ,JlSI .n:aJlbHetimero UCCJJe.lJ,OBaHU5I. B rrOC,IIe.D.Hee BpeMH
npHHHTHe TaKoro e.lJ,HH006pa3I-lOrO MeTO.lJ,a HarrpaBJleHO K rrocTeneHHoMY
nepeKJIaJJ,bIBaHHIO Bcetl pa60TbI Ha BbIIJHCJIUTeJIbHYIO MamHHY.
O.D.Ha H3 oc06eHHocTeti 9TOt! KHHrH - HCnOJIb30BaHHe l1JJ,etl TeopHH
l1HcpopMall.HH. MO)l(eT 6bITb, 9TO npocTo rrpe.lJ,paccYJJ,oK ar3Topa, HO Ka)l(eTC5I
pa3YMHbIM paccMaTpHBaTb YHHBepCaJIbHYIO BbIlJHCJlUTeJIbHYIO MaIllHHY KaK
«fIepepa6aTblBaTeJIb» HHcpopMaUMI1. MHorHe H3 H.lJ,etl TeOpHl' HHcpopMall.HI1
TeCHO CB5I3aHbI C CPYHKlI.H5IMH C orpaHu4eHHbIM crreKTpoM, KOTopbIe pac-
CMaTpl1BalOTC5I B TpeTbeti lJaCTH 9TOti KHI1rH. XOT!!, KaK MbI BH.D.eJIH
B r JlaBe 2, TeopHH, KOTopaH paCCMaTpHBaeT mYM KaK OCHOBHOti 9J1eMeHT,
em.e He.lJ,OCTaTOIJHO pa3BI1Ta, HeKoTopbltl C.D.BHr B 9TOM HarrpaBJleHHH Ha-
*) MHorolJJleHbl Ln onpe.QeJleHbI B 8.3; HX He CiIe.QyeT nyraTb C HHHeHHblM
onepaTopOM L a paaeHcTae (7.2-1).
,7.31
Y3JIOBbIE T04KI1. I1H<1>OPMAU'I151
95
MelJeH B HOBbIX KHHraX I-! ny6JIHKaUH5!X. J1YlJlllee, lJTO MbI MO)l{eM C.n:eJIaTb
B HaCT05!llLHtI MOMeHT, 9TO 3anOMHHTb, lJTO n04TH Ka)l{.n:Oe lJHCJIO, KOTOpOe
MbI nOJIYlJaeM, 6y.n:eT HMeTb HeKOTOpOe «IllYMOBOe 3arp5!3HeHHe». IT09TOMY
He06xo.n:HMO HCCJIe.n:OBaTb CPOPMYJIbJ, KOTOpbIe 3.n:eCb npHBO.n:5ITC5I, B CBeTe
Toro, KaK OHI-! 6y.n:YT pacnpOCTpaH5!Tb lllYM.
Ynpa}j{HeHHe 7.2-1. BblBecTH npaBHno Tpaneu,HH nepBblM, BTOpblM H TpeTbl-!M
cnoc06aMI-!.
7.3. Y3J10BbIe TOIIKH, HH$opMaUHH
ITepBbIM H3 4eTblpeX BonpocoB, 3a.n:aHHbIX B KOHue 7.1, 6bIJI Bon-
poc «lealClle Y3/lbl .Mbl 6yoe.M UCnO/lb30Bamb»? B npI-1HUVme 9TO - Bon-
poc CTan1CHIKI-!, oc06eI-1H0 06JIaCHI, Ha3blBaeMoti «nJIaHHpOBaHl1eM 9Kcne-
pHMeHToB». Ha npaKTHlze Y3JJOBbJe TO'-lKI1 '-IaCTO 3a.n:aHbI BHeIllHHMH
06CT05!TeJIbCTBaMH HJIH Mbl nOJJb3yelVic5! MHO)l{eCTBOM paBHoYL1:aJIeHHbIX T04eK.
B IJOCJJe.n:HeM cJJY'-Iae MbI JJHIJ,OM K J1Imy CTaJJKHBaeMC5I C meope.Moii.
8bt6opleu H3 TeopHH I-1HcpopMaIJ,HH, KOTOpa5! npHBo.n:HTC5! B TpeTbeti lJaCTH,
HO Ha caMOM L1:eJJe 061>5!CH5IeT !>\Horoe 113 MaTepHaJJa BTOpOti lJaCTH. TaKHM
06pa30M, B HeKOTopOM OTHOilleHI-!H MHoroe BO BTOpOti lJaCTH 5IBJJ5!eTC5!
HeCKOJJbKO nOBepXHOCTHbIM.
YT06bI YBH.n:eTb, '-ITO np06JJeMa BbI60pa Y3JJOB peaJJbHa, npe.n:CTaBbTe,
'-ITO Bac npOC5IT BbI'-IHCJJHTb HHTerpaJJ
1
I (x) dx
o
.1lJJ5I HeKOTopotl CJJO)!{HOtl <PYHKU.HI-I, KOTOPYIO BbI He MO)l{eTe npOHHTer-
pl1pOBaTb aHaJJHTl1lJeCKH. Bac nOnpOCHJJH B .n:etlCTBI-ITeJJbHOCTI-I oueHHTb
nlIOIlI,3L1:b 06JJ3CTH nOA KpHBOtl, OCHOBbIBa5!Cb Ha p5l.n:e Y3JJOB (cKa)l{eM,
I1X n), HlIH, '-ITO TO )l{e caMoe, OIJ,eHI-!Tb cpe.n:Hee 3HalJeHI-!e I (x). H.n:eaJJbHO,
HY)!{HO 3HaTb 0 CPYHKUHI-I OlJeHb MHoro, '-ITo6bI pacnOJJO)!{I-ITb Y3JlbI HaR-
lIY'-IIllI1M 06pa30M. D.onYCKa5! pa3JJI-IlJHbIe CBotlCTBa CPYHKIJ,I-!H, MbI nOJIYlJaeM
pa3JJHlJHbIe CI1oco6bI pacnOJJO)!{eHH5! Y3JIOB (BCJJetJ:CTBl-le npHBbIlJKH 3BTopa
CJJOBa «Y3JJbI» H «HHcpopMaUH5!» IJaCTO 6y.n:YT npI1MeH5!TbC5I B3aI-!M03aMe-
H5!eMO ).
D.PYrI1M npl1MepOM 3a.n:aIJI-!, B KOTOpOtl B03HI1KaeT Bonpoc «2oe /leJICum
UHrj'Jop.Mau,llfl?» - 5IBJJ5IeTC5! 3a.n:a'la HaXO)l{.n:eHI15! np0l13Bo.n:Hoti B TOlJKe
(XOT5I, B npOTHBOnOJJO)l{HOCTb HHTerpaJJY, npOH3Bo.n:Ha51 - JIOKaJJbHa51 xa-
paKTepHCTI-!Ka). 3.n:paBbJtI CMbICJJ .n:I1KTyeT I-!CnOJJb30BaHI1e I1HcpopMalUIH,
6JJH3KOtl K TOIJKe, B KOTOpOtl HaAO oueHHTb npOH3Bo.n:HYIO. C .n:Pyroti
CTOpOHbI, eCJJH Y3JJbI B351Tb CJJHIlIKOM 6JJH3KO, TO IlIYM B BbIIJHCJIeHHI1 I1JJH
H3MepeHHH Y3JJOBbIX TOIJeK 6y.n:eT MernaTb BbIlJHCJJeHl-!1O C xopoiliefi TOIJ-
HOCTblO. TaKHM 06pa30M, COOTBeTCTBYIOIlI,HtI OTBeT Ha Bonpoc «r.n:e B35!Tb
Y3JJbI?» Tpe6yeT OIJ,eHKH IlIYMa, KOTOpbIti nopn-IT BCIO HHcpopMaUHIO. l{aK
6bIJIO 3aMelJeHO paHbIlIe, COOTBeTcTBylOIlI,etl TeopHH IlIYMa HeT, HO pa3BI-!-
nle Me.n:JleHHO H.n:eT B 9TOM HanpaBlIeHHI1.
96
BBEJJ.EHI1E B MHO fOYJlEHHblE npI1BJll1lliEHI1
[f Jl. 7
7.4. KJIacc i»YHHU"H
BTOpOt! li3 4enIpex BonpocoB 6I.lJ/: «KaKoli K/laCC annpOKCU.MUPYlOlIi,UX
rjJYHKl1Uii Mbt 6yiJe.M npu),leHflmb»?
Cyw.ecTBYIOT TPH KJ/acca HIIH rpynnI.I CPYHKLl.Hll, llIHpOKO npHMeH5Ie-
MI.IX B 4HCJ/eHHOM aHaJ/H3e. TIepBa51 rpynna BKJ/104aeT B ce651 mmellHl>le
KOM6HHauHH CPYHKUHfi
I, X, X2, ...J x n ,
4TO COBna.lJ,aeT C KJ/aCCOM Bcex MHor04J/eHOB CTeneHli n (HJ/H MeHbwe).
BTOpofi KJ/acc 06pa3YlOT CPYHKLl.HH
cos ajx, sin ajx.
3TOT KJ/acc HMeeT OTHOWeHl-le K pHJ1.aM <l>YPbe 1-1 HHTerpaJ/Y <l>YPbe. OHH
paccMaTpHBalOTC5I B TpeTbefi 'laCTH. TpeTbSl rpynna 06pa3yeTc5I CPYHK-
UHSlMH e- ajz . 3TI1 CPYHlW,HH 4aCTO BCTpe4alOTCSI" B peaJIbHblX cHTyaLU1S1x.
K HHM, HanpHMep, npHBOJ1.SlT 06bl4HI.le 3aJ1.aYH HaKOnJJeHI1S1 H pacnaJ1.a.
OHH TaKllie pacc JaTpHBalOTcH B TpeTbell 'lacm: (CM. r JJ. 26).
KalliJ1.aSl 1-13 9THX Tpex rpynn 06J/aJ1.aeT OJ1.HHM BalliHbIM cBollcrBoM:
KOHellHoe MHOllieCTBO CPYHKU,HtI TaKoti rpynnbI nepexOJ1.HT caMO B ce611,
Kor J1.a x 3aMeWJeTCH Ha x + k. HanpHMep, eCJ//J P (x) - MHor04J/eH
CTeneHH n, TO P (x + k) - TaKllie MHorO'lJ/eH cTeneHH n no x, XOTII,
KOHe'lHO, HX K09cpcpl1u,MeHTI.l pa3J//JLJHI.l. To llie caMoe OTHOCliTCSI K ll,BYM
J1.pyrnM rpynnaM.
3TO CBOtlCTBO BalliHO, TaK KaK OHO nOJ1.pa3YMeBaeT, 4TO, KOrJ1.a MI.l
BI.I6HpaeM MHOllieCTBO annpoKc/JMHPYIOllI.HX CPYHKU,Ht!, HaM He Tpe6yeTcSI
HMeTb CBeJ1.eHHtI 0 Ha'laJ/e OTCyeTa. VlcnOJ/b30BaHHe J/1060ro J1.Pyroro
KOHe4Horo MHOllieCTBa annpOKCHMHPYIOW.HX CPYHKU,I1t1 (KpOMe KOM6HHaUI1H
9T11X Tpex) nOJ1.pa3YMeBaeT cyw.eCTBOBaHHe eCTeCTBeHHoro HaLJaJJa OT-
CLJeTa, TaK KaK Bbr60p Ha'laJla OTCLJeTa 6YJ1.eT BJJHSlTb Ha pe3YJJbTaT *).
ECJ/H B 3aJ1.a4e eCTI. oc06eHHOCTb, aTO aBTOMaTH4eCKH onpeJ1.eJ/5JeT eCTe-
CTBeHHoe Ha'laJ/O OTCLJeTa; 06I.I'lHO xapaKTep oc06eHHOCTH nOJ1.CKa3bIBaeT,
KaKYIO rpynny annpOKCHMHPYIOIl HX CPYHKU,HtI HaJ1.0 HCnOJ/b30BaTb
(CM. TpeTblO qaCTb, r JJ. 27). Ho B 60J/I.IllI1HCTBe 3aJ1.a'l HeT eCTeCTBeH-
Horo HaLJaJ/a OTC4eTa, H MI.l BblHYlliJ1.eHbl BI.l6HpaTb OJ1.HY H3 9THX Tpex
rpynn HJIH KOM6nH8UHH HX, eCJ/H BI.l60p He J1.0J/llieH nOBJ/HHTb Ha OTBeT.
MHOllieCTBO J/HHetlHI.lx KOM6HHau,Hll 1, x, x 2 , ..., x n HMeeT ewe
OJ1.HO BalliHoe CBOtlCTBO, a HMeHHO: KaK MHOllieCTBO OHO TaKllie He H3Me-
HHTC5I npH 3aMeHe x Ha kx. .ilpyrHe ,lJ,Be rpynfIbl 9THM CBOtlCTBOM He
06J/aJ1.aIOT. TaKI-IM 06pa30M, eCJ/H B 3aJJ.a'le HeT eCTeCTBeHHOrO MaCIllTa6a,
TO MbI BbIHYlliJ1.eHbl HcnOJ/b30BaTb MHOrOLJJ/eHbl I-IJ/I-I B03J1.etlcTBoBaTb Ha
*) MbI He 6YJJ,eM ,IJ.OKa3blBaTb 3)J:eCb 9TO YTBepJK)J:eHlIe, HO npOCTO HCnOJlb-
ayeM ero, 'H06h1 06'bJlCHHTb, JJOqeMY paCCMaTpHBalOTcR B OCHOBHOM 9TH TPH
rpyrlUbI.
7.51
COrJ1ACI1E
97
OIBeT caMBM Bh160pOM MaCWTa6a. Ho .n:a}/(e eCJJll MHO}/(eCTBO armpOKOI-
MHPYIOW.HX CPYHKlI.HI1 06J1a.n:aeT CBOI1CTBOM He3aBBCHMOCTH OT MaCWTa6a,
BO MHOrBX 3a.n:a4aX, OC06eHHO TeX, B KOTOpblX <j:JYHKlI.BB 3aBBC5IT OT
BpeMeHB, ecn eCTeCTBeHHblI1 MaCWTa6. f109TOMY npe06J1a.n:aHBe MHoro-
4JIeHOB Ha.n: .!LPyrHMB .!LBYM5I rpynnaMB BeCbMa npHCKOp6HO. CpaBHBTeJJbHO
3anyw.eHHOe COCT05IHHe BCnOJlb30B3HH5I .n:BYX .!LpyrBx rpYIm 03Ha4aeT, 4TO
reOpHH HX He 6hJ.TIH pa3BBTbl TaK }/(e .!LaJleKO, KaK .!LJl5I MHOr04JJeHOB, II
B pe3YJlbTaTe Jl.Ibl C411T3eM 9TH Teopml 60Jlee TPY.!LHbIMH, 4eM OHH ecn...
OnIOWeHH5I .n:BYX MHOr04J1eHOB, HJlH, npow.e, pall.HOHaJlbHble <j:JYHKlI.BH,
TaK}/(e 06J1a.!LaIOT CBOtiCTBOM He3aBHCHMOCTH OT MacwTa6a H .n:Jl5I HBX
cyw.eCTByeT nOCTeneHHO pa3BHBalow.a5IC5I H BIlOJJHe Y.!LOBJleTBopHTeJlbHa5I
TeOpH5I (CM. r JJ. 20). Mbi 06bl'JHO He CTaJJKHB3eMC5I C KOHCTaHTaMH, B03-
Be.!LeHHbIMH B MHor04J1eI!HYIO CTeneHb, HJIH MHOr04JIeHaMH, B03BeJi,eHHbIMil
B MHor04JJeHHYIO CTeneHb; Ka}/(.!LblI1 113 9TH X KJJ3CCOB TaK)J{e 06JJa.!LaJJ 6bl
CBOtlCTBOM He3aBHCHMOCTH OT MacJJJTa6a.
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COl/taCUfl .Mbl 6yoe.M nOJ!b30I3ambCfl»? KJI3CCH4eCKl1M 51BJI5leTC5I OTBeT:
«mottl-we COl3naOefllle 8 Y3/l08blX mottTCax», KOTOpbltl HMeeT npeHMY-
llJ,eCTBO npOCTOTbI TeOpUI1 H BbIIlOJIHeHH5I BbI4HCJJeHl1ti, HO TaK)J(e 11 He-
Y.!L06CTBO H3-3a HfHopHpOBaHH5I WYMa. B OCHOBHOM BTOpa51 4aCTb KHHrI1
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pHI1 IlpeHe6peraeT oc06eHHO Ba}/(HbIMI1 aCileKTaMH TI1IlH4H0I1 BbI4I1CJII1-
TeJIbHoti cHTyaQHH. Mbl y}/(e BH.!LeJJH, 4TO KpIJTeputi T04Horo COBII3.n:eHH5I
B Y3JIOBhlX T04Kax B CJIY4ae HCIJOJib30BaHH51 MHOr04JIeHOB .n:aeT TO }/(e,
4TO IlOJIY4eHI1e <j:JOPMY JIbI, T04HOti .n:JI51 I, x, x 2 , . . ., x n .
.IlpyrOI1 OTHOCI1TeJlbHO XOPOWHt! KpHTep1-1I1 - 9TO «Hall.MeHb1l1lle
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TOliKaX .!LOJl)J<Ha 6bITb C.!LeJlaHa H31-1MeHbweti B03MQ)KHOti HJJH, npOw.e,
MHHHMH3I1pOBaHa. STOT KpwrepHti HCIlOJlb3yeT H36bIT04HYIO HH<j:JOpMall.fHO,
4T06bl nOJlY4MTb HeKOTOpOe CrJJa}/(HBaHHe WYMa. OH nonYJl51peH Y MaTe-
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OTBeTcTByeT, 1-1 y cpI13HKOB, IlOTOMY 4TO Omi IlOJJaraIOT, 4TO HaHMeHbWHe
KBa.n:paTbJ - 9TO TIpHHlI.HIl npHpO.!Lbl. Ho 4eJlOBeK, 3aHHMalOIJUJtlC5I BhI4H-
CJleHHHMH, HMeeT rrpaBO CMOTpen Ha 9TOT KpHTepHt! C nO.!L03peHHeM
B Ka}/(.!LOM KOHKpen!OM CJlY4ae (CM. r JlaBbl 17 H 18).
TpenJti Kpl1TepHti CBH3bIBaeTC5I C HMeHeM 4e6blweBa. OCHOBHa5I H.n:e51
ero COCTOHT B TOM, 4T06bl YMeHbWHTb MaKCHMaJlbHOe OTKJJOHeHHe .!LO
MHHHMYMa. f1oCJle.n:HIlf! KpHTepHti CTaHOBHTC51 oc06eHHO norrYJJHpHhlM
B HaWM .!LHH (CM. r JI. 19). 04eBU.!LHO, B03MO}/(Hbl H .n:Pyrue KpWrepl1H.
Bhl60p KaKOrO-Jm60 H3 HHX 06bI4HO orrpe.n:eJI51eTCH <j:JH3H4ecKoti 3aLl,a4eti,
113 KOTOpOti B03HHKJIa BbI4HCJlI1TeJIbHa51. Mbl IlOilhITaeMC5I .!LaTh KpaT!ZYlO
xapal<Tepl1CTHKY pa3J1114HbIM KpIlTepUHM, KOrLl,a 6YLl,eM BBO.!LHTb HX, HO
4 P. B. XeMMIIHr
98
BBEllEHI-1E B MHOf04JJEHHbIE OPI-1BJJI-1}/{EHI1S1
{rJJ.7
1 aK KaK 9TO - YlJe6HHI< no lJHCJleHHbIM MeT01laM, TO MI.I He CTaHeM
TpaTHTb BpeM51 Ha TO, lJTo6bl 06YlJI1TI. lJHTaTeJl51 YMeHl110 HaX01lHTb HY}/{HI.Ii1
KpHTepHtt OLlHaKO Bbl60p KpwrepH51 lJaCTO CI1J1bHO BJJI151eT KaK Ha TO,
ljTO 1l0Jl}/{HO 6I.ITI. C1leJlaHO, TaK H Ha pe3YJII.TaT, KOTOpI.Ii1 no.'IY4HTC5I;
n03TOMY 9TOT Bonpoc 3aCJJY}/{HBaeT BHHMaTeJlbHOrO paCC IOTpeHH5I nepell
Ha4anOM BWlJI1CJleHHi1.
7.6. T01.JHOCTb
DOCJle1lHHM H3 lJeTbIpex 6blJI norrpoc: «KaKYlO mOttHOcmb _Mbl xomu.t!
l1oAyttUmb»? BHewHI1i1 BH1l OTBeTa, KOTOpbIi1 06hPJHO LIaeTC5I, TaKOB:
«3 llAU 4 oecflmUttHblx 3HaKa 8 om8eme».
Bo MHorHX 3a1la'JaX 9Toro He1l0CTaTOlJHO: TaKoi1 OTBeT HrHopHpyeT
OCHOBHYIO CTOPOHY Borrpoca. PaCCMOTpHM, HanpHMep, pellleHHe CHCTeMhI
JJrmetJHblX aJlre6pal14eCKHX ypaBHeJ-lHt! 1lJIR Hel13BeCTHbiX XI' HaM HY}/{HO
OTBeTHTb Ha C,le,lJ,YIOWHe Borrpocw:
I. LlOJl}/{Hbl JII1 6I.ITI. T04HI.lMH Xi?
2. Llon}/{Hbi JlH 6I.ITb MaJleHbKHMH OCTaTKH ypaBHeHHi1 rrocJle TOro,
1<aK BI.IlJHCneHHhle Xi n01lCTaBJleHhI B HHX?
3, LlOJl}/{Ha JlH LIaHHa51 rpynna ypaBHeHI1i1 6b1Tb 6J1113K0i1 K rpynne
'ypaBHeHll;i1, 1lJI5I KOTOpOi1 Xi CYTb TO'iHI.Ie OTBeTbI?
OlJeBI11lHO, 4TO B onpe1leJIeHHoi1 KOHKpeTlIOt! cHTyaIlHH MO}/{eT 10-
JlHTbC51 eme KaKa5l-J1I160 Llpyra5I Mepa T04HOCTH. 06bIlJHO OTBe4alOT
)'TBep1lHTeJIbHO Ha nepBbIi1 Bonpoc; HO J/HlJlb HHOr1la TaKOH OTBeT SIBJIIleTC51
C3MI.IM rr01lX01l5IWHM.
BTopotJ KpHTepHi1 4acro 51BJJ5IeTC5I Ha1lJIe}/{aJIlHM, nOTOMY 4TO crpe-
M51TC51 06paTlHb B HY.1Ib ypaBHeHH5I H BbI1.fHCJJr1Tb Xi' KOTopbIe 1lalOT 9TO
06paweHHe B HYJJb.
TpeT!li1 KpHTepHi1 TaK}/{e OlJeHb I10Jle3eH H nocreneHHO Bce 60JlhlUe
11 60.1IbllIe HcnoJlb3yeTc5I B npl1J10}/{eHH5IX, TaK }/{e KaK 11 B cpopMaJIbHOM
aHaJlH3e 01lIH60K. B H3BeCTHOM CMbICJJe rOBOp51T: «TaK KaK HIl BalliH
1'13MepeHI151, HH Ballia cpH3U4eCKa51 TeOpH5I He 5IBJI51IOTC5I TOIJHbIMH, TO Ka1<
6J1H3KO K HUM 51 1l0JJ)KeH n01l0t1TH B CBOI1X BbllJHCJJeHH5IX?» Bo MHornx
OTIlOllIem15lX 9TO - caMbIt1 OCHOBHOi1 KpHTepHt1, KpwrepHt1, 113 KOToporo
JleJ/aIOTC51 rrpeLInOJlO}/{eHH51 1l.1/51 1lpyrHx KpHTepHeB. OlliH6KI1 B H3BeCTHOM
Hece CHapR1la HJlH nJJaHeTbl HJIH 1la}/{e B BeJIl-jlJI1He CBeTOBoro 1laBJleHI151
1>WrYT npeBpaTHTbC51 B paBHOIleHHbIe 01lI116Kf!, 1l0nYCKaeMble Ha Ka}/{1l0M
mary BhIlJHCJleHI151 TpaeKTOpl1l1, 11 Y4eT 3Toro 06cr05lTeJlbCTBa HMeeT 60,1Jb-
mee 3Ha'leHl1e, 4eM 3a1laI-!He npOH3BO,lbH0t1 «1l0nYCTI1Moi1 OWH6KI1», KOTO-
:poe 6Y1leT 1l0CTHfHYTO B KOHIle TpaeKTOpHI1. TaKuM 06pa30M, MI.l BI11lHM,
-no nYTb, Ha KOTOpOM Mbl OTBe'laeM Ha 4eTbIpe Bonpoca:
1. KaKue Y3Ao8ble mOttKU Mbi 8bl6epeM?
2. KaKou K//aCC annpoKcu.ttupylOUJ,UX rjJYHKl1,lltl Mbi 8bt6epe.tt?
3. KaKou KpUmeplltl C02/taCllfl 3{bi 8bI6epe.tt?
4. f{aKYIO mottHocntb Mbi xomuM?
8.21
MHTEPnOJlHUl10HHWE MHOr04J1EHW
99
MOiKeT 3Ha1JHTeJIbHO BJIl-HlTb Ha OTBeTbl, KOTopbIe nOJIYlJaTCH Pl3 BbllJHCJle-
HHH. TaK)/{e .lJ,OJIJKHO 6bITb HCHO, lJTO OTBeTbI lIa 9TH BonpOCbl .lJ,OJIJKHbl
6bITb H3n.lJ,eHbJ B nepROH3lJ3nbHOn 33.lJ,3lJe, 3 lie B M3TeMan14eCIUlX TP3K-
Tanx HJIH .lJ,3)j{e B KIIHr3X no 4HcneHHOMY ali3nH3Y. 30paBaH Bbl'tIlCJlll-
meJlbHafl npaKmU1ca mpe6yem nocmOflHH020 UCCJleOOBaHUfl u3y'weAfOii
3aoa'tU He mOJlbKO nepeo op2aHU3al1,ueii Bbi'tUCJleHlUl, HO mmCJlCf
B npol1,ecce no pa3811mllfl II oco6eHHO Ha moil cmaOllll, K020a nOJlY-
tleHHbie IfllCJla nepeBOOflmCfl 06pamHO II UCmOJlK08biBalOmCfl Ha fl3biKe
nep8oHa'taJlbHOii 3aoa'tll.
rJJABA 8
HHTEPnOJISlUHSI MHOr04JIEHAMH. D.AHHbIE
C IIPOH3BOJIbHbIMH IIPOME)I{YTKAMH
8.1. <l>HJIoco4JHH
CYllI.eCTBYIOT .lJ,Ba rJI3RHbJX npHMelieHI1H HHTepnonHUHOHHbJX cpOPMYJJ.
rIpe)j{.n:e Bcero, OHH npHMeHHIOTC5I .lJ,n51 lI:enen 3aMeHbJ rp3cpH4eCKI1 3a.n:aH-
HOn cpYHKlI:I1I1 aH3JII1TI14eCKon. V13pe.n:K3 OHH I1cno.%3YIOTCH H .n:n51 TOro,
lJT06bJ BbJ.n:eJII1Tb 33BHCHMOCTb OTBeT3 OT nap3MeTp3, K3K B npHMepe
R 1.9.
BTopoe r JI3BHOe npHMeHel-lHe HX - .n:JIH Hl-lTepnOJJHlI:HH B Ta6JIlJU3x.
B H3WH .n:HH H3 YHI1BepC3JIbHblX lI:I1cppOBbJX BbJ4HCnI1Te.%HbJX M3WHH3X
HHTepnOJJ5!UI1et! nOJIb3YIOTC5I He C. I1WKOM OXOTHO. BMeCTO Toro lJT06bl
Haxo.n:I1Tb 3H3lJeHI151 H3 Ta6. l1l1:, 33nI1C3HHbIX B M3WI1HY, npe.n:nOlJI1T3IOT
nOJIh30B3TbC5I cpopMyn3MH. <l>OPMYJIbI .n:JI5I BbI4HCneHI151 3HalJeHI1t! 4aCTO
H3XO.n:51T nOJIY9MnI1pI1lJeCKH H He3aBHCHMO OT TeXHHKI1 I1HTepnOJHIUHH
MHOrOlJJJeHaMI1. C .n:pyrot! CTOpOllbI, BbICOKopa3BHToe HCKYCCTBO I13rOTOB-
JleHI151 Ta6JIHlI: *) .n:eJIaeT nOH5ITHbIM p33pa60TaHHOCTb HHTepnOJIHIJ.I10HHbIX
MeTo.n:OB. C06HpalOill.eMYCH COCT3BHTb Ta6JIHlI:bI peWHTenbHO He peKOMeH-
.lJ,yeTcH npI1CTynaTb K .n:eny, He H3YlJI1B npeJK.n:e XopOWO I13BecTHylO Tex-
HI1KY 11 B03MQ}J{HbIe JIOBYWKI1.
8.2. HHTepnOJIHlJ.UOHHbIe MHOrO'lJIeHbl
MHOrOlJJJeH CTeneHI1 n
n
y (x) = ao + alx + a2x2 + . . . + anx n = L akxk
k=()
(8.2-1 )
I1MeeT n + 1 K09cpcpHlu1eHT. ECTecTBeHHo 1I0naraTb, 'HO n + I yeno-
BHe, HaJIO)KeHHoe 11a MHOrOlJJle11 B 06ll1.eM BI1.lJ,e, n03BOJII1T 0.n:H03Ha4HO
or) CM., H3I1pHMCp, [to].
4'
100
nAHHblE C ITPOYl3BOJJbHblMYI ITPOME}!{YTK.-'.lIIYI
r f.TJ. 8
onpe.n:eJIHTb K09cjJcjJl1lJ.I1eHTbl. B '-IaCTHOCTI1, MO}!{HO rroTpe601:!aTb, l:IT06bl MHOrO-
4JJeH rrpoxo.n:I1JI 4epe3 n + I T04KY (Xi' Yi) (l = I, 2,..., n + I)
C Xi =1= Xj' To, 41'0 MHOr04JIeH rrpOXO.n:HT 4epe3 T04KI1 (Xi' Yi) o3HacraeT
BWrrOJIHeHl1e yCJIOBl1fi
n
Yi= a k x 7
k=O
(/=1,2,..., n+l).
Orrpe.n:emneJIb .n:JI5I 9TI1X n + I JII1HefiHbix ypaBHeHl1fi OTHOCI1Te.1JbHO He-
I13BeCTHbIX ak eCTb orrpe.n:eJIHTeJJb BaH.n:epMoHlI.a
XI xi ...x?
w=
X
x4 ...x
= I(x" X ,..., X n +\),
X n +, X +I"'X H
KOTOpbIfi, KaK MbJ cefi'lac rrOKa}({eM, He paBeH Hymo, eCJIH Xi:;t: Xi lWH
i#j.
51cHO, '11'0 orrpell.eJIHTeJIb eCTb cjJYHKlJ.I151 OT XJ, X ,..., x n +\. ECJ1H
C'IHTaTb ero cHacraJIa cjJYHKlJ.Hefi 01' X n +" TO OH eCTb MHOrO'lJIeH CTerreHH n
11 06paIll,aeTC5I B HYJIh BCHKHfi pa3, KOrll.a Xn+1 = Xj (.n:JI51 j = I, 2, ... n).
TaKHM 06pa30M, I (XI' X ,..., X n +,) COll.ep}!{HT MHO}!{I1TeJII1
n
II (X n +, - Xi) = (X n + 1 - X,) (X n +, - X ).. . (X n +\ - X n ).
i=l
PaCCMaTpI1Ba51 orrpe.n:eJII1TeJIh KaK cjJYHKUI1JO OT X n ' MhI BHlI.I1M TOcrHO
TaK )J{e, 41'0 CYIlI,eCTBYJOT MHO}({HTeJIH
n-l
II (X n - Xi) = (X n - X,) (X n - X )... (X n - X n _ 1 )
i=l
H Boo6Il1,e BCe MHO)J{I1TeJIH
n+l
11 (x; - Xi)
;>i=l
BCTpe'laJOTC51 B HarneM orrpe.n:eJII1TeJJe. n pOH3Be.n:eHHe BCeX 9TI1X MHO}({H-
TeJIefi eCTh MHOrO'lJIeH CTerreHH
, rt(n+l)
(/ -1 lfl - 1) + t (/ - 2) + . .. + 1 = 2 .
8.21
HHTEPf10JJ5Il\HOHHbIE MHOr04.TIEHbl
101
Ho Onpe.1l:eJIl-ITeJlb TaKme eCTb MHOr04J1eH TOt1 me caMot1 CTeneHH; cJle-
.n:OBaTeJlbHO,
n
W=c II (X;-Xi)'
;>i=1
(8.2-2)
r tI,e C - HeKOTOpa51 KOHCTaHTa, KOTOPYIO eru.e Ha.1l:0 Hat1TH. 4T06bl onpe-
tI,eJlI-ITb C, paCCMOTpHM BblpameHlle, KOTopoe nOJlY l JaeTC5I npl1 YMHomeHHH
4J1eHOB rJlaBHot1 .1I:HarOHaJ!H
1 . X2 . X . . . X +l'
PaCKpblBa51 npOH3Be.1l:elme, Mbl HaXO.1l:HM T04HO TaKot1 me llJleH, 1'1, cJle-
.1I:OBaTeJlbHO, C= 1. TaKHM 06pa30M, onpe.1l:eJJHTeJlb.BaH.1I:epMOH.1I:a He paBeH
HYJlIO, eCJlH Xi "1= Xi .11:.151 1"1= j.
B03Bparu.a5lCb K Halllet1 rJlaBHot1 3a.n:a4e 0 HaXOm.1l:eHHH MHOr04J1eHa
no n + 1 T04Ke (Xi' .Yi)' Mbl BH.1I:HM, 4TO ee BCer.1l:a MomHO pelllHTb H
Hat1TI1 K09cpqmUHeHTbI ak no npaBHJlY KpaMepa HJlH KaKHM-HH6Y.1I:b .1I:PyrHM
cnoco60M. rIO.1l:CTaB'HB 9TH 3Ha4eHH5I B 06ru.Ht1 BI1.11: MHOr04J1eHa (8.2-1),
Mbl MomeM npe.1l:CTaBHTb pe3Y JIbTaT B BH.1I:e
Y X X2 ... x n
Yt Xl xi ... x
Y2 X2 x ... x =0. (8.2-3)
Yn+\ I xn+l X +1 ... X +1
Bnp04eM, 9TOT pe3Y JlbTaT MomHO npoBepHTb, paccym.1l:a51 CJle.1l:YIOru.HM
06pa30M: Bo-nepBbIX, Bblpamelme (8.2-3) .1I:OJ!mHO 6bITb MHOr04J1eHOM
CTeneHH n no X, TaK KaK onpe.n:eJIHTeJlb MomHO pa3J10ml-lTb no 9J1eMeH-
TaM BepxHet1 CTp04KH, 1'1, BO-BTOpbIX, OH npoxo.n:HT 4epe3 n + I T04KY
(x i, Yi)' TaK KaK nO.1l:CTaHOBKa 9TI1X 3Ha4eHHt1 B BepXHlO1O CTPOKY .1I:eJlaeT
.1I:Be CTpOKI-I onpe.1l:eJlIITeJl51 O.1l:HHaKOBbIMH. B03MomHO, 4TO K09cpcjJHlJ.HeHT
npH x n eCTb HYJlb (YCJlOBI-I5I TOro, 4T06bI 9TO 6blJlO TaK, .1I:OCTaT04HO
51CHbl) 11 4TO c-reneHb MHOr04J1eHa MeHbllle, 4eM n. 4T06bl OXBaTI-ITb H
9TOT CJIY4at1, YTBepm.n:eHI-lIO, 4TO MHOr04JJeH HMeeT CTeneHb n, 4aCTO npl1-
.1I:alOT CMbICJI: C Ten e H b n 1-1 JI H Me H b III e. B BbIpOmtI,eHHOM CJlY4ae,
l<Of.n:a Bce Yi paBHbl, MHOr04JIeH I-IMeeT CTeneHb 0 11 Y (x) = C.
Bce 1-13,'IQmeHI-IOe MomHO nO.1l:bITOml-lTb, CKa3aB, 4TO eCJlI-I eCTb n + 1
Y3JIOBall T04Ka cpYHKIJ.I-IH, TO MomHO Hat1TI-I MHOr04J1eH CTeneHH n, KOTO-
phlt1 COBna.1{aeT (npeHe6pera51 0llJl-l6KaMH oKpyrJleHH5I) C cpYHKUl-let1 B Y3J10-
BblX T04Kax. HacKoJlbKO 6J1113KH o6e cjJYHKUl-I1-I Mem.!1:Y Y3J10BblMH T04-
KaMI-I, 6Y.1l:eT HCCJletI,OBaHO B 8.6, HO, npe.!1:nOJlOmHB, 4TO OHH 6J1H3KH,
Mbl MomeM I-ICnOJlb30BaTb MHOfOl[JIeH BMeCTO cjJYHKUHH B .1I:aJlbHetllilHx
102
D.AHHbJE C np0I13BOJJbHbIM npOME/KYTKAMI1
I fJI. 8
aHaJlHTH4eCKHX npoueccax: HHTerpHpOBaHHI1, D.H<pcpepeHUHpOBamm, OTblCKa-
HHH HYJlet! H T. .11..
BMeCTO Tpe60BaHH5I, 4T06bl MHOr04J1eH npOXO.1l.HJI 4epe3 HeKOTOpble
.1I.aHHble T04KH, MO/KHO nOTpe60BaTb, 4T06bl OH B HeKOTOpbIX 3a.1l.aHHblX
T04KaX HMeJl .!I.aHHbIt! HaKJlOH. TaK, ypaBHeHHe
Y X X2 X3
Y. I XI xi xl
Y2 I X 2 2 X =0
X2
Y' 0 1 2x, 3xi
Y2 0 1 2X2 3x
onpe.1l.eJl5JeT MHOr04J1eH TpeTbet! CTeneHI1 no X, npOXO.1l.51IlLHtI 4epe3
(XI' YI) C HaKJlOHOM Y; H 4epe3 (x 2 , Y2) C HaKJ/OHOM Y2 (TaK KaK .1I.J/51
.1I.HcpcpepeHUHpOBaHH5J .9Toro onpe.1l.eJ/HTeJl51 no X .1I.ocraT04Ho npO.1l.Hcpcpe-
peHUHpoBaTb n04J!eHHO ero BepxHlOlo CTPOKY),
ECJI!-I B T04Ke D.aHO 3Ha4eHHe y', TO BOBce He 065J3aTeJlhHO B Het!
LtOJl/KHO 6bITb YKa3aHO 3Ha4eHl-le Y; TO }J{e OTHOCI1TC5I H K 60Jlee BbICO-
KI-IM npOH3BO.1l.HbIM. OrpaHH4eHHeM Ha Bbl60p yCJ/OBHtI 51BJl5IeTC5J Tpe60Ba-
HHe, 4T06bl MHHOp 4J1eHa Y 06513aTeJlbHO He 6blJJ paseH HY, IO. B npo-
THBHOM cJlY4ae MHOr04J1eHa MO/KeT He cyw.eCTBOBaTb. 3TO 03Ha4aeT, 4TO
.1I.OJDKHO 6bITb .1I.aHO no KpatiHeti Mepe OLtHO 3Ha4eHHe CPYHKUHH Yi' no
KpatJHet! Mepe .1I.Ba YCJ/OBH5I Ha Yi H YI, Tpl1 Ha Yi' YI, YI H T. .11.. Bll/IOTb
LtO npOl-l3S0LtHOti caMoro BblCOKoro nOp5l.1l.Ka.
T OJlbKO 4TO CCPOPMY JJHpOBaHHoe yCJloBl-le He HBJl5IeTC5I .1I.OCTaT04HbJ:\l.
PaCCMOTpHM, HanpHMep, Tpl-l paBHOOTCT05Jw.l1e T04KI1, KOTopble Mbl 060-
3Ha4HM - 1, 0, I. B Ka/K.1I.otJ T04Ke HaM LtaHbI CPYHKUI-I5I Yi H BTOpa51
npOH3BO.!l.Ha51 (B cpH3H4eCKI1X TepMHHax nOJlO/KeHHe 1-1 YCKOpeJII-Ie).
MHHOp npH y paBeH
1 -I I -1 I -I
1 o 0 0 0 0
1 1 1 I 1 1
02-6 =0.
0 12 -20
0 o 2 0 0 0
0 o 2 6 12 20
T3KHM o6pa30M, 9TI1 weCTb YCJIOBHtI He OlIpe.!l.eJ/HIOT MHOr04JleHa n5lToti
CTeneHM.
II 8.3]
METO!l I1HTEPn0n5IU1-111 JJA fP AH}f{A
103
Ynpa>KHeHHSJ
8.2-1. HalIHwHTe MHoro'weH, npoxoJ\JJW.Hfi 'Iepe3 (XI> YI) C TaHreHCOM yrJla
HaKJlOHa Y; H BTOpoi1 npoH3BoJ\HOfi Y7 H HMelOuJ,f1H TaHreHC yrna HaKJlOHa
Y B X 2'
Y I X X2 X3 I
YI I XI X2 xf
1
o T B e r: Y; 0 I 2x I 3XI 1=0.
yr 0 0 2 6x I I
y 0 1 2x 2 3x
8.2-2. OOKa3aTb, '1TO eCJlH HCIlOJlb30BaTb YCJlOBHR Ha y, y', y",. . . , y(n- u
B O!lHOH TO'1Ke xl> TO MHOrO'l.1eH 6YJ\eT npeJ\CTaBJlRTb 060pBaHHblH pRJ\ TeHllOpa.
8.2-3. HarlTll HeTpHBHaJlbHbIH MHOrO'1J1eH Y (x) TJlKofi, '110
Y (-I) =y (0) =y (I) =y" (-I) =y" (0) =y" (1) =0.
vlcc.1eJ\oBaTb era.
OJ\HH 113 B03MO*<HblX OTBeTOB:
y (x) = 3X5 -IOx 3 + 7x.
(3 a M e 'I a H H e: sin 1tX TaKme o6J1a.aaeT 9THMH CBoHcTBaMH.)
8.3. MeTOIl. HHTepI10JUI HH J]arpaHlKa
.ilpyrot! nO!lXO!l K 3a!la'1e HHTepnOJI5ILI.J.HI - MeTO!l J1arpaH}f{a. OCHOll-
Ha51 l1!le51 9Toro MeTO!la COCTOHT B TOM, 4T06hl npe}f{!le lIcero Hal1TH
MHOr04JIeH, KOTOphlt! npHHHMaeT 3Ha4eHHe 1 8 O.n;HOt1 Y3JI080t1 T04Ke
H 0 80 8cex !lpyrHX. J1erKo 8H.n;eTh, 4TO CPYHKIlH5I
L (x)= (X-Xl) (x-x 2 )... (X-Xj_l) (x-Xj+1)'" (X-Xn+l) _
j (XI-Xl) (X/- X2)'" (Xi-Xl_I) (X/-X/H) ... (Xi-X/HI)
n+1
rr (X- Xi)
i=1
- n+l
IT (X/-Xi)
i=1
(me WTpHX Y 3HaKa np0I13Be.n;eHH5I 0311a4aeT «nCKJII04a51 j-e 3I1a4elme»)
51BJI5IeTC5I Tpe6yeMhIM MIIOr04JIellOM CTenellH n; 011 paBell I, eCJII1 x == Xi'
11 0, KOr.n;a X=Xi' i oF j.
MIIOr04JIell Lj(x)Yj npHHHMaeT 3Ha4eHHe Yi B i-fi Y3JIOBOtl T04Kt: H
paaeH IIYJIIO ao BCeX !lPYrHX Y3JI8X. 113 &Taro CJle.n;yeT. 'ITa
n+l
y(x)= L/(x)y/
1=1
eCTb MHOr04JIeH CTeneHII n, IlpOXO.LI.HU HA 'Iepea n + 1 TOliKY (Xi' Yi)'
104
.D.AHHbIE C DP0J.13BOJJbHbIMI1 DPOME)J{YTKAMJ.1
I I'll. R
MO)!{HO CnpOCI1Tb, COBna.D.aeT JII1 MHor04JIeH J1arpaH)J{a C MHOr04JIe-
HOM (7.2-3), nOCTpoeHHblM B npe.D.bl.D.Yll.J,eM naparpaQJe. I1x pa3HocTb-
MHOr04JIeH CTeneHl-J He Bblilie n II OHa 66pall.J,aeTC5I B HYJIb B n + 1
Y3JIOBOt! T04Ke x = xi> CJIe.aOBaTeJIbHO, OHa TOJK.aeCTBeHHO paBHa Hymo.
TaKIlM 06pa30M, MO)l{HO C.D.eJIaTb CJIe.D.YlOll.J,ee Ba)J{Hoe 3aMe4aHI1e: eCJII-J .aaHa
n + 1 Y3lIOBa51 T04Ka, TO COOTBeTCTBYIOIl.J,Ht! MHOr04JJeH CTeneHI1 n, npo-
XO.Il.5I1l.J,I1tJ 4epe3 9Tll T04KH, 0.aH03Ha'lHO (B npe.aeJIax OIlIll60K oKpyrJIe-
H1l5l) onpe.aeJJeH, He3aBllCI1MO OT TOro, KaK OH CTpOI1TC5I II KaKHe 060-
3HaqeHI151 I1CnOJJb30BaHbI. BTO He06xo.aHMO no.a4epKHyTb, nOToMY 4TO He-
KOTopbIe KHI1rll no 4I1CJIeHHoMY aHaJIl-I3Y Mor JIH 61,1 HaBeCTH 4HTaTeJI5I Ha
MbICJIb, 4TO pa3JJH4HbIe QJOPMYJIbl H306paJKalOT pa3Hbie MHOr04JIeHbI, I1JIH
H3-3a Toro, 4TO OHI1 nOJIY4eHbi pa3HblMH cnoc06aMI1, HJIH H3-3a pa3JIH41151
B 0603Ha4eHH5Ix. ECJJI1 I1CnOJIb3YIOTC5I pa3Hbie Y3JIOBbIe TO'lKH, TO, KO-
He4HO, MHOrO'lJJeHbl MorYT 6bITb pa3JIH4HbIMll, HO O.D.HHaKOBbIe Y3JIOBbIe
T04Kll .D.O JIJKH 1,1 npllBO.D.llTb K O.D.I1HaKOBbIM MHOr04JJeHaM (B npe.D.eJIax
Oll1ll60K oKpyr JIeHH5I).
I1HTepnOJI5Ill.HOHHa51 QJopMYJIa JlarpaHJKa HeY.D.o6Ha .D.JJ5I npaKTH4eCKoro
HCnOJIb30BaHlUI. ITpe06pa30BaHHH n03BOJIHIOT npllBeCHI ee K HeCKOJJbKO
60JIee Y.D.06HOMY .D.JIH BbI411CJJeHHH BH.D.Y. PaCCMOTpUM 4aCTHbItJ CJJY4atl,
KOr.D.a Bce Yi = 1. Tor.D.a Y (x) = 1 .D.JI5I Bcex x, TaK '1TO TO)l{.D.eCTBeHHO
n+1
1= Lj(x).
j=1
Mbi MO)\{eM renepb pa3.n.eJIHTb npaBYIO '1aCTb QJOPMYJIbi JlarpaH)\{a Ha 9TO
Bblpa)l{eHHe, nOJIO)l{HB
1
Aj= ,
IT (X;-Xi)
nocJle .D.eJleHHSI 4UCJIHTeJI5I H 3HaMeHaTeJI5I Ha n (x - Xj) nOJIY4HM
n+1
[A;y;/(x-x;)]
j=1
Y (x) = n + 1
[At/(x-x;)J
j=1
I1Hor.D.a 9TO Bblpa)J{eHI1e Ha3bIBalOT «6apHll.eHTpH4eCKotJ QJOPMYJIOt!». Ee
JIer4e npllMeH5ITb, 4eM QJOPMYJIY JlarpaHJKa.
PaCCMOTpHM Tenepb 3a.n.a4Y HaXO)l{.D.eHIl51 IJHTepnOJJ5lIU-IOHHOrO IvIHoro-
4JIeHa, KOTOpblt! B KaJK.D.ot! r04Ke Xi npllHHMaeT 3Ha'leHHe Yi H HMeer
npOH3BO.aHble y; (t = 1, '" , n + 1). TaKot! MHOr04JJeH Ha3hIBaeTC5I MHO
r04JJeHOM 9pMllTa. 5ICHO, '1TO OH HMeeT CTeneHb 2n + 1 (TaK KaK .D.OJI)I{Hbi
Owrb YAOBJleTBOpeHbl 2n + 2 YCJIOBHH).
8.31
METOll VfHTEPTIOJl5lwm JlA fP AH)j{A
.1 05
Hac 60JIbille HHTepecyeT MeT( JI., npHMeHHeMbIt! npH nOCTpoeHHH MHO-
rO'lJIeH3, 'IeM OKOH'IaTeJJbHbJt! p6lYJJbTaT. STOT MeTO.n: eCTb eCTeCTBeHHoe
npOltOJJ)j{eHHe MeTo.n:a JIarpaH)j{a. MbI Hlu:eM TaKHe MHOrO'lJIeHbI H j (x) Ii
h j (x) CTeneHH He BbIIlIe 2n + 1, '1TO npH l =1= j
Hj (Xj) = 0, h j (Xj) = 0,
Hj (Xj) = 0, hj(xj)=O,
H;(x;)=l, hj(Xj) = 0,
Hj(xj)=O, h;(xj)=1.
TIoCJIe Toro KaK 9TH MHOrO'lJJeHbl 6YltYT H a t!.n:eHbl, peIlIeHHe nOCTaBJIeH-
HOt! 3a.n:a'lH MO)J{HO 6y.n:eT 3anl1CaTb B BH.n:e
n+l
Y(X)= [Hj(X)Yi+hj(X)Yi].
j=l
TIpe)j{lteBCerO rrOCTpOHM hj(x). ECJIH hj(Xj)=OH h;(Xj)=O, Tohj(x)
.n:OJI}J{eH cO.n:ep)j{aTb MHQ)KHTeJIb
(x - Xj)2 (.n:JIH l =1= j).
ECJIH h j (x j) = 0, HO hj (x j) =1= 0, TO cyrn.eCTByeT npOCTOt! MHO)J{HTeJJb
(X-Xj)' LlT06w nDJIY'lHTb hj (Xj) = 1, CJIeltyeT B3HTb
h.( ) _ {X-X I )2 (X-X 2 )2 ... (X-Xj_I)2 (x -Xi) (X-XiH)2 ... (x-x n +d 2
J X - (X/-X I )2 (X/-X 2 )2 ... (X/-X/_ 1 )2 (X/-Xf+I)2 ... (X;-X nH )2
LlHTaTeJIb JIerKO y6e.n:HTCH, '1TO h j (x) 06JJa.n:aeT Tpe6yeMbIMH CBotlcTBaMH.
TIoCTpoeHHe Hj (x) rrO'lTH TaKoe me. OnHTb Mbl HMeeM MHO)j{HTeJJH
(x - Xj)2
HO B TO }J{e BpeMH Tpe6yeM, 'ITo6bI
(.n: JIH l =1= j),
Hj(Xj)=I,
Hi (Xj) = o.
6ylteM HCKaTb Hj (x) B BH.n:e
H. ( ) _ (X-X I )2(X-X 2 )2 ... (X-Xi_I)2 (ax+b)(x-xiH)2 ... (X-Xn+I)2
J X - (X;-X I )2 (Xf-X 2 )2 ... (X;-Xi-I)2 (x;-xi+1)2 ... (x;-X nH )2
Y CJJOBHe Hj (x j) = 1 Tpe6yeT, 'ITo6bl
aXj +b= 1.
B3RB npoH3Bo.n:HYIO OT Hj H nOJJO}J{HB B Het! x = x j, nOJIY'lHM
H{x)- + + + 2 +
J J - Xi-Xl X;-X 2 ... X/-Xf_l
2 2
+a+ +...+ =0.
X/-X;+l xf-x n +1
106
D.AHHblE C npOVl3BOJJbHblMI1 npOME)!{YTKAMI1
[Tn. 8
TaKHM 06pa30M,
n+1
a=-2 !'
;=1
Xi-Xi'
b=l-aXj
11 Tpe6yeMa51 cpOpMYJla fIOC'rpOeHa.
choT MeTO.rr !IOCTpOeHI151 MHoro'weHOB, 06J1a.rralOllU-IX cneuncpWJeCKHMH
cBoticTBaMH, HCfIOJlb3yeTC5I B TeopnH 'H!CJJeHHhIX MeT01l.0B BeChM3 'Jano
H nM cJle.rryeT OB.TJa.rreTh, MhI fIpnMeHnJIn ero B lI.BYX KOHKpeTHhlx cJ/Y-
'1a5lX - npn fIocTpoeHHH nHTepfIOJI5IUHOHHhIX nOJlnHOMOB JlarpaHiK3 11
3pMnTa; HO '1nTaTeJ/h .lI:OJl)!{eH Y5lCHI1Th ce6e o6w.nti MeTO.ll: n01l.60p3
Vi' y;, ..., y)n\ B Xi' KaK 6Y.ll:eT BH.rrHO npn nCCJle.ll:OB3HHn norpeWHOCTI-I
I-!HTepfIOJ/51UI10HHhIX cpOPMYJl, HCnOJlh30B3Hne fIpOH3BO.ll:HhIX B 6mf3KHX TO'l-
Kax BeChMa 'YBeJII14HBaeT T04HOCTb cpOPMYJlhI.
Ynpa>KHeHHe 8.3-1. YKa3aTb clloc06 110JlYQeHHR HHTepnOJlHI.J,l1oHHort 4Jop-
MYJlbl, KOTopaR I1MeeT 3HaQemW Vi' Y; H Y; B KaJK,IJ.OH TO'lKe Xi'
8.4. MHTepDOJUlu,noHHaH <J:IopMy.iIa HblOToHa
PaCCMOTpeHHhIe MeTo.rrbl HaxmK.rreHJJ5I HHTepnO.15Iu,HOHHoro MHOrO'lJleHa
no n + I TO'lKe nO.ll:pa3y teBaJlJJ, 41'0 MHO)!{eCTBO I1CfIOJIb3yeMblx Y3J/OB
JJ3BeCTHO. Yano H3BeCTHbIM. 51BJl5IeTC5I JIHWb Tpe6yeMa51 T04HOCTh, a MHO-
)!{eC1'BO Y3JJOB, KOTopbIe CJle.rryeT HCfIOJ/h30BaTb, onpe.ll:e.TJ5IeTC5I HHcpopMa-
lI.JJen 0 TOM, KaK BhI4JJCJl5IeTC5I cpYHKUH5I. VlHTepnOJl5lUHOHHa51 cpopMY;la
HbIOTOH3', KOTOpa51 6Y.ll:eT H3J10)!{eHa, npe.ll:CTaBJl5IeT c060ti fIpOCTO lI.pyrotJ
CfIoc06 HanHCaHH5I JJHTepfIOJl5lUJJOHHoro MHOrO'lJleHa. QHa nOJle3Ha, fIOTOMY
4TO '1HCJlO HCnOJIb3yeMhIx Y3JIOB MO)i{eT 6bITh JlerKO YBeml'leHO nJ/n
YMeHhweHO 6e3 fIOBTOpeHH5I Bcero Bbl'lnCJIeHI151. <t>OPMY Jla HhlOTOHa 113
1.5 eCTb !IpOCTO ee '1aCTHhIti cJlY'Iafi, KOr.ll.a Y3J10BhIe TO'lKH paBHO-
Yll.aJJeHhI.
KaK n paHhwe, 0603Ha'll1M MHOrO'lJ/eH, npOXOll.51W.Hti 'Jepe3 n + 1 T04KY
(Xi' Yi) (i = 1, 2, ..., n + 1), 'Jepe3 Pn = Pn (:c). MO)l{HO HanHcaTh
Pn (x)= YI + (x - XI) Pn_dx),
r.ll.e Pn-I (x) - HeKoTopbIti . MHOrO'lJleH CTeneHI1 n - L HCHO, 4TO
Pn (XI) = YJ n CJle1!yeT 3a60THThC5I J/HWh 06 n TO'lKaX (I = 2, 3,
.. ., n + 1). Ope.rrhIllyw.ee ypaBHeHHe MO)l{eT 6hITh saDHcaHO B BJJ.lI.e
P ( x ) = Pn (x) - YI
n-I X-XI '
1-1, CJle.rrOBaTeJlhHO, Tpe6yeTC5I, 4T06hI
P (x.)...= Pn(X;)-Pn(Xd_Y;- YI
fl-I I Xi-Xl Xi-XI
(i = 2, 3, .." n + I).
, .41
HHTEPnOJTSHU10HHAS'I <t>OPMYJJA HblOTOHA
107
TeM CaMblM HY)KHO, 'lT06bJ P n-l (x) npOXO.ll.HJJ 'Iepe3 rO'lKR
\ (Xi, Yi-YI ) (l=2, 3, ..., n+ 1).
Xl-Xl
BeJlH'II1Hbl
Yi-YI_ [ ] - [ 1
-- Xl' Xl - Xl' Xl
Xl-Xl
H3:3bIB3IOTC5I pa3aeAeHflbt.Mll pa31WCmfl.Mll H 06bJtcIHO 3anHCblBalOTC5I B KBa.ll.-
paTHbrx cKo6Kax *).
H a CJJe.ll.YIOrn.eM ware Ha.ll.O HanHcaTb
Pn-l (X) = [Xl> X2] +(x - x 2 ) Pn-2 (X)
H Tpe6cBaTb, 'IT06bl P n-2 (X) npHHHMaJI 3Ha'leHl-HI pa3.l1.eJIeHHblX pa3HocTeR
OT pa3JleJleHHblX p33HocTel1
I ]J - [Xi, XI] - [X2, xd _ [
[Ix;, Xl], X2, Xl - - Xi> X2, XII.
Xi- X 2
HeTPY.lI.HO BH.lI.eTb, 'ITO pa3.l1.eJJeHHbIe pa3HOCTH nepBoro nOp5l.ll.Ka He
33BHO!T OT nOp5l.ll.Ka 3prYMeHTOB B KB3.l1.paTHbJX cKo6Kax. DOKa)KeM
Tenepb, 4TO 9TO BepHO H .lI.JISI pa3.l1.eJIeHHbJX pa3HOCTel1 BToporo nOp5l.ll.Ka.
ECJJI1 H343Tb C Tpex T04eK
(Xl> YI), (X2, Y2), (xa, Ya),
TO nOJJy'mM e.ll.l1flCTBeHHbl11 MHOr04JJeH BTOpOI1 CTeneHH, IlPOXO.ll.SUlI.HR '1epe3
Tpl1 T04KH. Ero MO)l(HO 3anHcaTb TaK:
Y = Yl + (x - Xl) {[X2, xI! + (x - X2) [Xa, X2, Xl]}.
ECJJH Tenepb B351Tb TO'lKH B TaKOM nOp5l.ll.Ke: (XlV Va), (X b ' Yb)' (xc, Y c )'
TO IlOJJY4HM
Y = Ya + (X - Xa){[X b ' xal + (X - Xb)(X c , X b , Xa/).
TaK K3K 06a 9TH X ypaBHeHHSI Onpe.ll.eJJ5IIOT O.ll.HH H TOT )Ke KB8.l1.paTHbll1
TpeX4JJeH, TO K09<pqmuHeHTbl npH X2 .lI.OJJ)KHhJ 6bITb O.ll.HHaKOBbJ, TaK '1TO .lI.Ba
CHMBOJJa
[Xa, X 2 , xtl = [Xc> Xb' aal
BbIP3)!\3IOT O.ll.HO H TO )Ke.
Boo6rn.e Onpe.ll.eJJ5IeM
lXI' X2, Xa, ..., xnl= [Xl> X 2 , ..., Xn_2' Xn_d-[Xl> X 2 , ..., X n _ 2 , Xn]
X n _ l - X n
11 B r04HOCTH TaKHM )l(e cnOCo60M IlOK33bIB3eM, 4TO p33HOCTH He 33BI1C5IT
OT nOp5I.lI.K3 Xi' 33MeTI1M, 4TG 3H3MeHaTeJJb eCTb pa3HOCTb HerrOBTopmo-
UI.HXC5I 3H3'JeHI1I1 X, B351TbIX B TOM )l(e IlOp5l.ll.Ke,
*) VJCIJOJlb3YlOTCR 11 n:pyrne 0603HaQeHHiI, HanpHMep f lXi, xd f1 p lXi, xI!.
J08
IlAHHbIE C flP0l13BOJJbHbIMI1 flPOMEMYTKAMI1
11.'1. II
OIlI-IH 1-13 cnOC060B nOCrpOeBI-I5I ra6J1J.1l.l,bl Be06xo.ll.I1i\1bIX 3Ba4eBI-IH
c. e.ll.YIOIUl1tJ:
Xl I yt
i [X 2 , x l 1 *
x 2 I Y2 [X 3 , X 2 , xd *
I [x 3 , Xl] lx" X a , X 2 , xd *
Xa ! Yo
I [x 4 , xd
x, i Y4
'" I ...
[X 4 , X 2 , x l 1
3Be31l04KI-I 03Ha4alOr 4I1CJla, I-ICIJOJlb3YIOIUl1ec51 KaK onopBbIe 3Ha4eHH5I npH
BbI'1I1CJleHI11-1 3Ba4eH\1t\ MHOrO'lJIeH3.
1-13 3rOtJ Ta6J1I1Ubl MOMBO H3nl-lCarb
Y(X)=YJ +(X - X,)([X2, Xd +(X-X2) {[X3' X2, Xd+
+ (X - X 3 ) [.. .]}].
B Ka4eCTBe npm'Iepa paCCMOrpl-lM ra6J1l-1uy JlOrapl-lcpMOB
x I Igx \ I,] I [,,1 I I,,,]
I ( 0,0000*
0,30100*
2 0,3010 -0,06245*
0,23855 +0,01230*
3 0,477\ -0,05015
0,20070
4 I 0,6021
3Be3Il04KH 03Ha4alOr orropHbIe 3Ha4eHH5I. OTCIO.lI.3 rrOJJY4aeM
y(X)= 0 +(X - 1) {0,3010 + (x - 2) [(- 0,06245)+
+ (x - 3) (0,0 I 230)]).
B 4aCTHOCTH,
y(2,5)= { 0,3010+ ; [(-0,06245)+(- ; )(0,01230)]}=
= 0,40001.
BepHoe 3Ha4eHne
Ig 2,5 = 0,3979.
YnpalliHeHHe 8.4-1. YKa)J(HTe c/Joc06 Jl:06aBJleHHR ellJ,e OJJ,HOH Jl:aHHOH TO'1KlI
B KOHu,e Ta6.1Hu,bJ pa3,IJ.e.1eHHbIX pa3HOCTeI1. rIoKa)J(HTe TaK)J(e, KaK Jl:06aBf1'fb
OIlHY TO'lKY HaBepx. B uepBOM cJlY'Iae KaKoe 113MeHeHHe CJleIlyeT CJJ,eJlaTb B MHO-
rOIMeHe HbloToHa?
8.51
IlPYr A5J <!>OP MA JUI5J T ABJIVlLJ,bI P A3IlEJJEHHbIX P A3HOCTEH
109
8.5. D,pyraH l}JopMa JJ:JIH TaOJIlHl.bI pa3J1.eJIeHHblX pa3HocTeif
Ta6mJUa pa3JJ:eJIeHHbIX pa3HocTefi, KOTopaH JIe)f{Wf B OCHOBe HHTep-
nOJI5JQHOHHoti I}JOPMYJIbI HbIOToHa, MO)f{eT 6bJTb HanHcaHa B llPyroi!,
HHor lla 60JIee nOJIe3HotJ 1}J0pMe. 3aMeTI1M, qTO
[ ] [Xl' X 2 , ...,
XI,X2,..., X n =
xn_d [Xl> X 2 , ..., Xn_2' Xn!
Xn_I-X n
_ [Xl' X 2t "0' X n _ 1 ]-[X 2 , XSt ..., Xn]
XI-X n
B qaCTHOCTI1,
[Xl' Xa, X3] = [XI' X2J - [(Xl> XS)] = [XI' X 2 ] - [X 2 , XS]
- - -
TaKI1M 06paSOM, MOJKHO Hanl1CaTb
(,]
[,,]
[",]
Xl YI
[X2' xI!
X2 Y2 [X3, X2, Xl]
[X3' X2] [X4,' X3' X2' Xl]
X3 Y3 [X4,' X3' X 2 ]
X4, Y4, [X4, X3]
HeCMOTpH Ha TO, qTO qnCJIa B 9TOi! Ta6mme OTJIHqalOTCH OT qHCeJl
B npellbIllYIQeM pa3JJ:eJIe, ql1CJIa B BepXHeM PHllY, KOTOpbIe TOJIbKO M
I1CTIOJIbSYIOTCH B cpOpMYJIe HbIOTOHa, Te )f{e caMbIe.
B35JB TOT )Ke caMbIt! TIpHMep llJI5J Ig X, nOJIyqaeM
x I Igx , [,I I [,,1 I [,,,1
I
1 0,0000
0,3010
2 0,3010 -0,06245
0,1761 +0,01230
3 0.4771 -0,02555
0,1250
4 0,6021
B TaKOM BHlle MbI JIerKO MO)f{eM 1l06aBHTb CTPOKY H K TOMY H K llpy-
rOMY KOHQY Ta6JIHQbI H HalleHTbcH, qTO qHCJIa B ra6JIHQe H3MeH5JTCH
r JIaJJ:KO. XOTH TeOpeTl1qeCKH HeT Heo6xollMMOCTH pacnOJJaraTb TOqKH
B TIOp5l.lJ.Ke B03paCTaHH5J (HJII1 y6bIBaHI1H) xi' rJIallKOCTb B Ta6JII1Qe Hapy-
maeTcH, eCJIH TaKoro nopHllKa HeT. TIepBbIe pa311eJIeHHbIe paSHOCTH -ce-
110
JlAHHblE C f1POV!3BO.TIbHbIMV1 f1POME)I(YTKAMV!
IfJl.S
1();UJ,He JlHHHH 11, CJleJlOBaTeJlbHO, 6JIH3KI1 K nepBbIM np0l13BOJlHbIM B HI-ITep-
HaJle (x n , Xn+l)' AHaJlorH'IHO BTopbIe pa3JleJleHHbIe pa3HocTH 51 fW5IIOTCSI
JIOKaJJbHbIMI1 npI16JIH)I(eHH5JMH KO BTOpblM np0l13BOJlHbIM H T. JJ..
YnpamHeHHSI
8.5-1. VlCnOJlb3)'R Ta6J1Hu,y
x
2
o
3
y
8
o
27
BblqHC.1HTh l1f!Tepno.QRu,HOHHbltl MHOrOqJ/eH HbJOToHa. nOKa3aTb, qTO y = x 3 .
8.5-2. Bbl'1I1CmHb Ta6mIUY pa3HocTeti 060UMH cnoc06aMH ( 8.4 U 8.5) Mil
Y = sin x C warOM B 30':
x
o
30'
60'
90'
sin x
0.0000 -! 0,5000
0,8660
1,0000
HJihH annpOKC!1MaUHOHHbIH MHOrO'lJleH.
8.5-3. KaK npH6aBHTb OJ\HY TO'lKY BHH3Y Ta6J1\!llbI H BbI'IHCJlI1Tb cJleJlYJOw.yJO
pa3J\eJleHHYJO pa3HocTb, eCJlH B naMHTH 1Jbl'1HCJlHTe,1bHOH MalllHHbI xpaHHTCJJ
JlHmb BepxHui1 pRJl ra6.1Hu,bl pa3JleJleHHblX pa3HOCTeil? KaK npH6aBHTb OJ\HY
TOqKY HaBepxy H HaHTH HOBYJO CTPOKY?
8.6. norpeillHOCTb MHOrOqJJeHHOH annpOKCIIMalJ,lI II
3aJlaB <jJYHKlJ,HIO f (x), MbI 6paJJl-J n + 1 Y3JIOBYIO TO'lKY (Xi' Yi)
(l = I, 2, ..., n + 1) H HaXO.1UIJIH MHOrO'lJIeH P n (x), npOXOJlHlltEti '-Ie.
pe3 9TH TO'lKH. 3aTeM MbI C06HpaJIJ-JCb I1CnOJIb30B3Tb MHOrO'-IJIeH BMeCTO
nepBOHa'-IaJIbHOfi <jJYHKUHI1, H n09TOMY Ba)I(HO paCCMOTpeTb Bonpoc 0 TOM,
KaK CI1.'lbHO MorYT OT JJI1'-IaTbC5J <jJYHKUH5J H MHOrO'lJJeH B T04Kax, OT JIH'I-
HhlX OT Y3JIOBhlX (r.lLe OHl-J COBna.1l,aIOT B npeJleJIax 0lllH6KH oKpyrJIeHH5J).
B Ka'leCTBe npHMepa paCCMOTpl1M <jJYHKUHIO Y (x) = Ig x. B 8.4 H
8.5 Mbl H3XO.lLHJIH MHorO'lJIeH, anl1pOKCHMHpYlOlltHti <jJYHKUHIO. 1'1ccJIe.lLyeM
3Ha'-IeW15J pa3HoCTH Ig x - P (x) B Y3JIOBhIX TO'lKaX 11 Cpe.lLHHX TO'lKaX
Me)I(.lLY HHMH, rJle MO)j(HO O)j(HJlaTb, '1TO OHa .lLOBOJIbHO BeJIHKa.
x I Igx I P(XI I 19 x - P (X) x I Igx I P (X! jig X - PIX) I
1,0 I 0,0000 0,0000 0,0000 3,0 0,4771 0,477\ I 0,0000
1,5 I 0.1761 0,1707 0,0054 I 3,5 0,5441 0,5414 I 0.0027
2,0 0,.3010 0,3010 0,0000 I 4,0 0,6021 0.6021 0,0000
2,5 I 0,3979 0,4000 -0,002 J II I
I I
8.61
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Iwl1JJ,eHO, ecml 3aMen1Tb, 4TO pa3HOCTb paBHa HYJIIO BO Bcex Y3<10BhIX
TOlJKaX, H Hanl1CaTb
y (X) - P(x)=(x - XI)(X - x )... (X - Xn+l) K(x),
r.ll.e K (X) BbI6paHO COOTBeTCTBYIOUlI1M 06pa30M. LIll51 np0l13BOJIbHOrO x*
I1MeeM
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Tenepb paCCMOTpl1M (PYHI(IlI1K)
(j) (X) = y(x) - P(x) - (X - Xl) (X - X2).' .(x - xn+d K(x*).
Eum y (X) I1MeeT (n + 1)-10 npOH3BO.ll.HYIO, TO rpYHKlll110 (j) (X) MOjj{110
npoJ'u1q}rpepeHIU1pOBaTb n +] pa3. Tal( KaK P (x) - MHor04.'leH cre-
neHH n, a K (x*) - KOHCTaHTa, TO
<!>(n+I) (X) = y(n+I) (x) - (n + I)! K (x*).
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X2' ..., Xn+l)' CJJe.ll.OBaTeJJbHO, no TeopeMe 0 cpeAHeM SHaLIeHl111 (j)' (x)
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)j{allJ,eM Bce 3Ha4eHH5I (BKJII04a51 x*). Ilpo.ll.oJJ)j{a51 npHMeH5ITb :.HY Teo-
peMY, HaXO.ll.I1M, LITO $(n+l) (x) o6pamaeTC5J B HYJIb no KpatlHetl Mepe
n + 2 - k pa3 H 4TO cp(n+I) (x) 06pmllaeTC5J B HYJIb 110 KpatlHetl Mepe
O.ll.Ha)j{.lI.bI. TaKI1M 06pa30M, B mrrepBaJJe 3Ha4eHHtI X CY[lleC'fByeT Ta-
Koe :?, LITO
y(n+l) (x)=(n + I)! K(x*).
OTClO.lI.a MO)j{HO 110JJYlJHTb 3HalJeHHe KOHCTaHTbl K (x*). IloJ1cTaBI1B ero
B nepBOHalJa.libHOe Bblpa)j{eHHe, 110JJYlJaeM
( * ) _ P( * )+ (x*-xtJ (x*-x 2 )... (x*-X n +1)yW+11 (x)
y X - X (n + I)!
TaK KaK x* npOH3BOJJbHO, TO B leCTO x* MO)j{HO HanHcaTb X:
( ) _ P ( ) + (x - Xl) (x -x 2 )... (X- X M1 )yin+ll (x)
yx- X ( +1 1
n ).
(8.6- I)
VlCflOJJb3Y5I 9TO Bblpa)j{eHHe, 4T06bl Ollelll1Tb oWH6KY, c.n:eJlaHHYIO npl1
mrrepnOll5l11HH B Ta6JJHlle JJOrapl1rpMoB, nOJIY4V!M
(x-I)(x-2)(x-3)(x-4) i!VI ( - )
41 Y x.
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(')2) (_1/ 2 ) (_3/ 2 ) (_5/ 2 ) 6 _ 15
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112
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o 3H3'JeHHI-J X. J.13BeCTHO JIHlllb, 'JTO 1 X 4. B xy.n:llleM CJIY4ae
15
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CTBHTeJIbHOCTH )Ke OHa OKOJIO 0,0054, 'JTO nOKa3bIBaeT, C KaKot1 MaJleHb-
!wti TO'JHOCTblO MbI MO)KeM Ouel-!11Tb OlllH6KY, I1CnOJIb3Y5I nOJJyqeHHOe
BbJpa)KeHHe.
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B .n:eticTBI1TeJIbHOCTI1 .n:JI5I KaKOrO-JIH6o 3Ha'JeHH5I x MO)KeT cym.ecTBoBaTb
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CPYIIKUH5I OT X. 3TOT nOCJIe.n:Hl1ti cpaKT HJIJIIOCTpJ.1pyeTC5I npHMeHeHI1eM
TeOpeMbJ 0 CpeIJ,HeM 3Ha4eHI1H K CPYHKUHI1 y (x) = x (I - x) 11 BbJ60pOM
a = 0 B 06bJlJHoti cpopMe TeOpeMbJ 0 cpe.n:HeM 3HaqeHl1I1:
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x-a x
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1/3 .n:o 0, a Kor.n:3 x CTaHOBJ.1TC5I 60JIbme 2, nepenpbIrHBaeT 'Jepe3 HHTep-
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CPYHKUH5I OT x.
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I1MeeT .n:BOtiHOti HYJIb B Ka)K.n:oti Xi H MO)KHO nOJJO)KHTb
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npO.Q0J])Ka5J KaK H BbIme, HaXO.QHM
( ) _ P( )+ (X-Xl)2(X-X2)2",(X-Xn+,)2 ( n+2) ( - )
y X - X (2n + 2)! y x .
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<POpMa OCTaTOLJHOrO LJJJeHa, KOTOpOt! MbJ nOJIb30BaJIHCb, COOTBeT-
CTByeT OCTaTOLJHOMY LJJIeHY B p5I.n:e TetiJJopa, H3BeCTHOMY KaK OCTaTO'JHblti
lJJIeH B cpOpMe J1arpaH)]{a.
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f\JYHKQHR H rrepBbJe mi - I npOH3BOJ\Hble, paBeH
fill fIl2 fIl y(fIll (x)
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(3 a M e 'I a H H e: 9TO He caMbIH 06UJ,Hfi CJlYlJaH; B JII060H TO'lKe, rJJ:e J\aHa
rrpoH3BoJ\HaR, Mbl TaK}/{e Tpe6yeM, lJT06bl 6blJlH 3aJJ,aHbl H Bce rrpoH3BoJ\Hble
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aBTOpy, f\JOPMYJIb/ OCraTOlJHorO lJJleHa B 06w.eM cJlY'Iae orry6i1I1KOBaHO He
Db/liD.)
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113
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ITpl-lH5JTO, KaK 9TO CJJ:eJIaHO B npeJJ:bIJJ:YIlI.eM Tlapa-rpacpe, BbIpa)j{aTb
olliYJ6KY npl16JIll)j{eHllH MHOrOlJJIeHOM B TepMI1HaX COOTBeTcTBYIOIlI.etl npo-
113BOJJ:HOtl OT CPYHKJ,J,I1H, KOTopaH 6bIJI3 annpOKCI1MHpOBaHa. 06wIH0 JJ:Y-
MaIOT, lJTO «JJ:JIH 60JJbllil1HCTBa pa3YMHbIX CPYHKJ,J,HI1» TaKl-le BbIpa)j{eHHH
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nJIOCKOCTH, K3K tg x, log x, pall.HOH3JIbHaH CPYHKJ,J,HH OT X H T. JJ:., TO
pHJJ: TeHJIOpa JJ:OJI)I{eH HMeTb KOHelJHbII! pa.n.HYc CXOJJ:HMOCTI1 R, 3 9TO
B CBOIO OlJepe.n.b 03HaQaeT, lJTO AJIH 6eCKOHelJHOrO lJHCJI3 3HalJeHHI! n
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B K3lJeCTBe npHMepa,p3ccMOTpHM
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y - x n .
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B6JI1']311 HeKOTopbIX 3H3lJeHHH x, TeM He MeHee, Kor.n.a n CTaHOBHTCH
114
nAHHbIE C npOM3BOJJbHblMVI npOME}/{YTKAMVI
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npOfJ3BO.n:HbIX 60JJee BblCOKOrO nOp5J.n:Ka HMeJOT TeH.n:eHUHJO paCTH KaK n!.
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110 B .n:etlcTBHTeJlbHOCTI1 OKa3blBaeTCH, 4TO eCJJI1 cpYHKUH5J 5JBJl5JeTC5J ueJJo!!,
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cpYHKUI15J, KaK npaB!1JJO, Be.n:eT ce65J lIeCKOJlbKO XY)Ke.
B Ka4eeme 060CHOBaHH5J MHOr04J1eHHbIX npI16.111}/{eHHt! 4aCTO UHTH-
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rpy60 rOBOp5J, 4TO HenpepbIBHa5J cpYHKl(fJ5J Ha 3aMKHYToM oTpe3Ke MO}/{eT
6bITb paBHoMepHo npH6J1H}/{eHa MHOr04J1eHaMI1. O.n:HaKo MeTO.n: T04Horo
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BblX T04Kax. 06'b5lCH5IeTC5J 3TOT cpaKT, KOHe4HO, TeM, 4TO np0l13Bo.n:l1ble
paCTy-r CJ/I1WKOM 6blCTpO.
B Ka4eCTBe npl1Mepa paCCMOTpHM cpYHKUI1IO PmVIaHa, 3a.n:aHHYJO Ta6J1H-
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4TO npOHcxo.n:I1T H3-3a 04eBH.n:HO!! oco6eHHOCTH npH x = 1.
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no OTHOlUeHHJO K .n:pyrotl Ta6J1Hue TO!! }/{e caMO!! cpYHKUIIII, TO ee nep-
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Hl1lJeHbI. HanpHMep, eCJlu y = xex, TO y(ni (0) = n + 1.
8.71
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115
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BepOHTHO, 6bIJI B3HT CJIHWKOM MaJIeHbKHtI war, a Ta6JIHUa CJIHIllKOM
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k=1
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I
2 1,6449a
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3 1,20206 0,3231a
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4 1,08232 74a5 0,20023
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5 I,Oa69a 2580 3a35
-0,01959 -1520 -2403
6 1,01734 1060 9:32
-0,00899 -588 -596
7 1,00835 472 336
-0,00427 -252 -199
8 1,00408 220 137
-0,00207 -1\5 -75
9 1,00201 105 62
-0,00 \02 -5a -35
10 1,00099 52 27
-0,00050 -26 -16
II 1,00049 26 II
-0,00024 -15 -0
12 1,00025 11 11
-0,00013 -4 -11
13 1,000\2 7 0
-0,00006 -4 +3
14 1,00006 3 3
-0,00003 -I -4
15 1,00003 2 -I
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16 1,00002 0
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17 1,00001
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pa3HocTet! MorYT 6bITb BeCbMa rrOXO}j{ilMH Ha 3HayeHil51 rrpOH3BO.lJ,HOt!, il
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pa3Hocret! KaK HH.lJ,I1KaTOpa 3Ha4eHI1t! rrp0I13BOJl.HOt! I1HOr.lJ,a onaCHO; TaK,
Jl.JJ5I lI.eJIoro YI1CJJa X Bce 3HayeHil51 sin 'ltX paBHbl HYJIIO, TaK qro Ta5mIlI.a
pa3Hocret!, TaK}j{e cocr05lUJ,a51 H3 HYJIet!, 3aCTaBJI5IeT npeJl.nOJJo}j{ilTb HY-
JIeBYIO ollJH6KY B allnpOKCilMall.lIl1 sin 'ltX = 0, 4TO e.lJ,Ba JII1 BepHo. 3.lJ,eCh
ollJil6Ka OqeBI1.lJ,Ha, HO B HeKoTopbIX cJIyya5lx OHa MO}j{eT 6hlTb npony-
UJ,eHa.
YnfJamHeHHe 8.7-1. BW-/HCJlHTb n-IO npOH3BO,IJ.HYIO OT
--. ( - )
I + x 2 - 2i \ x - i x + i
(i = V - 1 ) .
8.8. 0 Bbloope Y3JIOBbIX TOqeK
Dp06JIeMa Toro, KaKl1e Y3JIOBbIe TOQKI1 BbI511paTb, B03HilKaeT BC5IKlltl
pa3, KOr.lJ,a npilXOLJ:I1TC5I I1HTeprrOJIHpOBaTb. Ecm1 6bI Mbl 11 MemI nOJIHYIO
CBo60Jl.Y Bbl6opa, TO Bbl6paJ!il 6bI HMeHHO TO 3HayeHile x, .LJ:JI5I KOToporo
c06HpaeMC5I mITepnOJIHpOBaTb, C.LJ:eJIaB, TaKI1M 06pa30M, 3a.lJ,ayy TpHBl1aJ1b-
HOt!. Ho QaCTO CJIYl.JaeTC5I, l.JTO MbI HMeeM 06llJ11PHYIO rpynny 3HayeHHt!
(Xi' Yi)' KOTopble MQ}j(eM ilCnOJIb30BaTb 11 HI1 O.LJ:HO 113 KOTOpblX He CO-
Bna.LJ:aeT C }j{eJWeMbIM 3HayeHI1eM X. ECJII1 0 BeJIl1l.Jl1He np0I13BO.LJ:Hott,
BCTpeQaIOUJ,et!C5I B Bblpa}j{eHI1I1 .LJ:JI5I OllJH6Kil, HI1Qero He H3BeCTHO, TO
MQ}j(HO iIHllJb BhI6paTb HallJH Y3JIOBbIe TO'lKI1 TaK, tJT06bI MHHHMH3ilpOBarb
K03cpcjmll.HeHT (CM. ypaBHeHHe (8.6-3) B ynpa}j{HeHHil 8.6-2)
(x - Xl)m 1 (x - X2)m 2 ... (x - xn)m n
rrepe.LJ: BbIpa)j(eHI1eM .LJ:JI5I rrpOH3BO.LJ:HOt!.
3.LJ:paBblt! CMblCJI 11 MHHI1MH3all.il51 3Toro MHO)j(IUeJI5I Tpe6YIOT O.LJ:HOro
il TorO )j(e: ilCnOJIb30BaHil51 ilHcpOpMall.ilil, 6JIH3KOt! K MeCTY, B KOTOpOM
HY)j(HO IlHTeprrOJJllpOBaTb. TIpH rrpilMeHeHilil HHTeprrOJI5IUilOHHott CPOPMYJIbl
HbIOToHa
y (x) = Y (Xl) + (X - Xl)([X2, Xl] + (X - X2) {[Xa, X2, Xl]})'"
B03HllKaeT )j(eJIaHlle .LJ:ep}j{aTh x 6JJI13KllM HMeHHO K Xl' 2:ho }j{eJIalllle
CJIeJl.yeT C.LJ:ep}j{HBaTb; BbIrO.LJ:Hee paBHoBeclle Bcex MHo)j(HTeJIett, BXOLJ:mHilX
B ollJil6KY, TaKHM 06pa30M, CJIe.LJ:yeT BbI6I1paTh 3HaQeHil51 Xi crpynnHpo-
BaHHbIMil, eCJIil B03MO)j(HO, no 06e CTOpOHbl OT Tpe6yeMoro 3HaQeHil5I X.
D.1]
<l>OPMYJlA HblOTOHA JJ:Jl5I I1HTEPDOJJVlPOBAHVl5I
117
V!HorJJ:a B03HI1KaeT JJ:pyroti BOnpOC (OTBeT Ha KOTOpblti He MOlKer
6hITb .ll:aH 3.11:eCb npoCTO). J1onYCTHM, 4TO Mbl XOTHM HHTep110JIHpOBarb
3Ha4eHH5J Ha BceM HHTepBaJIe. f,Qe HY)J(HO Bbl6paTb Y3JIOBble TO'-1KH, 4T06bl
MI1HI1MI13I1pOBan M8KCI1M8ilbHYIO OWI16KY? OTBeT H8 9TOT Borrpoc .lI.8eTCH
TeOpI1eti MHOrO'-1JIeHOB Ye6blweBa, H3JIO)J(eHHoti B r JJ. 19.
[J1ASA 9
I1HTEPI10JI5IIlH5I MHOr04J1EHAMH. PABHOOTCT05lIII.HE Y3JIbI
9.1. <I>opMYJla HbloToHa Jl,J1H HHTepnOJUIpOBaHHH
04eHb 4aCTO I1MeIOw.a5JCR fmcpopMali.I1H 0 <PYHKli.I1I1 (B BI11I.e 3H84eHI1tt
B Y3JIOBblX TOLJKaX) 3a.ll:aHa Ha MHO)J(eCTBe paBHOOTCT05JIl\HX 3Ha4eHHt! x.
B 9TOM cJIY4ae60JIbWa5J 4aCTb CPOPMYJI, BbI4HCJIeHHt!, KaK, Bnp04eM, H
3aTparHBaeMblX H.II:et!, 3aMeTHO ynpOw,aeTC5J.
J1JI51 pa3HocTet! npH paBHOOTCT05JW,HX 3Ha4eHH5JX 06bl4HO npHMeH5JlOT
0603Ha4eHH5J l1epBoti 4aCTH:
n = Yn+l - Yn'
3TO - 06bl4Hoe .ll:JJ5J HC4HC.JIeHH5J pa3HocTet! o603Ha4eHHe, HO Ax = h
8.11:eCb CPHKcHpoBaHo:
AXn=Xn+l -x n =h.
Mbl HMeeM TaK)J(e
A2Yn = Yn+2 - 2Yn+l + Yn
H T. .11:.
3TH pa3HOCTH COOTBeTCTBYIOT pa3.n:e.'1eHHblM
[ X X ] = Y2 - Y, = Yl
2, 1 x 2 - X, h'
pa3HOCT5JM
[ ] _ lx" x 2 1- [x 2 , xd
X 3 ' Xo , X l -
. - x a -x 1
Y2 Yl
h--h _ 2Yl
2h - 2!h 2
H Bo06w.e
n-ly
[XI' X2' ..., X n ]= (n_l)lh n - l .
Pa3HOCTH rrpH6.rrI1)J(8IOT np0I13B01I.Hble B T04K8X, JIe)J(aw.I1X rrocepeJJ:I1He
Me)J(.II:Y HcnOJIb3yeMblMH Y3.JIaMU:
dy (x + )
Aylrvh dx '
A 2 h 2 d 2 y (x+ h)
Yl rv dx 2 ,
dny ( X + )
Any! rv h n
dx n
j 18
V1HTEPIlOJJ5!IU1S1 MHOr04JJEHAMV1. P.J\BHOOTCT0S1l1H1E Y3JIbi
[r JI. 9
3TfJ COOTIiOWeHI1H AalOT B03MO)l{HOCTb annpOKCI1Ml1pOBaTb BbJpa)l{eHHII
R L1.l1cpcpepeHlI.l1aJIbHOM I1C<JI1CJIeHI1I1 KOHe<JHopa3HOCTHbIMH.
<l>opMYJIa HbIOTOHa B 9TI1X HOBbIX 0603HalJeHI1HX AJIH paBHOOTCTOH-
IIU1X Y3JIOBbIX TO<JeI< HMeeT BI1A (CM. ypaBHelme (1.5-4) H 8.4)
y= Yo +(x - xo) Il o +(x - xo)(x - Xo - h) 2h{:. +
f!8yo
+ (x - xo) (x - Xo - h) (x - Xo - 2h) 3! h B + '"
IloJJarall Xo = 0, nOJIY<JHM
y=yo+X +X(X-h) o +x(X- h){x- 2h) + '" ,
H eCJII1 AaJIbIlJe nOJIO)l{l1Tb h = I, TO nOJIY4aeTCII (cp. C ypaBHe-
Hl1eM (1.5-4)):
1l 2 y 11 8 Y
Y=Yo+xAYo+x(x-l) T +x(x-1)(x- 2) -F+ ...
YnpamHeHHe 9.1-1. HanH/mlTe OCraTO'lHbIH '1J1eH 4JOPMYJlbl HblOToHa, r.Qe
'IIH4JopMau,!1!I 3a,IJ.aHa B TO'lKaX -n, - (n-I), '" , 0, I, 2, ... , n H nCnOJlb-
3Y/OTCR pa3HocTH 2n-ro nOp!I.QKa. Ou,eHI1Te MaKCI1MYM K094J4JHQ!1eHTa I1PH npo-
H3BO.QHOH.
9.2. HHTepnOJIHpOBaHHe B Tao.rnm,ax
OAHI1M 113 r JJaBHblX npHMeHeHHt1 I1HTepnOJISIlI.I10HHbIX CPOPMY JIC paBHblM
marOM IIBJIHeTC5I I1X npl1MeHeHl1e Ami HHTepno.7Il1pOBaHHH B ra6JJI1lI.ax
C paBHoOTcTOHUJ.I1MH apryMeHTaMI1. Tl1nH<JHbIM npl1MepOM IIBJJHeTCH HHTe-
rpaJI OWH60I<, ra6JIHlI.a pa3HocTet1 I<OTOpOro AaHa B Ta6JIHlI.e 9.2-1. Pa3-
HOCTH BeAYT ce651 XopOWO B TOM CMbICJJe, <JTO OHI1 CTpeM5ITCH I< HYJJIO,
OCTaBa5lCb .lI.OBOJIbHO r JIa.ll.I<I1MI1 B cToJJ6l1.e TpeTbl1X pa3Hocret1. Ha411Hall
C <JeTBepTblX pa3HOCTet1 11 oco6eHHO B n5lTblX pa3HOCT5IX npeo6JIa.ll.aeT Y)I{e
mYM oI<pyr JleHH5I, IT09TOMY AJIH 9T0t1 Ta6mmbl HeJIb351 Ha.ll.eHTbC5I, npl1-
MeH5IH I<aI<YIO-HH6Y.ll.b HHTepnoJI5IlI.l1oHHYIO CPOPMYJIY, H.lI.TI1 AaJIee 4eTBepTblx
pa3HocTeti,
B np!1HlI.l1ne, !11HepnOJJHlI.l10HHbltJ MHOr04JIeH MO)l{eT 6bITb nOJIY4eH no
CPOPMY Jle HblOToHa .lI.JIH HHTepUOJII1pOBaHI1H 11, He C4I1Ta51 9cpcpeKTOB
oKpyr JIeHI151, OTBeT 6YAeT T04HO TaKI1M )I{e, I<aI<ot1 nOJIY411JICH npl1 I1CnOJJb-
30BaHI1!1 I1HTepnOJI5IlI.I10HHOrO MHor04.7IeHa, nOJIytcIeHHoro .lI.PYfHM cnoco-
60M, !1CnOJJb3YIOIlI.I1M Te )l{e caMble Y3JJOBble T04KI1. TeM He MeHee Ha
npaKTI1Ke npHMeHHeTCH MHoro Apyrl1x CPOPMY JI, H .lI.JliI o6UJ.ero 06pa30-
BaHI1H, 4T06bl 411TaTeJIb Mor nOHHTb MeTOAbl, rrpI1BO.ll.Hu,HeCH B npeAI1CJIO-
BI1HX I< 60JIbWliHCTBY ra6JIl1l1., Mbl pa3.gepeM HeKOTopble 113 HI-IX, HaI160JIee
110Uy JIHpHbIe.
9.31
POM60BI-1llHMI 11I-1AfPAMMA
119
T a 6 JI \I U a 9.2-1
[ 0 2
HHTerpaJi OWHOOK V 1t e - -2 dB
o
t I f (1\ I A I 6,2 I A3 I A4 I a'i
I
! I I
I 0,00 0,0000 I
987 I
0,25 0,0987 - 59
928 -50
0,50 0,19t5 -109 +19
819 -31 +4
0,75 0,2734 -140 +23
[ 679 -8 -10
1,00 0,3413 --148 +13
5at +5 +4
1,25 0,3944 -143 +17
388 +22 -11
1,50 0,43: 2 -121 +6
267 +28 -10
1,75 0,4599 - 93 -4
174 +24 I +4
2,00 0,4773 - 69 0
105 +24 -7
2,25 0,4878 - 45 -7 I
60 +17 +3
2,50 0,4938 - 28 -4
32 I +13 -4
2,75 0,4970 - 15 - 8
17 +5 +10
3,00 0,4987 - 10 +2
7 + 7 -9
3,25 0,4994 - 3 - 7
4 0 + 9
3,50 0,4998 - 3 + 2
1 +2
3,75 0,4999 I - 1
1 I
4,00 0,5000
9.3. POM6oBHllHafl nHarpaMMa
PmI60BH1lHa51 1l1'!3rpaMMa 51BJI51erC51 npHcnoc06JIeHI1eM nJI51 nOKa3aTeJIb-
CTHa roro, '-ITO orporvmoe '-IHCJIO <jJOPMY JI, KOTopbIe Ka)KYTCH pa3JJH'-IHhIrvIH,
B netJCTBI1TeJIbHOCTH Bce onHHaKOBhI.
Mhl onpeneJII1JJH 6HHorvlI1aJlbHbIe K03<jJ<jJHl.l.HeHTbI B 1.3:
C(u + k, n)= (u +k) (u+k- I) (u +k -2) ... (u +k - n+ 1)
nl
120
I1HTEPDOJ]5!lU15! MHO rol.fJJEHAMI1. P ABHOOTCT05!IlJ,I1E Y3JIbl
{fJJ. g
YHCJlI-ITe.% H 3HaMeHaTeJlb 9TOt! CPOPMYJlbl co.n:ep)J{aT no n COMHO)J{HTeJJet!.
f{oaCPCPI1lJ,HeHT C (11 + k, n), eCJlH ero paCCMaTpHBaTb KaK CPYHKUHIO OT 11,
eCTb MHOrO'lJleH CTeneHH n. Ha pHC. 9.3-1 11306pa)J{eHa pOM60BH.n:Ha51
1I:HarpaMMa. JIHHH5I, Ha'lI1Ha5lCb B TO'lKe Ha JleBOM KpalO H cJle.n:Y5I no
1I(-3)-:::i.(U+3,I) >iJ?;(-4) (LL+4,J) => IJi;(-J}
J iJ!f(-J) e-(U+J,6"J iJ3!J(-4J C(u+4,4J
. /J(-ZJ-:::i.(U+2J) >iJ?!(-3J-:::!..(lL t J,J) >iJjr(-4J
I ----= iJ!f(-2) C(u't2,Z) i!/!/(-3) C{ u+3,4)
!J(-/) (U+l,!) >iJl;(-ZJ0.(U+2,J) >iJ1{-J)
I -1!J{-/) C(u+I,2) J/;{-Z) C(u+2,4J
!J(O) (Ll,I) >iJ}t(-I)-:::C({JtI,J) iJ4!J(-2)
1 iJ!J(O) C{u,2) ...........iJ (-1) (U+l,4)
!!(J) (U-!,!) >iJ1;(OJc...C(u,J) '5-1 4 11(-1)
I J!/ (I) ({J-I,Z) IJ /; (0 J "-... C( {J,4)
!J(2) u-2,!) iJ (/J 8({J- J) iJ/;(O}
I iJ!J(2} C(u-2,2) ...........L1 (!} (Ll- 4)
!J (J) Ll-3, 1) > iJ ( 2) 0. ( {J -2,3) iJy (I)
1 A!I(3) (u-3,2) :1y(2) <!.(IJ-2,4)
,y(4)
PIIC. 9.3-1. POM60BHD,HaH .lJ.!larpaMMa.
'KaKotl-JlH60 .n:opo)J{Ke nonepeK CTpaHHUbl, onpe.n:eJl5leT IIHTepnOJl5lUHoHHYIO
{POPMY JlY, eCJlH npHMeH5ITb CJJe.n:YIOIlJ,He npaBHJla:
1 a. .QJl5I mara CJleBa HanpaBo - CKJla.n:blBaTb.
16. .QJI5I mara cnpaBa HaJIeBO - BbllJl-ITaTb.
2a. Ecm1 TaHreHC yr Jla HaKJJOHa mara nOJJO)J{IUeJlbHbltl, TO HCnOJIb30-
"BaTb npOH3Be.n:eHHe pa3.n:eJIeHHoti pa3HOCTH Ha K09cpcpHUHeHT, Haxo.n:SIUJ,I1t1C5I
no.n: Hett
26. ECJJH TaHreHC yr.rm HaKJlOHa mara OTpHuaTeJIbHbltl, TO HCnOJJb30-
BaTb npOH3Be.n:eHHe pa3.n:eJIeHHoti pa3HOCTH Ha K09cpcpHUHeHT, HaXOll -
UJ,HtlC5I Ha}J. Hetl.
3a. ECJJH mar ropH30HTaJIbHblti H npoxo.n:HT lJepe3 pa3HOCTb, 6pan
np0l13Be.n:ellHe pa3HOCTH Ha cpe.n:Hee apHcpMeTHlJeCKOe 113 KoacpcpHll.HeHTOB,
pacnOJlO)J{eHHbIX Bblille H HI1)J{e ee.
36. ECJJH mar ropH30maJlbHbiti H npoxo.n:HT lJepe3 K09cpcpHuHeHT,
6paTb npmJ3Be.n:eHHe K09cpcpHll.HeHTa Ha cpe.n:Hee apHcpMeTH'!eCKOe H3 pa3-
J!OcTeiJ, paCllOJlo)J{eHHblx Bblme 11 H(1)Ke ero.
9.31
POMBOBI1JJ:HMI lU1AfPAMMA
12 !
B KalfeCTBe npHMepa npHMeHeHHH npaBI1JJ I a H 26 paccMoTpHM nYTb,
HalfHHaI0ill.l1fiCH C Y (0) H I1.n:Yill.Hti HanpaBO BHI13.. Mbl nOJlYlfaeM cpop-
MYJlY
Y (ll) = Y (0) + C(ll, 1) Y (0) + C(ll, 2) 2y( 0) + C(ll, 3) 3y (0) +...=
Y (O)+Il Y(O)+ U(U;I) 2y(0)+ U(U- (U-2) 3y(0)+ ...,
KOTopaH HBJIHeTCH CPOPMYJJOti HbI0TOHa. ECJlH 6b1 Mbl nOlliJlH HanpaBO
BBepx, Mbl npl1MeHHJII1 6bI npaBI1J1a 1 a 11 2a 11 nOJIYlfl-lJJI1 6bI BTOPYI0 cpop-
MY JlY HbI0TOHa
Y(ll)=y(O)+C(ll, l) y(-l)+C(ll+ 1, 2) 2y(-2)+
+C(1l+2, 3) 3y(_ 3)+.. .=y(O)+ll y(-l)+
+ (U-1;I)U 2y(_2) + (u+2) +I)U 3y(_3)+ ... (9.3-1)
4T06bl nOJlYlfl1Tb CPOPMYJlY CTl1pJJl1llra, Half HeM C y (0) H notl.n:eM fO-
pl130HTaJlbHO HanpaBO, npl1MeHHH npaBI1J1a 3a H 36:
( ) _ (0)+ AYo+AY_I + C(U+l, 2)+C(u, 2) 2 +
Y II - Y 11 2 2 Y -1
A3 y + A3 y Au + Au
+C(Il+1,3) -2 2 -1 +...=Yo+ll ,-,, 0 2 '-" -1 +
+ u 2 2 + U(U 2 -1) A3 Y _ 2 + A3 Y _ 1 +
2 y -1 31 2 .. .
(9.3-2)
ECJII1 MbI Half HeM nOcpe.n:I1He Me)l{.n:y y (0) 11 Y (1), TO rrOJlYlfl1M CPOPMY JIY
5eCCeJIH
( ) - 1 YO+YI + C(U, 1)+C(a-l,l) +C( 2 ) A2 Y _ 1 +A2yo +
y 11 - 2 2 :Yo 11, 2 . . .
_ YO+YI +( _ ) + u(u-I) . A2Y_I+A2yO + ( 9.3-3 )
.. . - 2 11 2:Yo 2 2 .. .
ECJlI1 MbI notl.n:eM 3l-1r3arOM, cooTBeTcTBYI0ill.HM 06pa30M, TO MO)l{HO
noJlY'-!HTb I1HTeprrOJIHJJ.I10HHYI0 CPOPMY JIY faycca
) + A + u(u-I) A2 )+ U(U 2 -1) A3 )+
y (11 = Yo llu. Yo 2 u. y ( - 1 31 u. Y ( - 1 . ..
(9.3-4)
MO)l{HO Bbl6HpaTb Bce BH.n:bl nYTetl 11 Ka)l{.n:bltl 6y.n:eT .n:aBaTb HeKOTO-
PYI0 CPOPMYJlY. HaM HY)I{HO TOJlbKO .n:oKa3aTb, lfTO Bce 9TO - .n:eticTBI1-
TeJIbHO I1HTeprrOJlHJJ.I10HHble CPOPMYJlbl. LI.JlH .n:OKa3aTeJJbCTBa nOTpe6yeTC5!
rrOKa3aTb CJle.n:YI0ill.ee: .
1. 4TO nOJIy'-!aeTCH no KpaIlHetl Mepe o.n:Ha npaBI1J1bHaH CPOPMY JJa;
TaK KaK 1-13 .n:l1arpaMMbI MbI HalliJlI1 I1HTeprrOJlHJJ.HOHHYI0 CPOPMYJlY HbI0TOHa,
TO 9TOT mar .n:OKa3aTeJlbCTBa Y)I{e c.n:eJlaH.
2. 4TO BKJla.n: no JII060MY 3aMKHYToMY rrYTl1 paBeH HYJII0 11, cJle.n:o-
BaTeJIbHO, '-ITO MbI MO)KeM rrpe06pa30BaTb O.n:I1H rrYTb B JlI060Il .n:Pyrott
122
I1HTEPnOJUIUI151 MHOf04JJEHAMI1. PABHOOTCT05lilll1E Y3JJbl
[fJ1. 9
3. YTO eCJII1 ,D,Be <POPMYJlbI KOH'-JaIOTC5I B O,lJ,HOM 11 TOM )l{e MeCTe, TO
DHH O,D,I1HaKOBbl. 2:ho Heo6xo,D,HMO .n:OKa3bIBaTb, TaK KaK TOllKH BXOn:a
B pOM60BI1.l1:HYIO tJ:l1arpaMMY He 065I33Hbl 6b1Tb ommaKOBblMH .lI:.1f! p33HhlX
<POPMYJI.
C(u-rS, -fl-f) Jny(S_!) C(u-s-l, 17+/)
/J-/ flf
L1 !I(S )-(fj-C( I1-S, 17 4/l+}r.s_1)
C(tI-.s-t Jt(S) -$' /14)
-.r
2 3 S
y(s') C(U/ 42 !/(,J-1 j
.¥(")
!I($+/) (/hS-!,1)
PHC. 9.3-2.
PHC. 9.3-3.
YTo6bl ,D,OKa3aTb BTopoe YTBep)l{n:eHl1e, B03bMeM pOM6 (pl1c. 9.3-2):
DYTb 1: C(ll-S, n) ny(s-l)+
+ C (ll - s+ I, n + 1) n+ly(s_I).
DYTb 2: C(ll-S, n) nY(S)+C(II-S, n+l) n+ly(s__I).
DYIb 3: C(u-s, n) nY(S-lJ+ ny(s) +
+ C(u-s+l, n+ld+C(U-S, n+l) n+ly(S_ 1).
BCTIOMI1Ha5I, liTO CTIpaBa HaJJeBO TIOJIY4aIOTC5J orpm.J.areJlbHble 4.lJeHbI
(npaBHJIO 16), MbJ .n:OJI)!{Hhl .rIl1llJh nOKa3aTh, 4TO Bce TPH TIYTI1 O.ll:HHa-
KOBhl. Ho
I1YTh I -fIYTh 2=C(Il-S, n)l nY(8- l)- ny(S)]+
+ [C(ll- S+ I, n+ 1) - C(ll- S, n+ I)] n+ly(S - 1)=
=C(ll-S, n)[_ n+1y(s_I)]+
+C( ) u-s+I-(u-s-n) ,\n+1 ( _ 1) -
II - S, n n + I U Y S -
=C(ll-S, n)[ - n+ly(s_l)+ n+ly(s_I)]=O.
KpOMe TOro, TIYTh 3 paBeH cpen:HeMY apH<pMeTH'leCKOMY TIYTet! J 11 2.
TaKHM o6pa30M, BTopon llJar .n:OKa3areJIbCTBa 3aKOHlIeH.
YTo6bl TIpOBeCTH TpeTHn llJar n:OKa3aTe.%CTBa, HCTIO.lJb3yeM pl1c. 9.3-3.
DYTb l=y(s+I)+C(Il-S-I, 1) Y(S)-C(ll-S, 1) y(s)+S.
I1YIb 2= y(S+1 +Y(S) + C(U-S-l' )l+fJE -S, I) Ay(s)-
- <--(11'- S, I)Ay(S)+S.
I1YTb 3=y(s)-+ S,
9.41
3AME4AHI15/ K BbIBE.1lEHHbIM </IOPMYJIAM
In
r.ll.e S eCTb OCTaBIlla5lC5! l!aCTI; cpOPMYJIbl. Tenepb, HCnOJlb3Y5I paBeHCTBO
yes + 1)- y(s)=Ay(s),
nOJIY'IaeM
DYTb 1= Y (s) + y (s) + (ll - s) y (s) - y (s)-
-(ll-s) y(s)+S=y(s)+S=DYTb ;)
AHaJIOrWIHO nYTb 2 MO)l{HO CBeCTH K nYTI1 3. TaKHM 06pa30M, MbJ
MO)l{eM C.lI.eJIaTb BbIBO.ll., 'ITO HHTepn0JI5IU.110HHaSl <popMYJIa 3aBHCI1T 01
OKOH'IaTeJlbHbIX 3Ha'leHI1tl 11 He 3aBI1CI1T OT nYTH, HCnOJIb30BamIOrO .lI..1S1
.lI.OCTI1)1{eHI151 I1X.
YnpaiKHeHHe 9.3-1. IlpoBepHTb 4JOpMy.lbl (9.3-2) - (9.3-4).
9.4. 3aMeqaHHH K BbIBe,aeHHbIM If>opMYJIaM
J{aK cpaBHHTb 3fI1 pa3HbIe <POPMYJIbI Me)l{.QY c060tl H C <POPMYJIOti
JJarpaH)I{a, HatJ.n:eHHot1 paHbille? 3Ha'leHI1e, nOJIY'IaeMOe IJpI1 I1HTepnOJJ5!"
U.HI1, 3aBHCHT OT I1CnOJJb3yeMoro MHOrO'lJIeHa, a MHOrOlJJIeH 3aBHCHT OT
Bbl6paHHbJX Y3JIOBbIX TOlleK. OCTaTO'lHbItl 'IJIeH HMeeT <POPMY (CM. 8.6)
(x - Xl) (X - x 2 ) ... (X - xn_d y'n+lI (x)
(n+ 1)1
J{03<p<pHIJ.HeHT npH npoH3BoJJ:HOtl MI1HHMI1311pyeTcSI, eCJII1 x 6epeTcSI B ce-
pe.ll.Ime I1HTepBaJIa Y3JIOBbiX TO'leK. D03TOMY, KOr.ll.a I1HTepnOJIl1pyeMaH
TO'lKa HaXO.ll.l1TCSI B Cepe.ll.l1He Hl-ITepBaJIa, 06bI'IHO I1CnOJIb3YlOT 'IeTIme
'II1CJIO Y3JIOBbIX TO'leK, a Kor.n.a OHa JIe)l{HT OKOJIO Y3JIOBOtl TO'lKH -
He'leTHOe,
<POp MYJIbI, nOJIYlJaeMbIe 113 pOM6oBH.n.HOtl .n.HarpaMMbI, co.n.ep)l{aT
B 5iBHOM BH.n.e pa3HOCTI1, H 3T11 pa3HOCTI1 .n.alOT HeKOTopoe npe.n.CTaBJIe-
HHe 0 TO'lHOCTH, HO 3T11 cpOPMYJIbI Tpe6YIOT 60JIbIlle apl1cpMeTl1'1eCKI1X
.lI.etlcTBI1t!, 'IeM MeTO.ll. HHTepnOJISlUl111 J1arpaH)I{a. 3aTo MeTO.n. J1arpaH)I{(j
Henocpe.n.CTBeHHO He I1CIJOJIb3yeT pa3HOCTH, TaK 'ITO HeT 110Ka3aTeJISI TO'l-
HOCTI1. 3TO - O.ll.I1H H3 napa.n.oKcoB 'IHCJIeHHOrO aHaJJH3a: HcnOJIb30BaHI1e
pa3HoC'rert .n.aeT HeKoTopbIe YKa3aHHSI Ha TO'lHOCTb npl1MeHeHHoro MeTo.n.a,
B TO BpeMSI KaK MeTO.n. J1arpaH)I{a MHHI1MH311pyeT KOJIHlJeCTBO apHC!JMeTH-
'IeCKI1X .n.ei1cTBHi1. B pe3YJIbTaTe pa3HOCTHbie MeTo.n.bl 06bI'IHO npI1Me-
HSllOTCSI npH I1CCJIe.ll.OBaHI151x, a MeTO.n. JlarpaH)I{a - B 06bI'IH0i1 nOBCe.ll.HeB-
HOt! pa60Te.
J{03<pcpHUHeHTbI J1arpaH)I{a .n.JISI paBHOOTCT05llUHX Y3JIOB IlJHpOKO npo-
Ta6YJIVlpOBaHbI [32J. HeKOTopbIe K03cpcp11lJ.HeHTbl lI.JI5I Ky6H'IeCKot! Hlnep-
nOJISlUl111 .n.aHbl B Ta6JIHue 9.4-1. 3aMeTHM, 'ITO CHMMeTpH51 IJ03BOJISleT HaM
'IHTaTb Ta6JIHIJ.Y cBepxy, HCnOJIb3Y51 3af111CH CJIeBa, HJIH CHH3Y, I1CnOJlb3Y51
3aIlHC11 cnpaBa.
124
I1HTEPfl0J151UI151 MHO r04J1EHAMI1. P ABHOOTCT05lilll1E Y3J1bl
[fJJ. 9
T a 6 JI II U, a 9.4-1
KYOH'IeCKaSi HHTepnOJlSIlJ.HSI J1arpaH}J{a
{(x) = A_I (x)f(-I)+Ao (x)f(O)+A I (x)f(l)+A2 (x)f(2)
x
A-I
Ao
Ai
A 2
. . .
0 0 1 0 0 1,0
0,1 -0,0285 0,9405 0,1045 -0,0165 0,9
0,2 -0,0480 0,8640 0,2160 -0,0320 0,8
I o,a -0,0595 0,7735 0,aa15 -0,0455 0,7
0,4 -0,0640 0,6720 0,4480 -0,0560 0,6
0,5 -0,0625 0,5625 0,5625 -0,0625 0,5
A 2 Al Ao A_I X
Ynpa}J{HeHHSI. 9.4-1. I-1cnoJlb3YR I1HTepnOJlRl.lHOHHble K034J4JIIl.lHeHTb! JJa-
rpaH)/{a, BblqHC,111Tb HHTerpaJI OIllH60K B TO'IKe 1,40 no Ta6J1Hl.le 9.2-1.
9.4-2. I-1cnoJlb3YR Ta6J1Hl.lY 9.4-1, BblqHCJlI1Tb 3Ha'leHl1e B Ta6.1Hu,e JJ,JlR x = 0,35.
9.5. CMemaHHbIe HHTepnOJIHlI,HOHHbIe I}>OPMYJIbl
I1HOr.lI.a 6bIBalOT rr0J1e3Hbl HeKOTopble .lI.Pyrne <P0PMYJ1bl, He rrOJ1Y'lalO-
Ul,HeC5I HerrOCpe.lI.CTBeHHO H3 pOM60BH.lI.Hott .lI.HarpaMMbI. OHH 06blqHO OnI-I-
palOTC5I Ha TO, 'ITO J1106a51 pa3HOCTb MO)IWT 6bITb HCKJ1IO'IeHa rrpHl'vleHe-
l!HeM <POPMY JIb!
Any (s + 1) - Any (s) = An+ly (s).
UeHott 9Toro HCKJlIO'IeHH5I 5IBJl5IeTC5I BKJlIO'IemIe .lI.OrrOJlHHTeJlbHott pa3HO-
eTH HeKOToporo .lI.pyroro rlOp5!.lI.Ka.
HarrpHMep, B HHTeprrOJl5!lJ.HOHHott <popMYJle 6ecceJl51 MO)l{HO I1CKJlIO'IHTb
Bce pa3HOCTH He'leTHOro rrOp5!.lI.Ka; 9TO .lI.aCT <POPMYJJY
Y(Il)=(1-Il)y(0)+IlY(l)+ (2 -U)(I;U)(-U) A2y(-1)+
+ (u+ 1) (u-I) A2 y (0)+
KOTOpa51 rI3BeCTHa KaK <popMYJJa BBepeTra H O'leHb nonYJl5!pHa, TaK KaK
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'-IeTHOrO rrOp5l.lI.Ka.
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pa3HOCTH JII06oro nop5!.lI.Ka (UeHOIO .lI.06aBJ1eHH5I pa3HocTeti rex rrOp5l11,KOB,
KOTopbIe MbJ OCTaBJI5IeM B <popMYJIe). MbJ MOrJIH 61,1, eCJIH 6bI XOTeJJH,
BCKJlIO'iHTb, CKa)l{eM, pa3HOCTH KaK BToporo, TaK H rpeTbero nOp5!.lI.KOB,
I1CrrO,Tlb3Y5I JIHlUb <PYHKll.HIO, pa3HOCTI1 rrepBoro nopS!.lJ:Ka, a TaK)I{e '-IerBep-
TbIe 11 60JIee BbICOKl1e pa3HOCTH. I1poBe.lI.eHHe 9TOti n.lI.el1 .lI.0 KOHua npH-
Be11,eT K <popMYJle JIarpalI)j{a, KOTOpa5! Boo6w.e He HCrrOJIb3yeT pa3HocTett,
10.1]
BBELlEHVlE
125
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TaHHH Ta6JIHIJ:, H .n:eJJaeTC5J 9TO 3a ClJeT .n:onOJIHHTeJJbHot! pa60Tbl co CTO-
POHbl nOTpe6HTeJI5J. Ha.n:JIe)J{alll.ee paBHOBeCl1e 3aBHCHT OT 06CTOneJIbCTB
11 He MO)J{eT 6bITb .n:OCTHrHYTO pa3 HaBcer.n:a.
BTOpOti npHeM, KOTOpblM lJaCTO nOJIb3yIOTC5J, Ha3bIBaeTC5J «oT6paCbl-
BaHHeM». 3.n:eCb I1CrrOJIb3yeTC5J TOT cpaKT, lJTO K09cpcpHliHeHTbI lJOCJIe.n:OBa-
TeJlbHbIX pa3HOCTet! B pa3J1HlJHbIX HHTeprrOJl5JIJ:HOHHbIX cpopMYJlaX, KaK,
Harrpl1Mep, 3BepeTTa, 6blBaIOT npOnOpIJ:110HaJlbHb1MH .n:pyr .n:pyry, 11 rro-
9TOMY, eCJlI1 BbICOKHe pa3HOCTH, B35JTble C COOTBeTCTBYIOIlI.HMI1 K09cpcpH-
IJ:l1eHTaMH, npH nelJaTaHI1H Ta6J111IJ: 06"be.n:HHHTb C HH3KHMH, TO 60Jlbrn85J lJaCTb
9cpcpeKTa rrpl1MeHeHH5J pa3HOCTet! BWCOKoro rrop5J.n:Ka B HHTepnOJlHpOBaHHH
8BTOMaTHlJeCKI1 .n:OCTHraeTC5J HCIIOJIb30BaHHeM CPOPMYJlbI HI13KOrO IIop5J.n:Ka.
TaK, B cpopMYJle 5eCCeJI51 OTHorneHHe K09cpcpl1IJ:HeHTa IIPH /14 K K09cp-
qmIJ:l1eHTY IIPH /12 paBHO
SIV _ (u+ I) (u -2)
[if! - 12 (0 ",;; II ",;; 1),
3HalJeHHe 9Toro Bblpa)J{eHH5J Jle)J{HT Me)J{.n:y _1/6 11 - 3/ 16 , CJJe.n:OBa-
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6eIJ: MOJJ,I1cpI1IJ:HpOBaHHbIX BTOpblX pa3HOCTet!, Mbl .n:onycTHM o ll1H 6KY
(BIV - CBIl) /14. 4HCJlO. C lJaCTO 6epYT paBHblM - 0,184, KaK cpe.n:Hee
J / 3 /
Me)J{.n:y - 6 H - 16'
Mbl OII5JTb HaIIOMHHaeM lJHTaTeJJIO, '1TO C03.n:aHHe Ta6J111IJ: - 9TO 60Jlb-
woe HCKyCCTBO, H OTCbIJlaeM '1HTaTeJI5J K <POKCY [10] HJlH K KJJaCCH'Ie-
CK11M PYKoBoJJ.CTBaM KonaJla [20] 11 XHJlb.n:e6paHJJ.a [14].
Ynpa>KHeHHSI
9.5-1. J-laHT11 4JOPMYJlY, KOTopaR HcnoJlb3yeT JlHIIIb 4JYHKU,HlO 11 TpeTblO II
lIIeCTble pa3HocTH, HCKJlIO'1eHHeM JJ,PYfHX H3 HHTeprrOJli/QHOHHOH 4JOPMYJlbl 6ecceJli/.
9.5-2. To JKe, '1TO B ynpaJKHeHHH 9.5-1, HO HcnoJlb3YR 4JOPMYJlY CTIIpJlHfJra.
rJJABA 10
EJU1HbIH METOJI. HAXOJKJI.EHHSI HHTEPllOJISlUHOHHbIX
<l>OPMYJI
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1. KalCoBbl Y31loBbie mO'IlCu?
2. KalCoB xllace cjJYHlC11,UU?
3. KalCOB lCpllmepllu e02llaeufl?
4. KalCoBa mO'lHOcmb?
126 ED.f1Hblti METO!l HAX O)!{D.EHf151 f1HTEPnOJJ51UI10HHbIX <pOP MY JJ I fJJ. 10
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OH He TOJIbKO D.eJIaeT 60JIee JlerKI-IM 113YlJelme eD.HHOrO pa3HOCTHoro
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npeD.Jl3raeMbIt! MeTOD. I1MeeT HeCKOJlbKO D.OCTOHHCTB. Bo-nepBbIX, OH
5IBJl5IeTC5I MOU!,HbIM cnlMynoM D.Jl5I D.3JIhHet!1IlI1X HCCJleD.OB3HHt!. BO-BTOPbIX,
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10.21
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l1a OTpe3Ke 0 x 1 rro n Y3naM. Eho - CTaTHCTD4eCKa51 3a.n:a4a no-
cne.n:oBaTenbHott BbI60pKH D rrnaHDpOBaHI151 9KcrrepHMeHToB, H CTaTI1CTHKH
Ha4HHalOT .n:YMaTb Ha,lJ, TaKHlIIH rrp06neMaMH. O,lJ,HaKO, TaK KaK HX pe3Y JIb-
TaTbl Ll.0 CDX rrop CKY JI,Hbl *), MbI BepHeMC5I K KnaCCH4eCKDM MeTO,lJ,aM,
KOTopbIe 3apaHee orrpeJI,en5llOT KaK 110nO)J{eHH5I Xi' rLl.e Ll.On)J{HbI 6bITb
B351TbI Y3JlbI, TaK D Beca Wi' COOTBeTCTBYIOw.He 9THM Y3JIalll.
He 6y.n:eM HCnOJIb30BaTb aHanHTH4ecKYlO 3aMeHY, T. e. nOCTpoeHHe
HHTeprrOJJ5IUHOHHOI'o MHOI'04JleHa, npOXOJI,51w.eI'o T04HO '1epe3 Y3JIOBbIe
TO'lKH, H rrocJle.n:YIOIlI.ero HHTeI'pHpOBaHH5I 9Toro MHOI'O'lJIeHa lIIe)J{Ll.Y rrpeLl.e-
!laMH HHTerpHpOBaHH5I; BMeCTO 9Toro Hat1.n:eM K09cpcpHUHeHTbI CPOPMY JIbl
l1errOCpeLl.CTBeHHo, He 6y.n:eM TaK)J{e nonb30BaTbC5I MeTo.n:OM p5lLl.OB Tefl-
!lOp a, a c.n:eJJaeM cpopMyny TO'lHOfl ,lJ,JI5I nOJJHHOMOB B03MO)J{HO 60JJee BbI-
COKOt1 CTerreHH. B 7.3 6bI!lO nOKa3aHO, '1TO 9TH TPH lIIeTo.n:a 9KBI1BaneHTHbI,
a TpeTHt1 MeTO.n: 51Bn5leTC51 caMbIM neI'KHM .n:JJ5I npaKTH'IeCKOrO rrpI1MeHeHH5I.
11 pO CTet1 III a51 HeTpl1Bl1anbHa51 CPOPMY na .n:n51 Bbl4HcneHl151 HHTeI'pana
Hcnonb3yeT O,lJ,HY Y3nOBYlO T04KY, a HMeHHO:
1
!(x) dx= Wl!(X 1 ),
u
*) CM. J. H. H a I ton, Thesis, Princeton University, Princeton, N. J.
128 ED.I1HblI1 METOD. HAXQ}KJJ,EHI15/ I1HTEPTIOJJ5/l\110HHbIX 4>OP MYJJ [Ill. 10
rD.e Wl H Xl 1l0,1J)1e)l{aT 011 pe.n:eJleHI1 10. 3a ClieT Bbl60pa .n:BYX CB060.n:HbIX
llapaMeTpOB MO)l{HO C.n:eJJaTb CPOPMYJJY TOliHOt! .n:Jl5I I (X) = 1 11 I (x) = x.
),1cnO.TJb3YH 9TH .n:Be CPYHKUI1I1, nOJ1YllI1M ypaBHeHHH
1 1]
1 dx= 1 =Wl' 11 xdx =')= WJXl,
o 0
1
X1=2' TaKHM 06pa30M, I1MeeM CPOPMYJlY
1
I(x) dX=/( +),
o
Pa3YMHO oueHHBaTb 01l1H6KY :HOt! CPOPMYJlbl, HCIlOJJb3Y5I CJJe.n:ylOUJ.YIO
60JJee BbICOKYIO CTelleHb X, B .n:aHHOM cJlYliae XZ:
1
x 2 dx = = +E2 I1J1H E 2 = /2 '
o
OTKYJJ.a W1 = I,
(10,2-1 )
B cJle.n:YIOUJ.et! r.laBe 6y.n:eT 1l0Ka3aHO, liTO 01l1l16Ka, 11J111 OCTaTOllHbIt!
'IJleH, KaK erO lIacro Ha3blBalOT, eCTb
h 3 Ed" (6)
2[
h 3 j" (6) _f" (6)
-24'
(10.2-2)
r JJ.e e - HeKoTopoe lIl1CJlO H3 I1HTepBaJla (0,1) H h - JJ.JlI1Ha I1HTepBaJla
(B .n:aHHOM cJJYllae h = 1).
CTaH.n:apTHa51 cpopMYJla TpalleUI1t! HCIlOJlb3yeT .n:Ba Y3JIa B KOHuax
HHTepBaJla:
1
Vex) dx = wo/(O) + wd(I).
o
nO.n:CTaBI1M Oll5lTb I (x) = I 11 I(x)=x
1
1 =WO+W1' 2=W1'
),13 9TI1X ypaBHeHI1t! I10JlYliaeM
1
C 1
I(x) dx =2 [/(0)+ ((1)1.
o
YTo6b1 OlleHI1Tb olllI16KY, B03bMeM OIl51Tb CPYIIKUI1IO x 2 :
}= ; +E2 I1J1I1 Ez=- ,
(10.2-3)
'ITO D.n:Doe OOJlbllle olllI16KI1 CPOPMYJlbl (10.2-:2) c e.n:I1HCTBeHHbIM Y3JIOM.
(10.2-4)
10.21
HECKOJJbKO THDI14HbIX <POP MY JJ I1HTEfPHPOBAHI151
129
MO)l{HO nOMeCTl-ITb .lI,Ba Y3JIa B npOl'l3BOJIbHbIe TO'IKH
I
f(x) dX=Wt!(Xl) + wd(x ).
o
VlMeSi '1eTbJpe CB060.ll,HbIX napaMeTpa, HCnOJIb3yeM <PYHKIJ,HH 1, X, x2, x 3 :
(10.2-5)
1 =Wl +W , I
; = T£'JXl + W x ,
I 2 + 2 \
"3 = W 1 X 1 w x ,
= U'IX + w x . ]
(10.2-6)
MO)l{HO peWHTb 9TH ypaBHeHHH HenOCpe.a.CTBeHHO, HO npOlije nepel1TI1
B .a.pyrylO CHCTeMY KOOp.a.HHaT, Ha'laJIO KOTOpOI1 HaXO.a.HTCH B Cepe.a.HHe
OTpe3Ka I1I-ITerpHpOBaHI1H. MO){{HO TaK)I{e H3MeHHTb MacwTa6, He MeHH5I'
yCJIOBHH, 'ITO <popMYJIa .a.OJI)I{Ha 6bITb T0'IH0l1 .a.JJH 1, x, X2, x 3 . [J09TOMY
npeo6p33yeM 33.l1,3'IY TaK:
I
f(x) dx = Wt!(Xl) + wd(x,J,
-1
(10.2-7)
'1TO npHBo.a.HT K yp3BHeHHHM:
2 =Wl +W ,
o = WIXI + W X2'
2 2 + 2
"3 = WI X 1 W2 X 2 ,
o = WIX + w x .
Tenepb HCKJJIO'IHM W2X2 H3 BToporo H 'IeTBepToro ypaBHeHl111 H 110JJY-
'1HM
o = WIX - WIXIX
(WIXI * 0) *)
HJm
xi=x , Xl=-X2'
(3HaK IIJlIOC .a.aJI 6bI JIHWb o.a.HY Y3JIOBYlO TO'IKY,) BTopoe H3 'IeTbJpeX
ypaBHeHHI1 Tenepb .a.aeT WI = w , a nepBoe ypaBHeHHe .a.aeT WI = W2 = 1.
HaKoHeQ, TpeTbe ypaBHeHHe .lI,aeT
2 2 2 1
"3= XI HJIH Xl= V3 .
*) W1X 1 = 0 npoTHBope4HT CI1CTeMe ypaBHeHHi!.
5 P. B. XeMMHHr
130 ED.HHbltt METOD. HAXO}!{D.EHYUJ HHTEPfJ01UIU,HOHHbIX <l>OPMYlI [fJ], 10
<vopMYlIa, Clle.n.OBaTellbHO, TaKOBa:
1
I(x)dx= I( - vIa ) + 1( /3 )'
-I
I+x
ECJJI1 nOllO}!{I1Tb y = , TO .n.llH nepBOHa'WllbHoro I1Hrerpalla (10.2-5)
nOllY4aeM
[ ( I ) ( 1 )1
1 1-- 1+ -
1 Va Va
/(y)dY=21 2 +1 2 =
u
I
= 2 [/(0,2113) + 1(0,7887 . ..)1.
(l0.2-8)
TaKoro Tl1na <jJ0PMYlIbl, B KOTOpbJX Mbl He <jJI1KCI1PyeM 3apaHee Y311bl
II Beca, Ha3bIBalOTCH zaYCCOBbtMU; BrrOCJIe.n.CTBl1I1 OHI1 6yn.YT 113Y4eI-lbI
OOlIee n.eTalIbHO. B Ka4eCTBe .n.pyroro rrpl1Mepa rrpe.n.nOlIO}!{I1M, 'lTO 113Be-
CTHbl KaK <jJYHKQI1H, TaK 11 ee rrpOl-I3Bo.n.HaH B KpaIlHI1X T04Kax oTpe3Ka.
Tor.n.a MO)!{HO Hanl1CaTb
1
/(x) dx=wo/(O) + wd(I) + wd' (0) + wJ'(I)
()
11, rro.n.CTaBJIHH 1 (x) = 1, x, x 2 , x a , rrollYlJaeM ypaBHeHI1H:
1 =wo+Wj,
1
2-
1
3-
1
4-
Wj + W2 +
Wa'
WJ
+ 2w a,
+ 3w a.
WJ
KOTOpbJe JIerKO pelllalOTcH:
1 1 1 1
Wa=- 12 ' Wl=2' WO=2' W2= 12 '
MeeM <jJOPMY llY
I
/(X) dx= [/(O)+/(1)J+- I (f'(O)-j'(1)].
o
(10.2-9)
10.21
HECKOJJbKO TI1rll14HbIX <l>OPMYJJ I1HTErpI1POBAHI1
131
1.IT06bJ OUeHHTb OWH6KY, nO.llC"raBInI f (x) = X4-;
1 I
5=Wl+4W3+E , E4.= 30 '
YlcnoJlb3Yl! pe3YJlbTaTbJ npeLlbmYIHett rJlaBbI, nOJlYlJaeM OWI16KY
h 5 f ilV) ( 6 ) = f ilV) ( 6 )
30 4! 720'
B KaqeCTBe nOCJJeLlHerO npilMepa, paCCMOTpHM HHTerpaJl
I
f(x) sin ; xdx=w_d(-I)+wof(O)+wd(I). (10.2-10)
-I
MeTO.ll. HHKOI1M 06pa30M He 3aBHCHT or CBOt!CTB <PYHKUI1H sin x 11 Tal,
llie xopowo npl1J10lliHM K HHTerpaJlaM BHLla
b
K (x) f(x) dx,
a
XOTI! Beca Wi 6YLlyr 3aBHCeTb or BbJ60pa K (x).
YlMeSi TpHrrapaMeTpa, rrOLlCT3BHM I, x, x :
O=W_l+ wo+w\,
8
n2 = - W_l +Wl'
o=w_\
+ WI'
ITOJ1YQeHHbJe ypaBHeHH!! JlerKO pewa!OTCII:
4
w\=- w-\= 1t 2 ' wo=O,
'lro .lI.aer .lI.JHJ (10.2-10)
I
I(x) sin ; x dx = :2 [/(1) -1< - 1)].
-I
(10.2-11 )
YnpamHeHHH
10.2-1. I10JlYlJUTb 4JOPMY JlY
"
ttx) sinxdx= (1- :2 ) (f(n)-f(-n)J + [t ( ; ) - f(- i)] .
-"
5.
132 E.1ll1HbIti METO.1l HAXO)!{.1lEHl1ft I1HTEPnOJlft 110HHbIX <f>OPMYJ1 IIJI. 10
10.2-2. DOJI}''lHTb 4JOPMY JlY
:ih
1 (x) dx = I h 5 [7f (0) + 16f (h) + 71 (2h)] + ; [I' ( ) - f' (2h)] + h 7 :;; (6) .
o
10.2-3. flOJlYlfHTb 4JOPMYJlY
1
I(x)dx= [3/(- )+2/(O)+3/( )]+ 1;20 hGi V (6).
-I
10.3. <l>MKcMpoBaHHble Y3J1bl
YlcCJIe.1l0BaHI-1e e.n:HHoro MeTo.n:a .n:JI5I HaXO)!{.n:eHI-ISI <POPMYJI Ha'-iHeM
c paccMOTpeHI151 npoCToro cJIYIJa5l, Kor.n:a Y3JIbI onpe.n:eJIeHbI 3apaHee.
DycTb L(j) - HeKoTopbIfi JII1HefiHbItI onepaTop (CM. 7.2), .n:efiCTBY!O-
w.Hfi Ha f (x); THIII1IJHbIM npHMepoM L (f) Mbl CIJI1TaeM HHTerpHpoBaHHe.
Do aHa,1IOrHH C Mero.n:OM, HCnOJJb30BaHHblM B 8.2, IJr06bI HafinI HHTep-
nOJJ5IlI.MOHHY!O cpOPMYJIY, MO)!{HO opocro HanHcaTb onpe.ll.eJJHTeJIb
I L (f) fJ f2 ... fn
mo 1 1 I
mj XI X2 X n =0,
m 2 xi X X
1 1Z xn-\ xn-I x n - J
n-I 1 2 '" n
r.n:e mk - STO k-t! MOMeHT orrepaTopa L, T. e. peSYllbT8T npMMeHeHtUI
oneparopa L K cpYHKUHH X",
mk = L (X").
Mbl rrpe.n:nOIJreM, o.n:HaKO, npHMeHHTb Mero.n:bI rrpe.n:bI.n:ym:ero naparpacpa.
!{ a}K.n:bIt! H3 npHMepOB rrpe.n:bI.l1.YIlI.ero pa3.n:eJIa HMeJI rrOXO}KYJO rpyrrrry
onpe.n:eJHIIOw.HX ypaBHeHHt! o6w.ero BH.n:a
mo =Wj +W 2 +Wa +...+W n '
m l =WJXJ + W2 X 2 + WaX a +...+WnXn'
m'l =wjx; + W2X + WaX +...+wnxi" (10.3-1)
m =w n. I+ W x n - I + W aX I+",+wnx -I.
n-l l X 1 2 2
10.31
<HIKCI1POBAHHbIE Y3JJbl
133
MaTpHua HeH3BeCTHblX X = (X ) HMeeT orrpe.n.eJIHTe.'1b (orrpe.n.eJl!1TeJIb
BaH.n.epMOH.n.a), IWTOpblt! OTJII-I'JeH OT HYJI5I, eCJIH Xi "# Xj (CM. 8.2).
Mbl HaMepeHbl HCCJIe.n.OBaTb aHaJlHTH'1eCKOe 06paw:eHHe MaTpHubl X,
MO)!{Ho, KOHe'1HO, o6paw:aTb MaTpv,HY Ka)!{.n.blt! pa3 '1HCJleHHO, HO eCJIH
6bl 9TO .n.eJlaJJOCb, TO B03HHKJII1 6bl BOrrpOCbl TO'1HOCTH, Ha KOTOpble
Tpy.n.HO OTBeTHTb; .n.et!c-rBHTeJIbHO, B KOHKpeTHblX 3a.n.a'1aX '1aCTO 6blBaeT
Tpy.n.HO 06paTHTb MaTpI-lUY, ECJJI-I )!{e .n.alla aHaJIHTI-I'1eCKa51 cpOpMa 06paTHOt!
MaTpHUbl, TO OTHOCHTe.'lbHO JIerKO OUeHHTb TO'1HOCTb Bbl'lHCJJeHHSI. YT06b1
HatJTH 06paTHYlO MaTpI1UY, BBe.n.eM cpYH.n.aMeHTaJIbHble MHOrO'1JIeHbl (Cp. 8.3)
'r.i(X)=(X-Xl)(X-X 2 ) ... (X-Xi_l)(X-Xi+l) ... (x-x n )=
n-l
= Ci,kx k (l = 1, 2, ..., n), (10.3-2)
k=O
r.n.e i-tI MHOrO'1JIeH lie co.n.ep)!{l1T i-ro MHO)!{l1Te.'l51 (x - Xi)' OTlVleTI1M TOT
Ba)KHbltI cpaKT, '1TO
'r.i (Xj) = 0
(i "#j) H 'r.i (Xi)"# O.
Eho 06CT05lTeJJbCTBO rro.n.CKa3blBaeT crroc06 06paw:eHHSI X. TaK KaK
X-IX = 1, TO rrepBa51 CTpOKa X-I .n.OJI)!{Ha COCT05lTb 1-13 9JIeMeHTOB BH.n.a
G ( l, k ) ' .lletlCTBHTeJlbHO, YMHO)!{eHHe Ha j-tI CToJI6eu X .n.aeT
1t 1 Xl
1: G 1 kxJ 1t 1 (X 1 { I, j = 1,
1t 1 ( l) = 1t 1 (X ). 0, j"# 1.
- Boo6w:e m-51 cTpoKa X-I MO)!{eT 6bITb HarrHcaHa B BI1.n.e
G m , k
1t m (x m ) ,
r.n.e Cm. k - Cl1MMeTpl1'1Hble CPYHKUHH cooTBeTcTBYIOW:HX Y3JJOB, KOTopble
JIerKO Haxo.n.SlTCSI rrpH rrOMOW:H YMHO)!{eHI1t1 H CJIO)!{eHv,t!; OHl1 SlBJI5IIOTCSI
MHOrO'1JIeHaMI1 OTHOCHT eJIbHO y3JIOB Xi'
YaCTHbltJ CJIY'1at! paBHOOTCT05lW:HX Y3JIOB, Kor.n.a OHH CHMMeTpv,'1HO
pacrrOJIO)!{eHbl OKOJJO Ha'1aJIa Koop.n.HHaT, CTOHT 3aTa6YJIHpOBaTb. 06paT-
Hble MaTpHHbI .n.JISI 9Toro CJJY'1aSl 0603Ha'1eHbl '1epe3 Sn H npoTa6YJIHpO-
BaIJbI Hl1)!{e. Yl1CJJa B cK06Kax, c.'1e.n.YIOW:l1x 3a S, YKa3bIBalOT nOJJO)!{eHv,e
( 3 1 1 3 )
Y3JJOB, TaK '1TO S4, - 2"' - 2"' 2"' 2" 03Ha'1aeT, '1TO y3JIbI Haxo-
3 I I 3
.n.51TCSI B - 2"' - 2"' 2"' 2"' ECJJH MaTpl1HY YMHO)!{HTb cnpaBa Ha CTOlI-
6eu BeKTopa MOMeHTOB, BeKTOp-CTOJI6eu pe3YJIbTara JJ,aCT Beca.
'YH.UBepCallbHbte MampUll,bt
134 Elll1HbIH METon HAXO}!{ll.EHI15I I-1HTEPTIOJJ5IUI-10HHbIX <l>OPMYJJ I rJJ. 10
s,= S, (- ; . ) (t - :) ;( : - ).
- ; ; ) ( 0 - 1
0-1= ; 20
I I 0 I
"2 "2
( - 3 2 12 - 8\
( 3 1 I ) = 27 - 54 - 12 24 )
S4= S4 -2' -2' 2' 2 48 27 54 -12 -24 '
- 3 - 2 12 8
S. S.(-l. 0. l)=(
-D.
o 2 -1 -2 1
0-16 16 4-4
I
SB=Sc(-2, -1, 0,1,2)= 24 24 0 -30 0 6
o 16 16 - 4 - 4
o -2 -1 2 I
S6 = S6 (- , - , ..., ) =
45 - 18 - 200
- 375 250 1560
I 2250 - 4500 - 1360
= 3840 2250 4500 - 1360
- 375 - 250 1560
45 18 - 200
o - 12 4 15
o 108 - 54 - 120
o - 540 540 195
1
S7= 720 720 0 - 980 0
o 540 540 - 195
o - 108 - 54 120
o l:l 4 - 15
80 80
- 1040 - 240
2720 160
- 2720 160
1040 - 240
- 80 80
-5 -3 1
60 12 - 6
- 195 - 15 15
280 0 - 20
- 195 15 15
60 - 12 - 6
-5 3 1
-32
160
- 320
320 '
-160
32
, 10.41
HEKOTOPbIE TIPVfMEPbI <I>OPMYJI
135
10.4. HeKoTopbIe npHMepbI $OPMYJI
!-1cCJle.ayeM pa3HOBH.aHOCTI1 cpOPMYJI, KOTopble MomHO rrOJIY'IHTb 113
SrHX MarpHll.. PaCClIlOTpHM S3 = S3 (- I, 0, 1):
S3 = ( - - ) \ .
o
2 2
I1pe.aTIOJlOmHM, Tpe6yeTcH BbIBeCTI1 cpOPMYJlY CHMTICOHa n:JI!I HHTerpHpo-
BaHHH
I
f(x)dx=w_J(-I)+wo/(O)+wJ(l).
-I
HymHo HatlTH MOMeHThl m o ' mj, m2' KOTopble oTIpeJJ.eJlHIOTCH KaK
I
\ 1/ 1+(-1)"
mk = x dx = k + 1 .
-I
C'IHTaH HX BeKTopOM-CTOJl611.0M m, YMHomHM ero CJleBa Ha S3' '1To6b1
1l0JjY'lHTb Beca Wi B BH.ae BeKTopa-CTOJl611.a 'w:
S,m= - -; ; = ! ) =w. (104-1)
'1TO, KaK H3BeCTHO, HBJlHeTCH TIpaBHJlbHblM OTBerOM (CM. (7.2-4».
1J.aJlee, paccMoTpHM HaXOm.1l.eHHe «TIOJIOBHHHOti cpOPMYJlbl CHMTICOHa .1I.JlH
o
/(x)dx=a_J(- l)+aof(O)+aJ(l).
-1
MOMeHTbl paBHbl
mo=l,
C/leJJ.OBareJIbHO,
0 I I
-2 2
0 -1
0 I I
2 2
I
mj = - 2'
I
m2=3'
5
12
8 I 8
- 12 - 12
I -1
- 12
(10.4-2)
136 E)].YlHbII"1 METO)]. HAXO}J{)].EHH5/ l1HTEPnOJJ5IUI10HHbIX <l>OPMYJ1 lIJI. 10
fIOnbITaeMCH Tenepb HatiTlI <jJOPMYJIY BlI.lI:a
: lx O=W-J'(-l)+wof(O)+WJ(I).
BeKTop MOMeHTOB 3.l1:eCb
11 Mbl I1MeeM
G)
(:- r -iK )=C i}
(10.4-3)
IToc. e.ll:HlIM npllMepoM 10.2 (ypaBHeHlIe (10.2-10» 6bIJJa cjJOPMY JJ3 AJJH
HHTerpaJIa
I
[(x)sin ; xdx,
-1
8
MOMeHTbl KOTo p oro 0, -. H O. CJJe.ll:OBaTeJIbHO, HMeeM
1t"
1 a 4
-"2
a -1 8 0
1t 2 -
1 4
-2
\.jTO COBn3.ll:aeT C (l 0.2-11).
ITpe.ll:nOJIo}J{HM, \.jTO HY}J{HO npoHHTepnOJIHpOBaTb B TO\.jKe x, HCnOJJh3Y5I
Te }J{e caMbIe TpH Y3JIa. Ha 9TOT pa3 BeKTop MOMeHTOB 3aBHCl1T OT
nOJIO}J{eHI151 TO\.jKI1 x, B KOTOpOti Mbl lIHTepnOJIHpyeM. TIOJIYljaeM
( 0 _ ; )( I ) ( X2_X\
: -1 . :. :.)
(10.4-4)
IIJJH 06bl\.jHoe:
!(x)= X 2 2 x f(- 1)+(1- x 2 )f(0) + x2iX f(I),
'iTO, KOHe'IHO, 51BJI51eTCH I1paBH. hHbIM OTueTOM.
10,51
3H....'TEHV!5T B <l>I1KCI1POBAHHhIX T04KI\X
137
3rOT rrOCJIeAHI1H IIpHMep HaBOAHT Ha MblCJlb, 4TO MOJKHO HattTH <POp-
MY Jlbl .QJJ5I HeOrrpe.QeJleHHbIX HHTerpaJIOB TaK JKe, KaK H .QJI5I onpe.QeJJeH-
HbIX. ECJIH rrO.nYl\IaTb eme HeMHoro, CTaHOBHTC5I 5ICHO, 4TO TaKHe HHTe-
rpa.Tb!, KaK
o
f(x) dx,
-I
5
V(x)dX,
o
x
K (X) f(x) dx,
o
MorYT 6bITb Hatt.Qellbl B TOM JKe caMOM BH.Qe npH YCJIOBHH, 4TO Mbl MO-
)KeM HattTH He06xOAHMble MOMeHTbI.
YnpamHetUlSI
10.4-1. nO. yqHTb pe3YJlbTaT ynpa)/{HeHHR 10.2-1.
10.4-2. HafiTH HHTepIJOJlilIJ.nOHHbl!% MHOrO'lJleH '1eTBepTofi CTeneHI1, COOT BeT-
CTB)!OIl\HH (10.4-4).
,10.4-3. HaHTH 4JopM)'Jlbl .1WR y' (0), y" (0), y'" (0), ylV (0), HCIJO.lb3YJl Sk'
t 0.5. 3HaQemUI <PYHKIUHf H npOH3BOJl.HOM
B tPHKcHpoBaHHblx TOqKaX
!..facTo CJIY4aeTC5J, 4TO B YS,110BbIX T04Kax Mbl JJH60 YJKe HMeeM KaK
3HaqeHH5I <PYHK1UHJ, TaK H 3Ha4eHH5I rrpOH3Bo.QHbIX, ,)JH60, Hat1.Q5I 3Ha'-leHH5I
<PYHKlI.HH, MOJKeM, 3aTpaTHB ,)JHllIb He60,)JbllIYIO .Qorro,)JHTne.lJbHYIO pa60TY,
HatJTH npoH3BoAHble. XOT5I MOJKHO HCnOJJb30BaTb HeCKO,)JbKO npoH3BoAHb1X,
OrpaHH'-IHM Hallie paCCMOTpeHHe C,)JY'-IaeM O.QHOt1 npOlI3BO.QHofi; K 60.iJee
C.lJOJKHbIM C,)JY4a5IM .lJerKO nepet1TH, eC.lJH nOH5ITa OCHOBHa5J H.Qe5I BbIBO.Qa
TaKHX CPOPMYJJ. j],,)J5I CTaH.QapTH3all.HH nOMeCTHM CHa4Ma Bce 3Ha'1eHH5I
CPYHKlI.HH, a Clle.QOM 3a HHMH 3Ha'1eHH5I npOlJ3BOJJ:HOtl, KaK B (10.2-9).
HeCMOTp5I Ha cnpaBe.QmmocTb 06mefo npaBH,)Ja «HlJKOf.Qa He HrHopHpo-
BaTb .1J106ylO HH<pop:Vlall.HIO, KOTOPYIO BaM CJJY4HTC5I HMeTb», HHOf.Qa
6blBaeT CllHllIKOM JJ:OpOfO lICnOJJb30BaTb Bce, '1TO lJMeellIb.
I-1TaK, npaBa5I 4aCTb <POPMYllbl HMeeT BH.Q
WJ(Xl) +W?!(X 2 ) +. . .+wn!(x n ) + Wn+J'(X j )+. ..+W2,J'(X n ).
TaKHM 06pa30M, nO.QlleJKama5I 06pameHI11O MaTpHlJ,a eCTb
. .. 1 0 0 0
Xl X2 ",X n ] 1 1
X9 x ... X 2x j 2X2 2x n
1
. .
x;n-l X n-l. ..X n--1 (2n - l)x;n-2 (2n - 1)x;n-2... (2n _ l)x n-2
Ha STOT pas Mbl XOTHM 11athH K03<PCPHUHeHTbI MHOr04.lJeHa, YAOB.TeTBO-
psuolll.ero BceM ypasHeHI15IM
P(Xi)=O,
P' (Xi) = 0,
(l = 1, 2, ..., n),
138 E.D.I1HbIA METO.D. HAXO}\{.D.EHI1j1 I1HTEl'nOJJ5IUI10HHbIX <!>OP MYJJ IIJJ. II)
3a HCKJIIO'-IeHl1eM O.D.HOrO (Cp. C 8.3 11 HIUeprrOJI5IuI1etJ 3pMI1Ta). l1CKJIIO-
'-IeHHOe YCJIOBI1e 3aBHCI1T OT CTpOKI1; KaK B 10.3, rrepBa51 CTpoKa
HCKJIIO'-IaeT ypaBHeHI1e
PI (XI) = 0, n-'iI Pn (x n ) = 0,
BTOpa'il P (x ) = 0, (n + 1 )-'iI P +I (xd = 0,
TpeTb'il P 3 (X3) = 0, .
2n-51 P2n (x n ) = O.
COOTBeTCTBYIOIIJ.I1e MHOrO'-lJIeHbl JIerKO rroCTpoHTb. llJI5I (n + k)-It
CTpOKH B03bMeM
n
P n + k (x) = (x - xk)IT (x - Xj)
j=1
(UlTpI1X 03Ha'-laeT nporrycK j = k). llJI5I k-tJ CTpOKIJ (k n) rrorrpo6yeM
rrpOIJ3BOJIbHbltJ JIHHetJHbltJ MHO}\{HTeJIb ax + b:
n
P k (x) = (ax + b) rr (x - Xj) '
j=1
OTHornemle a K b (a 9TO Bce, '-ITO HaM HY}\{HO) HaXO,lI}JTC'iI 113 ypaBHe-
111151 PI. (Xk) = O. l1Hor.n:a Hy:>KHbJtJ MHOrO'-lJIeH HMeeT CTerreHb HIJ}\{e; Ha
9TO YKa}\{eT HeB03MO}\{HOCTb orrpe.n:eJIHTb 9TO OTHorneHHe.
B Ka'-leCTBe npHMepa nOCTpOI1M 06paTHYIO MaTpHUY .u.JI'iI .n:BYX TO'-leK
x = 0 H X = 1. MHOrO'-lJIeH .n:JI'iI nepBotJ CTpOKH HMeeT BH.u.
P 1 (x)=(ax+b)(x-l)2,
P; (0) = 0 = a - 2b.
I10JJO)/(HM JlJJ5I YJl06CTBa b = 1, '-ITo6bJ I1MeTb P 1 (0) = I. 3TO JlaeT
a = 2 H P 1 (x) = 1 + Ox - 3x 2 + 2x 3 . I1epBa51 cTpoKa o6paTHofi
MaTpHubl eCTb, CJIeJJ:OBaTeJIbHO, 1, 0, - 3, 2.
llJJ'iI BTOpOtJ CTpO'-lKH nOJIY'-IaeM MHOrO'-lJIeH - 2x 3 + 3x , TaK '-ITO
CTpOKa B MaTpHue eCTb 0, 0, 3, - 2.
TpeTbetJ cTpoKe COOTBeTcTByeT MHOrO'-lJIeH X - 2x 2 + x 3 , '-IeTBepTOti
- x + x 3 . TaKHM o6pa30M, o6paTHa'il MaTpHua HMeeT BH.u.
( - - \
o 1 - 2 1 ) '
o 0 - 1 1
ECJJH nCnOJIb30BaTb ee lIJI'iI BbIBO.D.a cpOPMYJIbl (10.2-9) (MOMeHTbl paBHbJ
] ] 1 )
1, "2' "3' <[ , TO rrOJJY'lHM B TO'-lHOCTH TO }\{e Bblpa}\{eHHe.
10.6]
CBOBOJ],HbIE Y3JJbI; KBAJ],PATYPA fAYCCA
139
YnpamHeHHH
10.5-1. nOKa3aTb, 'ITa 06paTHaR MaTpHQa -.nJlR CJlY'IaR, Kor.na 3Ha'leHHR 4JYHK-
U,HH H nepBOH npoH3Bo.nHOH 3a.naHbI B TO'lKaX x = - I, 0, 1, eCTb
o 0 1/4 - 6/ 4 - 2/4
1 0 -2 0 I
o 0 1/4 6/ 4 - 2/4 - 3/ 4
o 0 1/4 - 1/4 - 1/4 1/4
o I 0 -2 0 I
o 0 - 1/4 - 1/4 1/4 1/4
10.5-2. npf1MeHI1Tb pe3YJlbTaT ynpa)/{HeHHR 10.5-1 K HHTerpaJlY B ynpa)/{He-
1/HI1 10.2-2.
10.6. CBOOOll.HbIe Y3JIbI; KBall.paTypa faycca
B 10.2 I1CCJIe.QOBaJIaCb napa rpOPMYJJ (10.2-1) 11 (10.2-8), B KOTO-
pbIX nOJIO)l{eHHH Bcex Y3JIOB, TaK )I{e KaK 11 Bce Beca, C411TaJIHCb napa-
MeTpaMH. 06maH rpopMYJIa TaKoro TUna
b n
k (x) f (x) dx = Wi!(Xi)'
a i -l
k (x) > O.
(10.6-1)
HMeeT 2n napaMeTpOB H MO)l{er 6bJTb C.QeJIaHa T04HOrt lI.JIH f(x) = 1,
x, ..., X 2n - l . MbI nOJJY4aeM onpe.QeJIHIOmHe ypaBHeHHH
mo = 'WI +W2 +Wa + +W n ,
fll l =WIXI + W2 X 2 + WaXa + + WnX n ,
m2 = WIX + W2X + WaX + +WnX ' (10.6-2)
m = W x2n-1 + W x2n-1 + . .. + W X 2n - 1 .
2n-j 1 I 2 2 n n
CymeCTByeT XOpOWO H3BeCTHbIt! MeTO.Q .QJIH peWeHI-IH 9TOt! 4aCTHOt!
CHCTeMbJ HeJJHHet!HbIX ypaBHeHI-It!. CHa4aJIa onpe.QeJIHM Y3JIOBOt! MHOr04JIeH
n n
'It (X) = II (X - Xi) = 2: CkX R .
i=! k=O
(10.6-3)
3aMeTHM, 4TO
'It (Xi) = 0
(i= 1, 2, ..., n).
140 ED.I1HbII1 METOD. HAXO}l{)J.EHI151 I1HTEPflOJJ5IL\110HHbIX <t>OPMYJJ 1m. 10
YMHO}l{HB nepBoe 113 ypaBHeHYlt! (l0.6-2) Ha Co, c.1e.ay!Orn.ee Ha C 1 YI T. p'.,
n-e Ha C n = 1 M CJIO}l{YlB, nOJJY'IHM
n
n
Ckmk = Wj"rr:: (Xi) = o.
k=O i=\
Tenepb c.n:mmeM Ka}l{.n:bIt! K09cpcpHlJ,HeHT BHH3 Ha O.n:HO ypaBHeHHe fI,
nOBTopfiB npOlJ,eCC, Hat!.n:eM
n n
Ckmk+l = WiXj"Ii: (Xi) = o.
k=O i=\
ECJIfi npo.n:eJIaTb Bce 9TO n pa3, TO nOJIY4HTC5I CHCTeMa JIHHet!HhIX
ypaBHeHHt! OTHOCHTeJIbHO C k (k = 0, 1, ..., n - 1)
n
Ckmk+j = 0
k=O
u = 0, 1, ..., n - 1).
(10.6-4)
Onpe.n:eJIHTeJIb Ha3BaH «nepcHMMeTpH4HhIM» (CM. [30], CTp. 419). ECJJH
iI mk+j II =1= 0, TO MO}l{HO pellIHTb CHCTeMY OTHOCHTeJIbHO C k (npHH5IB
C n = 1).
ECJm Oflpe.n:eJII1TeJJb paBeH Hymo, TO MO}l{eT CJIY4HTbC5I O.n:HO 113 .n:BYX,
Bo-rrepBblX, ypaBHeHI1£ MorYT 6b1Tb HeCOBMeCTHbIMH. ITpHMep:
\
xf(x)dx=wtf(XI)'
-I
2
mo = 0 = WI . 1, ml ="3 = WI' 0 - rrpOTI-IBOpe4f1e.
CJJe.n:OBaTeJIbHO, TaKot! CPOPMY JIbl HeT.
BO-BTOPbIX, pam MaTpHlJ,bl MO}l{eT OKa3aTbC5I paBllblM n - 1. Tor.n:a
B rpynny onpe.n:eJI5I!Orn.Hx ypaBHeHHt! (10.6-2) .n:06aBJJ5IeM ern.e O.n:HO H
BbIBOP.HM ern.e Op'HO yp8BHeHHe (l0.6-4), BKJI!O'I8!Orn.ee mk+j H C k . choT
npolJ,ecc rrOBTOp5leTC5I .n:o Tex nop, nOKa He nOJIY'lI1TC5I n JJHHet!HO He3a-
BHCHMbIX ypaBHeHHt! .n:JI5I C k , KOTopbIe MorYT 6bITb, a MorYT H He 6h1Tb
COBMeCTHbIMH. Ka}l{.n:oe .n:onOJlHHTeJIbHOe ypaBHeHHe YKa3blBaeT Ha nOBhl-
rneHHY!O T04HOCTb CPOPMYJIbI. TIo K03cpcpHlJ,HeHTaM C k Y3JIOBOrO r.moro-
'1JIeH8 (l0.6-3)
'It (x) = x n + Cn_1x n - 1 +...+ C o = 0
MOJl{HO Hat!TM HYJJM 'It (x), KOTopble 51BJI5I!OTC5I Y3JlaMH Xi'
T enepb Bce CBeJIOCb K npe.n:hl.n:yw.eMY CJIY4a!O C H3BeCTHblMH Y3JIaMH
H MO>KHO HCnOJIb30Ban nepBbIe n onpe.n:eJI5I!Orn.HX ypaBHeHl1t! (10.6-2),
4T06b1 HatlTH Beca Wi'
11 10.71
CMEWAHHblH CJJyqAH
141
B Ka4eCTBe npHMepa paCCMOTpHM ypaBHeHH5I (l0.2-6), KOTopble Mbl
He CTaJIH pemaTb B 10.2 H3-3a rpOM03.L1.KOCTH BbIKJIa.LI.OK. TaM nOJJY411-
JIOCb (10.2-5) H (10.2-6):
J
f (x) dx = wJ (x) + w<J (x ),
o
1 =Wj +w ,
I
"2 = W1XI + w x ,
1 ' + .,
3" = w1xi w x2>
1 3 + '
"4 = W1X 1 w x ,
Co
C t Co
C t
'It (x) = (x - Xt) (x - X2) = x + Ctx + Co.
MHO)l(HteJIH .LI.JI5I 06pa30BaHl-HI HOBbIX ypaBHeHHtt YKa3aHbI BO BTOpOM
CToJJ6ue cnpaBa. I1cnoJJb3Y51 HX, nOJIY4aeM
Co + ( ) C t + = 0,
( ) Co + ( ) C t + l = 0,
OTKY.LI.a
I
Co =6' C1=-1.
CJIe.LI.OBaTeJIbHO,
1
'It(X)=X -X+6
H
XI=
'1/
1- V I-a
2
I-Vi
2
1+ -V 1 1
x = 2
l+yi
2
KaK \1 paHbme (CM. (10.2-8».
10.7. CMemaHHbIH CJIyqaH
YaCTO CJIY4aeTC5I, 4TO B Ka4eCTBe y3JIOBbIX MbI XOTHM I1CnOJIb30BaTb
O.LI.HY I1JIH o6e KOHueBbIe T04KI1. PaccMoTpHM nCrrOJIb30BaHI1e JIHmb O.LI.HOti
li3 KOHlJ,eBbIX T04eK. ITpl1 3TOM, KOHe4HO, Tep5leTC5I O.LI.I1H 113 2n napa-
MeTpOB. Onpe.LI.eJI5IlOIll.He ypaBHeHI151 (IO.3-l) 3an\1meM B BHIle
n
mk = wta ll + WiX
i 2
(k = 0, 1, ..., 2n - 2),
r.LI.e a - .LI.aHHa51 Y3JIOBaH T04Ka.
142 EllVlHblD METOll HAXO)!{llEHVl5I VlHTEPTIOJU1Ul10HHblX 4>OPMYJI [Tn. 10
4acn14HO HCKJII04HM a, YMHO)!{a51 Ka)!{.n:oe ypaBHeHHe Ha a H BbI4HTa51
ero H3 cJle.n:YIOillero ypaBHeHH5I:
n
mk=mk+l-amk= Wi(Xi - a) x:
i=2
(k=O, 1,..., 2n-3).
Tenepb 06pa3yeM Y3JIOBOti MHOr04J1eH, HcnOJlb3YR JlHlllb HeH3BeCTHble
Y3JIOBble T04KH
n n-!
1t(X) = IT (x - Xi)= CkX k ,
i=2 k=o
1 110BTOpHM npouecc HCKJlI04eHH5I, npHMeHeHHbltl B npe.n:bI.n:YIlleM pa3.IJ.eJle:
n-! n
Ckmk+j= Wi (Xi -a)1t(Xi)=O
k=o i=2
(j=0, 1,..., n-2).
nOCJle 9Toro MO)f(HO HatiTH C k H HeH3BeCTHbIe Xi' KaK H paHbllle.
YnpamHeHHSI
10.7-1. PaccMorperb cJlY'Iafi, Kor,Qa B qJopMYJle ,QomKHbl 6blTb HCnOJlb30BaabJ
06e KOHl.\eBblC TO'lKH.
10.7-2. PaccMoTpeTb 06 yJO 3a!J,a<IY HCKJlJO'ICHlHj JlJ060ro 'IHCJla 4JHKclfpoBaH-
HblX Y3JIOB.
10.8. 3aMeqaHHH
B 9TOM naparparpe 6y.n:YT paccMoTpeHbI nOn&ITKH npl'IMeHHTb pa30-
6paHHbIe Bblille MeTo.n:bI. TaKHM 05pa30M, OH He 51BJJ5IeTC5I 4aCTbIO npo-
rpaMMbI .n:Jl5I HaXO)f(.n:eHH5I rpOPMYJlbl, HO 51BJ/SleTC5! I1CCJle.n:OBaHHeM TOro.
'-ITO nOJ/Y411TCSI B npouecce OTbICKaHH5I rpOPMY JIb!.
HeT COMHeHH5I, 4TO MO)f(HO nepetl'fH OT onpe.n.eJl5lIOllUIX ypaBI-IeHHtI
(10.6-2) K ypaBHeHll5lM (10.6-4)
n
Ckmk+f=O.
k=o
ECJlll paHr MaTpuUbl (mk+) He.n:OCTaT04HO BbICOK, 4T06bl n03BOJJUTb
HatlTll C k , TO Mbl ,Q05aBJ/SleM ern.e O.n:HO onpe.n:eJl5IIorn.ee ypaBHeHlle If
On5lTb np06yeM pellll1Tb CllCTeMY 11 T. .n:. B KOHue KOHUOB, Mbl JlI160
110BblCllM paHr .11.0 Tpe6yeMoti Beml'JHHbl U peWllM CHCTeMY, .rIll60 nOJJY411M
HeCOBMeCTHble ypaBHeI-lll5l, nOKa3blBaIOrn.He, 4TO TaKot! rpOPMY JJbI HeT
(JJ1l60 HaM Ha.n:oeCT). Ka)f(.n:bltl pa3 .n:06aBJl5IeTC5! ern.e O.n:HO onpe.n:eJl5lIOllI.ee
ypaBHeHHe H nOJIY4aeTC51 rpopMYJla, KOTOpa51 51BJI5IeTC5I T04HOti .n:JJ5I cJ/e-
lI.YIOllI.etl CTeneHH X; MaJIOBep05lTHO, KpOMe caMblX TpaBHaJIbHbIX CJIY4aelJ.
'-ITO npouecc He OCTaHOBHTC5I 01JeHb 6blCTpo.
CJIe.n:YIOllI.a5J TPy.n:HOCTb B03HHKaer I1PH nOnbITKe Hat!TI-! KOpHH Y3JJO-
BbiX MHOr01JJIeHOB. IloKa)f(eM. 4TO BO MHorHX CJIY1Ja5lX OTbICKHlJaeMble
10.8]
3AME4AHVl5I
143
Y3J1bI n:efiCTBHTeJlbHbI, pa3J1I-JtIHbl H Jle)l{ar B UHTepBaJle UHTerpilpO-
BaHH5I.
TIpe)l{.lI.e Bcero, paCCMOTpilM cJlyqati, KorJ(a Y3J1hl He <PUKcupOBaHhI
1-1 K (x) O. TaK KaK <popMYJla TOqHa .lI.JHI Bcex CTerreHeti x BrrJlOTb
.lI.O 2n - 1, TO I1MeeM ['It (x) 3.l1.eCb TO )l{e, qTO H B ypaBHeHill1 (10.6-3)]
b n
K(x)'It(X)Xkdx= 2: WiX 'It(Xi)=O (k=O,..., n-I)
a i=1
(10.8-1)
(TaK KaK Bce 3HaqeHI151 'It (Xi), BCTpeqalO HeC51 CnpaBa, paBHbI HYJIIO).
Terrepb rrpeJI.rrOJlO)l{I-IM, qTO 'It (X) HMeeT MeHbllle, qeM n pa3J1l1qHbIX HYJJett
B 06J1aCTH I1HTerpl1pOBaHH5I. TIYCTb
XI' X'!, ..., X m
(m < n)
6YJI.yr HYJlil HeqeTHOfi KpaTHOCTH
MHorO'lJIeH
Ha OTpe3Ke HHTerpl1pOBaHHSI.
m
p (X) = I1 (X - Xi)
i=1
HMeer CrerreHb m n - 1, CJlen:OBaTeJlbHO, fIO (10.8-1) rrOJIyqaeM
b
K(x) 'It (x)p (X) dx = O.
a
Ho B HHTepBaJle ilHTerpHpOBaHil51 nO.ll.bIHTerpaJlbHa51 <PYHKUH5I ilMeer
nOCT05lHHbiti 3HaK H He MO)l{eT 6bITb TO)l{.lI.eCTBeHHblM HYJJeM. CJleLl.OBa-
TeJlbHO, MbI npHlllJJI1 K npOTilBOpeqUIO. TaKHM 06pa30M, B HHTepBaJle
HIuerpHpOBaHYt51 eCTb n .lI.eticTBilreJlbHbIX pa3J1U4Hb1X HYJIett 'It (X) il MbI
rapaHTupOBaHbI, 4TO HCKOMble .lI.etiCTBHTeJlbHble KOpHH B caMOM lIeJle
CYllleCTBYIOT.
B CJIY4ae, KorlIa <PI1KCilpOBaHa O.ll.Ha KOHUeBafi TOqKa a, Mbl HMeeM
Y3J10BOi! MHOrOqJleH 'It (X) CTerreHil n - 1. 06pa3yeM
b
K(x) (X - a) 'It (X) xkdx= 0
a
(k=O, 1,..., n-2)
H JJ:onycTH I, 4TO 'It (X) H:'vleer JlHlllb m < n - 1 pa3Jl11'IHbIX KopHett He-
qeTHorO rrOp5l.ll.Ka B HHTepBaJJe. TIOCTpOI1B, KaK H paHbllle, npoH3Ben:emle
p (X), nOJlyqaeM
b
K(x) (X - a) 'It (x)p (x) dx = O.
a
OTO rrpHBOJI.UT K rrpOTHBope4111O.
144
ED.I1HbIH METOD. HAXO}!(D.EHI151I1HTEPITOJJ5IIJ:110HHbIX cf>OP MYJ]
IflI. \0
B KatJeCTBe ynpa}!(HeHI15J npeD.ocraBJI5!eM 'JHTaTeJIlO ITOKa3aTb, qTO
,JHaJIOrnqHbIe co06pa}!(eHI-]5! npHMeHl1MbI H B CJIyqae, KOr.n:a rpHKCHpOBaHbI
06e KOHueBhle TOqKH.
MO}!(HO TaK}!(e nOKa3aTb, qTO BeCa Wi B CB06o.n:HbIX Y3JIaX nOJIO}!(H-
TeJIbHbI. 9TO - I1IITepeCHbIH pe3YJIbTaT; OH ITOKa3bIBaeT, qTO rpOpMYJIa
6y.n:eT CTpeMHTbC5J COnpOTHBJI5!TbC5I nOrpeWHOCT5IM B HCXO.n:HbIX .n:aHHbIX,
raK KaK He 6Y.QeT B3aHMHbIX YHHqTO}!(eHHH H3-3a BeCOB C nponmonOJIO}!(-
1!bIMH 3HaKaMI1 (XOT5! B03MO}!(HbI YHH'JTO}!(eHH5! H3-3a rpYHKUI1I1, I1MeIOlli,ef:1
nporHBOnOJIO}!(HbIe 3HaKI1 B pa3JIH'JHbIX Y3JIax). D.JI5I raYCCOBoro CJIY'Ja5!
CB060.il.HbIX Y3JJOB Y3JIOBOtl MHOrOqJIeH orrpe.n:eJJ5IerC5I paBeHcrBOM (cp.
(10.3-2»
7tk (x)=(x - XI) (x - x ).. .(x - Xk_l)(x -Xk+I)" .(x - x n ).
TaK KaK 7tk (x) !-HlIeeT CTeneHb 2n - 2, TO
b n
K (x) 7tk (x) dx = Wi 7tk (Xi) = Wk 7tk(x k ).
a i=1
CJIe.D.OBaTeJIbHO Wk> 0 npH YCJIOBHH, qTO K (X);;-: O. I10.n:06HbItt }!(e
npHeM npI1MeH5IeTC5I B CJJyqa5lX, KOr.n:a HCnOJIb3YIOTC5I O.D.Ha HJJ\-! 06e
KOHueBbIe TOqKH. Kor.D.a rpHKCHpOBaHHbItt Y3eJI JIe}!(HT BHYTPH HHTepBaJIa,
9TOT pe3YJJbTaT, Bo06lli,e rOBOp5l, HeBepeH, XOT5I MOrYT 6bITb Haii.D.eHbl
qaCTHbIe CJIY4aH, KOr.D.a OH BepeH.
He Y.D.aeTC5! nOKa3aTb, 'JTO Beca nOJIO}!(HTeJIbHbI, KOr.D.a rpHKcHpoBaHbI
Bce Y3JIbI, nOToMY qro B .D.etlcTBHTeJIbHOCTH 9TO IIeBepHo: OHH MorYT
6bITb H orpHuaTeJJbHbI.
ECJIH nonp060BaTb 0606lli,Hrb rpOPMY JIY rayccoBoro THna, HCITOJIb3Y5I
3HaqeHH5I rpYHKUHH H ee nepBott npo\-!3BO.D.Hott B Tex }!(e Y3JJaX, TO MbJ
Haii.n:eM (no KpafiHet! Mepe B HeCKOJIbKHX CJIyqa5lX, KOTopbIe HCCJIe.D.OBaHbI),
'HO HYJJH Y3JIOBOrO MHOrOqJIeHa - KOMnJIeKCHbIe 'JHCJIa.
YnpamHeHHe 10.8-1. I-1cCJleJJ,oBaTb c. y"aH, KOr.I\a 4JHKCl1pOBaHbl 06e KOH-
IJ,CBbIC TOljKI1. floKa3aTb, 'no Bce HYJlH .I\eHCTBI1TeJlbHbI, pa3.111'1HbI 11 HaXO.I\RTCR
B HHTepBaJle HHTerpHponaHHRH 'ITa Bce Beca nonOlliHTeJlbHbI.
10.9. JIHHefiHbIe OrpaHH'IeHHH Ha Beca
V!1!or.n:a CJJyqaeTC5I, qTO HaM .D.aHbI HJIH }!(eJIaTeJIbHbJ JJrIHefiHbJe orpa-
HHtJeHH5I Ha Beca. TaKHe OrpaHHqeHlI5I, Bep05lTHO, Moryr CB5I3bIBaTb Beca,
COOTBercrBYIOlli,He rpHKCHpOBaHHbIM YSJIaM, HO OHH Bp5lJJ: JIH MorYT Ha-
JIaraTbC5I Ha Beca, COOTBerCTBYIOlli,He CB060.D.HbJM Y3JIaM: TPY.D.HO SHaTb,
B KaKOM nOp5!JJ:Ke OHH 6Y.D.YT 0603HaqeHbI, - ynop5l.D.OQHTb I1X no BemI-
QHHe 5IBJI5IeTC5! TPYltHOti MaTeMaTl1'leclwti 3aJJ:aQeH. rJIaSIWe nCKJJI04eHHe:
KorJJ:a O.D.HO H TO JKe OrpaHI1QeHHe npl1MeH5IeTC5I KO BceM 1J.3BeCTHblM
BecaM.
10.91
JII1HEHHbJE OfPAHI1'1EHI15! HA BECA
145
Xoporno H3BeCTHb!t1 npOMep OrpaHO'lelmt1 Ha Beca n:aeT HHTerp"pOBa-
HI-Ie no '-Ie6bJIUeBY, rn:e
W J =W2="'=W n =W
M Bce Y3JIb! CBo60n:HbJ. 3TO n:aer n + 1 napaMerp " COOTBeTCTBeHHO
n + I onpen:eml!omee ypaBHelIHe
n
'\, k
mk=w L.JX;
;=1
(k = 0, 1,..., n).
nOJIara51 k = 0, HaXOJJ:\1M
W= m o
n
OcraJJbHble ypaBHeHH5I npHHHMalOT BHn:
n
nmk -
-=mk= . x .
mo '
;=1
T O)J{1!:eCTBa HblOTOHa n03BOJIHlOT Bblpa3I-Irb CYMMbI CTeneHet! KopHet!
'1epe3 K09<jJqJHlI.OeHTbI C k Y3.7JOBOrO MHorOlfJJeHa. OrClO1!:a nOJIY'IaeTC5! Y3JIO-
BoM MHor01weH, '1TO n03BOJJHeT Hat!TI1 HY JIO X;.
l13BecTHO, '1TO, Korn:a K (x) = 1, Bce Y3JIbl .!I.et!cTBIUeJIbHbl, pa3JJH'IHbl
11 Haxon:HTCH B HfITepBaJIe I1HTerpnpOBaHI1H n:JI5I n=l, 2,..., 9 II
TOJIbKO 11)151 9Tl1X n.
PaJIbCTOH *) orry6mIKOBaJI IIHTepecHblt! rrpl1Mep CMernaHHoro Tl1rra
I1Hrerpl1pOBaHHH, B KOTOpOM <jJI1KCl1pOBaHbl ,1LBa Y3J1a II O,1LHa JlIIHet!Ha5!
CBH3b. l1MeHHo, <POPMY JJa
1 n
f(x)dx= WJ(Xi)
-1 i=1
non:'1I1HeHa YCJJOBllHM:
xl=-I, xn=l, WI=-W,..
3lJ,eCb HMeeTC5I, CJJelJ,OBaTeJIbHO, 2n - 3 napaMeTpa H <POPMYJIY MQ}!(HO
ClJ,enaTb T04H0t1 lJ,nH 1, X,..., X 2n - 4 . OnpelJ,eJI5IlOw.He ypaBHeHIU! CYTb
n-I
Wi'
i=2
0=- 2w I + 2.: WiXi,
2 0
3= k. WiX1 ,
0= - 2wI + 2.: WiXI,
2=
2 W;X}n 4.
2n - 8 = "'"
*) A. Ralston, J. Assoc. Computing Machinery, vo1. 6, pp. 384-394, July 1959.
146 E.1JJ1HbIH METOI\ HAXO)!(I\EHl-HII1HTEPn0JJ51U110HHbIX <POPMYJI [Tn, II)
4T06hl C,lI,eJIaTb nOCJJe,ll,YIOUJ,YIO BbIKJIa,ll,KY 60JJee JIemot! 1-1 B TO )Ke
BpeM5I He BIlOJIHe TpHBHaJJbHot!, B03bMeM CJlY4at! n = 5, KOTOpblt! .n:aeT
2n - 3 = 7 ypaBHeHHti. Y3JIOBOti MHorOl)JJeH eCTb
(x)=(x-x )(X-X3)(X- X4)=X 3 + C X2+ C I x+ Co'
06hl4HhIti Ilpouecc HCKJII04eHH5I IlpHBO,ll,HT K n - 1 (B 9TOM CJIY4ae,
4eThlpeM) ypaBHeHHlo;
I
Co +3G2 - - W I G 1 - Wi>
I I
"3 C 1 +s=-wIC O - U'IC 2 ,
1 I
:rCo +5C2 - - W1C 1 - wi>
I I
'5 C 1 +7=-W 1 C O -W1C ,
JIerKo BH,lI,eTb, 4TO MO)!(HO 06pa30BaTb n - 3 (=2) ypaBHeHH5I, KOTO-
phle He BKJIIOIJalOT IJJJeHOB BToporo nopH.aK3 WiCi (i = 0, 1, 2),
2 2
:rCo+ 15 C =O,
2 2
15 C 1 + 35 = O.
YlMeeTcH, CJIe,ll,OBaTeJJbHO, n - 3 JIHHetlHhlx Heo.aHopo.aHhIX ypaBHeHTJ51
C HeH3BeCTHblMH Co, C 1 ,..., C n _ 3 HJIH Ha O.aHO HeH3BeCTHOe 60Jlbme,
4eM ypaBHeHHtt. Pema51 ypaBHeHT!5I B SaBHCI1MOCTH, CKa)!(eM, OT CO,
nOJlY4aeM
3
C 1 = - 7'
C 2 =- 5C o .
TIO,ll,CTaBHB HX B nepBbIe D:Ba ypaBHeHH5I cHcTeMhl, 1l0JJY4HM
I 2
-:r CO =-7 W1 '
I
35 = + 2C O W I'
B o6UJ,eM CJJY4ae, TaK )l{e KaK npH n = 5, HCKll104aH Co' npHXO,ll,HM
K KBa.apaTHoMY ypaBHeHl110 D:JlH WJ. B CJIY4ae n = 5
V T
-+ -
W1- - 60'
Bhl60p aHaRa WI Ollpe,ll,eJlHeT OD:HY li3 ,lI,BYX CHMMeTpH4HhlX qJOPMYJl.
3Ha51 WI' MO)l{HO JlerKO HatiTH C 1 li YSJlOBOti MHOr04.11eH
3 5 -.;T 3 + -. ;T
(x)=x - JI 245 x -7 x JI 245 '
TaKlie )!(e C006pa)!(eHH5I, KaK B 10.8, nOKa3hlBalOT, l)TO HY.IIrJ D:etlcnm-
reJlbHbl, paSJlrJ4HbI li HaXOD:HTC5I B HHTepBaJle HHTerpHpOBaHI151.
10.IOJ
<l>OPMYJJA rPEropl-f
147
Ynpa>KHeHHe 10.9-1. nOKa3aTb, '1TO CJlY'IalO PaJlbCTOHa npH n = 6 COOTBer-
<TByer Y3rroBo MHOrO'lneH
4 + 4 2 2 4 + I
1t(X)=X ---x -- -
3}l6x 83 7V6x 21'
10.10. <1>opMYJla rperopH
8 060l1x paccMoTpeHHbIx CJIY'Ia5lX, cO.1l:ep}({allLHX JIHHet!HbIe OrpaHl-I4e-
111151 Ha K09cpqmlJ:l-IeHTbI, 60JIbllHIHCTBO Y3JJOB 6bIJIO CB060.1l:HO. 8 9TOM
naparpacpe paCCMaTpl-IBaeTCH CJIY4aii paBHooTcTOHllLHX cp\1KCHpOBaHHblx
y3JIOB co MHOrl-lM\1 CB5I351MH. 8BO.1l:\1TC5I HOBa51 KOHllerrllHH CPOPMY Jib!,
I-IMelOllLeii rrpOH3BOJIbHOe 4HCJJO Y3,10B H orrpe.1l:eJIeHHOe 06llLee pacrrOJIO-
}({eHl-Ie BeCOB. .QJI5I orrpe.1l:e,1eHHOCTH HCCJJe.1l:yeM CPOPMYJIbI, Beca KOTOpbIX
pacrrOJJO}({eHbI TaK:
abbbb...bbba,
abeee ...e eba,
abcdd...dcba
(10.10-1 )
(10.10-2)
(10.1 0-3)
11 T.'.n.. TaK KaK Ka}({.n.aH cpopMYJIa HMeeT Ha O.1l:HH napaMeTp 60JJbUle,
<JeM rrpe.1l:bI.1I:YllLa5l, TO MO}({eM Ha.1l:e5lTbC5I HattTH ceMet!CTBO CPOPMY JI C B03-
paCTalOllLett T04HOCTb!O. <popMYJIa (10.10-1), HarrpHMep, rrpe.1l:rrOJIaraeT,
<JTO C HCrrOJIb30BaHHeM n + I Y3JIa HHTerpaJI
nh
Vex) dx = af(O) + bf(h) + bf(2h) +... + bf[(n - 1) h] + af(nh)
o
MO}({eT 6bITb Haii.1l:eH T04HO KaK .1I:JI5I f(x) = I, TaK H .1I:JJ5I f(x)=x.
81'0 rrpHBO.1l:HT K .1I:BYM ypaBHeHI-I5IM (HCOOJIb3Y51 paBeHcTBo (3.1-3»:
nh=b(n-1)+2a
(l
n 2 h 2 _ bh n (n - 1) + h
2 - 2 a n,
KOTopoe COBna.n.aeT C rrepBblM. Ilo.n.CTaBHB Tenepb f (x) = x 2 , nOJIY4aeM
n8t = bh 2 n (n - i (2n - I) + ah 2 n 2 .
PeweHHe 9TH X .1I:BYX ypaBHeHHtt .1I:aeT
_ n h=!!..- h
a- 2 (n+l) 2 2(n+1) ,
b= n 2
n 2 -1
h
h=h+ 1 .
n-
TaKHM 06pa30M, cpopMYJIa 51BJIHeTCH T04HOt! .1I:JI5I MHOr04JIeHa BTOpoii
CTeneHI1. f{03cpCPHllHeHTbJ 3aBHC5IT OT 411CJIa HHTepBaJIOB n, a 9TO 4aCTO
5I8J15leTC5J Hey.n.06CTBOM. MO}({HO f1oTpe60BaTb, 4T06bI rrepBoe BblBe.rr.eHHoe
148
EDYfHbII1 METOD, HAXO}!{D:EHI151 I1HTEPnOJJ51lU1OHHbIX <POPMYJI
I fJJ. to
ypaBHeHHe 6bIJlO BepHO LI)]SI BCeX n, H He f1bITaTbCSI c.u.eJIaTb cpOPMYJIY
TO'-lHOti D:JISI x 2 . 9TOT f10D:XO.u. .u.aeT
h
b=h, a=2'.
T. e. H3BeCTHoe npaBHJIO Tpaf1eUHti.
CxeMbI pacnOJIO)!{eHI1I! BeCOB BbI6paHbi CHMMeTpH'-IHbIMH; 9TO 03Ha'-laeT,
'-ITO eCJIH KaKa5!-HH6y.u.b H3 TaKHX cpOPMYJI 5!BJISleTC5! TO'-lHOti .a.JI5! Bcex
CTel1eHet1 x Bf1JIOTb .11.0 HeKOTopoti '-IeTHot1, TO OHa 5!BJI5!eTC5! TO'-lHOti .a.JJ5!
CJIe.a.YlOrueJ1 60JIee BbICOKOJ1 He'-leTHoJ1 CTeneHH. 9TOT 9cpcpeKT y)!{e Ha-
6,11O.II:aJICfI B npe.ll:bI.II:yrueM f1pHMepe, KOr.l1.a \1 I, \1 X .l1.aBaJI\1 O.ll:HO l! TO
)!(e ypaBHeHHe. 4T06bl y6e.a.fJTbCfI B Cf1paBe.a.JIY!BOCTH 9Toro 3aMe'-laH!15!
B 06meM CJlY'-Iae, 3aMeTI1M, '-ITO npe06pa30BaH\1e X = x - nepeBOJUJT
Ha'-laJlO Koop.a.HHaT B Cepe.ll:\1HY 06JIaCTH HHTerpl1pOBaHHfI. ITocJle 9Toro
Bce He'-leTHbIe CTeneHH aBTOMaTH'-IeCKH HHTerpHPYIOTCfI TO'-lHO, TaK KaK
06e '-IaCTH ypaBHeHl-lfi BCJle.u.CTBHe CHMMeTpHH 06pa!ll.aIOTC5! B HYJlb.
ITp\1 06paTHOM npe06pa30BaHHI1 Ha'-laJJa He'leTHa5! CTeneHb .ll:aeT H Bce
60JJee HH3KHe CTef1eHH.
ECJIH cpOPMY Jla flBJl5!eTC5! TO'-lHOt! .l1.Jl5! 9THX 60Jlee HH3KHX CTef1eHet!,
TO .ll:Jl5! HHX 06e '-IaCTH paBeHcTBa YHH'-ITO)!{aIOTC5! H cpopMYJla OKa3bIBaeTC5!
BepHoti D:JlSI OCTaBllIetlC5! CTapllleti He'-leTHoti CTef1eHH. OCTaeTC5! paccMOT-
peTb TOJIbKO '-IeTHbIe CTef1eHH x, TaK KaK cHMMeTpl1'-1HOCTb 3a.u.a'-lH aBTO-
MaTH'leCKH o6eCne'-lHBaeT TO'lHOCTb .u.JJfI He'leTHbIX CTef1eHeJ1.
ITocTapaeMC5! HatlTM cpOPMYJIbI, B KOTOpbIX cxeMbI paCf10JIO}!{eHHfI Be.
COB He 3aBHC5!T OT KOJIH'-IeCTBa B35!TbIX Y3.TIOB, f1pl1 YCJlOBHH, KOHe'-lHO,
'-ITO liX HaCTOJlbKO MHoro, 'lTo6bl n05!BHJlHCb Bce f1ap aMeTp bI. 9TH cpop-
MYJIbI MOrJlI1 6b] 6bITb Hatl.u.eHbI pa3 HaBCerD:a He3aBHCHMO APyr OT APyra,
HO OAHO npOCToe 3aMe'-laHHe CBO.ll:HT 3a.l1.a'lY K TOMY, '1T06bI, npe06pa30-
BbIBa5! Ka}!{.II:YIO cpOPMYJIY no o'-lepe.u.H, nOJIY'IaTb CJJeJJ:YIO!ll.YIO H3 ,Horo
ceMetlCTBa.
TpeTL5i cxeMa (10.10-3), HarrpHMep, MO}!{eT 6bITb 3arrHcaHa B BHD:e
nh
I(x) dx = (f1paBHJIO TpaneUI1t1) + Al [Af(O) - A/(n - 1») +
o
+ A 2 [A2f(0) + A2 fen - 2)).
.ilo6aBJleHHe pa3HOCTHbiX '1JJeHOB He rrOBJlH5!eT Ha TO'-lHOCTh 1jJ0PMYlibI
J.I)I5I 1 H x. 04eBHAHo TaK)Ke, 'lTO cyu ecTBYIOT TaIme KOHCTaHTbI Al \I
A2' '-ITO ljJopMYlia cBe.a.eTC5! K cxeMe (10.10-3).
B 06!ll.eM cJIY'Iae cpopMYJla
nh
I (x) dx = (f1paBI1110 Tpaf1eUI1t1) + Ad Af (0) - Af (n - 1)] +
a
+ A2IA2f(0) + A2f(n - 2)J + Aa [A3f(0) - A2f(n - 3)] +
+ Ad A4 f(0) + f1.4f(n - 4)] +...
\\ 10.101
<POP MY JJA fPE fOPI-!
149
J.J,aeT MeTOJ.J" npI1 KOTOpOM BCe K03cpcpI1lJ,I1eHTbJ MOrYT 6hlTb, OlJeBI1J.J,HO,
Hati.n:eHhI no OlJ'OpeJ.J,I1. B caMOM .n:eJJe, napa K03cpcpHIJ.I1eHTOB onpeJ.J,eJJ5leTC5J
o.n:HOBpeMeHHO BCJJe.n:CTBl1e CI1MMeTpHI1, KOTOpa5J 06eCne'lHBaeT TOlJHOCTh
cpOPMYJJbl .n:JJ5J He'leTHoti CTeneHl1. TaKI1M 06pa30M, '1T06bJ onpe.n:eJIHTb A J
11 A2' nOTpe6yeM, lJT06bl cpopMYJJa 6bIJJa TOlJHOti .n:JJ5I x 2 . l1cnoJJb3Y5I pe-
sy JJbTaThJ 3.1, I1MeeM (.n:JJ5I h = 1)
3 = n(n-I)6(2n-l) + n2+Ad1-(2n-1)]+Ad2+2].
TIpl1paBHI1Ba51 K03cpcpI1lJ.I1eHTbJ npI1 o.n:I1HaKOBbIX CTeneH5IX B o6el1x '1aCT5IX
paBeHcTBa, rrOJJY'laeM
n 3 . . 1 1
3-3'
n 2 .
3 1
0=-6+2 1
I
O=--2A 1
6 '
n:
I;
O=2A 1 +4A 9 .
TIoCJJe.IJ:HHe .IJ:B3 YP aBHeHI15J .n:arOT
I
Al = 12'
1
A 2 =- 2-4 '
TIo.n:CTaBJJ5I5I I (x) = x 4 , nOJJYlJI1M nOJ.J,06HbJM 06p a30M
A 19 A 3
3 = 720 ' 4. = - 160 '
a npl1 I(x) = x 6 I1MeeM
863
AD = 60480 '
275
A 6 =- 24192 .
l1TaK, MbJ rrOJJYlJaeM cpOPMYJJY fperopl1
nit
I (X) dx = ( 10 + 11 + 12 + .. . + In-l + ; In) +
+ 11 2 [ /o - In-d - ;4 [ 210+ 2In_2j +
+ ;;0 [ 310- 3In_3]- 1:0 ( 4/0+ 4In_4)+'"
YaCTO 60JJee nOJJe3Ha cpopMYJJa fperopI1 B cpopMe J1arpaH)I{a, B KOTOpOti
JJI1HetiHbJti onepaTOp BhII1HCaH '1epe3 3HalJeHI151 CPYHKIJ.I1I1; I1MeHHO C
Hee MbJ HalJaJJI1 STOT naparpacp. f{03(pcpl1lJ.l1eHTbJ STOtt cpOpMbl .n:aHbI B
150
EJU-1HbIH METOD. HAXO)J{D.EHI1511-1HTEPn0J15IUYlOHHbIX <l>OPMYJJ
[fJJ. 10
OleJ1.YlOl11,et! Ta6JIHlJ,e, r J1.e CTol/6elJ, Cl/eBa .lI.aeT K03rpqJHlJ,HeHT Ak HaHBbIC-
,wero nOpH.lI.Ka. llo CHMMeTpllH HaM HY)J{eH JIHWb O.ll.HH KOHelJ, <POPMYJIbl:
Ao: 1
"2
A,: 5 13
12 12
A : 9 28 23
2424 24
Aa: 251 897 633 739
7 20 720 720 720
'COOTBeTCTBYlOl11,aH <POp MYJIa, KOTopaR HCnOJIb3yeT np0113BO.lI.Hble BMeCTO
pa3HocTet!, H3BeCTHa leaK pa3l/0)J{eHHe 3t!l/epa - MaKJIOpeHa 11 HHOr.ll.a
6blBaeT nOJIe3Ha, HO npH HCnOJIb30B3HHH Ha MaWIIHe OHa Tpe6yeT npo-
rpaMMHpOBaHHH np0I13BO.lI.HbIX (CM. ypaBHeHHH (4.8-2)).
Ynpa>KHeHHSI
10.10-1. !-1cnOJIb3YR MeTOJJ,bl 9Toro naparpa4Ja, BblBeCTIi COOTBeTCTBYlOInHe
.4JOPMY Jib! .D:Jljj
n
f(x) dx = at ( +) + bf ( ) + cf ( ) + ... + cf ( n - ) +
+bf(n - )+af(n- ; ).
10.10-2. I-1ClIOJlb3YR Te )/{e Y3J1bl, '1TO 11 B ynpa)j{HeHHH 10.10-1, Bbl1TOJlHHTb
;pa60TY no HaXO)/{)!eHHIO K094J4JHI.I,HeHTOB nepBblx 'IeTblpex pa3HocTHblX '1J1eHOB.
. 10.11. BbIBOJl.bl
OCHOBHOti J,J,eJIblO ,notl rJIaBbI 6b1JIO nOKa3aTb, KaK BbIBOJI:I1Tb pa3JJHlJ-
Horo pO.lI.a <POPMYJIbl; MbI He CJIHWKOM yrJIy6JI5lJII1Cb B paCCMOTpeHHe
I-!MelOUJ,HXC5I pe3YJIbTaTOB. OJI:HOt1 113 npHlJHH 3Toro HBJI5IeTC5I 061-!JIHe
<POp MYJJ. VIx CYUJ,eCTByeT 3HalJI1TeJIbHO 60JIbWe, lJeM MO}J{HO nOMeCTHTb
B OJI:HOt1 KHI-!re, H HCCJIe.ll.OBaHHe Bcexpe3YJIbTaTOB MQ)KeT 06paCTH Ta-
KOt! Maccotl JI:eTaJIetl, lJTO 06ecKypa}J{HT JI:a}J{e caMoro nbIJIKOrO JII06HTeJIH
1Il1CJ/eHHoro aHaJIH3a. J{or.lI.a MeTO.lI.bI HaXO)J{JI:eHI151 <POPMYJI nOH5ITbI, JIerKO
Hat!TH KOHKpeTHY!O <POPMYJIY, He06xoJI:HMYIO B KOHKpeTHOM CJIYlJae.
Mbl OTMelJaeM TaK}J{e B03MO}J{HOCTb JI:etlCTBI-!TeJIbHO BbInOJIH5ITb Ha Ma-
,rIHlHe 60JIbWY!O lJaCTb pa60Tbl no BblBe.ll.eHl-J!O <POp MYJI. YT06bl nOMOlJb
3TOMY, Mbl .lI.aeM 6JJOK-cxeMY (pl1c. 10.11-1) MeTOJI:a BbIBO.lI.a <POp MYJI, He
HMe!Ol11,l-!X JIHHetlHblx OrpaHl1lJeHHtI Ha Beca.
3.l1.eCb n - KOJIHlJeCTBO Y3JIOB Xi' nl - KOJHIlJeCTBO <plIKClIpOBaHHb!X Xi'
'2n - nl - MI1HHMaJIbHOe KOJIl1lJeCTBO MOMeHTOB mk' 3a.ll.3HHblX MaWlIHe
{npl-! CJIYlIae MaWlIHa 6Y.lI.eT Tpe60BaTb JI:aJIbHetlWI1X MO leHTOB).
10.111
BbIBonbI
15Jt
COOTBeTCTBYIOlli.a!l 6JIOK-CXel\1a nJIR CJJY4a!l 3anaHHbIx JII-IHetlHhIx CB5I-
aeiJ He BKJII04eHa clOna H3-3a HeOTIpeneJIeHHOCTH THTIOB cpOPMY J}, KOTOp hie
I!Ol/tJllO
I!OI/t7llbHbIB JIfO'leH!/J1 { 1/!/CIlO OCflltJ5lUUXCJ1 Y.JIl08bIX fllOW/( =:]l;:!1'
TO'lIfOCfllb f/lOjJMYllb/ =/7=/7"
11c/f/lIO'lUfllb
He177 OOllfJ !l38eC177110e
X; 11170/l0.JffII177b
/7'= /7'-1.
J!l/70/WHllfllb
.31770 .1:i
( 10. 7)
,4000Bil!77b
xi )( pOHee
8b1I1!JCOffHbIA
!l !looeiitl-
1770mb 8ce
Xi J Wi 11 /7"
CfllfJl7
5'bfllLfCIlLf177b
1701/2 MOfll-
jJiliJbI = r
!l0/7fJII0i770177b: '
ffY.JI{elf eLI/I!
.JIOMellfll
Iloclle
TlOllYW!I!/.It
Tl0l10.Jl{IITb
/7"= 17"+ 1
l1e'l!l177b:
fllO/foil
f/I[1jJMYl1b/
118m
Cmol7
PHC. 10.11-1.
MOrJIH 6bI TIOHano6HTbC5I. f{aK 6bIJIO 3aMe4eHO paHbIlle, JIlmetlHhIe CB5I3W"
aepoRTHo, 110 mKHbl HaJIaraTbC5I Ha sera nJI5I cpilKCHpoBaHHblx Y3JIOB ilJII1
KaK npI1 ilHTerpilpOBaHilil no 4e6bIIlleBY, enilHoo6pa3HO Ha Bce Beca cso-
60.D.HblX Y3JIOB.
152
o HAXo nEHHH OCTAT04Horo 4nEHA $OPMYn
[HI. 11
I'JIABA 11
o HAXO)l{llEHHH OCTATOlJHOrO lJJIEHA OPMYJIbI
11.1. nOTpe6HOCTb B OCTaTO'lHOM 'IJIeHe
OnHcaB no HeKoTopoti CTeneHH enHHbIti MeTOn HaXO neHH5I CPOPMYJI,
p accMoTp HM Tenepb enHHblt1 MeTOn HaXOll{lleHHH cooTBeTcTBYIOW,HX OCTa-
T04HbIX 'meHOB. KaK H B CJIY4ae HaXO)KneHHH cpOPMYJJ, enHl-lblti MeTon
He Bcerna CaMbJti KOpOTKHi1 11 :meraHTHbJti, HO ero nOCTOHHCTBa, nO-BH-
nHMOMY, nepeBenHmalOT 60JIblllYID HHor na TpynoeMKOCTb.
B :HOti r JIaBe Mbl cocpenOT04HM BHHMaI-IHe Ha HaXO)l{neHHH CPOPMYJJbl
OCTaT04Horo 4JIeHa, I1rHopHPYH 9cpcpeKTbJ oKPyr JJeHHH. KaK H B CJJY4ae
HHTepnOJJHQHI1 MHOr04JIeHOM, OCTaT04Hblti 4JJeH 6yneT 3anHcaH B BHne
HeKOTopoti np0l13BonHot1 (HJJH npOH3BonHbIX) Bblcwero nopHnKa. 9ro-
rp3JJ:HQHOHHaH cpopMa, HO OHa HMeeT Bce HenocrarKH, o6CY)I{naBWHeCH
B 8.7.
3necb Mbl 3atiMeMCH BbIBonOM OCTaT04Horo 4JJeHa H OTJJO HM 06cY)I{-
)J,eHHe nOJIY4eHHbJX pe3YJJbTaTOB no cJJenYIOw,et1 rJIaBbl, rne c06paHbl
BMeCTe HeKoropbIe CPOPMYJJbl H I1X OCTaT04Hble 4JJeHbl. 04eBHnHo, 411Ta-
reJIIO nOJI)I{Ha 6bITb HCHa He06xonHMocrb OCTaT04HOro 4JIeHa. TIpe)l{ne
4eM npenn04ecTbi O)J,HY CPOPMYJIY npyroti, MbJ XOTeJJH 6bl 3HaTb HX OCT a-
T04Hble '.!JIeHbI, XOT5I BeJIH411Ha OCTaT04Horo 4J1eHa 11 He HBJlSIeTCH enHH-
CTBeHHblM KpHTepHeM MH Bbl60pa CPOPMYJJbl.
11.2. nOpHJJ:OK OCTaTO'IHOrO 'IJleHa
B Ka4eCTBe npHMepa JJHHetiHoro orrepaTopa 6yneM CHOBa HCrrOJJb30-
BaTh CPOPMYJlY npH6J1I1)1{eHHOrO BbI4HCJleHf1SI onpeneJleHHOrO HHTerpaJIa,
XOTf! MerOn pa60raeT 11 B CJJY4ae HeOnpeneJJeHHOrO HHTerpHpOBam1H,
HHTepnomHl.HH 11 HeKOTopblX npyrHx JlHHetlHblx orrepaTopoB. 0603Ha'lHM
'Jepe3 E" = E" (x) OW!J6KY CPOPMYJJbl, Kor.u:a Mbl rrOnCraBJJHeM x" BMeCTO
I(x). Ham MeTOn BblBo.u:a CPOPMYJlbI rrOKa3blBaJl (KpoMe CPOPMYJlbl rpe-
ropH), 'JTO E" = 0 .u:JJH k = 0, 1, ..., m - 1, r.u:e m - KOJlH4eCrBO CBO-
60.u:PbJX rrapaMeTpoB B cpopMYJIe. Mbl O H.u:aeM, 4TO Em:::j:: 0, HO rOTOBbJ
K TOMY, 4TO OHO paBHO HYJ1IO, TaK KaK MbI Y)I{e 3HaeM, 'ITO B HeKOTOpblX
CJIY'Ja51X, TaKI1X KaK cpOpMYJJa CHMrrCOHa, TO'JHOCTb BblWe, tJeM KO.1JH'IeCTBO
CB060.u:HbJX napaMeTpOB.
TIpoBepHM, paBHO JJI1 Em HYJJIO. B TOM CJJYtJae, KOrna BCe Y3JJOB e
r04KH aanaHbJ, Em ecn paQHOHaJJbHa51 CPYHKQHH naHHblX MOMeHTOB H .u:aH-
H X Y3JJOBblX "J 04eK. Ho B CJJYtJae KBanpaTypbI raycca, tW-
npHMep, He06xo.lU-lMO peWHTb aJJre6paHtJeCKoe ypaBHeHl1e nJIH Y3-
JJOBbIX T04eK X j' Y3JJOBOti MHorOtJJJeH, KOpHH KOToporo HY HO HatiTH,
4T06bJ rrOJJY4Hrb Y3JJOBble r04KH, naeT COOTHoweHHe, n03BOJJHIOw,ee Bbl-
pa3Hrb crerreHH Xj' 60JIbWHe n (KOJJHtJeCTBO HeH3BecrHb1X. Y3JIOBblX ro-
11.3]
<l>YHKW151 BJJI151HI151
153
'1eK), 4epe3 CTeneHH, MeHbWHe n. Eho BbIpa}J{eHHe BKJIIO'1aeT K03<pqJH-
UHeHTbI C k y3JIOBOrO MHOrO'1JIe!Ia. Tenepb CTeneHH Xj' MeHbWHe n, YMHO-
}J{aIOTC5I Ha HX Beca W j H,CJIe.n.OB3TeJIbHO, SbIpa)l{aIOTC5I '1epe3 MOMeHTbJ
mk' B pe3Y JIbTaTe On5lTb nOJIY4aeTC5I paUHOHaJIbHa51 <jJYHKUIUI OT MOMeH-
TOB *). fIpOBepKa Em = 0 51BJI5!eTC5I, TaKI1M 06pa30M, paUI10HaJIbHOli one-
palJ,l1eli Ha.a .aaHHbIMI1 MOMemaMH H .aaHHbIMH Y3JIaMI1, 11 eCJIH MO)l{HO
BblITOJJHHTb .n.OCTaTO'1/JO M/Joro aJIfe6paW-IeCKHX .aeliCTBHli C palJ,HOHaJIb-
HbIMH Bb!pa)l{eHH5IMH, TO MO}J{/JO peWI1Tb BOnpOC 0 TOM, paBHO HJIH /JeT
Em !Iymo, 6e3 nOMex oKpyrJIeHH5I, KOTopble B03HI1KalOT npH HaXO)l{.aeHHH
KOpHeit Y3JIOBOrO MHOrO'1JIeHa.
ECJIH MbI Hatt.n.eM, '11'0 O.aHO 3Ha4eHHe Em paBHO HYJIIO, TO nonp06yeM
CJIe.n.YlOmee, H TaK .n.o Tex nop, !lOKa He Hatt.aeM TaKoe m, '11'0 Em::j::. 0,
11JJ1-1 }J{e MbI YCTaHeM H 6pOCHM; nOCJIe.aHee MaJIOBep05lTHO, KpOMe Tpl1-
BHa,1bHbIX CJJY4aes, TaK KaK o.aHa HJII1 .ase CTeneHl-l .n.onOJJHI1TeJIbHott
T04HOCTH - 31'0 06bI4HO BCe, 41'0 MO}J{HO nOJJY4HTb.
Ynpa)l{HeHl-lSl
11.2-1. OIJpeiJ.em!Tb m .1I..1R paBeHCTBa (10.2-8), HcnOilb3YH MeTO,IJ.bl 11.2.
11.2-2. Onpe.aeJlHTb m .aJlR <jJOpMY i1bI CHMIJCOH3 (10.4-1).
11.2-3. Onpe.aeJlI1Tb m )(JIR YIJpalliHeHHR 10.2-1.
11.3. <l>YHKU.HH BJUIHHHH
YT04HHM CHa'1aJla, KaKot1 Knacc <jJOPMYJI Mb! co6HpaeMC5I HCCJIe.aOB31'b.
.II.JI5I onpe.n.eJIeHHOCTH 6y.n.eM rOSOpl1Tb 06 OnepalJ,I1H I1HTerpl1pOBaHH5I, XOT5I
.n.onYCTHMb! H .n.Pyrl1e JII1Het1!{b!e onepaTopbI.
B npaBoit 4aCTH paBeHCTBa HCnOJIb3YIOTCH 3HaqeHHH <jJYHKlJ,HI1 H ee
np0I13Bo.aHbIX; HO bI 6y.aeM Tpe60saTb, 4T06bI CTapWa51 npOH3Bo.aHa51
HMeJIa nopH.n.OK MeHbwHtI, 4eM T04HOCTb <jJOPMYJIbI, KOTOpa51 B npe.n.bI-
.n.ymeM naparpa<jJe 0603Ha'13JIaCb 4epe3 m - 1. TaKHM 06pa30M, paCCMaT-
pHBaIOTC5I <jJOPMY JJbI BI1.n.a
b n n n
f(x) dx = 1: akf(xk)+ 1: bkf' (Xk)+'''+ 1: m k f(m-2) (Xk) + R,
a k=! k=l k=l
(11.3-1)
r,lLe HeKOTopbIe H ..Ak' b k , ..., mk MorYT 6bITb HYJI5IMI1. CJIe)l,yeT 3aMe-
TI1Tb, '11'0 Y3JIOBble T04KI1 X k MorYT JIe}J{aTb BHYTpl1 I1JII1 CHapY}J{H 06-
JIaCTH HHTerpHpOBaHH5I. I1pe.anOJIaraeTC51, '11'0 31'a <jJopMYJIa T04Ha .n.JI5I 1,
x, ..., x m - 1 11 Em::j::. 0, r.ae m onpe.n.eJIeHO, Ka\( B npe.n.blJJ,ymeM na-
parpa<j)e.
*) MOMe1-ITbI mk He 06R3aHbI, KOHe'lHO, 6bITb paQHOHaJlbHbIMH, H, Bo06/lJ,e
rOBop>l, MOllieT B03HHKHYTb Bonpoc 0 TOM, YHHllTO)/{aIOTCH IWI1 HeT Bce ll,1eHbl
B BhlpallieH\111 Em l!epe3 MOMeHTb!; HO Ha npaKT\1Ke 9TO 06bI'IHO He npH,IlIBJJeT
6ecnOKOHCT!!a.
154
o HAXO nEHKK OCTAT04HOrO 4nEHA OPMYJJ
[rn, 11
4TOGbI nOJJY4HTb BbIpa eHHe .11)151 OCTaTO'lHOrO '1JIeHa R, Ha'lHel\l C
,pH,lI,a Tet!JJopa
I(x) = I(A) + (X - A)f' (A) + (X-;!A)2 r (A)+...+
x
+ (X - A)m-l I(m-l) (A) + I S I(m) (S) (x - S)m-l ds. (11.3-2)
(m -1)J (m-I)!
A
cha cpOpMYJJa MO eT GbITb npOBepeHa HHTerpHpOBaHHeM nOCJIe,lI,HerO
"'IJIeHa no '1aCTHM m pa3 (TahHM c6pa30M, OHa npe,lI,CTaBJIHeT c06ot! npoCTO
TO ,lI,eCTBo). MbI EbJ611paeM A haK MHHHMYM H3 a H Bcex Xk' TIO,lI,-
DaBHM renepb BbJpa eHlle (11.3-2) B CPOPMYJIY (11.3-1) B 06e '1aCTH,
T. e. BMeCTO I(x) H BMeCTO f' (X k ), Iff (Xk), ..., l(m-2) (Xk)' OillHeh(a
R (j) B cpopMY.1Je (11.3-1) eCTb pa3HocTb Me ,lI,Y ,lI,BYMH '1aCT5JMH paBeHcTBa.
CYMMa Bcex '1.1JeHOB pa3JJO eH}Uj Tei1.1Jopa, KpoMe HHTerpaJlbHOrO, eCTb
MHOrO'lJIeH aT x CTeneHH He Bblille m - 1. A TaK KaK CPOPMY JTa (11.3-1)
HBJl5JeTC5J TOlJHOi1 ,lI,JUI 1, x, ..., x m - J (T. e. Ek = 0; k = 0, 1,..., m - 1),
TO nOCJ1e nO,lI,CTaHOBKH oCTaHYTcH JJHillb HHTerpaJJbHble '1J1eHbI. MbI HMeeM,
JIe,lI,OBaTeJJbHO,
b
R=R(/)=
a
x
I \ Jlm) (s) (X - s)m-J ds dx-
(m-l)! i
X k
- (m I)! } j\m) (s) (Xk - s)m-I ds-
x k
b k d \ I (m) ( ) ( ) m-J d
- (m-l)! dXk X s Xk- S S-...
x
m dm-2 k
... - 1: (m I)! dx'J) - 2 f(m) (S)(Xk - S)m-J ds. (11.3-3)
A
YnpOCTI1M (11.3-3) C nOMOw,blO CJle.ll.YlOilLero npl1eMa. Orrpe.ll.eJII1M LUfH
i>O
{ 0 ,
(x - s) = }
ex - s) ,
ECJII-I j = 0, TO
eCJII1
x - S 0,
x - s ;;:: O.
II U-4)
eCJJI1
{ 0,
(X-$) =
1,
eC.1JI1 X-S<O,
eCJIH x - S> 0,
HO j = 0 He Mo eT BCTpeTHTbCH B HaweM CJJY4ae, TaK KaK CTapwaH
npOH3BO.LI.HaH HMeeT llOpH.LI.OK m - 2
% 11.3\
<l>YHKUH$I BJJH5IHH5I
155
!lemo BH.n.eTb, lJTO
d
dx (X - S)'f-l =(m - 1){x - S)'f-2,
d 2
dx 2 (X-S)'f-l=(m-1){m-2){x-s)'f-3
(11.3-5)
H T. .n.., .n.O TeX nop, nOKa nopSl.n.OK npOH3Bo.n.HOti MeHbwe, lJeM m - 1.
KpoMe TOro,
b
5 (b-s)'f-(a-s)'f
(x-s)'f-ldx= In
a '
(11.3-6)
Cn0J1b3YSl 9TH HOBbIe 0603HalJeHHIJ, MbI MO)\{eM YBeJ1HlJHBaTb Bepx-
Hl1ti npe.n.eJI HHTerpHpOBaHHSI no s .n.o B B ypaBHeHHH (11.3-3), r.n.e B
BbI6HpaeTcSI KaK MaKcHMYM H3 a H Bcex Xk' cho YBeJIHlJeHHe B BepXHel\!
npe.n.eJIe He BJIHSleT Ha 3HalJeHl'le HHTerpaJla, TaK KaK Ka)\{JlbIti lJJleH eCTb
HYJIb npH S>Xk'
YlJHTbIBaSl 9TH 3aMelJaHHSI 11 BbInOJlHSISI JlH<p<pepeHllHpOBaHHe C HCIlOJlb-
30BaHHeM (11.3-5) H HHTerpHpOBaHl1e C HCIlOJJb30BaHHeM (11.3-6), nOJIY-
qaeM JlJ1S1 (11.3-3)
B
R=R(j)= I 5 rm(S) [ (b-s)'f-(a-s)'f -
(m - 1)1 m
A
n n
- ak(Xk-S)'f-1- b k (m-1){Xk- s )'f-2_...
k=\ k=1
n
. .. - mk (m - 1)(m - 2). .. 3 . 2 (Xk - s)+J ds. (11.3-7)
k=l
I
IlpoH3Be.n:eHHe (m _ 1)1 Ha BbIpa)\{eHHe B KBa.n.paTHbIx cKo6Kax 3aBHCHT
OT S H HCCJJe.n.yeMoti <P0PMYJIbI, HO He 3aBMCHT OT <PYHKllHH I(s). TaKHM
06pa30M, IlHweM
B
R=R(j)= I(m) (s) O(s) ds;
A
(11.3-8)
O(S) Ha3bIBaeTCSI «<PYHKllHeti BJIHSlHHSI». B lJaCTHOCTH, I(x) = x m .n.aeT
B03MO)\{HOCTb BbI4HCJIHTb HHTerpaJI OT a (s) H3 (11.3-7):
B
R (j) = R (x m ) = m! O(s) ds = Em *- O. (11.3-9)
A
3aMeTHM, lJTO 9Ta <PYHKlJ,HH BbIl!eJJ.eHa AJiH (11.3-1) H HenpnMeHHMa K
.n.pyrHM BI1.n.aM <POPMY JI.
156
o HAXOM.lI.EHHH OCTATOqHOrO qJJEHA $OPMYJJW
[rJ1. !I
MCCJJellOB3HIJe 0(8) rrOK33b1B3eT, '1ro OH3 Orrpe.D.eJleH3 Ha [A, B] IJ
RBJIReTCR MHor01weHOM CTeneHI1 m H3 K3MJWM oTpe3Ke, Orp3H!1'1eHHOM
COCellHIJMIJ Y3JIOBblMIJ TO'lK3MO IJ I1peJJ:eJl3MO oHTerpopOB3HOR a, b.
11.4. CJIyqau, Kor.n:a a (5) "MeeT nOCTOHHHbIU 3HaK
B 6CJJbllll1HCTBe B3MHblX CJlY'!3eB OKa3bIB3eTC5!, '1TO tpYHKll.1-H1 BJIl-HIHH5I
0(8) I1MeeT 110CT05!HHblt! 3HaK. ECJm 9TO T3K, TO OCTaTO'1Hblti '1JIeH (<Pop-
MYJIa (1\.3-8»
B
R = R (f) = f(m) (s) 0(8) ds
A
YJl.OBJleTBOp5!eT yCJJOBWJM reopeMW 0 CpeL\HeM 3H3'!eHI1I1, II, HCrrOJIbsY51
(11.3-9), MOMHO H3nHC3Tb
B
R=j(m) (8) \. 0(8) ds= Em f\m) (8) (A 8 B).
J m!
A
Tenepb BCTaeT 3a.D.a'1a - y3HaTb, HMeeT JlH 0 (s) rrOCT05JHHblJ1 SHaKo
llJl5J HaqaJIa paCCMorpHM oIlm6KY rrpaBHJJa Tp3neUI1tt
1
R(f)= f(x)dx- 1(I)t/(O) ,
o
KOTOpa5J 5JBJI5IeTC5J TOlfHOti Jl.JJ5I f(x) = 1 H JJ:JIR x. TaKoM 06pa30M, m=2.
Tor.u.a H3 (11.3-7) rrOJlY'1aeM
110(s)= (I-S)t2-(-S)
(I-s)++(-s)+
2
B 9TOM CJlY'13e A=a=x 1 =O; B=b=X2= 1; O(s) RBJJ5JerC5J MHO-
rOlfJleHOM Ha BceM orpe3Ke [A, B], 3 HMeHHO:
_ (I-s)2-(I-s) _ -S+S2 _ S(l-s) <
O(s)- 2 - 2 - 2 0,
KOTOpblt!, 0'1eBf1.D.HO, HMeer nOCT05!HHblti 3H3K (plJc. 11.4-1).
B K3'1eCTBe BToporo rrpHMep3 paccMorpHM rpOPMYJIY CUMrrcoHa
I
j(x)dx= 1(-1)+4 (0)+/(1) .
-1
KpaTKoe UCCJIe.1l.0BaHUe nOKa3bIBaeT, '1TO m = 4; CJIe.1l.0BareJIbHO, HCnOJlb-
BY5! (11.3-7),
, (l-s)t-(-l-s)t (-I-s)t+4(-s) +(l-$ )!I.
310(8)= 4 - 3
1\ 11.4]
_ CJJYLJAH, KOf.QA a (s) I1MEET nOCT05IHHI.IH 3HAK
157
3.n:eCb A = a = Xl = - 1; B = b = X3 = 1; X9 = O. CJIe.n:oBaTeJIbHO,
HY)j{HO paCCMaTpl-lBaTb .n:Ba OTpe3Ka: [-1, 0] 11 [0, + I]. 1 >8>0:
3!0(S)= (I S)4 (I 3 s)3 =(1-8)3( -\2 3s )<0,
o>s>-l:
3!0(8)= (14 S )4
(1- S )3
3
4 (- S)3 - I +6s 2 +8s 3 +3s 4
3 - 12 =
=(1 +8)3 ( -1 ;t3S ) < °
(pI1C. 11.4-2).
9T1-1 npHlv!epI.I nOKa3I.lBa1OT, 'ITO eCJJH cpopMYJJa CHMMeTpI14Ha, TO 0(8)
TaK)j{e CI-IMMeTpH4Ha. lI.oKa3aTeJII.CTBO BblTeKaeT 1-13 3aMe4aHU5I, 4TO eCJJU
(;(8)
-1/8
' [J(S} " 7
1
-1/J2
-J
1 $
PHC. 11.4-1.
PHC. 11.4-2.
cpopMYJIa CI-IMMeTpI-l4Ha, TO MI.l .n:OJI)j(HI.I nOJJY411Tb TY )j{e caMYIO OIJ1I16KY,
npOl-IHTerpHpOBaB f (X) H f (- X). A TaK KaK MI.l HMeeM CB060.n:Hblli
Bbl60p CPYHKll.HU f (x), JII060e OTCYTCTBHe CUMMeTpHI1 B 0 (x) MO)j{eT 6I.ITb
HCnOJIb30BaHO, 4T06bl nOJIY411Tb pa3Hbie OTBeTbl.
<t>YHKll.HH BJIHHHl'lH 0(8) onpe.n:eJIeHa Ha OTpe3Ke [A, BJ, KOTOpblli
pa36HT Ha I-IHTepBaJIbl Y3JJaMU Xk 11 npe.n:eJIaMI1 I1HTerpHpoBaHHH a H b.
OHa HenpepblBHa Ha [A, BJ H Ha Ka)j{.n:OM no.n:blHTepBaJJe 51BJJHeTCSI: MHO-
r04JIeHOM CTeneHH m (.n:JJH o.n:HoKpaTHblx HHTerpaJIOB; .n:JJH nOBTopHoro
uHTerpHpOBaHH5I 9Ta CTeneHb MO)j{eT 6blTb H BI.lllIe).
3a.n:a4a CBO.n:HTC5I K TOMy,4T06bl onpe.n:eJJHTI., HMelOT JJH 9TH lHoro-
4JIeHbl HYJIH B CBOHX no.n:I.IHTepBaJIax. ITpHMoli no.n:xo.n: COCTOI1T B TOM,
4T06bl HaliTH I-IyJII1 9THX MHOr04JIeHOB H 3aTeM npOBepUTb, JIe)j{aT JIH OHH
B COOTBeTCTBYIOill,HX no.n:I.IHTepBaJiax. O.n:I-IaKO 9TOT cnoc06 MO)j{eT no-
Tpe60naTb MHoro MaIllI1HHOrO BpeMeHH. C .n:pyrotl CT, OpOllhI .n:JI5I OOJJbIllHX
npOMe)j{YTKOB BI.l411CJIeHl-le 0(8) B 6JIH3KnX TOLJKaX H BI.l4ep411BaHHe rpa-
cpHKa e,1Ba JIH B03MO)j{HI.I, XOTH npOCTOTa 9Toro MeTo.n:a 04eBH.lU-la. AJI-
re6pau4ecKuI1 MeTO.n:, KOTOpI.ltl OWl npI1MeHeH B npI1Be.n:eHHbIX npI1MepaX,
Bep05JTHO, Tpy.n:eH .n:JI5I nporpaMMHpOBaHHH Ha MallIHHe, XOTH .n:JIH PY4HOti
npOBepKH OH 4aCTO npHro.n:eH. TaKHM oopa30M, BnOJIHe y.n:OBJIeTBopHTeJJb-
HI.It! cnoc06 npOBepKH, MeHReT JJH 3HaK 0(8), Bp5l.n: JJI1 CYill,eCTByeT. Ho,
BO BCHKOM CJJY4ae, CJIe.n:yeT HCHO npe.n:CTaBJJRTb ceoe, 4TO 0 (8) 3aBHCHT
JJHIlib OT CPOPMY Jlbl H, CJIe.n:OBaTeJJbHO, npOBepKa JJ.OJl)j{Ha npOBOJJ.I1TbC5I
AJII! JI106ot\ JJ.aHHot\ CPOPMY JIbl JIHIlib OAHH paa.
158
o HAXQ)KlIEHI1I1 OCTAT04HOro 4JJEHA <POPMYJJbI
1m. It
Ynpa}/meHHII
11.4-1. flpoBepHTb, '!TO !jJopMYJla B ynpa)!meHHH (10.2-1) lJ,JlR I1HTerpaJl&
1t
5 t (x) sin xdx
-1t
HMeeT 4JYHKU,HIO BJlHRHHR a (s) nOCTORHHoro 3HaKa, XOTR R,IJ.pO K (x) = sin
MeHileT 3HaK.
11.4-2. flpoBepHTb !jJOPMYJlY ynpa1KHeHHR (10.2-2).
11.5. CJ1YQafl, KorJl,a «!iYHKD.MH BJ1MHHMH MeHHeT 3HaK
ECJII1 0 (s) MeHHeT 3HaK, TO CJIeAyeT OTMen1Tb Tpl1 06CT05!TeJIbCTB3.
Bo-nepBblx, TOr.n:a cyrn.eCTBYIDT CPYHKIJ,I1H f(x), .n:JIH KOTOpbIX HeT TaKOrO
e, 4T06bl 6bIJJa BepHo!i TeopeMa 0 cpe.n:HeM 3Ha4eHI1I1. YT06bl nOKa3aTh
3TO, rrpe.n:rrOJIQ)!{I1M, '1TO O(s) ilBJI5JeTCiI nOJIO)!{HTeJIbHo!i BCID.n:y, KpOMe
MaJleHbKOro I1HTepBaJla, B KOTOpOM OHa OTpl1uaTeJlbHa, 11 '1TO I1HTerpan
OT O(s) TIOJlO)f{I1Te.lJeH. PaCCMOTpl1M TaKYID CPYHKIJ,HID f(x), '1TO f(m) (x)
nOJlO)f{HTeJIbHa H HenpepblBHa, HO MaJIa BHe ::noro ManeHbKoro IIHTepBal13
11 04eHb BeJlHlW BHYTpl1. Tor.n:a I1HTerpaJI
13
f(m) (s) O(s) ds
A
(11.5-1 )
OTpl1lJ,aTeJJeH, HO f(m) (6) n HHTerpaJI OT O(s) 06a nOJIO)f{I1TeJlbHbl. Clle-
.n:OBaTeJIbHO, He cYIl eCTByeT HY)f{HOrO e. TI@.ao6Hble aprYMeHTbJ MorYT
6bITb rrpHBe.aeHbJ .aJlil .apyrnx o (s).
BO-BTopblX, B aTOM CJlY4ae MO)f{HO OUeHI1Tb cBepxy MO.aYJlb HHTer-
paJIa (11.5-1). JlJIiI I O(s) I BepHa TeopeMa 0 cpe.aHeM, oTKy.aa JIerKO
nOJIY'IaeTCiI oueHKa:
13 13
I V(m) (s) O(s)ds I Itt m ) (s) II O(S) Ids=
A A
13 13
= If(m) (e)I' I O(S) I ds max I f(m) (s) I. I O(S)! ds,
A A
r.lle MaKCI1MYM B3HT no BceM x (A x B). V1HTerpaJI
13
I O(s) I ds
A
MOjf(HO Bbl'll1CJIHTb aHaJIHTH4eCKH IIJIH '.JI1CJIeHHO. Eho OTIRTb Heo6xolIHMO
c.aeJIaTb TOJIbKO O.aHH pa3 .ll}1iI .lIID6otl .aaHHo!i cpOPMYJlbl.
B-TpeTbI1X, XOTH BHaK O(s) MeH5IerCiI, Mb! Bce jf(e MOjf(eM Hanni
OCTaT04Hblt! 4JIeH. B Ka4eCTBe I1JIJIIDCTpaUI-1H paCCMOTpl1M CPYHKUI1ID 0 (s)
11.51
CJlY4AI1. KOniA <PYHKUI151 BJJY!51HY!51 MEI.mET 3HAK
159
Ha pl1c. 11.5-1, T04Ka x = C BbI6paHa TaK, 4TO ABe 3aUlTpHXOBaHHbIe
flJIOllJ.a.l111 paBHbJ no BeJJU'-IHHe, T. e.
B
O(s)ds=O.
C
Mbi HMeeM (ClII. ypaBHeHHH (11.3-8) HJJH (11.5-1))
B C B
R=R (f)=V(m)(s)O(s)ds= + .
A A C
s
ITpol1luerpHpyeM BTOpOti '1JJeH no '1aCTHM H nOJJO)!{HlII O(s) ds = H (s).
(3aMe'1aHHe: H (B) = H (C) = 0.) C
C{ S 1
A c WVg
C{ S
. /\
A '--" B
PHC. 11.5-2. j],pyroJ1 BHJ.t a (s).
PI1C. 11.5-1. 4aCTHbJrl BHJ.t a (s).
TorL!.a
C B
. R = j\mJ(s) O(s) ds + j\m) (s) H(s) I - j\m+l) (s) H (s)ds =
A C
C B
_=-.V (m) (s) O(s) ds - )t(m+1) (s) H(s) ds.
A C
Tenepb MO)!{HO npHlIIeHHTb TeopeMY 0 epe.llHelll 3Ha'1eHI1H K Ka)!{L!.OlllY
IHTerpaJ]Y
C B
R = j\m) (6 1 ) O(s) ds - f(m+1) (6 2 ) H(s) ds.
A C
EeJJll O(S) TaKaH, KaK ITOKa3aHO Ha pHe. 11.5-2, TO lIIO)!{HO
110JJO)!{I1Tb
npOCTO
s
H(s)= O(s)ds
A
11 I1HTerpl1pOBaTb (11.5-1) ITO lfaeTHM
B
R(j)=f(m) (s)H(s) I - j\m+l) (s)H(s)ds=
A
}j
= JIm! (B) H (B) - flm-I) (6) H (s) ds.
A
160
o HAXOlliLl.EHVII1 OCTATOYHOrO YJJEHA <POPMYJJbI
[rJJ. 11
MbI He 6YLl.eM aHaJJH3HpOBaTb 06I1 HH CJIY4aH, TaK KaK 9TH pe3YJIb-
TaTbI He HCnOJIb3YIOTC5I B TeKCTe. 3aMeTHM TO.%KO, 4TO TaKHe <pOpMbJ
OCTaT04Horo 4JIeHa MorYT 6bITb HHor Ll.a 110JIe3HbI.
11.6. CJIa60e MeCTO B MeTOJ1.e pHJ1.0B TeiiJIopa
TaK KaK cymecTByeT OCTaT04HbIH 4JIeH Ll.JI5I p5ILl.OB TeliJIOpa B BHLl.e *)
(x a)2
f(a+x)=f(a)+(x-a)f'(a)+ -; j"(a)+...
+ (x _a)'n-l f(m-l) ( )+ (x-a)m I'm) (a + ex)
... (m-l)! a ml ·
eCTeCTBeHHO nOnbITaTbC5I HCflOJIb30BaTb MeTOJJ, pHJJ,OB TetlJIopa (BTOpOtl
MeTOJJ, 7.2), 4T06bI flOJIY4HTb 01IlH6KY <POPMY JIbI.
DYCTb HCl10JIb3yeTcH MeTO.Q pH.QOB TetlJIopa. K09cpcpHIU'leHTbI CPOPMYJIbI
onpe.QeJIeHbI TaK, 4T06bI CTenelH1 x YHH4TO)!(aJII1Cb C 06eHx CTOpOH
BnJIOTb .1J:0 CTeneHH m, KOTopaH flO I1pe.QnOJIO)!(eHHIO He YHH4TO)!(aeTCH.
3acpHKcHPyeM MbICJJeHH9 3Ha4eHue x. Tor.1J:a 110JIY4UM pH.Q H3 4JIeHOB
f(m) (a + BiXi) Ll.JJH pa3JIJ1l!HbIX B i
rJ.tf(m) (a + Bl Xl) + rJ.d(m) (a + B!Jx!J) +.. . + tJ.k f(m) (a + Bk Xk),
r.1J:e, KORe4HO, B i 3aBI1CHT OT Xi, B i = 0i (Xi)' MbI XOTHM Tenepb 33MeHlHb
9TO BbIpa)!(eHHe TaKHM:
(a, +a!J+' ..+tJ.k)f(m) (a+ OX)
.1J:JI5I KaKHX-TO HOBbIX 3HalJeHHft 0 . (H 0)!(H.1J:3eM, 4TO MHOrHe 4JIeHbI at
YHI14TO)!(aTCH). DpH K3KHX 06CTOHTeJIbCTB3X cymeCTBYIOT T3KHe O?
KaK H3BeCTHO H3 MeTOJJ,a CPYHKUHH BJIH5IHH5I, eCJIH O(s) coxpamreT
3H3K, TO T3KOe 0 cymeCTByeT, 3 eCJIH y 0(8) 3HaK He nOCTOHHHbI.t!, TO
CYIJJ.eCrBYIOT HeKOTopbIe CPYHKII.HHf(x), .QJIH KOTOpbIX TaKoro 0 HeT. TaK!1i\!
KOCBeHHbIM I1YTeM MO)!(HO OTBeTHTb Ha Bonpoc. Ka)!(eTCH, npHMoro eno-
c06a H3ftTH OTBeT HeT, HO eCJII1 OTBer HaN.QeH, TO eCTb B03MO)!(HOCTh
.1J:eftcTBoBaTb Ll.BYMH pa3JIU4HbIMH cnoc06aMu, a TaKOH BbI60p HHOr.1J:a no-
Jle3eH.
B CJIY4ae CPOPMYJIbI raYCCOBoro BH.1J:a C HeCKOJIbKUMH CB060Ll.HbIM\1
y3JIOBbIMH TOlJKaM\1 cymecTByer CneIl.HaJIbH351 TeXH\1K3 **) onpe.QeJIeHHH
oCTaT04Horo 4JJeHa npow.e, lJeM lJepe3 CPYHKlI.HIO 0 (x), \1 9TO HaBO.QI1T
Ha MblCJIb, lJTO eCTb HeKOTOpa5I H3L1.e)!(Ll.a HaliTH Ll.pyroi! e.1J:\1HbIH MeTOLl..
*) 3TO BblpallieHHe Ha3blBalOT 4JOpMOl! JJarpaHllia JWH OCTaTO'lHOrO 'I. e!la, a
I1HOrJ1a Bce BblpallieHHe Ha3blBaCT.'R paCIJII1peHHOH TeopeMotl a CpeJ1HeM.
**) CM. XII.1b.ll.e6paHJ1 [14] 11m! KOllaiI [20].
12.11
BBEJ.l.EHI1E
\6\
I'J1ABA 12
<t>OP M Y JIbl lI.JI5I OI1P EJl.EJIEHHbIX HHTEfP AJIOB
12.1. BBe.n;eHHe
Me)\{.n;y Bbl'll1CJJeHl1eM onpe.n;eJIeHHOrO I1HTerpaJIa
tJ
V (x) dx
a
H Heonpe.n:eJIeHHOrO I1HTerpaJJa, KOTOpblft MO)\{HO 3armcaTb B BHJ.l.e
x
f(x) dx,
a
I1MeeTC5I cymeCTBeHHOe pa3JIH'Il1e. Pe3Y JIbTaTOM Bbl'lI1CJJeHI151 nepBoro 51B-
JI5IeTC5I 01lHO '1HCJIO, Tor JI:a KaK pe3Y JIbTaTOM BToporo 51BJI5IeTC5I Ta6JJ11ua.
B 9TOft rJJaBe 6y.n:eT paCCMOTpeHO Bbl'lHCJJeHI-Ie onpeJI:eJleHHblX I1HTerpaJIOB,
CJIeJI:YIOill.a51 r JJaBa nOCB5IUleHa Heonpe.n:eJJeHHbIM.
Onpe.n:eJIeHHblft I1HTerpaJI MO)\{HO BbI'IHCJJ5ITb, KaK npHMeHHH eJI:l1HYIO
cpOPMYJIY Ha BceM OTpe3Ke, TaK H pa36l1Ba5J OTpe30K I1HTerpHpOBaHI15J H
np!1MeH5J5J cpOPMYJIY K Ka)\{JI:oft 113 ero'lac-ret!. Dpl1MepOM nOCJIe.n:Hero 5JB-
JJ5JeTC5J IlIHPOKO I1CITOJIb3yeMa5J cpopMYJJa CHMnCOHa
a+2nh h --
f(x)dx =3 [tea) + 4/(a + h) + 2/(a + 2h)+ 4/(a+ 3h) +...
a
...+/(a+2nh)], (12.1-1)
KOTOpa51 BhIBO.n;IHC5I nOCJIe!J.OBaTeJIhHh!M npl1MeHeHI1eM tjJOPMY JJbl
a-j-.2h
I(x) dx = 4 [I (a) + 4/(a + h) + I(a + 2h)]
a
K Ka}KIJ:OMY !J.BOt\HOMY HlnepBaJIY.
06w.moe C006pa)\{eHHe B nOJIb3Y npHMeHeHlU! eLl,liHoft <POp MY JILl J.l.JI5J
BCerO OTpe3Ka 3aKJIIOLlaeTC5! B TOM, '1TO Ll,JI5! 01lJIH3KOBOrO 4v,CJIa Y3JJOBbtX
T04eK TaKa51 cpOPMY JIa HMeeT OCTaIO'lHbtt! 'WeH 60JIee BblCOKOro f10p5!!J.Ka.
Ho eC.1H B KOMTIJIeKCHOt! rIJIOCKOCTH B6JIl1311 UHTepBaJJa lJHTerpUpOBaH1H1
eeTL Oe06eHHOeTH f10JI:bIHTerpaJIbHOt! lj"JYHKUHM, TO OCTaTO'1HbJt! '1JleH Bbl-
COKOrO nOp5!JI:Ka CKOpee BCero 6Y.lleI 60JIbIllUM. KOHel.fHO, npM h - 0 OC-
raTOl.fHblt\ </JIeH 60JIee TO'IHOt! lj"JOPMYJIbl CTpeMI1TC5! K HYJIIO 6b1CTpee, </eM
.1(JI5J (POPMYJIbl HH3KOtl TO'lHOCTH. !{ CO)\{aJIeHIlIO, Mecro, rJI:e 9TO npel1MY-
UtecIBO Ha</MHaeT pa60TaTb, 06bI</HO HeH3BeCTHO, .n:a)\{e eCJJH I1HTerpaJi
BbIIH1CJ1eH npn n:BYX pa3JII1'4HbIX paS6HeHH5IX, KOr,l{a, HanpHMep, BTopoe
6 p, B, XeMMllur
162
4>OP MY JJbl LlJJfI OnPELlEJJEHHblX HHTE rp AJJOB
[W. 12
BKlIIO t l8eT rrepBOe. D09TOMY Ha rrpaKTHKe '1aCTO rrpe.arrO'lHTaIOT COCTaBHYIO
<POPMY JIY.
LlpyrHM OCHOBaHHeM .amI rrpe.arrO'lTeHHH O.aHOti cpOPMY lIhJ .apyroti HB-
lIHeTCH BJlHHHl1e oKPyr JJeHHH Y3J10BbIX 3Ha'leHHti rrOLlblHTerpaJIbHoti CPYHK-
UHI1 Ha pe3YJlbTaT HHTerpHpOBaHHH. (ho BlIHHHHe H3MepHeTCH CYMMOt!
KBa.n:paTOB BeCOll. EClIH cpopMYJJa LlOJl)lma 6bITb TO'lHOti, KorLla f(x) KOH-
CTaHTa, TO CYMMa BeCOB Wi nOCTOHHHa. Tpe6yeTcH MHHHMH3HpOBaTb
n
wi npH YClIOBHH, '1TO CYMMa Wi rrOCTOHHHa. KaK H3BeCTHO, JlY'illle
i=l
Bcero B3S1Tb Bce Wi paBHblMI-I Me)\(LlY c060t!. JlI060e OTKlIOHeHHe OT pa-
BeHCTBa Bbl3bIBaeT HeKOTopoe YBelIH'ieHHe 01lm60K OKpyr lIeHHH CBepx
MHHHMYMa. OLlHaKO 3TOT 3cpcpeKT He TaK BelIHK, KaK MO)\(HO 6blJJO 6bl
rrpe.n:nOJIaraTb. HanpHMep, B COCTaBHot! <popMYJle CI-IMnCOHa (12.1-1) KO-
JJe6aHHH B Becax BLlBoe LlalOT cpeLlHeKBaLlpaTH'iHblt! 3cpcpeKT
(4(3)2 + (2(3)2 _ 10 _ I 111
2 - 9- ,
no cpaBHeHHIO C e.n:HHHD.et!, KOTopaH nOlIY'IaeTCH, Kor.n:a Bce Beca paBHbI.
EClIH HeKOTopble Beca OTpHl.LaTeJ/bHbl, TO CYMMapHa51 florpeIilHOCTb MO)KeT
6blTb 3Ha'iHTeJlbHO 60JJbIile, KaK, HanpHMep, B cpopMYJIaX HblOTOHa - Ko-
Teca (CM. 12.2) BblCOKoro flOp5lLlKa.
3aLla'ia '1HCJIeHHOrO HHTerpHpOBaHH5I, KaK 6blJ/0 3aMe'ieHO B 7.3,
eCTb 3a.n:a'la Bbl60pa Y3J10BblX TO'leK, flO KOTOpblM MO)KHO OD.eHr-!Tb cpe.n:-
Hee 3Ha'leHHe nOLlblHTerpaJIbHoti <PYHKD.HH. 113J/o)KeHHble LlO CHX nop
B 9TOt! KHHre MeTo.n:bl COCT05lJlH B anflpoKcHMaIUIH MHOrO'lJleHOM flO.n:blH-
TerpaJIbHoti CPYHKD.HH, HJlH MHO)\(HTeJl51 nOLlblHTerpaJIbHot! <PYHKD.HH, 11 no-
CJle.aYIOUl,eM IIHTerpHpoBamm MHOrO'llIeHa. TaKHM 06pa30M, fIoLlblHTerpaJlb-
Ha51 <PYHKD.H5I f (x) paccMaTpHBaeTC5I KaK npOH3Be.aeHHe .n:BYX COMHO)\(II-
relIe1t: K (x) 11 g(x). DpH 9TOM MbJ Tpe6yeM BbInOJlHeHHH Tpex CBOtiCTB,
Bo-nepBblx, <PYHKD.II51 K (x) He LlOJI)\(Ha H3MeH5ITbC5I 3a Bce BpeM5I nOJJb-
30BaHH5I CPOPMYlIOti; OHa MmKeT, KOHe'lHO, 6b1Tb rrpOCTO e.aI-IHIIILetj,
BO-BTOPhJX, MOMeHTbl mk
b
mk = K (x) Xk dx
a
(12.1-2)
tl.OJl)KHbl HaXOLl/HbCSl aHaJIHTH'leCKH. B-TpeTbHX, CPYHKD.II51 g(X) LlOJl)\(Ha
LlOrrYCKaTb LlOCTaTO'lHO XOPOIilYIO arrrrpOKCHMaUHIO MHOrO'lJIeHOM CTeneHII n.
Dpll BbIlIOJ/HeHHI1 3TIIX Tpe6oBaHHt! MaIilHHa MO)KeT BblBeCTH <POPMYJIY
LlJl51 Bbl'lHCJlem-!51 HHrerpaJla. MeTO.n:bl r JI. 10 flOKa3bIBaIOT, '1TO IICKOMa51
<pOpMYJIa CYUl,eCTByeT H MO)\(eT 6bITb Hat!LleHa MalllIlHOt!, eCJlII 3a.n:aTb
Y3J1bI, rrpH YCJlOBHI1, '1TO 1I 10 6 ail rrpOH3BO.ll.Ha51 B y3J10BOt! TO'lKe 3a.n:aeTC5I
BMeCTe co BceMH npOH3Bo.n:HblMH HH3IilHX nOp5l.n:KOB. ECJlH HeKOTopble HJlH
Bce Y31IbI He 3a.ll.aHbI H Bbl6HpalOTCH MeTo.n:OM HHTerpllpOBaHH5I, TO CYUJ,eCT-
eyer PHCK, 'lro KaKHe-JJII60 Y3JIOBble r04KI1 MorYI OKa3aTbC5I KOMnJleKC-
12.1]
BBEJlEHI1E
163
HbJMH '1HCJ!aMU. ::ho, OJlHaKO, He !l.OJ!)I{HO 6bITb HenpeO.ItOJlUMblM npemn-
CTBHeM K npl1MeHeHHIO CPOPMYJIbl, XOTH, KOHe4HO, HeCKOJlbKO 06eCKY-
pa)l{l1BaeT.
B KaIJeCTBe npHMepa npl1MeHeHI1H BblIlleynoMHHYToro MeTO!l.a paCCMOT-
pHM B 4HCJIeHMe MHTerpaJla
J
\' lnx
1= - I + eX dx.
o
(12.1-3)
MHO)l{l1TeJJb In x MMeeT oc06eHHocTb npu x = 0, 11, CJIe!l.OBaTeJJbHO, BOI
nO.ItblHTerpaJIbHaH CPYHKll.I1H He npl16JII1)1{aeTCH MHOrOIJJIeHOM B 3TOM I1HTep-
BaJIe. TIOJIO)l{I1B K (x) = In x, nOJJY4MM MOMeHTbl
I
\ k 1
mk= - J x Inxdx= (k+ 1)2 '
o
(12.1-4)
I
TIoCJIe 3Toro OCTaeTCH annpOKCHMupOBaTb MHOrOIJJIeHOM CPYHKll.MIO 1 + eX '
4TO C.IteJIaTb HeTpY!l.HO, npoCTo B3HB I1HTepnOJIHll.110HHbII! MHOrOIJJIeH, KaK
9TO ,lJ.eJJaJIOCb BO BTOpOI! IJaCTH, I1JIU pa3J10)KHB 3TY <PYHKll.HIO B pHJ.\
MaKJJOpeHa 11 npoHHTerpl1pOBaB nOIJJIeHHO.
K CO)KaJIeHI1IO, HeJ!b3H B HBHOM BM!l.e C<POPMYJIHpOBaTb YCJ!OBU5I, I1PH
BbInOJ1HeHl1H KOTOpbIX CPYHKQI1H MO)KeT 6blTb JIerKO annpOKCI1MHpOBaHa
MHOrOIJJIeHOM. OCTaTOIJHble IJJ!eHbl Ha npaKTHKe 05bI4HO TPY!l.HO HaIlTH,
H HaM npH!l.eTCH pemaTb Bonpoc 0 TOM, KaK pa3511Tb f (x) Ha !l.Ba
MHO)KHTeJl51 K (x) H g(x), U3 !l.p yru x C006pa)KeHH!t
TIpe)l{.Ite Bcero, eCJII1 f(x) I1MeeT oc06eHHOCTb, TO, OIJeBI1!1.HO, OHa
,lJ.OJl)l{Ha Bollnl B K (x). Lipyroe C006pa)l{eHV1e COCTOI1T B TOM, IJTO MHO-
r04JJeH CTeneHI1 n He MO)l{eT I1MeTb 60JJbme 4eM n KopHet! 11 n03TOMY
He MO)l{eT !l.OCTaT04HO xopomo npH6J1M)I{aTb CI1JIbHO KOJIe6molll.YIOC5I
CPYHKll.I1IO. YI HaKOHell., CJIe.n:yeT YIJeCTb, IJTO MHOrOIJJIeH He OCTaeTC!!
OrpaHlPjeHHblM B OKpeCTHOCTI1 6eCKOHeIJHOCTI1. Lio HeKoTopot! CTeneHI1
9TI1 paCCY)I{!l.eHI1H BMeCTe C yCJIOBI1!!MM, 4TO K (x) He !l.OJI)I{Ha MeHHTbC5J
OT pa3a K pa3Y 11 4TO MOMeHTbI K (x) HY)KHO YMeTb HaXO!l.l1Tb aHaJII1-
TI1IJeCKI1, onpe.n:eJIHIOT BbI60p K (x). Yl3JJO)l{MB 3TI1 06lll.l1e co06pa)l{eHI1!!,
06paTI1MC!! K paccMOTpeHl11O HeKoTopbIX KOHKpeTHbIx CPOPMYJI. TIepe4l1c-
JIeHl1e KOHKpeTHbIx CPOPMYJI OT4aCTI1 !l.aeT OCHOBY !l.JIH BbIBO!l.a HOBbIX
CPOPMYJI. C !l.Pyrol! CropOHbl, I1X 06hIIJHO XBaTaeT !l.JIH perneHMH OTHO-
CI1TeJJbHO rrpOCTbIX 3a!l.aIJ.
Ynpa>KHeHH5I
12.1-1. I-1cnoJlb3YH ymmepcaJlbHble MaTpll!J.bI 10.3, uafiTll nHTllTO'le'luylO
(j>opMyny jl,JI!!
1
- ) f (x) In x dx.
o
(3 a M e 'l a H H e: rrpHBecTH 06J1aCTb HHTerpHpoBaHHII K IiI:lTepBaJlY (- 2, 2)
11 3areM Bbf1JHCIII1Tb MOMeHTbl, HClIOJlb3YJI (12.1-4).)
6'
164
<POPMYJlbl AJI$! OnPEAEJJEHHblX I1HTEfPAJlOB
[fJJ. 12
12.1-2. HafiTI1 K091jJ4JIU.J,HeHTbJ .MII
1
C 1 (x)
VI-x 2 dx = a_1f_1 +a_ I //'_1/ 2 + 00/0 + a l / 2 /1/ 2 +a1h
-I
12.1-3. Pacc lOTpeTb 1jJ0PMYlIY ynpalliHeHHR 10.2-3 KaK cOCTaBHYIO. CpaBHHTb
C QJOPMYJlOl1 CHMIJCOna.
12.1-4. )].OKa3aTb, 'ITa eCJlH }:; Wi = C, TO Wl = min Tor,aa H TOJlbKO TOr.1la,
Kor.aa Bce Wi paBHbl Me)/{,ay c06ofi.
t 2.2. <l>OPMY JIbl HbloToHa - KOTeca
JIofll'leCKH caMble npOCTbIe <POPMYJIbI ,LJ)H1 HHrerpnpOBaHn51 - Te, KO-
TopbIe HCrrOJib3YlOT MHO)l(eCTBO paBHOOTCT051IllHX Y3JIOB, - nOJIY4alOTCH
l11uerpHpOBaHHeM MHOr04JIeHa, npOX01lHIllerO 4epe3 Y3JlbI Y3JIbi *)
0, h, 2h, ..., nh rrpHB01l51T K <POPMY JIaM HblOTOHa - f{OTeca
nh
f(x)dx=wof(0)+w1f(h)+W2f(2h) +... +w,J(nh).
o
nepBbIti 4aCTHblti CJIY'Iatt <POPMYJlbl HblOTOHa - f{oTeca - aTO XOpOIllO
H3BecTHoe npaBHJJO TpaneuHtt
h
/(x)dx= [/(0)+f(h»)- 112 h3!'(0), (12.2-1)
o
BTOpOn - <popMyna CHMncoHa:
2h
f(x) dx = [f(0) + if(h) + f(2h)] - ;0 h''/v (0); (12.2-2)
o
TpeTHt! - npaBHJJO Tpex BOChMbIX:
3h
I(x) dx = 3t [f(0) + 3f(h) + 3f(2h) + f (3h)] - ;0 hi't V (8). (12.2-3)
o
T a6JJI1l1.a 12.2-1 1laeT K09<p<PHUHeHTbI 1lJI51 <POPMY JIbI
nh
/(x)dx= Ah [Bo/(O) + BJ(h) + BJ(2h) +... + Bnf(nh)] + Kn!
o
r1le n - q H C JI 0 H H T e p B a JI 0 B, a He 411CJJO Y3JIOBblX TO'leK. f{o9rpqm-
UHeHTbJ CHMMeTpH4HbI, H, CJie.u:OBaTeJlbHO, n:OCTaT04HO .u:aTb JIHIllb 4aCTb
Ta6JIHUbl.
*) 3aMeTHM, 'ITa 3,D,eCb HCIlolIb3yerCij n + 1 Y3J10Uaij TOl/Ka, a HI: n, KaK.
610 JJ,eJlaJlOCb paHbllle.
II 12.2J
<l>OPMYJlbI HbIOTOHA-KOTECA
165
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<POP MY JJbJ )lJJ5I OnPE)lEJJEHHbJX I1HTErP AJJOB
[r JJ. 12
HeKOTopbIe K03cjJ!}JHUHeHTbI CT3HOB5!TC5! OTpHI-L3TeJlbHbIMH npH n = 8.
LLm! n = 9 OHH Bce nOJJO)!(HTeJIbHbJ, HO .D:JI5! n? 10 cYU-LeCTBYJOT H OT-
pl1U3TeJIbHbIe K03!}J!}JI1UHeHTbJ. 3TO npHBO,lJJIT K POCTY OWI160K oKPyrJIe-
HH5!; no 3THM npl1'1I1H3M cjJOPMYJIb! HbiOTOH3 - KOTec3 60JJee BblCOKoro
nOp5!.l.LKa HCnOJIb3YJOTC5! pe.l.LKo. TIOp5!.l.LOK OCTaTO'lHblX '1JIeHOB B03p aCTaeT
H3 .D:Ba npl1 nepeXO.D:e OT He'leTHOrO HOMepa K CJJe.l.LYJOlll.eMY 'leT-
HOMY, TaK '1TO 'cjJOPMYJIb! '1eTHOrO 110p5!.D:Ka npe.l.LnO'lTHTeJIbHee.
06p3TI1M BHHMaHI1e Ha 1105!BJIeHHe CTeneHet! h B OCT3TO'lHOM '1JIeHe_
ECJJ\1 6paTb He e.l.LI1HH'IHOe pa36I1eHHe, TO CTeneHI1 h aBTOM3TH'IeCKH B03-
HI1KHYT B onpe.D:eJI5!JOIll.I1X ypaBHeHH5!X (10.3-1). MO)!(HO JJI160 npOBeCTI1
BbIKJIa.l.LKH C HaCT05llll.HM h, JII160 C4HTan, 4TO h = 1, 11 cooTBeTcTBYJO-
lll.I1e CTeneHI1 h BI111C3Tb 110TOM (.l.LJI5I paBHOOTCT05l1ll.I1X Y3JIOB); 06bl4HO
JIY4we 110CJJe.l.LHee.
ECJIH Ha.l.LO cpaBHHTb 3TI1 !}JOPMYJlbI J1.Jl5I OJ1.HOrO H TOro )!(e I1HTep-
BaJIa, TO CJIe.D:yeT 110MHHTb, 4TO h paBHO .l.LJIHHe 'HHTepBaJIa HHTerpI1pO-
BaHH5!, .l.Le.JJeHHoti Ha n, H BHeCTH COOTBeTCTBYJOIll.He 110rrpaBKH B OCT3T04-
Hble 4.JJeHbJ.
<l>OPMY.JJbl HbJOTOHa - KOTeC3 MorYT 6bITb nO.JJY'leHbI MHOrHMH CI10CO-
63MH. Bep05lTHO, C3Mblt! I1pOCTOt! nyn H3XO)!(.l.LeHI15! K03cjJcjJHu.HeHTOB
OCHOBaH H3 TOM 06CT05!TeJIbCTBe, '1TO cjJopMYJIa f'peropH, Ham1CaHHa51
C BK.JJJO'IeHHeM Bt:ex pa3HocTei1, JWTOpble MorYT 6bITb BbI'IHC.JJeHbl 110 Y3-
.JJaM, 51BJI5IeTC5I TO'lHOt! .D:JI5I MHOrO'lJIeHOB MaKCHMaJIbHot! CTeneHH n, CJIe-
.l.LOBaTeJIbHO, B cjJopMe J1arpaH)!(a OI-ia .D:OJJ)!(Ha 6bITb TaKoi1 )!(e, KaK eCJIH
6bI OHa 6blJIa BblBe.l.LeHa HenOCpe.l.LCTBeHHO. TaKI1M 06pa30M, .l.LJI5! CJJY4a51
n=4 HMeeM
4
l' 1 I 1
/(x) dX=2 10+ 11 + 12+ 13+2 I. +12(/1/ 0 - /1/3)-
o
- i4 (/1210 + /1212) + (/13/0 - /13/J) - 1 0 (/1!'lo + /1'/0) =
14 64 24 64 14
= 45 10 + 45 11 + 45 12 + 45 13 + 45 f..
Ynpa)ImeHHSI
12.2-1. flpHMeHHTb 4JOPMYJlbl HbJOToHa - KOTeca ,IJ.JljJ n = 2, 4, 6, 8 K
I
e-xdx. CpaBHHTb c npaBHI1bHbIM OTBeTOM.
o
12.2-2. PaccMOTpen pOCT norpewHocTH oKpyrJleHHR B cjJopMYJIaX HbJOToHa-
KOTcca.
12.2-3. BblBeCTH cpOPMYJlY HbJOToHa - KOTeca H3 4JOPMYJlbI fperopl1 npH n = 6.
12.3. McnoJIb30BaHHe «l>OPMYJIbl rperopH
KaK 6blJl0 nOKa3aHO, OT6Hpa51 HY)!(HLIM 06pa30M lJJIeHbl B cjJopMYJIe
rperopH, MO)!(HO 110.JJY4HTb K03cp<!mu.neHTbI cjJOPMYJlLI HbJOTOH3 - KOTeca.
9TO 06"b5!CH5IeT, nO'leMY <POPMY JIa rperopl1 TaK nOJIe3Ha Ha npaKTHKe;
II 12.3J
I1CnOJJb30BAHI1E <POP MY JJbI rPE rOPI1
167
OHa H rn5KaH 1-1 T04Ha5l. YaCTO, 4T06bI 1136e)\{aTb OTpIUJ,aTeJIbHbIX K034J-
<pHIUJemOB (KOTopbIe .n:a!OT 60JJbliIOt! pOCT norpeliIHOCTH oKpyr JIeHH5J)
H npOH3Bo.n:HbIX BbICOKoro nop5J.n:Ka B OCTaTO'lHOM '1JIeHe (KOTOpbItJ MO-
)\{eT 6bITb O'leHb BeJIHK), npHMeH5J!OT COCTaBHbIe CPOPMYJIbI, KaK, HanpH-
Mep, COCTaBHY!O <POp MY JIY CHMIlCOHa
2nh
I(x) dx= (jo+ 4f1 + 2/2 + 4/a + ... + 4/2n_l + f n)'
o
ECJJl1 B cpopMYJIe rperopH orpaHWmTbC5J BTOpblMI1 pa3HOCT5JMH, TO
OCTaTO'lHblt! '1JIeH 6y.n:eT Taro )\{e nOp5J.n:Ka, a 01iIH6KH oKpyrJJeHH5J MeHbliIe,
TaK KaK 60JIbliII-IHCTBO K09cpcpHlJ,HeHTOB 6y.n:yr 6JJ1-J3KH no BeJIH'IHHe H
CYMMa KBa.D:paTOB 6Y.D:eT 6JlH)\{e K MHHI-IMYMY.
T a 6 JI H U, a 12.3-1
co
BbI'lHCJIeHMe MHTer_p_ilJIa S e- x2 dx
o
h I I
0,5 0,88622 69524 5 C TO'lHOCTb!O JJ,O II-ro
.n:eCRTH'IHOrO 3HaKa
0,6 69254 8
0,7 69285
0,8 0,88622 72808
0,9 23 598
1,0 32 0
1, I 0,88674
TaKHM 06pa30M, WHpOKO HCnOJJb3yeMa5J cpopMYJIa fperopH
nh
C I I 1
J l(x)dx=2 10+ 11 +... + In-l +2 In+ 12 (f::../o- f::..ln_l)-
o
- d4 (f::..210+ f::..2In_2) + ;;0 (f::..3fo - !l3In_a) - I O (f::..410 + f::..4fn__4) +...
.n:JJ5J paBHoy.n:aJJeHHWX Y3JJOB 5JBJI5JeTC5J HaH60JJee nOJIe3HotJ H rH6Kot!. II pH
BbllJHCJIeHHH TaKOro HHTerpaJIa, KaK
co
e- x2 dx,
- 00
KOHu.eBble nOnpaBKI-I K CYMMe paBHOOTCT05J1lLI1X paBHOB3BeWeHHb1X 3Ha4e-
HHIJ CPYHKll.HH (pa3HOCTH, CT05JliJ,l1e Ha 060nx KOHll.aX <POPMYJJW rperopn)
168
<l>OPMYJJbl iVI5I OfJPEJ1EJJEHHbIX I-1HTErPAJJOB
I fJJ. 12
He .llalOr HHKaKoro BKJJaJ1a. 3Tor pe3YJlbrar yCTaHOBJleH KaI( reOpeTl1Ye-
CKH, TaK H 9KcnepHMeHraJlbHO npH YCJlOBIIII, yro paCCr05lHlIe Me)l(J1Y
Y3J13MII He CJJHIllKOM BeJJI1KO. X3pTpl1 *) .LJ:3.TJ T36JJvu.W pe3YJlbr3rOB
(CM. r36JJ!1lJ.Y 12.3-1), nOJJYlJ31OUJ,I1XC5I nplI BbllJHCJleHHI1 no <POPMY Jle
00 00
1= e- x2 dx=h(-}+ 2>-h 2 k 2 ).
00 k=l
1 .
TIplI h =2 COBn3.LJ:eHlIe npeKp3CHoe. Ho, K CO)!(3J1eHIIIO, HeII3BeCTHO,
K3K C YBeJllIlJeHHeM h n3.LJ:3er rOlJHOcrb. AHaJlH3 OCrarOlJHOrO 4JIeH3
<POPMYJlbl fperopH - CJlHIllKOM rpY.LJ:Ha51 reMa.
YnpalliHeHl1e 12.3-1. HanHcaTb 6J10K-cxeMY JJ,JlR ijJOPMYJlbl I'peropH C aB-
TOMaTl1'leCKI1M OrpaHH'IeHHeM nOpR,IJ.Ka pa3HocTefi. 061>HCHHTe YCJlOBllH 9THX
OrpaHHlJeHHH.
] 2.4. OTlipbITble rPOPMYJIbI
1>OPMYJlbl HblOTOH3 - KOTec3 HHor.LJ:3 H33bIB31OT 33MKHYTbIMH, IIOTOMY
lJro OHH BKJllOlJalOr KOHelJHble TOYKH HHrepBaJla, B npOTIIBOnOJlO)l<HOCTb
OTKpbITblM <popMYJ/aM, Koropble He HCIIOJlb3YIOT KOHelJHbre 3H3lfeHH5I:
nh
j(x)dx=wd(h)+w j(2h)+ ... +wn_Jl(n-l)hj.
o
TIepenlIllleM <POPMYJ/Y B BH.LJ:e
nh
j(x)dx= Ah [BJ(n) + Bd(2h)+...+ Bn_J[(n - 1) h]] + Kn-
o
K09<p<pHUlIeHTbI .LJ:aHbl B r36J1HUe 12.4-1.
T a 6 1I11/J, a 12.4-1
n I A I B 1 I B 2 I 8 3 I B4 I K
n
3 I 3/2 I 1 1 I (3h 3 /4)f" (0)
4 4/3 2 -I 2 (14/45) h 5 / v (0)
5 5/24 II I 1 II (95/144) h 5 / V (0)
6 3/10 II -14 26 -14 (41/140) h7fvI (0)
3TH <POPMYJlbl l1HOr.ll.a HCnOJlb3YIOTC5I npl1 lJHCJleHHOM rmrerpHpOB3HlIH
.lI.H<p<pepeHllHaJlbHbIX ypaBHeHHtt TIOTpe6HTeJllO cJlenyer 6bITb Ocropo)!(-
*) CM. raK)]'!: R. A. Fisher, Phil. Trans. Roy. SOC., London. A 222,1922.
Ii 12.51
KBAilPATYPA fAYCCA
169
HbIM OpB HCn0.1Ih30BaHBH (j.JOPMY.rJ OTKpbITOrO THna, eC.rJB BblqHC.rJI1Tb
3HaqeHHe nO,lJ,blHTerpaJlbHOfi <PYHKlJ.HI-I B KOHlJ.e I-IHTepBaJJa He Y,lJ,aeTC51,
TaK KaK TPY,lJ,HOCTH BblqHC.rJeHH5J, 6b1Tb MO)f{eT, 03Ha'laIOT, 'lTO TaM HMeeTC5J
oc06eHHOCTb, a TOr,lJ,a JJ,onyw.eHHe 0 B03MO)f{HOCTH alJlJpOKCHMalJ.HH IJOJJ,bIH-
Terpa.1lbHOfi <PYHKUI-IH MHOrO'l.rJeHOlll MO)J{eT OKa3aThC5J HenpaBH.rJbHbIM.
Me)f{JJ,y <popMY.rJaMH OTKpblTOrO H 3aMKHYToro T11/Ja CYlu,eCTBYIOT npo-
Me)f{YTOQHble <POPMYJJbl (npHl-laAJ1e)f{aw.He MaKJlOpeHY), KOTopble HenOJ1b-
SylOT YSJ1b1 B cpeJJ,HHX TO'lKaX n HHrepBa.rJOB
nh
f(X)dX='(C!d( )+w f( 3 )+... +w n f ( 2\ 1 h).
o
Ynpa}f{HeHHe 12.4-1. BbIBecTH 4JOPMYJlbJ Tfl1Ja MaKJlopcHa B cjJopMe JJa-
rpaHJKa H3 ynpaJKHeHHH 10.10-1.
12.5. KBaJI.paTypa faycca
Ha3BaHl-le «raYCCOBa KBaJJ,parypa» accoUfmpyerc51 co C.rJYQaeM, KorJJ,a
Bce YSJ1bI CB06oJJ,Hbl. Cyu,eCTBYlOr rpH Wl-lpOKO HCnOJJb3yeMbIx C.'IYQaSl:
orpeSOK [- 1, 1] e nOJJ,bIHTerpa.TlbHbIM BeeOBbIM MHO)f{l-Ire.rJeM, paBHbIM
eAHmme (rayce - JIe)f{aHAp); o6J1acTb lO, 00) C nOJJ,bIHrerpaJJbHblM Be-
COBbIM MHO)J(HTeJ1eM e- x (raycc - JIarepp) H o6J1acTb (- 00, 00) C 1J0JJ,bIH-
Terpa.rJbHbIM BeCOBbIM MHO)f{HTe.rJeM e- x2 (raycc - 3pMwr). B Ka)f{JJ,OM
CJIY4ae BTopoe HM5J npHcoeJJ,HH51eTC51 nOTOMY, 4TO Y3J10BbIe T04KH OKa-
SbIBalOTCSI HY JI51MI1 eOOTBeTCTBYIOill.HX opTOrOHaJIbHblX MHoro'-weHOB CTe-
neHH n. CJJYQaH JIe)J{aHJJ,pa (n OT 1 JJ,O 16) !-I JIareppa (n OT 1 .11,0 15)
saTa6YJ1HpOBaHbI HaUHOHa.rJbHbIM 61OpO CTaHJJ,apTOB [31] *).
Mbl nOKaSaJJH, 4TO Y3JIOBbIe T04K!-I Bce paSJIHQHbI, AetlcTBHTeJIbHbJ
!-I .1Ie)f{aT H HHTepBaJIe HHTerpHpOBaHYJ5J, a Heca Bee nO.llO>KHTeJIbHbI.
OCTaT04Hb1fi QJIeH nponopUHOHaJIeH npOH3BOAHofi nOp5JJJ.Ka 2n
b
_f'2nl (8) C
E -(211)1 .) 7t (x) dt,
a
fAe r. (X) = (X - Xl)(X - X ) ... (X - X n ).
Y!cCJIeJJ,onaHO MHoro JJ,pyrHx llaCTHbIX C.rJY4aeB, B KOTOpblX 5J1I.pO K (X)
I1MeeT oc06eHHOCTH Ha OJJ,HOM I1ml Ha 060l1XKOHUax 06JJaCTH HlHerpH-
pOBaHH5J. OC06bltl HIHepec npeJJ,CTaB.II51eT CJJYllatl raycca - Ye6blWeBa
1 n
\ f(x) "
.) Vf=X2 dx = kiWJ(Xi)'
-1 i=I
*) AHaJlorHIJHbJC Ta6mlu,bl H )])111 60JlbWHX 3HaIJeHHti n HMClOTC!I B KHHraX:
B. 1-1. K P bl JI 0 B H JJ, T. ill y . b r H H a, CnpaSO'lHaH KHl1ra no IJHCJleHHOMY
HHTCrpl1pOBaHl1lO, HaYKa», M., 1966 11 A. C. K P 0 H P O.ll., Y3JIbi 11 Beca KBa.ll.-
paTypHblX 4JopMyn, "HaYKa), M., 1964. (flpUM. peiJ.)
170
4>OP MY JJbI L\JJ5I Of1PEL\EJIEHHbIX HHTErp AJIOB
[rJJ. IZ
Y3JJOBbIe TOqKI1 OKa3bIBalOTC5I TaKI1MI1:
2j-1
Xj=COS 1t,
Tor,Qa KaK BCe BeCa OKa3bIBalOTC5I paBHbIMn
1t
Wi=/i'
Llpyrl1e qaCTHbIe CJJyqan, Bep05ITHO, JIyqllle BCer9 nOCMOTpeTb B [20}
11 [28].
rayccoBbI MeTO,QbI I1HTerpl1pOBaHI15I BeCbMa 3cprpeKTI1BHbI npn npn6JIn-
)j(eHHOM BbJqnCJJeHl1n I1HTerpaJJa no HeMHOrI1M Y3JIOBbIM TOqKaM npn
YCJIOBHI1, qTO CPYHKUH5I (HCKJIlOlJaH H,QPO) MO)j(eT 6bITb XOpOliIO arrrrpoKcH-
MHpOBaHa MHOrOlJJJeHOM. cho cnpaBe,QJII1BO ,QJI5I 60JIbllll1HCTBa npHMepOB
113 yqe6HI1KOB; MHoroe 3aBI1CHT OT 6JJI130CTI1 oc06eHHoCTeti K 06JIaCTH
I1HTerpl1pOBaHH5I.
. Wl1pOKO pacnpOCTpaHeHO MHeHHe, qTO TaK KaK MeTO,QbI KBa,QpaTYP
raycca CTOJIb XOPOIllI1, OHH qYBCTBHTeJIbHbI K He3H8qI1TeJlbHbIM 01llH6-
KaM. He60JIbillOe P8CCY)l(,QeHl1e nOK83blBaeT, 4TO eCJIH Y3eJI 38,Q8H C
01ll116KOt!, TO npH6JJI1)j(eHHoe 3H84eHHe HHTerpaJI8 H3MeH5IeTCH Ha BeJII1-
4HHY 3TOti 0Illn6KI1, YMHO)l(eHHYIO Ha np0I13BO,QHYIO nO,QbIHTerpaJIbHoti CPYHK-
UI1I1 11 Bec B 3TOM Y3JJe. To )I(e caMoe" KOHe4HO, BepHO ,QJI5I JII060ro
MeTO,Qa l1I-ITerpI1pOBaHI15I. TOT cpaKT, 4TO Bce Beca rayccoBoti KB8,Qparypbl
nOJIO)l(I1TeJIbHbI, 06eCne4I1Baer ,QJIH Hee oTcyrcTBHe I1HTepcpepeHIJ.I-II1.
Ynpa>KHeHl1e 12.5-1. floKa3aTb, 'ITa 4JopMYJla
1
V 2 dX= ; [f(- V 2 3 )+f(0)+f( V; )]
-I
TOt:fHa JlH MHOrO'lJleHOB nRTOH CTeneHH.
12.6. <l>OPMYJIbI HHTerpHpOBaHHH CMemaHHoro rayccoBoro THna
9cpcpeKTI1BHOCTb rayccoBoro nO,Qxo,Qa K IIHTerpl1pOBaHl11O H I1CnOJIb-
3yeMbIe 3TI1M MeTO,QOM MaTeMaTI14eCKI1e co06pa)l(eHI15I n03BOJI5IIOT pellll1Tb
MHorne KOHKpeTHbIe 3ana411. B HaCT05IIlI.eM naparpa<jJe 6Y,QYT nepe'mc-
JIeHbI HeKOTopbIe 113 HI1X. Pa,Qo I1CCJIe,QOBaJI CJJY4ati, Kor,Qa HeKOTopbIe
Y3JIbI 3apaHee cpl1KCl1pOBaHbI *). OCHOBHOti I1HTepeC npe,QCTaBJI5IeT CJJY4ati,
Kor,Qa B 411CJIO Y3JIOB BKJII04eHbI 06e KOHueBbIe T04KI1. TIpl1411Ha 3TOro
I1HTepeCa COCTOI1T B TOM, qTO, Kor,Qa COCTaBHa5I rpopMYJIa npl1MeH5IerC5I
KO BceMY OTpe3KY, 3Ha4eHI15I CPYHKUI1I1 B 06Il1.HX KOHlI.eBbIX T04Kax
*) AHaJlofHlJHblH CJlY'IaH paccMaTpHBaJlCH TaK)/{e A. C. KpOHpO OM H 3a-
Ta6Y.ijHpOBaH HM JlR n aT I o 40 B KHl1re, ynoMHHYTOt! Ha CTP, 169.
([JPUM. pea.)
12.7)
CYMMVlPOBAHVlE P5I110B
171
,lJ,OJJ)I{HbI BbI4HCJI5ITbCH JIHlllb O,lJ,HB)I{,lJ,bI. f{onan [20 J 3aTa6ynHpOBaJI 9TOT
CJIY43ti ,QJJH Bcex 3Ha4eHHti n OT 3 Jl,0 11.
PaJIbCTOH (CM. 10.9) HCCJIeJl,OBaJI HHTepecHbIti cJIY'Iatl, Kor Jl,a Beca
Ha KOHuax oTpe3K3 OTJIH431OTCH TOJIbKO 3H3K3MH. B COCT3BHOt! 4Jop-
MYJIe Beca B3aHMHO YHI14TO)l{aIOTCH, 3a HCKJJI04eHl1eM ,lJ,BYX KpatiHHx KOH-
UOB. TaKHM 06pa30M, T04HOCTb YJIY4111aeTCH 6JIarOJl,apH CB060Jl,HOMY
BeCOBOMY nap aMeTpy, a uel' 3 BbI411CJJeHHH nOBbIlllaeTCH JIHlllb CJJema,
H 4eM Jl,JIHHHee 06JJ3CTb HHTerpHpOBaHHH, TeM 9Ta Jl,onOJIHI-ITeJIbHa51 ueHa
OTHOCIITeJIbHO MeHbllle.
Bo Bcex STI1X CJIY4aHX OCT3T04HbIti 4JIeH 06bI4HO HMeeT BHJI,
b
f: 6) k (x) 'lt 2 (x) Q'dx,
a
r.n:e Q' - npOH3Be.n:eHHe MHO)l{l1TeJIeti B 4JHKcHpOBaHHbIX Y3JIOBbIX T04Kax.
Ynpa}f{HeHHe 12.6-1. PaccMoTpeTb npHMeHeHHe <tJOPMY Jib! B ynpa)/{HeHHH
10.2-2 KaK COCTaBHOH 4JoPMYJlbl. CpaBHHTb ee C COCTaBHOH 4JOPMY JlOH CHMncoHa. .
12.7. CYMMHpOBaHue pH!l.OB
<POPMYJI3 fperopH BbIpa)l{aeT HHTerpaJJ KBK CYMMY 3Ha4eHl-Iti nOJJ:blH-
TerpaJIbHoti 4JYHKUHH, BbI4HCJIeHHbIX B p3BHoy.n:aJIeHHbIX T04Kax, C paB-
HbIMH BecaMH, nmoc nonpaBKH, 3aTparHB31OI1U-Ie JIHlllb KOHueBbIe T04KH.
ECJJH 05JJacTb I-IHTerpHpOBaHHH YXO,lJ,HT B 6eCKOHe4HOCTb, TO CYMMa CTa-
HOBHTCH 6eCKOHe4HbIM PH.n:OM.
MO)l{HO, Ha060pOT, BbIpa3HTb CYMMY 6eCKOHe4HOrO p5l.n:a KaK He-
C06CTBeHHbIt! HHTerpaJI, nJIIOC nOnpaB04HbIe 4JJeHbI B Ha4aJIe (TaKHe)l{e no-
npaBKI-I H3 6eCKOHe4HOCT!-I 05p3lll.3IOTC5I B HyJIb BCJIe.n:CTB!-Ie CXO.n:I-IMOCTH
pH.n:a). HanpHMep,
ro ro
_ \ dx + ( l ) _ ( _ )+ ( _ + ) _
'- x 2 - x 2 2 1 2 12 2 2 1 2 24 3 2 ",2 1 2 ' .. -
,\;=1 \
I 3 11
= 1 + 2 + 48 + 432 ... = 1,588 ... (BepHbIt! OTBeT = 1,64 .. .).
4aCTO 6bIBaeT BbIrO.n:HO CJJO)l{l1Tb HeCKOJJbKO nep BbIX (CKa)l{eM, 10) 4JIe-
HOB OT.n:eJIbHO H np!-IMeHHTb HHTerpaJI K OCTaJIbHbIM.
MO)l{eT CJJY4HTbCH, 4TO HHTerp3JI, B03HHKlllHt! H3 PH.n:3, He MQ)KeT
6bITb BbI4HCJIeH aHaJII1TH4eCKH. TorJl,3 MO)j{HO !-IJIH npOHHTerpHpOB3Tb ero
4I1CJIeHHO, HJIH c.n:enaTb 3aMeHY nepeMeHHbIx H npH6JIH)I{eHHO BbI4HCJJHTb
HHTerpaJI no 4JopMYJIe fperopH. TIpH nepexo.n:e OT pH.n:OB K HHTerpaJIY
H 06paTHo .n:Ba)j{.n:bI B03HHK31OT nonpaBo4HbIe 4JIeHbI Ha KOHuax. BblHr-
pbIlll, KOTOpblt! MO)j{HO uo.nYlJHTb UpH TaKHX npe06pa30BaHHHx, COCTOHT
B TOM, 4TO UOJIY4alOlll.Ht!CH HOBbl11 pHJI, CXOJl,HTCH ropa3.n:o 6blCTpee nep-
BOH3'l3JIbHoro.
172
<!>OP MY JJbI llJJ5I ODPEllEJJEHHblX !1HTE [P AJJOB
(fJJ. 12
12.8. 3iP<f>eKTbl 3aMeHbJ nepeMeHHoH
B llBYX <popMYJlaX raycca
00
00
e- x f (X) dx
o .
e-x"f(x)dx,
-00
H3MeHeHI1e MaclllTa6a 5lBJI51eTCH e.n:HHCTBeHHotJ creneHblO cB06o.n:b1. Vlc
nOJlb3YH BTOpOtJ I1HTerpaJl B Ka'JeCTBe npHMepa, 3anHllleM ero KaK
00
1= e- ax [e-(I-<x)X I(x)] dx.
o
ECJJH IIOJIOJKHTb r:J.X = e (a. > 0),
en
1= e- 0 F (B) de,
o
r.n:e
F (B)= e-[O-a)/aj6 I( ),
TO MOJKHO npHMeHHTb TY JKe <POPMYJlY HHTerpHpOBaHHH K 9TOMY HOBOMY
HHTerpaJJY, KaK H K nepBOHa'JaJlbHOMY. Bbl60p 3Ha'JeHHH a. 3aBHCHT OT
Toro, HaCKOJlbKO XOpOlllO COOTBeTcTBylOlll.a51 F (B) annpoKcHMHpyeTC51
MHoro'weHOM 3a.n:aHHoti creneHI1. Bo06lll.e rOBopH, eCJIH 60JIblllHHCTBO
CJJaraeMblX Wi F (Xi) OKa3aJIHCb MaJIbIMH, TO CJle.n:yeT .n:YMaTb, 'JTO a. Bbl-
6paHo Hey.n:a'JHo.
TaKoe H3MeHeHHe MaclllTa6a CTaBHT Bonpoc 0 3aMeHe nepeMeHHol1.
CHOBa HCnOJIb3YH
en
e-XI(x) dx,
o
nOJIY'IHM X = - In B. Tor.n:a npH.n:eM K HHTerpaJIY
I
1(- In B) dB.
G
B03HHKaeT Bonpoc: «KaKYIO <PYHKUHIO I (x) HJIH 1(- In B), COOTBeT-
CTBeHHO B 06JIaCTHX annpoKcHMauHH (0 X co) HJJI1 (0 B I)
JIer'Je npH6JIH3HTb MHoro'JJIeHoM»?
3aKJIIO'JHTeJIbHOe 3aMe'JaHHe Ha TeMY 0 3aMeHe nepeMelIHblx: 06bl'JHO
3aMeHotJ nepeMeHHblx MOJKHO HCKJIIO'JHTb oc06eHHocTH. B npHBe,lJ,eHHOM
paHee npHMepe ( 1.9) paccMaTpHBaJIC51 HHTerpaJI
v
d /(x)
g(y)=([ .) V - dx .
Yo y-x
13.11
OI1I1CAHI1E COIJ,EP)!{AH!,HT TJJABbI H CHCTBMA 0603HA'IEHI1H
173
DO.ll.CTaHOBKa x = y sin 2 e rrpI-IBO.ll.I1T ero K BH.lI.Y
,,/2
g= d 2 Vj/ f(y sin 2 e) sin e de,
o
T. e. YCTpaH5!eT oc06eHHocTb.
12.9. HHTerpaJIbI c napaMeTpoM
llaCTo 6bIB8eT He06xo.lU1MO BbI4HCJIHTb HHTerp8JI BH.lI.8
00
F ( ) = f(x, A) dx
o
.nJI5! HeKoTopbIX 3Ha4eHHt! rrapaMeTpa A.
O!1bIT pa60TbI rrOKa3bIBaeT, 4TO 06bI4HO rrpH .lI.HrprpepeHQHpOB8HHH rro
rrapaMeTpy MO)!{HO Hat!TH IJ,\1cprpepeHIJ.HaJIbHOe ypaBHeH\1e )JJII! F (A), \1 91'0
ypaBHeHHe MO)!{eT 6bITb peweHO B TOM CMbICJIe, 4TO OHO CBO.ll.HTC5! K
I1HTerpaJJY BI1.l1.8
i-
F (A) = A (A) + g(A) dA.
D
B 9TOM BI1.ne Ka)!{.lI.bIt! !lIar Brrepe.ll. rro A .lI.8eT IJ.pyroe perneHHe F (A).
B 60JIblliI1HCTBe CJIY4aeB 91'01' MeTO.n: \!BJI5IeTC5! 60JJee 9rprpeKTl1BHbIM
rrOIJ.xo.n:OM K 3a.n:a4e.
IIpyroe co06pa)!{eHHe COCTOHT B TOM, 41'0 48CTO I1HTerpaJJ 01' CHJJbHO
OClJ.l1JIJIHpYIOw.et\ I10,lI,bIHTerpaJIbHol1 rpYHKlJ.l1H MO)!{eT 6bITb rrpeBpaw.eH B IJ.PY-
rot! C 6bICTpO y6bIBaIOIlJ,eli HeKOJIe6JIIOIIJ,et!C5I rrOIJ.bIHTerpaJIbHot! rpYHKQl1et!.
MO)f{HO CKa3aTb, 4TO 9TY! HeCKOJIbKO 38Me48HI1t! c.n:eJIaHbI Ha TeMY,
KaK He BbJlIHCJI\!Tb I1HTerpaJI. B .lI.et!CTBI1TeJIbHOCTI1 rrpaKTHKa ocyw.eCT-
BJIeHI1\! 9THX COBeTOB OTHOCHTC5! K TOt! 06JIaCTH KJIaCCH4eCKoro «rpop-
MYJIbHO-aHaJIHTH4eCKoro» MaTeMaTH4eCKoro aHaJIH3a, KOTOpOt! pe.ll.KO
BJIa.ll.eIOT B HaWH .lI.HH H KOTOpa\! He OTHOCHTCH K rrpe.ll.MeTY Hawet! KHHrI1.
r JJ A B A 13
HEOnPEJI.EJIEHHbIE HHTEfP AJIbI
13.1. Om-lCaHHe cOJIepmaHHH rJIaBbI H CHCTeMa 0003HaQeHHii
OCHOBHOt! QeJIbIO 9TOt! rJIaBbI HBJI\!eTC5I BBe.n:eHI1e .lI.BYX HOBbIX rrOHH-
nt!. f1epBoe - 9TO HeYCTot!4I1BOCTb, B03HHKaIOw.a51 I1Hor.ll.a rrpH I1CrrOJIb-
30BaHHH y)J{e BbI4HCJJeHHbIX 3HalJeHHt! (pYHlmI1H .lI.JI5I 1J0JIyqeHHR ee CJIe-
.n:YIOIIJ.erO 3HaqeHH5!. ChOT 9cpcpeKT MOiKHO cpaBHHTb C Heycrot!4I1BOCTbIO
B CHCTeMax C «o6paTHol1 CB5I3blO».
174
HEOITPEJ1EJJEHHbIE HHTE fP AJJbl
[fJJ. ta
06paTHa51 CB5I3b eCTb HCrrOJIb30BaHHe 4aCTH BbIXO.n:a B HeKOTopbllt
MOMeHT B Ka4eCTBe BXO.n:a B HeCKOJIbKO 60JIee rr03.n:Hee BpeM5I. !1HOr.n:a
:HOIt 3a.n:epJKKOIt BO BpeMeHH MOJKHO rrpeHe6pe4b, a HHor.n:a HeJJb351.
Bep05lTHO, rrpOCTeltlllHM cpH3H4eCKHM npHMepOM 5IBJI5IeTC5I 06paTHa51 CB5I3b
YCI1JJ!ueJI5I (pHC. 13.1 1). BbIXO.n: eCTb cYMMa .n:BYx BXO.n:OB CJJeBa, YMHO-
)KeHHa51 Ha K03cpcpI1lJ.HeHT yCHJIeHH5I, B351Tbllt paBHbIM - 10 9 . Mw I1MeeM
(y + I X) (- 109) =x
I1JIH
\OlOy _
x= - 109+ 10
10y
1 + \0-8 - 10 y.
TaKI1M 06pa30M, BbIXO.n:HOIt CI1rHaJI paBeH BXO.n:HOMY, YMHOJKeHHoMY Ha - 10,
H COBepllleHHO He 3aBHCHT OT He60JIblllI1X H3MeHeHHIt B xapaKTepI1CTI1KaX
f/. m . b YCI1JIllTeJI5I (T. e. He4YBCTBI1TeJIeH
1I0:J'f'Wll/4l1t:/im YClIl1t:/fllR ;;::::'7[J
. K T04HOMY 3Ha4eHI11O K03cpcpHlJ.I1eH-
Ta YCHJJeHH5I).
IlpH HeKOTopbIX 06CT05ITeJIb-
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T04Ha51 3a.n:epJKKa rrpH B03Bparu:e-
HHH CI1rHaJIa C BbIxo.n:a Ha BXO.n:,
CI1CTeMa OClJ.I1JJJJHPyeT HJII1, KaK
4aCTO rOBOp5lT, «IlIYMI1T». TaKI1e
crnyalJ.HH 4aCTO B03HHKalOT B rry-
6JIH4HbIX ay.n:I1TOpH5IX, KOTopble HMelOT MI1KpOcpOH (Y) H BbIXOJI. YCHJIHTeJIb-
HOlt' CHCTeMbI (x). BbIXO.n: 113 rpOMKOrOBOpI1TeJI5I 3aJI.epJKHBaeTC5I Ha
BpeM5I, KOTopoe Tpe6yeTc5I 3BYKY, 4T06bI JI.06paTbC5I OT rpOMKOrOBOpl1-
TeJI5I .n:o MI1KpocpOHa, 11 eCJIH He rr03a6oTHTbC5I 0 perYJJHpOBKe, TO pe-
3Y;1JbTaT 6YJI.eT CJIblllleH ay.n:I1TopHI1 B BI1.n:e IlIYMa.
Teopl151 CHCTeM C 06paTHolt CB5I3blO XOpOIllO pa3BI1Ta H He MOJKeT 6bITb
113JJOJKeHa 3JI.eCb rro.n:po6HO. CJIe.n:yeT JIl1ll1b Y5lCHI1Tb, 4TO Kor JI.a pe3Y JIbTaT
HeKoero rrpOlJ.eCca I1CnOJIb3yeTc5I B Ka4eCTBe Bxo.n:a B Hero JKe Ha 60JIee
rr03.n:Helt CTaJI.HI1, TO eCTb pHCK rrOJIY411Tb :-ieJKeJJaTeJIbHble KOJIe6aHI151,
rrepHOJI. 11 peJKI1M KOTOpbIX 3aBHC5IT OT JI.aHHOrO npOlJ.ecca.
BTOpOIt HOBOIt l1JI.eelt 5IBJI5IeTC5I pa3BHTHe npHeMa, KOTOpblM MbI rrOJJb-
30BaJIHCb .n:JI5I BbIBOJI.a CPOPMY JI. 110 CI1X rrop HCnOJIb30BaJIHCb Bce I1MelO-
ru:l1eC5I rrapaMeTpbl, 4T06bI c.n:eJIaTb CPOPMYJIY BepHoti .n:JI5I B03MOJKHO 60JJb-
llIero 411CJIa CTerreHelt x. lIpyrl1MI1 CJIOBaMH, Mbl CTpeMHJJI1Cb CJI.eJIaTb
rrOp5lJI.OK OCTaT04Horo 4JIeHa B03MOJKHO 60JIee BbICOKI1M. KaK OKa3blBaeTC5I,
4T06bJ CrrpaBI1TbC5I C HeYCTolt411BOCTblO 11 JI.pyrl1MI1 3cpcpeKTaMI1, TaKHMI1
KaK, Hanpl1Mep, HaKOrrJIeHHe 0lllH60K oKpyr JIeHI151, Heo6xoJI.HMO OCTaBI1Tb
HeKOTopble K03cpcpHlJ.l1eHTbI 06uJ,elt CPOPMY JIbl Heonpe.n:eJIeHHhlMI1 JI.O HaXOJK-
JI.eHI151 OCTaT04Horo 4JIeHa H JIHllIb TOrJI.a Bbl6paTb JI.JI5I HI1X 3Ha4eHH5I,
KOTopble OKOH4aTeJIbHO H JI.a.n:YT HYJKHYIO CPOPMYJIY.
8%03 = !J
Bbi.rorl ;;;:c
PHC. 13.1-1.
13.2J
HECKOJ1bKO npOCTbIX 4>OPMYJI
175
BM4HCJIeHHe HeOrrpe eneHHOrO HHrerpaJJa
x
y(x)= f(x)dx
(13.1- I)
a
9KBHBaJleHTHO perneHHIO HcpcpepeHU\1aJlbHOrO ypaBHeHl-l5!
y' (x)=f(x),
y(a)=O.
(13.1-2)
3Ha4eHI1J1 rrO MHrerpanbHOfi cpYHKUHH 6y eM 0603Ha4arb 4epe3 y', a
OTBer - 4epe3 y. 3rH 0603Ha4eHH5I xoporno cornaCYIOTC5I C CHCTeMot1
0603Ha4eHI1t!, npHH5ITot! npl1 paCCMOTpeHHH HcpcpepeHll.HaJlbHbIX ypaBHe-
Hilt! 60nee 06lllero BH a
y'(x)=f(x, V),
y(a)=b
(13.1-3)
(CM. r n. 15). ECTeCTBeHHO, 4TO rrpH arOM KJJaCC cpOPMY JI, cpe H KoropbIX
CJIe yer HCKarb HY}jWYIO, cY}I(aerc5l. C eJIaeM TaK}I(e He3Ha4HTenhHbIe
il3MeHeHil51 B 0603Ha4eHI151x. LLo CI1X nop y (x) H Yx I1CrrOnb30BaJJ!1Cb ,TI5I
0603Ha4eHI151 O Horo l! Toro }l(e. Terrepb MbI 6y eM rrpHMeH5ITb HH eKC-
HbIe 0603Ha4eHli5I; T04KH, B KoropbIX Mbl BbI4I1Cn5leM OTBer, 6y eM HYMe-
pOBarb 0, 1, 2, ..., war rrO-rrpe}l(HeMY 0603Ha4aerC5I 4epe3 h, a Ha4aJJO
OTC4era - 4epe3 a. TaKliM 06pa30M, y (a + nh) II Yn ecrb O Ha H ra}j{e
BenH411H3. TIyraHHll.bI rrpl1 arOM He B03HI1Kaer, a 0603Ha4eHH5I CraHOB5ITC5I
3Ha4HrenbHO rrpOIlJ.e.
13.2. HeCKOJIbKO npocTblX Ij>OPMYJI 1l:JIH HeOnpe1l:eJIeHHblX HHTerpaJIOB
Bep05lTHO, caMaH rrpOCTa51 cpopMYJla arrrrpoKcllMaUHil O Horo wara
Heorrpe eneHHoro HHTerpana TaKaH:
Yn+l=Yn+hY'[a+ (n+ ) h ]=Yn+hY +l/Jj'
(13.2-1)
r e
y' =f(x)
I1JJH
x
y = Y'dx.
a
04eBI1 HO, 9TO - cpOPMY na cpe Het! r04Kll (10.2-1) H OHa .u:aeT oW1l6KY
B Boe MeHbrne, 4eM 06bl4Hoe rrpaBHno TparrelI.Htt (10.2-3)
+ h (y ' + "
Yn+l = Yn 2 n+l Yn).
<t>opMyna cpe Hett T04Kll nY4we raK}I(e C T04KH 3peHllH acpcpeKTa oKpyr-
JJeHl-l5I.
176
HEODPELlEJJEHHblE I1HTEfP AJJbJ
[fJJ. 13
GOJIee TOtJHa51 cjJOPMY JIa, KOTOpa51 MO)\{eT 6bITb HattlI.eHa npH HCrrOJlb-
aOBaHHH YHI1BepCaJlbHOn MaTpHllbl 10.3, eCTb
Vn+1 = Yn + (-Y -.l + 13y + 13Y +1 - Yn+2).
(13.2-2)
Ee OCTaTOtJHbJn tJJleH paBeH
7 1 do hV 5 ) (8) 0,0 153h 5 y(5) (8).
(13.2-3)
ECJIH J1J151 BbJlJHCJleHH5I HHTerpaJla I1CnOJlb3yeTC5I 9Ta cpopMYJJa, TO HY)\{HO
COC'IHTaTb <PYHKI.U!lO B .D:BYX ,lI,OrrOJlHHTeJlbHbIX TOCJKax, Jle)\{aw.HX BHe
06J1aCfH HHTerp!1pOB3HH5I (rro O.D:HOn Ha Ka)\{.D:OM KOHlJ,e). ECJlH 3HaqeHH5I
rrOJ1bIHTerpaJlbHotJ cjJYHKIJ.HH He MorYT 6bITb HanJ1eHbI B 9TI1X TOqKaX !13-3a
oc06e !HOCTetJ lex), TO rrpe,ll,rrOJlO)\{eH!1e, qTO rrO.D:bIHTerpaJJbHa51 cjJYHKllH5I
MO)\{eT 6bITb XopOWO armpOKCHM!1pOBaHa MHOrOqJleHOM, Bep05lTHO, HeBepHO.
llJI5I KOHIJ.OB !1HTepBaJIa MO)J{HO !1CrrOJIb30BaTb .D:PyryJO cjJOPMYJlY, a
!1MeHHO:
Yi = Yo +;4 (9yo + 19YJ - 5Y2 + Y3) *).
(13.2-4)
llJ151 BbIQHCJleH!151 HeOrrpe,ll,eJleHHbIX !1HTerpaJlOB WHpOKO !1CrrOJIb3yeTcSI
cjJOPMY Jla CI1MrrCOHa (10.4-1). B HaWHX TerrepeIlIHI1X 0603HaQeHlHlX OHa
rrpHHHMaeT BHLl
Yn+1=Yn-i + (y -!+4Y +Y +1),
( 13.2-5)
C OCT3TO'lHbIM 4JIeHOM
_ h"Y (0) A::; _ 0,011 ] h 5 y(5) (8),
(13.2-6)
KOTOpbIn HeCKOJJbKO MeHbwe, 4eM (13.2-3).
Kor,ll,a cjJopMYJla C!1MrrCOHa HcnoJlb3yeTc5I CTaH.QapTHbIM 06pa30M, B03-
HHKaeT 0J1H3 TPYlUJOCTb, KOTopaH Bo06we He OyeHb cepbe3Ha, HO rrpH
CJIyqae MO)\{eT ,lI,OCa.D:HTb. <PopMYJJa CHMI1COHa .D:aeT 3Ha'leHHe HHTerpaJla
Ha .D:Ba wara Brrepe.D:, a qTo6bl HaqaTb llerrOqKY .D:JlH HeyeTHblX Y3J10B,
HCnOJJb3yeTcH CrrellHaJlbHaH «rrOJJOBHHHaH <POPMY JJa CHMrrCOHa» (10.4-2).
Pe3YJIbTaTOM 9Toro rrpbl)J{Ka Ha .D:Ba wara Brrepe.D: SIBJJHeTCSI TO, 'ITO HaKarr-
24
*) Ha caMOM .ucJle Mb!, KOHe'lHO, Bbl'lHCJlReM Ii Yn H TOJlbKO B KOHu,e c'JeTa,
h
Kor.ua Bhme'laTblBaeM Ym YMHO)/{aeM Ha 24' n09TOMY I1PH TOM me KOJlI1'JeCTBe
BbJ'JHCJleHHH MbJ 1136eraeM HaKOIIJleHHR 01l1H60K oKpyrJleHHR.
1\ 13.31
06lli.11I1 METO)],
177
JlHB31Olli.HeC5I OIlll16KI1 no He'leTHI.IM H tJeTHbiM TOtJKaM .1(0 He-
KOTOpoii CTeneHH He 3aBI1C5IT .1(Pyr OT .n:pyra, oc06eHHO 113-3a pa3Hl1UI.l
B secax, C KOTOpI.IMH O.Lt:HH H TOT )I{e Y3eJJ BXO.n:HT B 4eTHI.le 11 He4eTHhle
nOCJIeAOB3TeJII.HOCTH. Bce 9TO crroc06cTByeT B03HHKHOBeHI11O KOJle63HHti.
XOT5I 3TI1 KOJIe6aHH5I npOI1CXOA5IT BCJJe.1(CTBl1e CAeJlaHHhlX OIllH60K fI no-
9TOMY .1(alOT HeKoTopylO Mepy T04HOCTI1 pe3YJlbTaTOB, HHor.n:a OHn MorYT
OK333TbC5I OtJeHI. Henpn5lTHh1MH. Ko.'Ie63HHt! MO)l{HO Im6e)l{3TI., eCJJI1 BI.I-
'1I1CJI5!Tb JJflIllI. uenOlJKY tJeTHI.IX TOlJeK, a .n:Jl5! Ka)l{Aot! HelJeTlIot! T04KH
HCnOJlI.30BaTb «nOJJOBHHHYIO cpOPMYJlY CI1MnCOHa».
Ynpa)J{HeHI151
13.2-1. BbIBecTH 4JOPMYJlbl (13.2-2) H (13.2-4) C HX OCTaTO'lHbIMI1 tJJleHaMH.
13.2-2. CpaBHHTb MeTOJJ, 4JOPMYJlbl CHMITCOHa C 4JOPMYJlOrt cooTBeTcTBYIO-
lllei1 TO'lHOCTH, OCHOBaHHOH Ha cpopMYJle fperopl1 ( 10.10),
13.3. OOIIl.I-IH MeTO.l\
MO)l{HO I1CCJleAOBaTI. MHoro CneUHaJlI.HhlX cpOPMY,'I B OTJJ:eJII.HOCTH, HO
MI.l BepHeMC5I K 06il eMY MeTOAY H HCCJIe,lJ,yeM l eJlI.It! KJIaCC cpa3Y. rIpH
npn6J1!1)1{eHHOM BhI'JI1CJleHHH CJJeAYlOilJ,ero 3HalJeHI15! I1HTerp3J13 Yn+l MO)l{HO
QhIJlO 6I.l rrOJlI.30B3TI.C5I CTapI.!MH 3H3lJeHH5IMH HHTerp3J13 Yn' Yn-l' 3 T3K)I{e
8H3lJeHH5IMH nO,lJ,I.IHTerp3JJI.Hot! cpYHKUHH Y -1> Yn> Y +l> ... 3AecI. MI.)
C03H3TeJlbHO Orp3Hl1lJHMC5I cpOPMYJJOt! BH.n:3
Yn+l = aoYo + atYn_l + a9Yn_2 + h (b_tY +1 +
h'y(5i (0)
+boy +blY -1+b9Y -2)+E5 51 ' (13.3-1)
KOTOpa5! HcnoJlI.3yeT TPH CT3pI.lX 3H34eHH5I I1HTerp3J1a H OAHO HOBoe H TPH
CT3pI.lX 3H34eHH5I nO.1(I.IHTerpaJlbHot1 cpYHKUHH.
HeMeAJleHHo B03HI1KaeT Borrpoc: «nOlJeMY 6I.l He nonp060B3Tb 60Jlee
pa3YMIII.lt! MeTOA, HcnOJlb3YIOIl Ht! 3HalleHH5I nOAhHlTerpaJII.Hot! cpYHKUHH
B TOlJKaX, pacnOJJO)l{eHHI.IX CHMMeTpl111HO OTHOClneJlbHO TOt!, 3HalJeHHe
B KOTOpOt1 HaAO BI.)llHCJIHTb»? TaKot! MeTOA, Bep05!THO, ,lJ,3J1 6I.] 60Jlee
TOllHI.le cpOPMYJlbl, HO MI.l He 6YAeM TaK ,lJ,eJlaTI. no npHllHHaM, KOTopI.le
IJhl5!CH5ITC5! npl1 HCCJJeAOBaHHH 06lliero AHcj)cpepeHUHaJlI.HOrO yp3BHeHH5I
(13.1-3) B rJl. 15.
H3 ceMb napaMeTpoB B cpopMYJle (13.3-1) H3J10)J{[1M nOKa JIHWI. n5lTI.
YCJIOBHt!, a I1MeHHO, llTQ cp9pMYJIa .Lt:OJI)I{Ha 6I.JTb TO'JHO!l ,lJ,JI5I Y = I,
x, ..., x n I1JIH, '1TO TO )I{e caMoe, TOlJIWt! ,lJ,JJ5I Y = 1, (x - h), ..,
..., (x -. h)4, H coxpaHHM ,lJ,Ba rrapaMeTpa JJ,Jl5I ,lJ,pyrHx ueJlet!. PaHI.we 1,
x, x 9 , ... 6I.lJll1 nOJJ,I.IHTerpaJII.HI.IMH cpYHKUI151MH, Tenepb )I{e noJJ, Y = 1,
x, x 2 , .,. MI.l nOHHMaeM OTBeT Ho HHTerpaJi OT CTeneHH x eCTb CTe-
neHb x Ha e,lJ,HHHUY 60JII.Wa5l, n03ToMY HY)l{HO ellie 06'b5ICHI1TI., 334eM MI.)
npI163BHJIH YCJlOBI1e TOllHOCTI1 JJ,Jl5I Y = I, 3AecI. Mbi He .n:a.n:HM 06'b5lCHe-
HH5I 9TOMY, OAHaKO YK3)1{eM, '-ITO HeBI.lnOJIHeHHe 9Toro Tpe6oBaHI151 BJlelleT
He)l{eJlaTeJlI.HI.)e CJIe.n:CTBH5I.
178
HEOnPEUEJJEHHWE WHTEIPAJJW
[IJJ, 1:
VlCIlOJIb3YH YKa3aHHbIe IlSITb YCJIOBl1il 11 BSSIB al 11 a2 B KalJeCTBe Ilapa-
MeTpOB, rrOJIYlJl1M
a 1 =al'
1
b o = 24 (19 + 13al + 8a2)'
I
b 1 = 24 (- 5 + 13al + 32a2)' I
b 2 = 2 1 4 (I - al + 8a2)' J f
Eo = i (- 19 + 11 al - 8a2)'
(13.3-2)
a o = 1 - at - a'l1
a2 = a2,
I
b -1 = 24 (9 - at),
MbI BbIlJHCJJI1JII1 Eo (11 .n:pyrHe K03cpqmUl1eHTbI) TaK, KaK eCJII1 6bI waf h
6bIJI paBeH e.n:r1HHue. Pas60p o.n:Horo rrpHMepa C pa36HeHHeM h rrOKa}KeT
lJHTaTemO, rrOlJeMY 6bIJIO peweHo, '11'0 TaKoti cnoco6 HaXO)/(.n:eHH5I K09cp-
cpl1UHeHTOB (KOTOpbIil .n:ettcTBHTeJIbHO HeCKOJIbKO MeH5IeT CMbICJI Eo) rrpe.n:-
IlOlJTI1TeJIbHee Ha rrpaKTHKe. E:ho He IlpI1IJO.n:HT K rrYTaHl1ue B IlOHHMa-
HHH Eo'
Ynpa>KHeHHe 13.3-1. BbIBecTH ypaBHeHH5I (13.3-2)
13.4. Omu6Ka BCJIetJ,CTBue oT6pacblBaHuH QJIeHOB
06Il1,Htt BI1.n: cpOPMYJIbI (13.3-1) HaIlI1CaH TaK, KaK eCJII1 6bI cpYHKUI151
BJJliSIHHSI O(s) (CM. 11.3) I1MeJJa rrOCTOSIHHbItt 3HaK ( 11.4). Heo6xo.n:HMO
ee I1CCJIe.n:OBaTb 11 HattTH, KaKHe 3HalJemIH al' a2 .n:eJlaIDT 31'0 rrpe.n:IlOJIO-
}KeHHe BepHbIM.
PYKOBO.n:CTBYSICb rJI. 11, Haxo.n:I1M, BO-IlepBbIx, 3HalJeHI1e m = 5 (3a
HCKJJIDlJeHHeM crreUl1aJIbHbIX al' a 2 (CM. 11.2». Vl3 cpYHKUHH BJII1SIHHSI
11.3 Haxo.n:HM A = - 2h, TaK '11'0 cpopMYJIa (11.3-2) rrpl106peTaeT BI1.n:
y(x)=y(- 2h)+(x+ 2h)y' (- 2h)+ (x-1;,2h)2 y" (- 2h) +
+ (X 112h)3 y'" (_ 2h)+ (X-j /h)4 y(4) (_ 2h)+
x
+ it y(o) (s)(x - S)4 ds. (13.4-1)
-2h
ECJII1 no.n:CTaBUTb 91'0 BbIpa}KeHl1e B 061l(YID cpOPMYJIY (13.3-1), TO OKOH-
lJaTeJIbHO rrpl1.n:eM K cpopMYJIe (11.3-8) C
h
R (y) = y(o) (s) O(s) ds,
-2h
13Al
OWI16KA BCJJELICTBVlE OT6PACbIBAHVl5I '-IJJEHOB
17Y
Ole O(s) 3a.rr.aHa CJIe,ll,YIOIUI1M paBeHCTBOM (3aMeTHM, 4TO ,lI,JI5I nOJIY4eHH5I
:Horo paBeHCTBa He Ha,ll,O HHTerpl1pOBaTb):
O(s) = ! {(h - s)+ - ao(- s)+ - a 1 (-h - s)'t- - a3(- 2h - s)+-
- 4h [h_1 (h - s)+ + ho (- s)+ +h1 (- h - s)++
+h3(-2h-s)+]}. (13.4-2)
ECJII1 O(s) HMeeT nOCT05IHHbIti 3HaK ( 11.4), TO
h
R (x 5 ) = 5\ ) O(s) ds = E5h5,
-2h
r.rr.e E5 eCTb pe3YJIbTaT no.rr.CTaHOBKH y (x) = x 5 .rr.JI5I pa30HeHH5I c h = 1.
3a,ll,a4a COCTOHT B TOM, tJTo6bI HatlTI1 TaKHe 3 Ia4eH115I a1' ai' npl1 KOTO-
pbIX O(s) I1MeeT nOCT05IHHbItt 3HaK. TaKI1M 06pa30M, HaM Ha,ll,O HattTI1
HYJII1 O(s) ,lI,JJ5I Ka)j(,lI,ott napbI a1' a2' TaK KaK 3TO C,lI,eJJaTb TPY,lI,HO, rrpH-
6emeM K OObltJHOMY npl1eMY H OOpaTl1M 3a.rr.a4Y: rrOJIO)j(HM O(s) = 0 H
HCCJIe,ll,yeM nOJIY4eHHbIe B pe3YJIbTaTe 3Ha4eHI151 a1' a i .
YlcnOJIb3Y5I (13.3-2), HarU!weM O(s) KaK JII1HettHYIO <PYHKlI.HIO OT a1' ai:
O(s) = 0 0 (s) + a101 (s) + ai03 (s),
rLIe
0 0 (s)= ! [(h-S)+-(-sA- h(h-S)+_ I: h(-S)++
+ h(-h-s)+- (-2h-sA].
0 1 (s) = [( - s)'t- - (- h - s)'t- + (h - s)+ - I: h (- s)+ -
- h(-h-s)++ (- 2h-S)+].
(]2(S)= I [(-S)+-(-2h-S)+- h(-s)+_ I: (-h-sA-
- h (- 2h - s)+ ] .
MbI LIOJI)j(Hbl HCCJIe.rr.OBaTb O(s) B CJIeLIYIOllJ,HX Tpex I1HTepBaJlax:
h s O, O s -h 11 -h s -2h.
B rrepBOM HHTepBaJIe paBeHCTBO
(h - S)3 [(h - s) - h + a1 ] = 0
Jl.aeT
,, + 68
al=<J h'
180
HEOnPEllEJJEHHWE HHTETPAJJW
f TJJ. 13
liTO OnI-IChlBaeT CeMetlCTBO BepnmaJlbHbIX npHMblX B nJJOCKOCTH (ai' a2)'
np5lMbie ABHraIOTC5I OT a 1 = 9 AO al = 3, KOfAa s npOXO.LI.HT OT h AO 0
(pHC. 13.4-1).
Bo BTOpOM HHTepBaJJe HMeeM
[(h-S)4_ S 4_ h(h-s?+ I: hs3]+
+ a 1 [ S4 + (h - S)3 + hs 3 ] + a 2 [S4 + ¥ S3] = o.
8TO ypaBHeHHe onHCblBaeT npSJMble,
az
PHC. 13.4-1. 06.1aCTb nOCTORHCTBa 3HaKa a (s)
(npiIMbIe - JlI1HHH ypOBHR a (s) = 0 .LVIiI YKa-
3aHHblX 3HaqemlH s).
COXpaH5IeT nocToSJHHblti 3HaK.
TOllHWtl lIJJeH HMeer BH.LI.
TaKHM
E5 h5 (5) (8)
51 Y .
KOTopb1e npO.LI.OmKalOr .n:BHfaTbC5I
BJJeBO, HO nOCTeneHHO HaKJJO-
HSJIOTCSJ .n:o Tex nop, nOK3
npH s=- h
al-l
a2= .
Eha npSJMaSJ npoxo.n:I1T lIepe3
TOllKY (1, 0) C H3KiiOHOM
1: 2.
a J B TpeTbeM HHTepBa. e nOJIY-
lIaeM npSJMYIO, Bpaw,alOUJ.Y!OcSJ
OKOJJO TOllKH (I, 0) .n:o no.
JJO)f(eHH5I
a 1 -1
a2 = ----.:r-
BbIlue 11 CJJeBa OT 9TOt! npH-
MOH (CM. pHC. 13.4-1) CPYHK-
II.H5I BJJ[JiIHI151 O(s) He HMeer
HYJJefi *) H, CJJeLI.OB3TeilbHO,
06pa30M, B 9Tofi 06JJaCTH OCTa-
BHe ,noro pat!OHa Mbl He HMeeM TaKOro ocraTotJHoro tJJJeHa. 1J3. bllle
9ror Bonpoc HCCJle.n:OBaTbC5I He 6y.n:er.
B 9TOH 06JJacTI-I 0lllH6Ka Bblpa)f(aerc5I lIepe3 E5 COrJJaCHO (13.3-2).
np5lMbie «nOCT05lHHOH 0IllH6KH» 11306pa)f(eHM Ha pHC. 13.4-2 CBerJJblMH
*) CYill,eCTByeT TaKllie 06J1aCTb CIIpaBa BBH3Y, B KoropoH a (s) He MeHRer
3HaKa, HO KaK BCnHqf!Ha OCTarOqHOfO qJlCHa, raK f! 94J4JeKTbI OKPYfJlCHlfR YJ(ep
lliHBalOT Hac aT HCIIonb30BaHHR raKHX 4JOPMYJI; B JJ,aJlbHeHlUeM MbI 6Y,IJ.eM HfHO-
pHpOBaTb 9ry 06J1aCTb.
13.51
YCT0J14I1BOCTb
18!
JlIHI11HMH, 3TH JIHHHH nOKa3bIBaIOT, lJTO TIPH np04HX paBHblx YCJIOBHHX Mbl
.n.OmKHbI nOIJbITaTbC5I .n.ep)l(aTbC5I HH)/{e H npaBee, r.n.e OCTaT04Hblti lJJleH
-/
c {1;
PHC. 13.4-2. JJHHHI1 paBHOH Oml16KH (E - 3HalJe-
HHe omH6KH B,IJ.OJlb KaJK,IJ.OH JlHHHI1).
MaJIeHbKHti. HaKJIOH JlHHIIti 11:8 nOKa3blBaeT H3MeHeHHe OTHOlIJeHH5I .n.BYX:
napaMeTpoB; eCJII1 YBeJlHlJl1BaTb a2 Ha 11 e.n.I1HI1U, a a1 Ha 8, TO 0llIli6Ka
OCTaeTC5I Toti )Ke caMott.
I 13 . YCTO qUBOCTb
YlcCJIe.n.yeM CHalJaJIa nOJlY4eHHble <POPMYJlbI Ha TpIIBHaJIbHOM CJJY'Iae
y'=0. (13.5-1)
ECJlU Y)I(e 3.n.eCb B03HIIKHYT 3aTpy.n.HeHH5I, TO B 60JIee 06w.eM CJIY4ae
y' =f(x)
(13.5-2)
HX CJIe.n.yeT O)l(II.n.aTb II no.n.aBHO.
ECJIH 'B KalJeCTBe TOlfKH nJIOCKOCTH (ai' a2) BbI6paTb al = - 1"
a2 = 0, TO <popMYJla npIIMeT BH.n.
Yn+l = 2Yn - Yn-t.
PacnlOTpHM, 'ITO CJlYlJHTC5I C pellleHl1eM Y = 0, eCJIH .n.onYCTl1Tb B Y.,
He60JlblllYIO olllu6KY e (npe.n.CTaBHM ce6e, lfTO OHa np0!130lllJIa OT oKpyr-
JIeHH5I). J]erKO BbIlJHCJIHTb Ta6JIHuy 13.5-1. TaKHM 06pa30M, olllH6Ka Bce
Bpel'vIIl B03paCTaeT.
B Ka4eC"fBe .n.pyroro npl1Mepa BbI6epeM at = I, a 2 = 1. YpaBHeHI1e
eCTb
Yn+1 = - Yn + Yn-I + Yn-2
!1 1l0JIYlfaeTC5I Ta6J1IIUa 13.5-2. Tenepb olllI16Ka paCKa'lHBaeTC5I H TaK)I(e:
pacreT.
182
[fJl. 13
HEODPEilEJJEHHWE I1HTEfPAJlW
T a 6 JI f1 U a 13.5-1
Y-2 = 0
Y_l=O
Yo=.
Yl =2.
Y2 = 3s
Ya = 4.
T a 6 II II U a 1:-1.5-2
Y-2 = 0
Y_l=O
Yo=.
Yl=-.
Y2 = 2.
Ya =- 2.
Y4 = 3.
Y. = - 3.
9TH JJ:Ba npHMepa nOKa3bIBarOT, ,-[TO 9TO 5IBJIeHHe Ha a:0 HCCJIeJJ:OBaTb
'ilO)].p06Hee; 5ICHO, lfTO HeKOTopbIe TOlfKI1 (al> G2) OlfeHb nJIOXO BbI6paHbI.
06w.a5I cpOPMYJI8 (13.3-1), KOTOpa5I HCCJle)].yeTC5I, 5IBJHleTC5I JII-IHe!!HbIM
p83HOCTHbIM ypaBHeHHeM C nOCT05IHHbIMH K09cpcpHUHeHTaMH, H MO)/{HO
npHMeHHTb MeTOJJ:bI rJI. 5. TaK KaK 3HalfeHI15I y' eCTb 3HalfeHH5I nOJJ:bIH-
TerpaJIbHo!! cpYHKUI1H, KOTopbIe BbIlfHCJI5IIOTC5I He3aBHCHMO OT 3HalfeHIj!! y,
TO paCCMOTpHM pa3HOCTHOe ypaBHeHHe B BHJJ:e
Yn+t - aoYn - atYn_t - a2Yn_2 = Fn- (13.5-3)
I1pe)/{)].e Bcero, peuIHM O)].Hopo)].Hoe ypaBHeHHe
Yn+t - (1 - at - a2)Y" - at.Yn_t - a2Yn_2 = O.
(13.5-4)
D.Jl5I oHoro nOJlO)/{HM Yn = pn H nOJIYlfl1M xapaKTepHCTHlfeCKoe ypaBHe-
HHe
flJIH
p3 _ (1 - at - a2) p2 - atP - a2 = 0
(p - 1) [p2 + (at + a2) P + a2] = O.
(13.5-5)
I1YCTb Pt H P2 - HYJIH KBaJJ:paTHlfHoro MHO)/{HTeJI5I. PeweHHe (13.5-4) npH
YCJIOBHH, lfTO HHKaKHe )].B8 KOpH5I He COBna)].aIOT, eCTb
Yn = C t (Pl)n + C 2 (P2)n + Ca. (13.5-6)
J{or)].a JJ:Ba KOpH5I paBHbI, KaK 9TO CJIYlfHJJOCb B )].BYX BepXHHX npHMe-
"pax, cpOPMYJIY (13.5-6) Ha)].o H3MeHI1Tb. ECJlH Ka)/{)].bIti. 113 )].BYX KopHen
'! 13.51
YCTOI1'-1V1BOCTb
183
B (13.5-6) no MO.n:YJJlO 60JlbllJe e.n:I1HHll.bl, TO pellJeHl1e 6y.n:eT paCTH no-
.lI.06HO reOMeTpl14ecKotl nporpeCCHH no Mepe Toro, KaK n YBem-l4HBaeTC5f
(npl1 YCJlOBI1I1, 4TO COOTBeTCTBYIO[lJ.l1e K03cpqmll.l1eHThI He HY JII1).
D03TOMY HCCJle.n:yeM B nJJOCKOCTI1 (at, a2) 06J1aCTb, .n:Jl5I KOTOpOti
I Pi I < 1. CaMbltl JlerKl1t1 nyn - onpe.n:eJlHTb rp aHI1 11.bl 3TOti 06JJaCTI1.
Oz
flp08U/lO J/8
"
)
II
C)..
7,0
OJ
-7,.P2,0
PHC. 13.5-1. 06J1aCTb YCTofi'lI1BOCTI1 Ha nJlOCKOCTH.
CHa4aJla I1ccJle.ll.yeM rpaHI1UY, B.lI.OJlb KOTOpOti KopeHb CTaHOBI1TC5I paB-
HblM 1. D.Jl5I 3Toro nOJlO)!{I1M P = 1 B KBa.n:paTI14HOM MHO)KI1TeJle. 9TO.
onpe.n:eJl5leT npSIMYIO
1 + at + 2a 2 = 0,
l+a 1
a2=- .
(13.5-7)
J],aJlee, HCCJJe.n:yeM, r.n:e P = - 1; nOJJO)KI1M B KBa.n:paTH4HoM MHO)KI1TeJle.
P = - 1. 9TO onpe.n:eJII-1T npSIMYIO
at = 1. (13.5-8}
Tenepb HCCJle.n:yeM, r.n:e KOpHH CTaHOBSITC5I KOMIIJIeKCHbIMH. rpam14HaSI
KpHBaSI onpe.ll.eJlSIeTCSI ypaBHeHl1eM
(aj + a 2 )2 - 4a2=O
I1JJI1
at = + 2 V a2 - a2 (13.5-9}
(:JTa KpHBaSI nOKa3aHa H3 pl1c. 13.5-1). BHYTPI1 napa60Jlbl KOpHI1 KOM-
nJJeKCHble, B3al1MHO COnp5l)!{eHHble 11 06a I1MelOT lIIO.n:YJJI1
lp!= .
'184
HEOnPEnEJJEHHWE MHTEfPAJlW
[f n. 13
TaKI1M o6pa30M, o6JIacTb, B KOTOpOtl I Pi I < 1, OrpaHI1'JeHa npHMotJ
a = I,
(13.5- 10)
KOTOpaIJ KaK pa3 nepeceKaeT napa60JIY (13.5-9) TaM, rJ\e OHa KaCaeTCIJ
npIJMbJX (13.5-7) H (13.5-8). 3TH TPH npIJMbJe 06pa3Y1OT TpeyrOJJbHHK
(pnc. 13.5-1), BH}'Tpn KOToporo 11 J\OJJ)!{HbJ JJe)J{aTb xapaKTepI1CTI14eCKHe
KOpHH, '-IT06bI MeTOJ\ HHTerpHpOBaHHH HMeJI oilm6KY, KOTopaH He pacreT
KaK reOMeTpH4eCKaH nporpeCCHH. BHe ':Hot! 06JIaCTH HeKOTopble I Pi I> 1.
JlerKo BHJ\eTb, '-ITO Jl,Ba HJIJllOCTpHpylOliWX npl1Mepa 3Jl,eCb COOTHeT-
'CTBYlOT TO'-lKaM, KOTopble JJe)!(aT Ha KpalO 06JIaCTH 11 Jl,aIOT Jl,BotJHbIe
KOpH11 B xapa1CTep11CT114eCKOM ypaBHelll1H, a TpOtJHOtJ KopeHb B CBOIO
oClepeJ],b npHBOJl,HT K JIHHetlHoMY poery. KorJ],a I Pi 1<1, MbI rOBOpI1M,
ClTO MeTOJ], YCTofl411BbItI; KorJ],a \ Pi \ = I, MbI rOBOpI1M, ClTO OH YCJJOBHO
YCT0t14I1BbItI (JII1Wb J],JIH npocTbIX KopHet1).
PeWeHl1e OJ],Hop0J],Horo ypaBHeHI1H (13.5-4) J],OJI)!{HO 6bITb LLo6aBJIeHO
K peWeHI1!O nOJIHOrO ypaBHeHI1IJ (13.5-3). KorJ],a pewaeTcH (13.5-3),
JII06aH oWH6Ka oKpyr JIeHI1H 6YJl,eT HaClaJIbHbIM YCJIOBHeM LLJIH peWeHI1H
OLLJ-JOpoLLHoro ypaBHeHBH 11 nopoJ\HT peweHHe, KOTopoe CTpeMHTcH K 6ec-
KOHe'-lHOCTH (eCJIH TOJIbKO He OKa)j{eTC5J, '-ITO Pi no MOJ\y.1J1O He 60JJbme
eJl,11HHUbJ) He3aBI1CHMO OT nOBeJ\emJH Fn.
He)j{eJIaTeJlbHble pemeHHH B03HHKaIOT OTToro, '-ITO Jl,HrpcpepeHlJJIaJIb!We
ypaBHeHf:!e nepBoro nopHJ],Ka 3aMeHHeTCH pa3HOCTHbIM ypaBHeHHeM Tpenero
nopHJ],Ka. KopeHb P = 1 cooTBeTcTByeT )KeJIaeMoMY peWeHI1IO, a J],pyrne
J],B3 Pi - He)!{eJIaTeJIbHbIM pellleHHHM. TIpH Tpe6oBaHl-!H i Pi i 1 OHB He
OKa3bIBa1OT 60JIblllorO BJIHHHH5J Ha OKOH4aTeJIbHbltJ OTBeT.
YnpamHeHHSI
13.5-1. PacClI!OTpHTe npHMep G j = - 1, G 2 = 0 H nO/(8)f(HTe, HcnoHh3YR
ypaBl!eHl1e (13.5-6), 'no Bbl'1HC. eHHaR Ta6J1Hu,a COBIJaJ],aeT C TeOpem'leCKoJ:i.
13.5-2. To )/{e caMoe, '1TO 11 B ynpa)/{HeflHH 13.5-1, .lI..1JR G 1 = 1, G 2 = 1.
13.5-3. Pacc !OTpeTb c.1JY'laH G 1 = 2, G 2 = O. f1oKa3aTb, 'ITa, B COOTBCTCTBHH C
TeopHci1,OIlIH6Ka paCTeT B reOMeTpH'ICCKOH nporpeccHH.
13.5-4. PaccMoTpeTb nOBe.ll.eHHe olll\16KH npH nepeCe'leHHI1 Ka}KlJ,OH H3 Tpex
rpaHH'IHbIX JlHHI1H 06J1aCTH YCroll'lHBOCTH.
S 13.6. lIlYM OKPYfJIeHRH
3Ha'-leHHH Yn' KOTOpbJe Mbl BbI'-InCJI5IeM, 6YLLYT, Boo6uJ,e rOBop5J, I1MeTb
HeKoTopbIt1 ypoBeHb lllYMa OKpYrJIeHI151 o. Ha rrepBbItI B3rJIHLL Ka)l{eTCH,
4TO K03cpcpI1U11eHT YCHJIeHH5I mYMa oKpyr JIeHI1H paBeH
Na=(ag+ aj + a )1/2.
OJ],HaKO, TaK KaK 3Ha4eHHe Yn+1 BbJ4HCJI5IeTC5I 113 3Ha'-leHHt1 Yn' Yn 1 11
Yn-2' WYM B Yn+/ CB5I3aH (KOppeJIHpOBiJH) C mYMOM B Yn' Yn-l' Yn :b no-
.aTOMY CJJeJ\yer OTHeCTHCb K Bonpocy 60JJee BHHMaTeJJbHO.
13.6]
mYM OKPYfJ1EHI151
185
PaCCMOTpHM pemeHl1e ypai3HeHH5I
y'=O,
KOTOpOe eCTb TO)\{.zreCTBeHHbIt! HYJIb; npe.zrnOJIO)\{I1M, '1TO B03HHKJJ3 MaJJeHb-
Ka51 Oml16Ka oKpyr JIeHH5I. PemeHHe (13.5-6) eCTb
Yn= C I (Pit + C (p t + Ca. (13.6-1)
ECJJH MeTO.zr YCTOt!lJHB 11 !Pil<:l, TO
Yn C a , Kor.zra n 00.
BbllJl1CJJHB C a 113 HalJaJIbHblX .zraHHbIX
y =O, Y I=O, Yo=S,
I1MeeM
C 1 + C 2 + C =0 C 1 + C 2 +C =0
PI P a , Pl P2 a ,
C 1 + C + C a = S .
Pewa51 CHCTeMY OTHOCHTeJIbHO C a , nOJIYlJaeM
e
C a = 1 - (Pl + P2) + P1P2 .
VlCnOJIb3Y5I KBa.zrpaTl14Hblt! MHO)\{HTe.'Ib B (13.5-5), Haxo.zrI1M
PI + P = - (a l + a ),
PIP = a .
CJJeAOBaTeJIbHO,
C a = e
I + a 1 + 2a 2
Nes,
(13.6-2)
r.zre
N
c
1
1 +a 1 +2a 2 '
TaKI1M o6pa30M, '1T06bl omH6Ka oKPyr JIeHH5I S He pOCJJa, HMO 4T06bl
I N c [= [ 1 +- al + 2a I-I I
11 4eM MeHbwe, reM JIY'Iwe. TaK KaK 06bl4HO npHxo.zrI1TC5I HMeTb AeJIO
c nOJIO)\{HTeJIbHbIM Bblpa)\{e!meM 1 + al + 2a , TO YCJIOBl1e, 4T06bl 1130J/H-
pOBaHHa51 oWH6Ka S He pOCJIa, COCTO[1T B TOM, 4T06bl T04Ka (aj, a )
JIe)\{aJJa Bblwe np5lMot!
a 1
a =-2-'
(13.6-3)
9ra 06JIaCTb orp aHH4eHa Ha p HC. 13.5-1 nYHKrnp HOt! JIHHHet!.
YnpalliHeHMe 13.6.-1. TeOpl1R oKpyrJleHHi/, npHBOJ],i!lIJ,aR K ypaBHcHHlO
(13.6-3), 6bI1la pa3BHTa Jl,JlR eJl,HHCTBeHHoH H30nnpoBaHHoi\ oll1H6KI1. OJJ,HaKo
Ka)I(LI,bIH mar 6YJJ,CT flOpOlliJl,aTb HCKOTOPYIO omH6KY, KaK H3MeHHTb aprYMeHTa-
UHIO npHMeHHTCJlbHO K 3TOMY cJlyqalO?
186
HEonPEAEnEHH E HHTEfPAnM
IIn.13
13.7 I'ITorn
Ta6J1l.JU.a 13.7-1 nepetJHeJIHeT pHA XOpOllIO H3BeeTHbIX MeTo.n:OB, KO-
TopbIe nOna.ll.alOT B 06llI.HtI KJJaee (ypaBHeHHe (13.3-1 )), HeCJJe.ll.ye-
MhItI HaMH.
T a 6 .1 H U, a 13.7-1
HeKOTopble MeTO.ll.bl IIHTerpHpOBaHIISI
A aMt - fIpaBHno 2/3
oaw$opT CHMncoH Tpex BOCb I 1/3 1/2 «s»
MbIX I
i \
a o I 0 0 1(3 [(2 0 9f8
a, 0 I 0 ](3 If2 213 0
a, 0 0 1 11 3 0 1)3 -1/8
b 91 24 1/3 3/8 13136 17/48 25(72 318
b o 19/24 4f3 918 39/36 51/48 91172 6f8
1 b , -.5124 1/3 9/8 15/36 3f48 43)72 -3f8
b 2 1(24 ° 3/8 51 36 If48 9/72 0
IE" -19/6 -4(3 -9(2 -3 -9,4 43(18 -3
jYOInCliHe
m)'MaN c I I I, 0, I, 0 I, 0, 0, 1, 0,0 1/2 2/3 3/7 1,13
N a I I I I 11 Va I(Vf V513 V 82f 8
TOtJKa al = 1, a2 = 0 AaeT MeTO.ll. CHMneoHa H paenOJJo)j{eHa Ha
-rpaHHlle 06JIaCTH yeTotltJHBoeTH.
TOtJKa a 1 = 0, a2 = 1 .lI.aeT npaBHJJO Tpex BoebMblX (12.2-3) H TaK)j{e
JIe)j{HT Ha rpaHHlle 06JIaeTH yeTofitJHBoeTH.
TOtJKa al = 0, a2 = 0 .lI.aeT CPOPMYJIY «A.uaMea - EallIcpopTa», Ha-
3BaHHYlO TaK nOToMY, llTO OHa HenOJIb30BaHa B MeTo.n:e A.lI.aMCa - EallI-
cpopTa HHTerpl1pOBaHH51 .lI.l1cpcpepeHllHaJIbHbIX ypaBHeHHJt (eM. r JI. 15).
Bee .apyme MeTO.a o6Illero BH.aa (13.3-1) 5JBJHllOTCJI JJHHefiHbIMH
KOM6HHallH5JMH 9TH X Tpex. PHe. 13.5-1 .aaeT eooTBeTcTBYlOllI.YlO HHcpOp-
MallHlO. Ka)j{.aaH TOtJKa B nJIOeKOeTl-I (ai' a2) eOOTBeTCTByeT MeTo.ay
HHTerpHpoBaHHH.
Bo-nepBblx, MbI crpeMHMe5J BbI6paTb TOllKY BHYTPH TpeyrOJIbHHKa
YCTofillHBoeTH, XOTH MO)j{HO OeMeJIHTbCH B HeKoTopbIX eJIYllaHx BbIfiTH
Ha rpaHHllY, KaK B MeTO.ll.e CHMTICOHa H npaBHJIe Tpex BOCbMbIX. Bo-
BTOpbIX, eJIe.n:yeT .n:ep)j{aTbeH HanpaBJIeHIUI BHH3 HarrpaBo, llT06b1 HMen
B03MO)j{HO MeHbllIee Eo (pHe. 13.4-2). B-rpeTbHx, MbI XOTHM .ll.ep)j{arbefi
BbIllIe rrYHI{THpHOJt JIHHHH (pHe. 13.5-1), llT06bI coxpaHHTb KaK MO)j{HO
MeHbllIl1fi llIYM.
TaKHM o6pa30M, 3a I1CKJIlOtJeH!-IeM 6eCnOKOHllI.ett TeH.lI.eHllHH K KOJIe-
6aHHlO, cpopMYJIa CHMncoHa - OJI.Ha 113 JIYtJUlI-IX B paCCMorpeHHotl o6.llI.etl
cpopMe (KorOpa51 He BKJIlOtJaeT (13.2-2)). TIoTpe611TeJIlO CJIeAyeT oT06paTb
ce6e c06cTBeHHbItt MeTO.lI., nOJI.XOJI.51llI.HtI JI.JIH KOHKpeTHoro COOTHOUleHH5I
13.81
HEKOTOPbIE OBllJ,\1E 3AME4AH\151
187
cpaKTopOB B Ka)l{.n:otl np06J1eMe; HeJlb3H npe.n:JJmKHTb e.n:I1HCTBeHHYlO, CaMYIO
XOPOlllYIO cpOPMYJIY npH BCeX 06CTOHTeJIbCTBaX. MeTOAbl H3 Ta6JJI1lJ,bl
13.7 -1, Ha3BaHHbIe 1/3, 1/2 11 2/3, HBJIHlOTCH CMeCbIO OCHOBHbIX MeTO.LJ,OB
11 I1HTepeCHbI B HeKoTopbIX npI1JIO)l{eHI151x.
YnpamHeHHe 13.7-1. Pa3BHTb Teopmo 13.3 110 13.7 (HC1IOJlb3YR Tom,KO,
011HH napaMeTp a j ) llMI 4JOPMYJl 061lJ,ero BHlla
h y(IV) (0)
Yn+1 = aoYn + alYn_j + h (b_jY -1 + boY + bjY -1) + E4 41
H cpaBHHTb C H3BeCTHbJMlI pe3YJlbTaTaMlI.
o T BeT:
\ al =0 CHMIICOH aj = t/2 I aj =0 I CHMIICOH la l =I/2'
a o = I - a j I 0 1/2 b _ -1+5a j -1/12 1/3 1/8
aj=a j 0 I 1/2 1- 12
5-a j E4=-I+a j -I 0 -1/2
b -I = -----r:z- 5112 1/3 3/8 Y CI1.1eHHC IllYMa
b _ 8+8a j 8/12 4/3 8/8 1 1 I, 0, 1, 2/3
0- 12 N c = I +a j 0, ..., I
N a I I I/V2
PI = - aj, c.1elIOBaTeJlbHO, 06.ijaCTb YCTOi\.4HBOCTH I < a j < I.
L
13.8. HeKoTopbIe oom:He 3aMeqaHHH
BHl1MaTeJIbHbItl 411TaTeJIb, BepOHTHO, 3aMeTHJI, 4TO 6bIJI IJCCJIe.n:OBaR
JJlllllb OAHH OC06bItl KJIaCC cpOPMY JJ; HO B03MO)l{HbI 11 .n:PyrHe cpOPMY JIbl.
HanpHMep, MO)l{HO paccMaTpHBaTb cpOPMYJIbI, B KOTOpbIX I1CnOJIb30BaJIHCb 6bl
KaK 3HalfeHI1e nOAblHTerpaJIbHOii cpYHKlJ,111l (y'), TaK 11 3HalfeHile npoH3-
BOAHOt! nOAblHTerpaJJbHot! cpYHKlJ,111l (y"). B 3TOM KJIaCCe eCTeCTBeHHO
O)l{HAaTb 60JIee XOpOlllHX cpOPMYJI B CMbICJIe 3cpcpeKTIlBHoCTil BbllfHCJIeHHfi,
TaK KaK 4aCTO np01l3Bo.n:HaH JlerKO BbI'lI1CJlHeTC5I 113 pa3J1H4HbIX KYCKOB'
BbI'lIlCJleHIlH nOAblHTerpaJlbHot! cpYHKlI.lll1. 3TO nOTpe60BaJlO 6b1 €)OJlblllet!
KO.n:llpOBKIl (60JIbllle Olll1160K) 11 AJIH CJlY'latlHot! 3a.n:alfl1, B03MO)l{HO, 6bIJIO 6bI
nJJOxoti CTpaTemetl. O.!!.HaKO .!!.JJH 4aCTO nOBTOpHIOllJ,eiiCH 3a.rr,;:i\lH TaKl1e
cpOPMYJJbI MOrJJIl 6b1 3Ha41lTeJJbHO COKpaTI1Tb Mallll1HHOe BpeMH 1:1 AOJJ)I{Hbl
6bITb HCCJJe.n:OBaHbI.
MOIlJ,HbIe cpOPMYJJbI TIma faycca He npl1cnoc06JIeHbl .n:JIH Heonpe.n:e-
JleHHOrO HHTerpaJIa 1l3-3a TPYAHOCTIl nO.LJ,rOHKIl CnelI.I1cpH4eCKoro pa3-
6I1eHH5I K nOCJle.n:OBaTeJJbHbIM rnaraM.
188
HEOnPEIJ.EnEHHbIE I!!HTE rp AJJbI
[rn, 13
I1pIlMeHeHHe <POPMY JI cpeIJ.Heii TO'-!KH (13.2-1) (3Ha'-!eHl-HI y' B CpeLLHI1X
TO'-!Kax, 3Ha'-!elm5! y B Y3noBblx), HaCKOJIbKO H3BeCTHO aB'ropy, HeLLOCTa-
TO'-!HO HCCJIeLLOBaHO. aHO TaK)!{e LLOJI)!{HO .lI.aTb HHTepecHble pe3YJIbTaThI.
BbI60p E5 B Ka'-!eCTBe OCTaTO'-!Horo '-!JIeHa HaWeJJ wHpoKoe pacnpo-
cTpaHeHHe B npaKTHKe H Y'-!HTbIBaeT CMeCb pa3JIHlJHbIX cpaKTopoB. CfIe-
Ull aJlbHhle CJIy'-!all, KOHe'-!HO, Tpe6YlOT .lI.pyroro BbI6opa.
r JIaBHOii ueJIhlO ,not! r JIaBbI 6bIJJO BBeCTII H.lI.elO YCTOt!'-!IIBOCTII H
MeTOLL BhI.lI.eJIeHl-I51 napaMeTpoB ,lJ,JI5I ee LLOCTH)!{eHH5I. I109TOMY MbI He
BLLaBaJIHCh B CJIO)!{Hhlt! Bonpoc 0 TOM, KaK Ha'-!aTh HHTerpHpOBaHl1e,
nOKa HeT npe,lJ,bILLYllI.HX 3HaQeHHt!, CT05lmHX B npaBot! QaCTII o6u IIX
'cpOPMYJJ. YlHor,lJ,a .lI.JI5I ,not! ueJJH MorYT 6bITb HCfIOJIb30BaHbI CPOPMYJIbI 13.2.
Ynpa>HHeHHSI
13.8-1. I-1cc. eJJ,oBaTb KJlaCC 4JOPMYJl
h 4 y (IV) ( 8 )
Yn+1 =Yn +h (b_1Y +1 +boY ) + h 2 (C_ 1 Y;;+1 +coY ) +E4
41
J-ICnOJIb3Yil b_ 1 B Ka'leCTBe napaMeTpa. 06paTHTb BHHMalme Ha HaQa.10.
o T B C T:
I 06U.(ee I I MeTOA I I I
pellleHI1e MHA.
! Wi *J
b_ 1 b_ 1 I 1/2 0 1/3 2(3
b o l-b_ 1 0 1/2 I 2(3 1/3
1-3b 1 -2/6 -1(12 1/6 0 -1/6
C_l 6
2 - 3b_ 1 -1/6 1/12 1/6 1/6 0
Co 6
E4 -1 +2b_ 1 1 0 -I -1/3 1/3
E. . . . . . . 1/6
*) CM. [26].
(]4 (8) MeHilCT 3HaK J],nR l/s < b_ 1 < 2/ S ' HO B c, Y'lae b_ 1 = 1/ 2 0"1ll1l6Ka
E5 h5 y(5' ( 8) _ y<51 ( 8)
5! - 720 '
TaK KaK
G (8) = (h -4 )2 8 2 :/=0
(0<8<h),
H 9TOT CJlY'IaH oco6eHRo npeLLIT04THTeJleH.
13.8-2. P a3BI1Tb MerOLL LLJliI HaqaJla HHTerpHpoBaHHiI B 06llJ,eM BI1JJ,e (13.3-1)
4: JJ,OJlJKHOti 3a60ToH a rOqHOCTIl.
,13.101
npVlMEP, VlJlJlIOCTPVlPYIOill.VI!i VlJ],EIO YCTO!i'-lViBOCTVI
189
13.9. 3KcnepHMeHTaJIbHaH npoBepKa YCTOMqHBOCTH
YI1TaTeJIIO He CJIe.lJ,yeT nyraTbcH pa3pa60TaHHoti BbIllle CJIO)l{Hoil TeOp11I1.
ECJIH Ha.lJ,O HCCJIe.lJ,OBaTb 06lllHpHbIti KJIaCC <jJOPMYJI, He06xo.lJ,HM DpHMepHO
TaKon MeTO.lJ" HO eCJIH paCCMaTpHBaeTC5I OT.lJ,eJIbHa5I <jJopMYJIa, TO 4aCTO
Y.lJ,OBJIeTBOpHIOTCH ee 9KcnepHMeHTaJJbHoti npoBepKoti.
Ilpe.lJ,noJIo)l{HM, 4TO paccMaTpHBaeTcH <jJopMYJIa
Yn+l = - Yn + Yn-l + Y'2- 2 + (l5Y +1 + 85y + 61Y -1 + 7y _ ).
J1erKo YBH.lJ,eTb, nO.lJ,CTaBHB Y = 1, X, x 2 , x a , x 4 , Y.lJ,OBJIeTBOpmoT JIH OHH
T04HO 9TOMY ypaBHeHHIO.
YTO KacaeTCH YCTOti4I1BOCTH, TO npoCTO 6epYT
Y_ =O, Y_l=O, yo=a
U BbI4HCJI5!IOT
Y1 = - a,
5
Y2 = "2 a,
7
Ya=-"2e.,
B .Q3HHOM CJIyQ3e \1CHO, 4TO MeTO.lJ, HeYCTOflQHB. YT06bl HMeTb YBepeH-
HOCTb, '1TO 9KcnepHMeHTbI OXBaThIB31OT Bce B03MO)KHble Ha4aJIbHble cHTya-
nmJ, HpKHO np05JBHTb HeKOTopoe BHHMaHHe (HC/JbITaTb Ka)l{.lJ,Oe Ci -::j::. 0
XOTH 6bI pa3). 06bI'IHO KOHKpeTHoe HCnhlTaHl1e Ha41maeTC5I C JIHHeflHoti
KOM61maUHH Bcex TaKMX 4JIeHOB.
DpOCT3H 9KcnepHMeHTaJIbHaH npOBepKa Toro, 4TO MeTO.Q HeCTa6HJJeH,
y6e.Ql1TeJIbHa H JIerKO npMMeHMMa. 06paTHoe 3aKJII04eHMe OTHOCMTeJIbHO
YCToti'lI1BOCTH Tpe6yeT 60JIblllerO BHMMaHH5!.
3TOT MeTO,lJ. MJIJIIOCTpMpyeTcH B CJIeJJ.YIOill.eM naparpa<jJe.
13.10. npMMep MHTerpaJIa cBepTKH, HJIJIIOCTPHPYIOUI.MH
HJJ.elO YCTOMqMBOCTH
l{OJIe6aHI15! C reOMeTpWJeCKMM pOCTOM aMnJJMTyJJ.hI 5!BJIHIOTCSI THnM4-
HbIMH JJ.JISI HenOJIa,lJ.OI( C YCTOti4I1BOCTbIO. CJJeJJ.YIOIIJ.MtJ npMMep nOKa3bIBaeT,
KaK pa3JIH4HbIe H.QeM MorYT npMMeH5lTbC\1 B KOHKpeTHbIx CJIYtI3S!x.
3a.Qa4a COCT05!JIa B TOM, 4T06bI HailTM <II (t), eCJJI1 (t) 6bJJIa 3aJJ.aHa
B T04Kax t i (t i = 10(-14.4 + io. ), l = 0, 1, 2), ypaBHeHHe, CBSl3h1BalOill.ee
<II (t) H (t), 6bIJIO
t
) <II(t- 't)H't) d't =t
o
(13.1 0-1)
lIJIM 9KBMBaJIeHTHOe
t
) <I> ('t) (t - 't) d't = t,
o
(13.1 0-2)
npM4eM t(t) 6bIJIa MOHOTOHHO BOSpaCTiHOIIJ.en.
190
HEODPE.lI.EJJEHHWE HHTEfPAJJW
[TJJ. 13
6hIJI BbI6paH H HCllbITaH MeTOn Bbl'1I1CJJeHHi1, BbIr JJHneBllJHt! pa3YMHblM.
Pe3Y JJbT3Tbl paC'IeTOB naJJH KO.1e6aHHH C paCTywet! aMIlJJHTY not!. [1epHon
KOJJe6aHHt! 6bIJI cpHKCHpOBaHHbIM, a pOCT aMIlJIHTynbI - 60JJee HJIH MeHee
reOMeTpH'IeCKHM; OTClOna 6bIJJO CneJIaHO IlpenIl0JIO)/{eHHe, '1TO 3TO - CJJen-
CTBHe HeycTot!'1HBOCTH. CCJJenOBaHHe npouecca BbI'IHCJJeHHH npOBOnl1JIOCb
.lI.JJH (f) = 1, HO C e.ll.HHCTBeHHbIM B03MyweHHeM; TaKHe naHHbIe npHBO-
.lI.HJIH K KOJIe6aHHHM.
TIocJJe HeKoToporo H3Y'leHHH 6bIJIa npHHHTa cJJe.ll.YlOwaH HOBaH cxeM3
BbIlJHCJJeHI1H. MbI nOJJO)/{I1JJH
I
t(f) = ('t) d't.
o
TaKHM 06pa30M, 1(0) = 0 H I' ('t) = ('t). llaJJbllIe BBeJIH 0603Ha'leHH5f
fi(i=O, 1,...,n) c fo=O, t 1 =10- 1U ,..., fi=101-14.6+0.2i) H no ny-
CTHJJI1, '1TO (f) Me)/{.lI.Y fo H f 1 HBJIHeTCH KOHcTaHTotl. l.JT06bl Bbl'lHCJJHTb
f (f), npHMeHHJJH nJJH HHTerpHpoBaHHH npaBliJJO TpaneUHtl
f(t n + 1 ) = f(tn) + + [ (tn+1) + (tnJ](tn+l - tn)'
TaK KaK naHHbIe He onpaBnbIBaJIH CKOJIbKO-HH6Y.lI.b 60JJee CJIO)/{HOrO
MeTona.
Tenepb H3 (13.10-2) nOJJY'laeTcH
I n+1 n li+l
tn+1 = <I> ('tH (tn+l - 't) d't = <I> ('t) (tn+1 - 't) d't.
o i= 0 Ii
B Ka)/{nOM H3 HHTerpaJJOB B cYMMe MO)/{HO CneJJaTb annpoKcHMaUHIO
1 1 I I
<I> ('t) (tn+1 - 't) d't = <I> (ti + 1/2) f' (tn+1 - 't) d't =
= - <I> (ti+ 1/2) [/(tn+l - t i + 1 ) - f(tn+1 - t i )},
I
r.ll.e t i + 1/ 2 eCTb cpenHee 3Ha'leHYJC:2 (f i + 1 + t i ). TaKHM 06pa30M,
n-)
f n + 1 = - <I> (f i + 1/2) If(tn+1 - t i + 1 ) - f(tn+l - ti)j +
i=o
+ <I> (tn+'/2)/(tn+1- tn).
PelliaH 3TO ypaBHeHHe OTHOCHTeJJbHO <I> (t n + 1/2)' nOJJY'IHJJH
n-l
t n + 1 - <p (t i + 1/2) [f (t n +! - ti) - f (t n +! - t i + 1 )]
i=fJ
<I>(t n +1/2)=
f (t n +! - tn)
(13.10-3)
\4.1]
DPI1POJJ:A H CMbICJI nH<f><f>EPEHUVlAJlbHhIX YPABHEHI-Ifi
191
Eho LLaer MeTOLL LLJIH rrOCJleLLOBaTeJIbHOrO Bbl4HCJJeHI1H Ka)f{LLOrO 3Ha4eHHH
1>(tn+1/2)' rLLe rrepBoe 3Ha4eHHe
cp t 1
1/ 2 = f (t 1 ) .
TIpH41ma Toro, 4TO ypaBHeHHH (13.10-3) H (13.1 0-4) OKa3aJJHCb
YLLa4HbIMH, a HeKOTopble .Qpyrne crroc05bl arrrrpOKCHMall.HH - HeT, KpoeTcH
B TOM, 4TO onm5Ka B Bbl4HCJleHIIH cp (t) (BCJJe.QCTBHe OlUH604HbiX .QaHHblx
HJJH LLa)f{e rrpoCTO 4HCJJeHHOrO oKpyr JJeHHH rrpOH3Be.QeHHH) He paCTer
rro BeJIH4HHe Ha CJJe.QYIOIlI.HX 9Tarrax BbI4HCJIeHHH. 'h06bl Har Jl5I.QHO rro-
Ka3an 9TO, Bbl4HCJII1M K09cpcpl1UHeHT rrpH cP (tn -1/ 2 ) B Bblpa)f{eHHH .QJJH
CP(t n +l/ 2 ). EhOT K09cpcpHUHeHT paBeH
f (tn+1 - tn_I) -I (tn+1 - tn)
f (t n +! - tn) - f (tn+1 - t n + I )'
TaK KaK [(0) = O. TIpHMeHHM reOpeMY 0 CpeLLHeM 3HaqeHHH K QHCJJI1'
reJJIO H K 3HaMeHareJIIO OTLLeJIbHO:
(13.10-4)
0/ (t n + 1 - 6 1 ) (t n - tn_I) _ [ 0/ (tn+l - 6 1 ) ] [ tn - tn_ I ] (13.10-5)
- 0/ (t n + 1 - 6 2 ) (tn+1 - tn) - - 0/ (t n + 1 - 6 2 ) t n +! - t n '
r.Qe t n > 8 1 > tn_I' t n + I > 8 2 > tn- Ka)f{.Qa51 I13 CK050K rro MOLLY JJIO MeHbUIe
€LLHHI1Ubl, TaK 4ro onm6Ka B cP (tn _1/ 2 ) YMHO)f{aerC5I Ha 4I1CJIO, MeHblllee
€.QI1HI1Ubl. TaKI1M 06pa30M, olUI16Ka KOJle6JJeTC5I H 6blCTpO I1C4e3aeT.
TaK KaK cpYHKUHI1 cP H BXO.QI1JIH B ypaBHeHl1e CHMMeTpWIHO (cp.
{I3.1 0-1) H (13.1 0-2», TO B Ka4eCTBe rrpOBepKI1 HCXOLLHa51 cpYHKUI151
QblJIa Bbl411CJJeHa TeM )f{e MeTOLLoM 113 nOJJYlJeHHotl cpYHKUI1H cP (t). Pac-
XO)f{LLeHHe rrOJIYlJHJIOCb B rrpeLLeJIaX 9KCrrepI1MeHTaJIbHotl Onm6KI1.
\13 9Toro rrpHMepa BnLLHO, lJTO HeYCToJ14HBOCTb MO)J{HO I1CCJIeLLOBaTb
H 6e3 xapaKTepHCTHlJeCKoro ypaBHeHI1H (cp. (13.5-5». O.QHaKo o 6 Ill. a51
u.Qe51 06paTHoJ1 CBH3H Bce )f{e HCrrOJIb3yeTcH. TIpe.QblLLYlll.ee HCCJIe.QOBaHI1e
{)XBaTHJIO JJHlllb 06paTHYIO CBH3b or O.QHoro lJJIeHa; Ha rrpaKTHKe 06bl4HO
DblBaeT He06xoLLHMO CLLeJJarb He \topoe HCCJIe.ll.OBaHJ1e, '-IT06bl y6e.QHTbc5I,
'-ITO LLOrrOJJHHTeJJbHaH o6paTHaH CB$tdb OT .npymx 'WeHOB He Bbl3bIBaer
HeYCToi14HBOCTH.
r JJ A B A 14
BBE EHHE B H EPEHUHARbHME YPABHEHH
14.1. ilpupoLLa U CMbICJI )l.U!pifJepeHlI.UaJIbHbIX ypaBHeHHH
HalUl1 cOBpeMeHHble B3rJIH.Qbl Ha cpYH.QaMeHTaJIbHbIe 3aKOHhi rrpnpo.QbI
<faCTO H3JIaralOTCH B BH.ne LLHcpcpepeHlI.HaJIbHblX ypaBHeHHJ1. HarrpHMep,
saKOH HbIOrOHa
d 2 x
!=ma=m dt 2
192
BBEllEHME B llM$$EPEHUHAJJbHWE YPABHEHMH
I rn. 14.
eCTb .n:fJ(pcpepeHUH3. bfIOe yp3BHeHHe BToporo nopR.n:K3. B K3'-1eCTBe !1PY-
1'01'0 npl1Mepa paCCMOTpHM KOJIOHHlO 631\Tepl1tJ, paCTYIUYlO B nHT3TeJlb-
HOtJ cpe.n:e. Kor.n:a KOJlOHI151 113YlIaeTC5I B UeJJOM, eCTeCTBeHHO npe.n:nOJJO-
>KI1Tb, liTO B MOMeHT t CKOpOCTb pOCTa nponopUHOHaJIbHa I1MelOmeMYC5I
KOJl!1l1eCTBY y (t), TaK 4TO
y' (t) = ky (t).
TaKI1M 06pa30M, .n:HcpcpepeHUHaJlbHble ypaBHeHI151 4aCTO .n:3lOT caMoe
npOCTce MaTeMaTH4eCKoe onHcaHHe 5IBJleHHH npHpo.n:w.
ECJIH B03MO>KHO, .n:I1cpcjJepeHUHaJlbHCe ypaBHeHHe peWaeTC5I B TClIHOM
BI1.n:e. Y4e6HI1KI1 .n:I1cpcpepeHUI1aJlbHhIX ypaBHeHHtJ 4aCTO OCTaBJl5IlOT nne-
4aT JleHHe, liTO B T04HOM BH.n:e MO>KHO peWHTb 60JlbWHHCTBO LlocpcpepeH-
UUaJlbHhlX yp3BHeHutJ, HO OnblT pa60TbI He no.n:TBep>K.n:aeT !HOrO. 11mI
nOJlY'leHH5I npH6J1H>KeHHbIX pewelmtJ npHro.n:HbI pa3J1H'JHble 3H3J1HT!14eCKHe
MeTOLlb!, H nepe.n: TeM, KaK 06paTi-ITbC5I K 4HCJleHHOMY peweHHlO, HaLlO
HCCJleLlOBaTb TaK>Ke n HX.
TIYCTb .n:aHO LlHcpcpepeHlU1aJlbHOe ypaBHelme
y'=x2_ y2.
(14.1- I)
I.JTO rrO.n:p33YMeB3eTC5I nOLl ero peweHHeM? OCT3BJI5I5I B CTopOHe cpop-
MaJJbHOe MaTeMaTH4eCKoe onpe.1l.eJleHl-le, HHTYIJTHBHO Mhl I1MeeM B BIJLlY
Kpl1BYlO Y = Y (x), B Ka>K.1I.otJ T04Ke (x, Y) KOTOpOtJ TaHreHC yrJl3 Ha-
KJlOHa Kpl'IBOtJ y' (x) LlaeTC5I ypaBHeHHeM (14.1- I). 3TO - JlOKaJlbHOe
CBOt1CTBO. PeweHne ypaBHeHH5I 03Ha4aeT pacnpoCTp3HeHHe 9Toro JlOKaJIb-
HOro CBot1cTBa Ha 60J/bI11YIO 06J/acTb. KOHe'-lHO, peI11eHHeM 51BJI5IeTC5I He
0.n:H3 KPHB351, 3 CKopee Bcero 4epe3 J/106yIO TO'JKY (xo, Yo) fJpOXo.n:Hl'
peweHHe ypaBHeHH5I (14.1- I).
14.2. nOJle HanpaBJIemd\
VlLleH npe.1l.hl.n:ymero naparpacpa Moryl' 6hITb H306p3>KeHhl rpacplJ'JecKH.
B nJIOCKOCTH (x, y) MI.I Bb!6I1p3eM p33JIH'-IHble TOllKH H CTpOHM T3HreHC
yr J/3 J-I3KJlOJ-J3, p3BJ-JbIIJ x 2 - y2. B Ka)!{J\ot1 T3KOtJ TO'JIce 'JepTHM KOPOT-
KYlO JIHHHIO C BI.l4HCJleHHblM T3HreHCOM yr JIa HaKJIOHa. 3TH JII1HHH nOKa-
3bwalOT JIOKaJILHOe HarrpaBJJeHHe peWeHH5I, H npl1 HeKOTopOM Boo6pa)Ke-
HI1H MO>KHO JJerKO HapHCOBaTb pa3JJH4Hble peweJ-JH5I npH YCJIOBI1I1, 4TO
T04KH. 4epe3 KOTOpb!e npOBOLlI1TCH JIHHH5I, pacnOJIO)KeHhI LlOCTaT04HO
4aCTO. O.lIJI8KO qaCTO 6blBaeT npome BOCnOJIb30B3TbC5I npyrHM MeTOl101\!.
MbJ Ha3bIBaeM JlHHHH, BJIOJlb KOTOpbIX Bce KopOTKHe WTPHXH HMeIOT
OimH H TOT llie HaKJlOH, U30lCJlllfta.Mll (JJHHH5IMH OJ\lmaKOBoro HaKJIOHa).
B paCCM3TpHBaeMOM npHMepe HaXO,lUIM, 'JTO H30KJIHHbl C T3HreHCOM
yr Jla H3KJ10Ha k - rHnep6oJIbI: x 2 - y2 = k. TIOJle' Ha]]p3B" eHHtJ, 1130-
6pa)KeHllOe l1pH nOMOU\H H30KJIHH, npHBe,lI,eHO Ha PHC. 14.2-1.
14.3]
4HClIEHHOE PEWEHI1E
193
CJJe.nyeT OTMeTiUb HeCKOJJbKO 06CTO lTeJJbCTB. ECJJI-I rpaqJl1K peWeHHS!
HMeeT MaKCI1MYM, TO OH .nOJJ)I(eH JJe)KaTb Ha HY JJeBotl 1130KJJHHe. AHaJJo-
rH'IHO TO'lKH neperl16a MO)l(HO 06Ha- II
PY)l(I1Tb, nOJJO)l(l1B y' = 0: 2
y'=0=2x- 2yy'=
= 2[x _y(x 2 _ y2)],
I Vf+4Y4
x= 2y
Ehotl Kpl1BOtl Ha pl1c. 14.2-1 HeT. -2 2 .1J
HC!lOJJb30BaHHeM nOJIS! HanpaB-
JJeHI1t1, HecMoTpS! Ha ero npoCTo-
TY, He CJJe.nyeT npeHe6peraTb.
YaCTO OHO BbIS!BJJS!eT !lpHpO.ny 3a-
.na'l 11 , TaK '1TO npaBI1JJbHbltl no.nxo.n -2
MO)l(eT .naTb .nOCTaTO'lHO TO'lHOe PHC. 14.2-1. floJle Ha/JpaBJleHHtI l(JlII
'II1CJJeHHOe peWeHl1e. B npaKTI1Ke ypaBHeHHR y' = x2 _ y2.
aBTopa HeCKOJJbKO pa3 11306pa)l(e-
HHe nOJJS! HanpaBJIeHHtI CI1JJbHO !lOMOr JJO peWeHl110 nOCTaBJJeHHotl 3a.na'lH,
a B O.nHOM cJJY'lae aKKypaTHbltl '1epTe)l(, c!!,eJJaHHbltl Ha '1epTe)l(Hotl .no-
CKe, .naJJ .nOCTaTO'lHYIO TO'lHOCTb, '1T06bI nOJJHOCTblO OTBeTiUb Ha Bonpoc
3a.na'lI1.
YnpamHeHHH
14.2-1. Ha'1epTI1Tb /JoJle Ha/Jp3BJleHI1t1 H pemeHl111 ypaBHeHHR y' = - siny.
14.2-2. Ha'1epTHTb /JOJIe H3/JpaBJleHHtI 11 pemeHIiIl ypaBHeHIiIl y' = x _ y2.
14.3. '-IHCJIeHHOe pemeHHe
ECJJI1 MbI XOTI1M nOJJY'lHTb JJI1Wb O.nHO peweHHe, npoxo.nmu.ee '1epe3
!!,aHHYIO TO'lKY, TO 6bIJJO 6bI 6eCnOJJe3Hotl TpaTotl BpeMeHI1 npOBo.nI1Tb
Bce nOJJe HanpaBJJeHHtI. CKopee CJJe.nyeT Ha'laTb C !!,aHHotl TO'lKH, Bbl-
'II1CJJI1Tb JJOKaJJbHoe HanpaBJJeHl1e, a 3aTeM BbI'IHCJJI1Tb TaHreHCbl yrJJa
HaKJJOHa B HeCKOAbKI1X 6JJI13KHX TO'lKax. TaK CJJe.nyeT npo.nOJJ)I(aTb,
npOBO.nS! HaKJIOHbl JJHWb TaM, r!!,e no HaWI1M npe!!,nOJJO)l(eHI1S!M .nOJJ)I(HO
6bITb peWeHl!e.
EhOT nYTb MO)l(eT 6bITb CBe!!,eH K apHcpMeTI1Ke. oepeM !!,aHHYIO Ha.
'1aJJbHYIO TO'lKY, BbI'IHCJJS!eM TaHreHC yr JJa HaKJJOHa H .nBH)I(eMCS! B STOM
HanpaBJJeHHH Ha He6oJJbWOe paCCTOS!Hl1e (CMbICJJ CJJOBa «He6oJJbWoe»
3aBI1CHT OT KOHKpeTHotl CI1Tyau.HI1, HO .06bl'lHO !!,OBOJJbHO O'leBH.neH) Ii
Bbl6HpaeM CJJe!!,ylOw.ylO TO'lKY. V1C!lOJJb3YSl ee KaK Ha'laJJbHYIO TO'lKY,
nOBTopSleM npou.ecc CKOJIbKO HY)l(HO.
PaCCMaTpIiBaS! pHC. 14.3-1, JJerKO 3aMeTHTb He.nOCTaTOK SToro Me-.
TO.na. ECJJI1 BepHoe peIlIeHi-le 3arH6aeTCSI BBepx, TO KpI1BaSl, KOTOPYIO MbI
BbI'II1CJJSleM, Bcer lla OTXo.nI1T BHI13 OT ElTOrO peWeHI1S1, nOTOMY '1TO AJJ5I
7 P. B. XeMMHHr
194
BBEnEHHE B nH EPEHUHAnbH E YPABHEHH
r rJJ. 14
nOJIYlfeHI1R Ka)!{.n.oiJ CJIe.n.YlOrneiJ TOlfKI1 I1CI10JIb3yeTcR TaHreHC yr JIa Ha-
KJJOHa KaCaTeJIbHotl B npe.n.bI.n.yrnetl.
CJJe.n.OBaJIO 6b1 6bITb YMHee: nOCMOTpeTb npe.n.Bapl1TeJIbHO Imepe.n.,
3aMeTI1Tb TaHreHC yr JIa HaKJIOHa TaM, a 3aTeM npO.QOJI)!{aTb .QBI1)!{eHl1e
B HanpaBJIeHI1H cpe.QHero apl1cpMeTl1lfeCKOro HalfaJIbHOrO 11 KOHelfHoro
TaHreHCOB. 8TO npe.n.JIO)!{eHHe JIerKO CH-
CTeMaTH3I1pOBaTb. 4T06bI Y3HaTb, r.Qe Mbl
c06HpaeMCR B3RTb CJIe.n:ylOw.ylO TO'lKY,
I1CnOJJb3yeM cpOPMYJJY
Yn+l = Yn-l + 2hy .
!J
!:!
?
f.
1).'(.'(. Q.,f,
GDI l}J
e'(.
BVl tJ.G
!J
PHC. 14.3-1. fpy60e
peIlleHHe.
'IHCJleHHOe
(14.3-1)
j:
(MbI 6epeM TaHreHC yrna HaKJIOHa Y
B cepe.QHHe y.QBoeHHoro HHTepBaJIa, lfT06b1
1-I36e)!{aTb olfeBH.QHotl cHcTeMaTHlfecKoti
01l1H6KI1.) OWH6Ka 6y.n:eT paBHa
3 y'" (6).
(14.:1-2)
HanHlUeM Tenepb KoppeKTHpYlOrnylO cpOPMYJIY
Y +l +Y
Yn+l=Yn+ h 2 '
olUI16Ka KOTOpOti
(14.3-3)
- ; y'" (6).
(14.3-4)
ITepBbIti Bonpoc, KOTOpbIti B03HI1KaeT, Kor.Qa MbI nOJIb3yeMc5I 9TOti
MeTo.n.HKoti, - C lfero HalfaTb? BbllfHCJIeHHe no nepBoti cpopMYJIe Tpe6yeT,
KpoMe TeKyrnero TaHreHca yrna HaKJIOHa, 3HaHHR O.n:HOiJ cTapoiJ TO'lKH.
O.n.I1H 113 cnoco60B nOJIYlfeHI1R nepBoti TOlfKH COCTOI1T B I1CnOJIb30Ba-
HHI1 pa3JIO)!{eHH5I peWeHI151 B p5l.Q TetiJIopa OTHOCHTeJIbHO HalfaJIbHoti
TOlfKH, K03cpcpHI1.l1eHTbI KOToporo HaXO.Q5ITC5I 113 ypaBHeHH5I Hero npOH3-
BO.QHbIX:
+ h ' + h2 " + h8 '" +
Yl=YO Yo 2! Yo 3f Yo ...
(14.3-5)
TaK KaK oWH6Ka Ha Ka)!{.QOM IlIary 3aBI1CMT OT TpeTbeti npOH3Bo.n.Hoti,
npO.QOJI)!{aTb p5l.Q TetiJIopa .QaJIbille lfJIeHOB C h 3 HJJI1 h" He CJIe.QyeT.
8Ta cxeMa npOfH03a 11 rrOCJIe.n.YlOrnero nCnpaBJIeHHSJ HMeeT HeCKOJIbKO
npl151THbIX CBOt!CTB. Pa3HOCTb Me)!{.QY npe.QCKa3aHHbIM H HCnpaBJIeHH M
3Ha'-leHI151MH paBHa
; [4y'" (6 1 ) + y'" (6 2 )] 51 3 y'" (6).
(14.3-6)
ECJII1 npe.n.nOJIO)!{I1Tb, '-ITO y'" He MeH5IeT 3HaKa B HHTepBaJJe, TO 9TI1 .QBa
<fHalfeHH5I HaXO.Q5ITC5I no pa3HbIe CTOpOHbI OT BepHoro 3HalfeHI151 (.QJI5I
'!\ 14.41
nPMMEP
195
9Toro wara). TaKHM 0t'5pa80M, MeTO.a. .a.aeT Cpa3Y H crroc06 OlJ.eHKH T04-
HOCTH. ECJm oWH6Ka CJIHWKOM BeJIHKa, HHTepBaJI h MO)KHO YMeHhWHTh
h
.11.0 h' = 2"' 4TO YMeHhWaeT 01l1l16KY rrpHMepHO B BOCeMh pa3.
4HTaTeJJh ,D.01I)KeH SICHO rrOHHMaTh, 4TO Mhl 06parn.aJIHCh C HaWHMH
CHMBOlIaMH HeCKOlIhKO BOlIhHO. MhI Ol.l.eHHJJH HOBOe 8Ha4eHHe <jJOPMY-
lIOt! (14.3-1), 01l1l16Ka KOTOpOt! (14.3-2). 9TO 3Ha4eHI1e HcrrolIh3yeTcSI
.a.JI5I Toro, 4T06hI BhI4HClIHTh y + h H, ClIe.a.OBaTelIhHO, XOT5I MhI Harrl1calIH
y + 1> KaK eC1II1 6hl OHO 6bIlIO BepHblM, 9TO He TaK. OWI16Ka BbIpa-
lKaeTC5I BTOpblM 4JJeHOM <jJOPMYlIhI
f(X, y_ 8 Y"'(6)) f(X, y)+ O!<;; Y) [_ 8 y'" (6)].
06bl4HO CHa4alIa CTpOSIT rrporH03, a rrOTOM HCl1paBlISIIOT, H eCJJI1 ,D.Ba
rrOJJY4eHHbIx 8Ha4eHHH He.a.OCTaT04HO 6JJH8KH, TO rrOBTopHIOT HCl1paBlIeHl1e
HeCKOlIhKO pa3, HCrrOJIb3YH caMoe rrOCJIe,D.Hee 8Ha4eHl1e, rrOKa H8MeHeHI1e
He CTaHeT .a.OCTaT04HO MalIbIM. B I1pHHlJ.l1rre, lIY4il1e COKpaTHTh HHTepBalI,
4eM np0I13BO,D.I1Th nOBTopHble BbI411ClIeHHH no KoppeKTHPYIOrn.et! <jJ_opMYlIe.
14.4. npHMep
PaCCMOTpHM ypaBHeHl1e
y'=y2+ 1, y(O)=O
,(O x l)
(14.4-1)
(BbI6paHHoe 110TOMY, 4TO peilleHHe y = tg x H3BeCTHO H MO)KHO HCrrOJIh-
30BaTh ero .a.JJH npoBepKH).
TIpe)K.a.e Bcero, Hat!.a.eM HeCKOJJhKO nepBhlx 4JIeHOB pH.a.a Tet!lIopa
(14.3-5), npO,D.H<jJ<jJepeHIJ.HpOBaB ypaBHeHue:
y' =y2+ 1,
y" = 2yy',
ym = 2yy" + 2 (y')2,
yIV = 2y'y" + 2yy'" + 4y'y",
y' (0)= 1,
y" (0) = 0,
y'" (0) = 2,
yIV (0) = 0.
ClIelI.OBaTelIhHO,
o'
y(x)=O+x.l +2,0+ 31 2 +4"!+'''=x+ 3 +..,
HCI10JIh3yeM rpy60e pa3611eHI1e h = x = 0,2. B Ka4eCTBe rrepBolI
T04KI1 HMeeM
0,008
Yl = Y (0,2) = 0,2 + a- = 0,203
Ii (c rrOMOIll.hlO JIOrapH<jJMI14ecKot! lIHHet!KI1) Haxo.a.HM
y =y(0,2)=(0,203)2+ 1 = 1,0412.
7.
196
BBEllEHME B llM$$EPEHUI1AJJbHWE VPABHEHM
[fJI. 14
Tenepb Mb/ rOTOBbl Ha4aTb nOCTpoeHHe pellleHI1S1
P2=YO + 2hy; = 0 + 0,4 .(1,0412)= 0,4165
H Bb/4HCJISleM, HCnOJIb3YSI ypaBHeHl1e (14.4-1),
p = 1,173.
D.aJIbllle BW4HCJIReM nonpaB04Hoe 3Ha4eHl1e H3 (14.3-3)
C2 = 0,203 + 0,1 (1,0412 + 1,173) = 0,424.
Cor JIaCHO (14.3-2) 113BeCTHO, 4TO P2 I1MeeT 01ll116KY
8 y'" (6 1 )
H nO (14.3-4) C2 HMeeT 0lllH6KY
h 8 If' ( l! )
-12 y V2 .
ECJIH y'" npH6JJ113HTeJIbHO nOCTOSlHHa B HallieM HHTepBaJIe, TO
P - c = 51 3 y"'. (14.4-2)
TaKHM 06pa30M, 0lllH6Ka B KoppeKTHpYIOw:eM 3Ha4eHHH npHMepHo paBHa
1 1
- 5" (p - c). CJIeJJ.oBaTeJIbHO, HaJJ.O JJ.06aBHTb 5 (p - c) = - 0,002,
4T06w nOJIY4HTb OKOH4aTeJIbHOe 3Ha4eHHe
1
y=c+5"(p-c)=0,422.
Tenepb MOiKHO nOBTopSlTb UHKJI, nOKa He nOJIYllHM x = 1 (CM. Ta6JIHuy
14.4-1 ).
T a 6 JI H U, a 14.4-1
qHCJIeHHOe peweHHe ypaBHeHHSI y' = y3 + 1 (y (0) = 0)
x y y' I p p' c p-c p -c
f)
0 0 1
0,2 0,203 1,0412
0,4 0,422 1,178 0,41648 1,173 0,424 -0,008 -0,002
0,6 0,683 1,466 0,674 1,455 0,685 -0,011 -0,002
0,8 1,027 2,047 1,008 2,016 1,031 -0,023 -0,004
1,0 1,546 1,502 3,256 1,557 -0,053 -0,011
KOppeKTHp. 1,557
3f1aqeHlfe
OWH6Ka 0,011
11 14.51
YCTO!1lJHBOCTb METO.n.A npocToro npOrH03A
197
B Ka4eCTBe npOBepKII HaXO.n.I1M, '1TO BepHoe 3Ha'ieHIIe Y (1) = 1,557,
Tor.n.a KaK Hat!.n.eHHoe HaMIl y(l)= 1,546. A6coJJ!OTHaSl 01lU16Ka 0,011
.n.aeT OTHOCIITeJIbHY!O oum6KY
0,011
1 557 X 100 = 0,7 npoueHTa,
,
KOTOpa\! lI.OBOJIbHO xopollla, eCJIU Y4HTblBaTb pa36Hemie H UcnOJIb30Ba-
Hl1e JlorapHCpMH'IeCKOfi JlHHeI!Kn.
Ynpa>HHeHHSI
14.4-1. PellIHTb '1HCJIeHHO ypaBHeHHe y' = y, y (0) = 1, B3R'B h = 0,2 !{JlR
O:o;:;x:o;:;l.
14.4-2. PellIl1Tb '1HCJJeHHO ypaBHeHHe y' = y- sins x, y (0) = 0, B3RB
1t
h ="6 .l(JlR 0:0;:; X E;; 21t.
9 14.5. Y CTOi\1JHBOCTb MeTo.n;a "pOCTOro nporHo3a
I1Hor.n.a npe.n.nOJlara!OT, '1TO CPOPMY JIY npOrH03a (14.3-1)
Yn+1 = Yn-1 + 2hy
MO}\{HO I1CnOJIb30BaTb 6e3 .n.aJIbHefirnel! KOppeKTl1pOBKU.
YCTOt!'1HBOCTb TaKoro MeTo.n.a.
TIyCTb z (x) - T04Hoe pelllemJe ypaBHeHU5I
z' = f (x, z), (14.5-2)
a Yn - Bbl41-1CJleHHOe pellleHl-le npl-l x = xn' ECJU-I 3Ha'leHHe Yn no.n.CTa-
BI1Tb B .l!.ncpcpepeHl.lnaJIbHOe ypaBHemle, TO OHO Y.l!.OBJJeTBOpI1T eMY, TaK
KaK Bbl4HCJleHHOe 3Ha4eHHe y' 6blJJO Hat!.l!.eHO H3 ypaBHeHIHI. TaKHM
o6pa30M, npeHe6pera51 oll1H6KaMH oKpyrJleHH5I, HMeeM
Y =f(x, Yn)'
Bbl411Ta51 (14.5-3) 113 (14.5-2), nOJlY4aeM
s =z -Y =f(x, zn)-f(x, Yn)= iJ/(;; 6) 5n=A5n' (14.5-4)
r.l!.e A = Z , a e JIe}\{I1T Me}\{.n.y Yn U Zn'
Korll.a Mbl nO.l!.CTaBJl5IeM BepHoe pellleHl1e B pa3HOCTHoe ypaBHeHHe
npOrHoaa, TO nOJlY4aeTC5I 01llH6Ka e (x), TaK KaK BepHoe pelllefme,
Bo06m.e rOBOp5l, He Y.l!.OBJJeTBOp\!eT 3TOMY pa3HOCTHOMY ypaBHeHH!O, T. e.
Zn+1 = Zn_1 + 2hz + en' (14.5-5)
Tenepb, Bbl41-1TaH ypaBHeHl1e (14.5-1) H3 (14.5-5), nOJlY4aeM
511+1 = £n-1 + 2he + en' (14.5-6)
Mbl 3aHl1MaeMCSI oueHRot! JIOKaJIbHOrO pOCTa ollll-l6RI-I 5n' n09TOMY
MO}\{HO npIlH\!Tb, 4TO en II = A nOCTOjjHHbl. ECJIII }\{e OHI1 H3Meli5l!OTC5I
(14.5-1)
Hcc.ile.l!.yeM
(14.5-3)
198
BBEJlEHl1E B JlI1<!><!>EPEHIJ,I1AJIbHbJE YPABHEHI1}'1
[fJJ. 14
He Me.D,JJeHHO, 3Ha t nn liar HHTerpHpOBaHH5I CJlHliKOM BeJlHK. npH aTHX
rrpe.D,nOJlOJKeHH5IX MOJKHO nO.D,CTaBHTb (14.5-4) B (14.5-6); nOJiY4aeTc5I
en+1 = en_I + 2hAe n + en'
HJlH
en+1 - 2Ahe n - e n _l = en'
(14.5-7)
4TO 51BJl5IeTC5I JlHHetlHbIM pa3HOCTHbIM ypaBHeHHeM C nOCT05lHHbIMI1 Koa<p-
<p n uneHTaMI1.
XapaKTepHcTwlecKoe ypaBHeHHe aToro pa3HocTHoro ypaBHeHH5I eCTb
p2 _ 2Ahp - 1 = O. (14.5-8)
CJle.D,OBaTeJIbHO, ero peweHHe
en = C 1 (Pit + C 2 (p2)n,
(14.5-9)
f.lIe
PI =Ah+ V A 2 h 2 + 1, P2=Ah - V A 2 h 2 + 1. (14.5-10)
of
ECJJHA = oy > 0, TO PI> I, 3 eCJIH A < 0, TO I P21 > 1. B JII060M
cJJY'Iae O.D,HH H3 .D:BYX 4JJeHOB p HJ1H P; CTaHOBHTC5I 60JlbWHM npH
n -+ 00. B STOM CJlY43e rOBOp5lT, 4TO MeTO.lI HeYCTott4HB.
Mbl BH.lIHM, qTO npH C.lIeJl3HHbIX npe.D,nOJlOJKeHH5IX nOBTopHoe npHMeHe-
HHe nporH033 (14.5-1) 5IBJl5IeTC5I HeYCTott4HBbIM MeTOJWM. (CM., O.D,HaKO, 14.6.)
Ynpa}lmeHile 14.5-1. rIoDblTaHTeCb '1l1CJlCHHO pellll1Tb ypaBHeHHe y' = - y,
y (0) = 1, h = 0,1 J1.J/JI. 0 x I, HCIJOJlb3YJl JlHmb rrpofHo3, H cpaBHHrc pe-
3YJIbTaTbl C rcopHelt
14.6. YCTOH'IRBOCTb KOppeKIJ.RR
npeJI.nOJlOJKeHl1e, '1TO KoppeKUH5I nOBTOp5lerCH .D:O rex nop, nOK3 He
npeKpaTHTC5I H3MeHeHHe y, MOJKeT 6bITb HCCJleJI.OBaHO TeM JKe cnoc060M,
K3KHM MbI nOJib30B3J1HCb JI.JlH rrpOrH03a. <POPMYJl3 KoppeKUHH
Yn+l=Yn+ (Y +I+Y )
(14.6-1)
npH80J!.HT 3H3J10fH'IHbIM nyreM K xap3KTepnCTH4eCKOMY ypaBHeHH£O
(1 - A2h ) P - (1 + A2h ) = 0,
Ah
1 + 2 (Ah)' (Ah)8
P=---xii = 1 + Ah+:r+-r+'"
1- 2
(npn YCJIOBHH, 4TO I A 2 h I < I).
(14.6-2)
14.61
YCTOHlJl-1BOCTb KOPPEKU,1-11-1
199
DpH paCCMOTpeHHI1 I1Tepa 110HHOrO npo ecca, npHMeHHeMOrO nH
Ah
KOppeKUHI1, I1ccne yeM CHatJana T" DYCTb Ci - 3Ha4eHHH Yn+l B t.Ft HTe-
paUHH, BbI4HcnHM nonpaBKY
Ci+l - Ci= [lex, ci)-f(x, Ci_l)]=
h of ' Ah
= 2 OCI (ci - CI_l) 2 (ci - CI_l)'
04eBH.lI.HO, npo ecc 6y eT CXO.n:HTbCH, ecnH K03cpcpH HeHT CXO HMOCTH
! A 2 h \ < 1.
(14.6-3)
B03Bparu.aHCb K (14.6-2), 3aMe'laeM, '1TO P .lI.aeTCH nepBbIMH TpeMH
'weHaMH pa3nO)!{eHH51 e Ah nmoc He3Ha"HiTenbHaH OU1H6Ka B '1eTBepTOM
4JJeHe.
B npOCTOM ypaBHeHl111
y'=Ay
(A> 0)
HMeeM
of - A
oy-
11 peweHHe B TO'lKe X n eCTb
Yn = Ce Ahn ,
'1TO COBna aeT C. (14.6-2) (ecnH npeHe6pe4b 4JJeHaMH C h 3 ).
TaKHM o6pa30M, BHJIHO, 4TO MbI 3a6nY)!{JIaeMCH, C4I1TaH KpHTepHeM
HeYCToFt4I1BOCTI1 BbInOnHeHl1e HepaBeHCTBa I PI I I e Ah I < 1. 04eBI1 HO,
ecnii peweHI1e paCTeT KaK reoMeTpH4ecKaH nporpeccHH, oWH6KH, KOTO-
pbIe paCTYT C Toll )!{e caMoll CKOpOCTblO HnH MeHbwe, He 06H3aTenbHO
6YJIYT ry611Tb peweHHe. C pyroll CTOpOHbI, eCJJI1 A < 0, TO OWl16KI1
JIOJJ)!{HbI y6b1BaTb no KpallHell Mepe TaK )!{e 6bICTpO, KaK e(Ah) n, HHa4e
OHI1 6yJJ.YT npe06JJaJIaTb B BbI4HcneHHOM pewemll1. TOT cpaKT, 4TO
oW116KI1 orpaHH4eHbI, He nOMoraeT B CJIyqae y6bIBalOru.ero peweHHH.
Boo6me rOBopH, Ba)!{HbIM cpaKTopOM npH o eHKe OWH6KI1 HBnHeTCH '1HCJIO
Kpl1BbIX - peweHHIl, KOTopbIe MbI nepeceKaeM.
Bce CKa3aHHoe HaBOJII1T Ha MbIcnb, 4TO OT peweHl111 .lI.HcpcpepeH Hanb-
HbIX ypaBHeHHI1 Hy)!{HO Tpe60BaTb OTHOClnenbHoll YCT0I14HBOCTH, KOTopaH
onpeJIeJJHeTCH KaK CKOpOCTb pOCTa oWH6KH no OTHoweHHIO K POCTY
peweHHH. Kor.n:a OTHOCHTenbHaH YCTOtJ4HBOCTb no MOJIynlO MeHbwe e.n:H-
HH bI, WYM BCJJeJICTBHe H30JJHpOBaHHoro oKpyr JJeHHH IWI1 pyrotJ oWH6KH
6Y.ll.eT paCTH He 6bICTpee, qeM peweHHe.
C TQ4KH 3peHHH 3Toro HOBoro KpHTepHH HaM HaJIO nepeCMOTpeTb BbI-
BOJIbI 14.5 06 Hcnonb30BaHHH npOC-fOro npOrH03a. ECJ)H npe nOJIar:aTb,
200
BBEnEHHE B nH EPEHUHAJJbHWE YPABHEHH
[W. 14
'no I Ah 1<1, TO XapaKTepI1CTW-reCKI1e KOpHI1 Pi ypa8HeHI1S! (14.5-10)
MorYT 6b1Tb rrpe.ll.CTaBJleHb! B BH.lI.e
PI =Ah+ V I + (Ah? =Ah+ 1 + (A;) - (Ah)4+...,
P2=Ah- VI +(Ah)2=Ah-[1 + (A 2 h)2 _-}(Ah)4+...].
I1J11i
- 1 + Ah + (Ah)2 + =-., Ah
PI - . . . "-=' e ,
[ (Ah)2 ]
P2= - l-Ah+-r+'" _e-Ah,
(14.6-4)
a caMO pemeHI1e paCTeT KaK e Ah , OTClO.lI.a BI1.l1.HO, 'JTO eCJlIi A> 0, TO
MO)/(HO Hane5lTbCS!, 'JTO nOBTopHoe HCnOJlb30BaHI1e npOfH03a .1I.aCT pa3YM-
Hble pe3Y JlbTaTbl, HO eCJlI1 A < 0, TO (P2t 6Y.ll.eT paCTfI no BeJJl1411He fI
KOJle6aTbCS!, B TO BpeMS! KaK BepHoe pemeHl1e y6b!BaeT. CJle.1l.0BaTeJlbHO,
MeTO.ll. nOBTopHoro npHMeHeHI1S! npOfH03a OTHOCI1TeJlbHO HeCTa611J1eH
TOJlbKO, KOr.1l.a A < 0 (CM. ynpa)/(HeHHe 14.5-1).
B03BpamaS!cb K I1Tepalll1oHHoMy npolleccy KOppeKTl1pOBKI1, MI.! BI1.11.I1M,
'JTO xapaKTepl1CTH'JeCKoe ypaBHeHl1e I1MeeT JJHmb O.1l.I1H KopeHb (ypaBHe-
Hfle (14.6-2» fI pemeHfle Be.ll.eT ce6S! TaK )/(e, KaK fI omfl6Ka; TaKflM
06pa30M, HTepalll10HHb!tI MeTO.ll. KOppeKlll111 OTHOCI1TeJlbHO cTa611J1eH,
Ynpa)l{HeHUe 14.6-1. rIoKa3aTb, lITO npH HcnOJlb30BaHHH npocroro nporH03a
pe3ynbTaT 411CJleHHOrO HHTerpl1pOBaHI1H
y'=y, y(O)=I, h=0,2 ,IlJlH O.s;x.s;1
cornacyeTcR c TCOpl1etl.
14.7. HeCKOJIbKO OOUJ.HX aaMeqaHHA
MI.! .lI.aJlI1 'JeTb!pe npl1Mepa aHaJlfl3a YCTOt!4HBOCTH: nepBblt! OTHOCHTCR
K Heonpe.ll.eJleHHb!M I1HTerpaJlaM, BTOpOt! - K OC060MY HHTerpaJlY co
cBepTKot!, nOCJJe.1l.HHe .lI.Ba - K .1I.HtPtPepeHllHaJJbHblM ypaBHeHHHM. 3aMeTl1M,
'JTO YCTOtl'JI1BOCTb .lI.Jl5I HHTerpaJJOB H .1I.Jl5I .lI.HtPtPepeHllHaJlbHb!X ypaBHe-
HHt! - He O.ll.HO 11 TO )/(e; .QJJS! .lI.HtPtPepeHltHaJlbHb!X ypaBHeHI1t! cymeeTBYIOT
rrym 06paTHot! CB5I3H OT npOH3BO.ll.HblX, 4ero HeT B cJlY'Jae HHTerpaJlOB.
He06xo.ll.I1MOCTb aHaJlH3a YCTOtllJ.HBOCTI1 B03HHKaeT BC5IKHt! paa, KOr.ll.a
craphle 3I1a'JeHH5I HcnoJlb3YIOTC5I .lI.JJ5I BbI'lHCJleHI151 HOBb!X, H TaKHM 06pa-
30M nOJlY'JaeTC5I 06paTHa51 CB5I3b omH60K. He cYlllecTByeT npocToro npa-
BI1J1a, OTHOC5ImerOC5I KO BeeM CJlY'Ja5lM, HO 5ICHO, 'JTO YCTOt!'JI1BOCTb TeCHO
CBS!3aHa C 06paTHot! CBR3b/O. Ha nOCJle.ll.HlO1O TeMY HanHcaHO MHoro
KHHr. ChlJJO nOKa3aHO TaK)/(e, 'JTO .lI.Jl5I .lI.HtPtPepeHLtHaJlbHblX ypaBHeHHtI
H.lI.e51 YCTOtlt[l1BOCTIi Be.ll.eT no JlO)/(HOMY nYTI1 HOT HOC Ii T e .II b H a \J
y C T 0 tI 'J H B 0 C T b SlBJJSleTCSI 60Jlee pa3YMHblM Kpl1Tepl1eM,
14.81
CMCTEMW YPABHEHHA
201
AHaJIH3 YCTOIt'JHBOCTH, KOTOpbl/1 6blJJ npOBell.eH Ha npOCTOM npHMepe
npell.bl1l.yrn.ero naparpacpa, 6blJI HenOJIHbIM. Mbl npOaHaJJH3HpOBaJIl-l npo-
fH03 H KOppeKUHIO OTlI.eJIbHO, npeHe6pera51 9cpcpeKTaMH HX B3aHMOll.e/1-
CTBI151 H HrHOpHPY51 3aKJIIO'JHTeJIbHYIO 'JaCTb MeTOll.a - ypaBHeHHe (14.4-2)-
11ml HCCJJe1l.0BaHH5I, KOHe'JHO, MO)/{HO HanHcaTb CJIO)/{HYIO cHcTeMY lI.Hcpcpe-
peHUl1aJIbHblX ypaBHeHl11t H nOCTpoHTb cooTBeTCTBYIOrn.ee xapaKTepHcTH-
<IeCKOe ypaBHeHHe, HO pe3YJlbTaTbi 6Y1l.YT O'JeHb 3anYTaHHbIMH. Mbl npe1l.-
nOlJTeM C1I.eJlaTb HeCKOJlbKO 3aMelJaHHIt 0 TOM, KaK B3aHM01l.e/1CTBYIOT
npOrR03 11 KOppeKUH5I. HeYCToltlJHBOCTb npOrH03a HMeeT OlJeHb MaJIeHb-
KOe BJ!H5IJ-lHe Ha OTHOCI1TeJlbHYIO YCTOltlJHBOCTb pe3YJlbTaTa nocJle npl1-
MeHeHHo/1 01l.HH pa3 KoppeKUHH. 3aKJlIOlJHTeJlbHbilt 9Tan TaK)/{e He Bbl3bl-
BaeT 60JlbUlHX 3aTPY1I.HeHHIt no cpaBHeHl-l1O C HCCJIe1l.0BaHHeM OTHOCI1TeJlb-
HO/1 YCTOltlJHBOCTH KoppeKUHH.
Ty )/{e H1I.elO, KOTOpa51 6bIJIa HcnOJlb30BaHa, 'JT06bl YHHlJTO)/{HTb
OUlH6KY CKoppeKTHpORaHHoro 3HalJeHH5I, MO)/{HO HcnOJlb30B(lTb, lJT06bl
nO.!J.npaBHTh npOrH03HpOBaHHoe 3HalJeHHe H, TaKHM 06pa30M, YMeHhUlWTh
OUlH6KY B oueHKe npOrR03HpOBaHHolt npOH3BOll.Ho/1, BX01l.5Irn.e/1 B OKOH-
<IaTeJJbHOe 3HalJeHHe. 4T06bl ClI.eJJaTb 9TO, CJJell.yeT npH6aBHTb K nporRo-
4
3HpoBaHHoMY 3Ha'JeHHIO - 5 (Pn - cn) nOCJJell.HerO UHKJJa. ECJJH cYlne-
CTByeT TeHlI.eHUI151 K KOJle6aHI1IO, 9TO TOJIbKO nOll.lJepKHeT ee. C 1I.pyrol1
CTOPOHW, HeT COMHeHI151, lJTO eCJII1 HeYCToltlJHBOCTb H oKPyrJJeHHe He3Ha-
QllTeJJbHbl, TaKolt 'JJJeH XOpOlllO BJII-ISleT Ha TOlJHOCTb BCero Bhl'JHCJJeHHSI.
YnpamHeHHe 14.7-1. Pa3pa60TaTb 4JOPMYJlbl f/pOrH03a, f/Of/paBKH, KoppeK-
ltHH H OKOH'IaTeJlbHOrO 3Ha'leHHR JJ:JlR HHTerpHpoBalIHR CHCTeMbI.
14.8. CHcTeMbl ypaBHeHHH
11HCpcpepeHuHaJJhHoe ypaBHeHHe n-ro nOp5l.!J.Ka
y(n) = f(x, Y, y', ..., y(n-l))
MO)/{eT 6bITh CBe1l.eHO K CHCTeMe n ypaBHeHH/1 nepBoro IIOp5l1l.Ka IIpO-
CTbIM 113MeHeHfleM B 0603Ha'JeHI151X. HalIHllleM
Y=Yo, y' =y"
", ,
y =Y2=Ya, ...,
y' =Y; =Y2>
y(n-1) = Yn-b
1-1 Mb1 IIMeeM 9KBI1BaJJeHTHYIO CIICTeMY:
y =Y"
y; = Y2,
y; = Ya,
y -I =f(x, Yo, y" Yg, ..., Yn I)'
202
o EUl. AS'! TEOP }'1 METOJ],OB npOrH03A 11 KOPPEKIU11iJ
r TJJ. 15
4T06bI npHMeHHTb H3JIO)!{eHHbItI MeTO.1l: HHTerpl1pOBaHH5I K CHCTeMe,
MbI rrpOCTO npHMeHHM OrrepaTOpbI O.1l:HOBpeMeHHO KO BCeM ypaBHeHH5IM.
CB5I3H Me)!{.1I:Y ypaBHeHH5IMH npOSIBJl5IIOTCSI TOJIbKO TIPH BbI4HCJIem1H npo-
H3BO.1l:HbIX. llpyrHMH CJJOBaMH, ypaBHeHI15I 06pa6aTbIBalOTCSI OT.1I:eJIbHO, HO
napaJIJIeJIbHO.
MeTO.1l: nOJISI HarrpaBJIeHHt! 06bI'lHO npHMeHSIeTC5I JIHlllb K OllHOMY
ypaBHeHHIO nepBOrO nOpSI.1I:Ka, HO B CJIY4ae CHCTeMbI .1I:BYX ypaBHeHI1t1
nepBOrO nOpSI.1I:Ka, B KOTOpbIX npaBa5I qaCTb He 3aBHCHT OT rrepeMeH-
HOI1 x:
y' =f(y, z),
z' = g(y, z),
.1I:eJJeHHe O.1l:HOro ypaBHeHHSI Ha .1I:pyroe .1I:aeT ypaBHeHHenepBoro TIOpSI.1I:K3
6e3 x. DOJJe HanpaBJIeHHt! .1I:JI5I SToro ypaBHeHHR .1I:aeT y B 3aBI1CI1MOCTH
OT Z, a HHOr.1l:a .1I:OCTaTOtJHO nOKa3aTb, KaK Be.1l:eT ce6SI y B 3aBHCHMOCTH
OT z, He onpe.1l:eJISISI x, .1I:JISI KOToporo T04Ka (y, z) BCTpeTHJIaCb H3
KpHBOii.
r JI A B A 15
ODIUA TEOPH METO OB npOrH03A H KOPPEKUHH
15.1. BBe.n:eHRe
B r JIaBaX 13 H 14 6bIJIH BBe.1l:eHbJ OCHOBHble rrOHRTH5I H YKa3aHbI He-
KOTopbIe TeXHH4eCKHe npHeMbI peaJIH3all.HI1 MeTO.1l:0B nporHo3a H KoppeI'-
lI.HH .1I:JISI 4HCJIeHHOrO I1HTerpHpOBal-lHSI 05bIKHoBeHHbIx .1I:WpepepeHlI.HaJJbHbIX
ypaBHeHnt!. Tenepb MbI HCnOJIb3yeM Bce STO, 4T06bI pa3BI1Tb o61lI.YIO Teo-
pHIO MeTO.1l:0B npOrH03a H KOppeKlI.HH. STa TeopHSI BKJII04aeT MeTO.l(hl
MHJIHa H A.1I:aMCa-6alllepopTa KaK 4aCTHbIe CJJY4aH. TaI'HM 06pa30M, Mbl
MO)!{eM cpaBHHTb STH llIl1pOKO npnMeH5!eMbIe MeTO.1l:bI C .1I:PYrHMH B paM-
Kax e.1l:HHol1 TeopHH.
MeTO.1l:bI nporHo3a 11 KOppeKlI.HI1 .1!:JI5! HHTerpHpOBaHH5I 06blKHoBeHHbIx
.1!:HepqJepeHlI.l1aJIbHbIX ypaBHeHHt! lllHpOKO HCnOJIb3YlOTCSI 6JIarO.1!:apSI CJIe-
.1!:YlOlllHM npeHMY1lI.eCTBaM:
1. Pa3HocTb Me)!{.1I:Y npOrH03HpOBaHHbIM H cKoppeKTHpoBaHHbIM 3Ha-
4eHH5!MH .1!:aeT oll.eHKY OlllH50K, c.n:eJIaHHbIX Ha Ka)!{.1!:OM wary H, CJIe.LJ.O-
BaTeJJbHO, MO)!{eT 5bITb I1CnOJJb30BaHa .1I:JISI KOHTpOJI5I BeJIH411HhI wara,
npHMeH5IeMOrO npH HHTerpHpoBaHHH.
2. Ha Ka)!{.1!:OM wary Ha.1l:0 Bbl4HCJJSITb npOH3BO.1!:HYIO O.LJ.HH HJJH .LJ.B3
pa33 no cpaBHeHHIO C 4eTbIpbMSI .1!:JISI MeTO.1!:a PYHre-KYTTa (r JI. ) 6),
4TO .1!:JISI CHCTeM BbICOKoro nOp5I.1!:Ka MO)!{eT 3Ha4HTeJlbHO SKOHOMHTb BpeM5I
Ha BbI4HCJJeHH5I.
3. JIerKo JIOBI1Tb pa3JIH4I-lbIe OWH6KH.
15.1]
BBEltEHHE
203
HeKOTopblt! HeltOCTaTOK COCTOnT B TOM, lITO MeTOltbl CJJO>KHbl .lMSI
npOrpaMMHpOBaHnSI n He SIBJISI!OTCSI <!:caMOHal/HHa!OIlI.HMnC SI». 3TOT Borrpoc
6YL(eT 06CY>Kn:eH B rJI. 16.
TIpn HCfJOJIb30BaHHn MeTOL(OB rrpOrH03a H KOppeKIJ.nn L(JISI nHTerpn-
pOBaHHSI 06blKHoBeHHblx n:mpcpepeHIJ.naJIbHbIX ypaBHemlt! B03HnKa!OT pa3-
Horo pon:a TPYL(HOCTH. BOT HX rJIaBHble HCTOl/HHKH:
1. O1l1l16KI-I OT oT6paCbIBaHI-I5I tJJIeHOB, KOTopble B03HHKa!OT rrpn
arrrrpOKCHMaIJ.HH rrpoH3Bon:HbIX.
2. PacnpoCTpaHeHHe 01lIl160K - HeYCTot!4nBocTb, KOTOpa51 B03HnKaeT
npH pellIeHI1H arrrrpoKcHMnpY!OllI.nx pa3HocTHblX ypaBHeHI1t!, He COOTBeT-
CTBY!OW.nx pellleHHSIM ncxo.n:HblX L(HcpcpepeHlI.HaJIbHblX ypaBHeHnt!.
3. Y CHJIeHHe 0llIn60K oKpyr JIeHH5I 1-13-3a KOM6HHall.Ht! K03cpcpnll.HeHTOB
B KOHe l IHopa3HocTHblX cpopMYJJax.
B 3TOt! rJIaBe Mbl L(OCTaTOl/HO rro.n:p06HO HCCJIeL(yeM IlInpoKnt! KJIaCC
cpOPMYJJ n nOKa>KeM, lITO Bbl60p KOHKpeTHot! cpOPMYJIbl eCTb KOMrrpOMncc
Me>Kn:y BeCbMa npOTHBopetJnBblMn >KeJIaHI-ISIMn. TaK KaK npn nHTerpnpo-
BaHHH 06blKHOBeHHblX n:ncpcpepeHlI.naJIbHblX ypaBHeHHt! cpopMYJIa KoppeK-
unH HrpaeT Han60JJee Ba>KHY!O pOJIb, eCTeCTBeHHO nCCJIen:OBaTb KoppeK-
UH!O CHal/aJIa n npnTOM 60JIee BHnMaTeJIbHO.
TIoCJIe HCCJIeL(OBaHHSI KOppeKIJ.HH Mbl 3at!MeMC5I n3Y4eHHeM npOrH03a.
XOpOIliO 03HaKOMHBIlIHCb KaK C rrporH030M, TaK n C KOppeKIJ.Het!, Mbl
CMO>KeM 3aTeM HCCJJeL(OBaTb 3aL(al/Y Bbl,6opa llIara n Oll.eHKH TOtJHOCTn
pellIeHH5I npH nHTerpnpOBaHnn 06blKHOBeHHblX L(HcpcpepeHIJ.l1aJIbHbIX ypaB-
HeHHi:j, HaKOHeu, Mbl cn:eJIaeM BbIBOL(bI n 06CYL(HM HeKOTopble 3KcnepH-
MeHTaJIbHble pe3YJIbTaTbl.
HCCJJen:OBaHHble MeTOL(bl HMe!OT 0llIn6KY OT oT6paCbIBaHH5I l/JIeHOB
nOpSIL(Ka h tl ; OnbIT pa60Tbl rrOKa3blBaeT, 4TO HMI1 rrOJIb3YlOTC5I HaH60JIee
IlIHpOKO. B rJI. 14, 6wI pa3pa60TaH MeTO.n: npOrH03a H KoppeKUHH rro-
p5lL(Ka h 3 . TIpen:ocTaBJJ5IeM tJHTaTeJIIO caMOCTOHTeJIbHO pa3pa60TaTb npo-
Me>KYT04HYIO TeopHIO MeTo.n:OB rropH.n:Ka h 4 .
MeToL( TpeTberO nop5In:Ka rJI. 14 He I1MeeT CB060n:HbIX napaMeTpOB;
MeTOL(bl l/eTBepToro rrOpHL(Ka, H3JIO>KeHHble B ynpa>KHeHH5IX, BBOL(51T n:Ba
HOBblX K03cpcpHuneHTa, OL(nH n3 KOTOpblX nCrIOJJb3yeTcH, 4T06bl rrOBblCHTb
rrOpHL(OK 01lIH6KI-I OT oT6paCbIBaHHSI tJJIeHOB, a L(pyrot! - n:JJ5I YCTOt!l/H-
BOCTH; MeTon:bl rr5ITOrO IlOpSIL(Ka HMelOT ellI.e ABa K03cpcpHIJ.neHTa, n3 KO-
TOpbIX CHOBa on:nH nCrrOJJb3yeTC5I L(JI5I oT6paCbIBaHn51 tJJIeHOB H OL(nH n:JISI
YCTOt!l/HBOCTH, n T. L(. L(JI5I MeTOL(OB 60JIee BblCOKoro nopHn:Ka.
B 60JIblllHHCTBe CJIytJaeB 01lI116Ka OT oT6paCblBaHHSI 4JIeHOB rrpn KO-
HetJHot! annpOKCHMaUHH npOl!3Bon:HbIX n:HcpcpepeHUHaJIbHOrO ypaBHem1H
HJIM CMCTeMbl n:HcpcpepeHlI.HaJJbHblX ypaBHeHni1 Ha Ka>K.L(OM llIary 3Ha4HTeJIb-
HO npeBocxon:l1T OlllH6KH oKpyrJJeHI1SI. On:HaKO rrpH rrOCTpoeHHH cneUHa-
JIH311pOBaHHblX Bbll/HCJII1TeJIblIbIX MallIHH CTpeMHTCH cn:eJIaTb n:mIHY tJl1ceJI
11, CJIen:OBaTeJIbHO, OCHOBHOi1 ypOBeHb T04HOCTH, KaK MO>KHO MeHbIlIHMH.
B TaKHX cHTyaUHHx 01lIV16Ka OT oT6paCbIBaHI1H 4JIeHOB MQ)KeT 6bITb npl1-
MepHO TOt! >Ke BeJII14HHbl, liTO H 01lI116Ka oKpyr JIeHI1H, H nOCJIen:HIolO
204
OBlllA5I TEOPVl5I METOnOB m'OfH03A VI KOPPEKUVlH
rrn. 1
HeJIb3SJ I1rHOpl1pOBaTb npl1 Bbl60pe MeTO.ll.OB HHTerpl1pOBaHI151 .lI.nSJ .lI.aHHot!
3a.ll.aIf11. KpoMe Toro, B HaUII1 .lI.HI1 Bce 60JIee lfaCTO CJIYlfaeTCH, lfTO B
HeKOTopbIX 3a.ll.alfaX, TaKI1X KaK BbllfHCJIeHHe TpaeKTopnt! JIYHb! 11 nJIa-
HeT, ClfeT C BOCbMI1- 11 .lI.a)1{e .lI.eCSJTI13HalfHot! TOlfHOCTblO MOJKeT naBaTb
3Halfl1TeJIbHbie OWI16KH oKPyrJIeHHSJ, 11 Torna HeKOTopble 113 Onl1CaHHbIX
3neCb MeTonOB CTaHOBSJTCSJ nOJIe3HbIMI1.
15.2. OlIliioKa OT oTopacblBaHliH 'IJ1eHOB
CaMaSJ 06ulaSJ epopMYJIa JIHHet!Hot! KOppeKIlI1I1, I1CnOJIb3YIOlllaSI I1HepOp-
MaUl110 0 epYHKUHI1 Ii nepBot! np0l13BOnHotl B nOCJIenHHX Tpex TOlfKax
peweHHSJ 11 oueHKY np0I13BO.ll.HOt! B TOlfKe, KOTopaSI nOJIJKHa 6bITb Bbl-
4HCJIeHa, I1MeeT BHn
Yn+1 = aoYn + alYn_1 + a2Yn_2 + h (b -lY + I + boY +
h 5 y(5' (6)
+blY -I+b2Y -2)+E5 51 .
(15.2-1)
MOJKHO \JcnOJIb30BaTb \J nPYrHe JI\JHetlHble epOpM!'I, HO 3Ta BKJJlOlfaeT
60JIbWOe lfHCJIO 06111HX MeTonOB 11, KpoMe Toro, elO MI,l YJKe nOJIb30Ba-
JII1Cb B r JI. 13 (ypaBHeHHe (13.3-1 », KOr.ll.a I1CCJIenOBaJII1 HeOnpeneJIeH-
H!,le IHlTerpaJIbl. OnHaKo B OTJIHlfHe OT r JI. 13 Tenepb MI,l I1MeeM 06paT-
HYIO CB513b He TOJIbKO 4epe3 3Ha4eHI15J CTapoto peWeHH51, HO 11 4epea
4JIeHbi C npOH3BonHbIMH; CJJenOBaTeJIbHO, H3Y4eHHe. YCTofilfl1BOCTH 6yneT
3.l1.eCb 60JIee CJIO}{(HbIM.
BbIITHCaHHa51 epopMYJJa IIpe.ll.nOJIaraeTC51 T04HOtl nJJ51 MHOrOlfJJeHOB .11.0
4eTBepTotl CTeneHH, T. e. nJISI y = I, X, X2, X3, X4. I-1cnOJIb3YSI 3TH nSITb
YCJIOBHtI H B3SIB al H a2 B Ka4eCTBe napaMeTpOB, nOJJYlfaeM (CM. ypaB-
HeHHe (13.3-2))
al = aI'
I
b o = 24 (19 + 13al + 8a2)'
I
b l = 24 (-5 + 13al + 32a2)'
I I (15.2-2)
b2=2:t(1-al + 8a 2)'
E 5 = (-19 + llal - 8a 2 ).
ao = 1 - al - a2'
a2 = a 2 ,
1
b_ 1 = 24 (9 - at),
D5ITb K09epepHUHeHToB H3 ceMH HCnOJIb3YIOTCSJ, lfT06b! YMeHbwHTb
01lm6KY OT OT6paCbIBaHHSI lfJJeHOB. DpH nOMOW:H nBYx npyrHx Mb!, BMeCTO
Toro lfT06bl ew:e naJIbIJle YMeHbwaTb OIJlH6KY OT OT6paCbIBaHH51 4JIeHOB,
nOCTapaeMcSJ YMeHbWI1Tb oWH6KH BToporo H TpeTbero THnOB. KaK H B
rn. 14, pa3HocTb nporH03HpoBaHHoro H cKoppeKTHpoBaHHoro 3HalfeHHt!
6Y.lI.eT ncnOJIb30BaHa K8K .lI.JI51 Toro, lfT06bl naTb oneHKY oum6KH, TaK H
.lI.JIH Toro, lfT06bl <YHHlfTOJKHTb» r JIaBHbifi IJJJeH OillH6KH.
1115.31
YCTOHlfl1BOCTb
205
06bJlfHblt! MeTO.l1 HaXO>K.l1eHI1S1 OCTaTOlfHoro lfJJeHa OCHOBaH Ha HC-
nOJlb30BaHHH pSlIla Tet!Jlopa (CM. (11.3-2) 11 (13.4-1»
(x A)2
y(x)=y(A)+(x-A)y'(A)+ -; y" (A) +...
x
.. . + (x -:,A)n yln) (A) + ! ylMl) (8) (X _ st d8
A
c A = - 2h 11 n = 4. CJleIlYSI r JI. 13, IIOJlYlfaeM
h
R (y) = y(3) (8) 0 (8) ds,
-2h
rIle O(s) - CPYHKUHSI BJJI1S1HI1S1:
O(s)=(h - s) - ao(- s) - al(- h - s) - a:a(-2h - s)t-
- 4h [b_ 1 (h - s) + b o (- s) + b 1 (-h - s) + b 2 (-2h - s) ] (15.2-3)
H (CM. (11.3-4»
{ (a - S)R, eCJlI1 a - 8:;::: °
(a-s) = 0, eCJlI1 a-s<O.
(k > 0),
ECJJI1 O(s) I1MeeT nOCToSlHHbltl3HaK, TO CY1l1.eCTByeT TaKoe 9 (-2h<9<h),
lfTO oIlIl16Ka MO)l{eT 6bITb 3anl1CaHa B BH.lle
h
y(3) (9) O(s) ds.
-2h
(15.2-4)
ECJlH O(s) MeHSleT 3HaK, TO Hat!.l1YTCSI CPYHKUHI1 I(s), .l1J1S1 KOTOpblX
oIlm6Ka He MO)l{eT 6bJTb npe.l1CTaBJJeHa B Bl1.lle (15.2-4).
MCCJle.l10BaHlle 13.4 llaeT nOKa3aHHYlO Ha pl1c. 13.4-1 11 13.4-2
06J1aCTb nOCTOSlHCTBa 3HaKa o (s). OHa-TO 11 npellCTaBJlSleT .l1J1S1 Hac
OCHOBHOt1 I1HTepec. Ha Tex )l{e pHCYHKax llJlSl Hee nOKa3aHbl JJI1Hlll1 paB-
HOt! OWH6Kll OT oT6paCblBaHI1S1 lfJJeHOB.
15.3. Y CToiiql1BOCTb
CJJellYSI 14.5, llonycTI1M, lfTO z = Z (x) eCTb peWeHl1e
z' =f(x, z), (15.3-1)
zn+l = aoz n + al z n_l + a2 z n_2 +
. + h (b_lZ +l + boz + blZ -1 + b2Z -2) + en' (15.3-2)
a y = Yn - Bbl411CJleHHOe peweHHe.
y =f(xn' y,,), (15.3-3)
Yn+1 = a!\Yn + alYnl + a 2 Yn_2 +
+ h (b -lY +1 + boY + blY -1 + b2Y -2)' (15.3-4)
206
05ll!A5I TEOP!151 MEronOB nporH03A !1 KOPPEKLt!1!1
[rJJ. 15
rIOJIO)I<fIM IOn = Zn - Yn H, BblqHTaH COOTBeTCTBeHHO paBeHCTBa (15.3-2)
H (15.3-4), rrOJJY4HM
€n+l = ao€n + a 1 €n_1 + a2€n_2 +
+ h (b_l€ +1 + bo€ + bl€ _1 + b2e _2) + en' (15.3-5)
Torna KaK (15.3-1) H (15.3-3), eCJIH HCrIOJIb30BaTb TeopeMY 0 CpeJLHeM
3Ha'IeHHH, JJ:aIOT
' /( ) I( ) af (Xn> 8)
IOn = X n , Zn - X n , Yn = iJz en'
(15.3-6)
r,a:e e - 06blqHOe Cpe,a:Hee 3HaqeHHe.
rIpH H3Y4eH!1H pOCTa OliIH6KH en pa3YMHO ,a:OnYCTHTb, qTO en H
= A o6e nOCTOHHHbl; Ha npaKTHKe OHH Me,a:JIeHHO H3MeHHIOTCH OT ruara
K ruary. rIo,a:CTaBHB (15.3-6) B (15.3-5), nOJIyqaeM
(1 - b_ 1 Ah) €n+l = (a o + boAh) IOn + (al + blAh) €n_l +
+ (a2 + b 2 Ah) €n_2 + en' (15.3-7)
T. e. JlHHefiHOe pa3HOCTHoe ypaBHeHHe C nOCTOHHHblMH K03rpqmll.HeHTaMH.
B ypaBHeHHH (15.3-7) He BCTpelIaeTCH nop03Hb HH h, HH A, HO BClOny
TOJIbKO BeJlH4HHa Ah. XOTH Ah MO)/{eT MeHHTbCH B rmlpOKHX rrpe,a:eJlax,
HanO pa3YMHo Oll.eHHTb 3Ha4eHHH, KOTopble MO)/{HO O)/{HnaTb. L{To6bI XOTb
KaK-TO rrpe,a:CTaBHTb ce6e HX, paccMoTpHM rrpOCTble ypaBHeHHH
y'= + Ay,
Y (0) = 1.
OruM6KY OT oT6pacbIBaHHH 4J1eHOB
h 5 A5 ( I
40 40 - 3TO THrrH4Hoe 3Ha4eHHe
npHxonHM K Ta6JJHl.l.e 15.3-1.
Ha Ka)l{,a:OM wary rrpHMeM paBHotl
,a:JJH ;: ; CM. Ta6JJHll.Y 13.7-1). MbI
T a 6 JI H lJ, a 15.3-1
OUIII'6Ka or oropacblBaHiHI QJIeHOB KaK $YHKIJ,HSI or I All I
I Ah I
0.1
0.2
0.3
0.4
0,5
h 5 y(5) I I I I I
-----;m- 2,50 X [0- 7 8,00 X 10- 6 6,07 X 10- 5 2,56 X 10- 4 7,88 X 10- 4
TaKHM 06pa30M, 4T06bl oWH6Ka aT oT6paCbIBaHfIH 4J1eHOB 6bIJ/a no-
5
CTar04HO MaJla, Heo6xoJI.HMO, 4ro6bl I Ah I 10 H ,a:JlH T04HbIX peweHHtI
2
iAhl m'
II \fi.3]
YCTOt\4HBOCTb
207
KOHe4HO, .n:pyrne ypaBHeHl15l 6y.n:YT I1MeTb .n:pyrHe rpOPMYJIbl .n:JI5I
60JIee BblCOKHX np0l13BO.ll.HblX (B03MO)f(HO, npOH3BO.ll.Hble 6Y.ll.YT paCTH
6b1ctpee, lJeM npOCTbIe CTeneI-UI), 11 Boo6w,e HeJIb35! CKa3aTb HH4ero KOH-
KpeTHOrO OTHOClneJJbHO Toro, KaKotJ 6Y.ll.eT I Ah I no cpaBHeHHIO C
E h 5 y(5)
. TeM He MeHee Mbl npHMeM .lI.JJ5I 60JIbllll1HCTBa I1HTepeCYIOw,l1x
Hac ypaBHeHHtJ, 4TO \ Ah \ < 0,4.
1
ilz
...... -*1
<:> Ah=o'
..... Iljltl!JlIllO 3/8
........... , ..... -* .......
<k. ......"''''' ? .......
K" .....- 'Ah=-OZ
""=:: ............... Ah=-04> ...... «2/3)) 1-
......: ..... Ah--Ofj ....0«(11.5')) tj
Ah= 0..Q
r"': i'....;:1oo.:" :--.. .......- ::;;;.- /y "J ...... lOll
4(;:;1 I a J
'_ I' ::i :::. ] ...... /...
AU!lMC- .......... :::::: ...........
Otl U/!/Jo '1l1 ,
.. ......... :::::.
........... ,
H
7
0.
-3
-2
-1
0.
1
PIIC. 15.3-1. 06J1aCTb YCTOi\'1I1BOCTH.
Ha pl1c. 15.3-1 nOKa3aHa 06JIaCTb YCTOt!4HBOCTI1 B CJIY4ae Ah = O.
}{or.n:a Ah:;i: 0, O.n:HopO.ll.HOe pa3HocTHoe ypaBHeI-Ule, cooTBeTcTBYIOw,ee
(15.3-7), npHBo.n:HT K xapaKTepHcTI14ecKoMY ypaBHeHHIO
3 2
Ah - P - aop - alP - a 2 (15.3-8)
- b_ I P3 + b O p2 + blP + b 2 .
E:">To ypaBHeHHe 60JIee CJIO)f(HOe, 4eM COOTBeTCTBYIOw,ee ypaBHeHHe B rJI. 13
113-3a .n:onOJIHl-lTeJIbHot! o6paTHot! CB5J311 OT 4JJeHOB C npOIi3BO.ll.HbIMI1. Pe-
WeHl1e (cp. C (13.5-6» eCTb
Sn = C 1 (PI)n + C 2 (P2t + C a (Pat (15.3-9)
PeweHlie nepBOHa4aJlbHorO .lI.l1rpcpepeHUl1aJIbHOrO ypaBHeHIi5J (15.3-1) HMeer
TeH.lI.eHUli1O paCTH KaK
Y n = Ce(Ah) n,
208
OBUl.Ml TEOPI151 METOIlOB rrpo m03A H KOPPEKIJJUf
rrJJ. t
11 Boo6U1.e TO, C '!eM Mbl 6Y,lI:eM I1MeTb ,lI:eJIO, - :JTO He a6COJlIOTHblt! pOCT
OllJl16KH, a ee pOCT OTHOCHTeJlbHO JlOKaJlbHorO pOCTa peweHIiSl, TaKI1M
06pa30M, MbJ ,lI:06liBaeMCH He YCTOtl'!HBOCTH, KOr,ll:a \ Pi I < L a OTHOCU-
reJIbHotl YCTOt!'!HBOCTIi, Kor.n:a
Ipe-Ahl 1. (15.3-10)
O.n:HH li3 KopHetl, KOTOpbltl fIO-fIpe)l{HeMY 6y.n:eT 0603Ha'laTbCH P3' Be.n:eT
ce6S! TaK
P3 = 1 + Ah + (A;)2 + (A )3 + ( )' + ki ). e AII (15.3-11)
,lI:JlS! MaJleHbKliX I Ah \. TaKHM o6pa30M,
P3 e - All 1
1-1, CJleJIOBaTeJIbHO, JIUWh PI Ii P MorYT fIpHBecTH K OTHOCIiTeJIbHott He.
YCTOtl'lHBOCTH.
0,2
az
-
;' - --
\ - 6- - -
r'\: " Ah;::;-l,
I Ii ........ -r-:,
"- ", --- ./
" ,- /'
..... T'-. / , /
I"- 11/
l/
,-......
o
0.1
0.2
0.3
Q,
0.1
o
-0,$ -0,5 -0.4 -0.3 -0,2 -0,1
PHC. 15.3-2. 06J1aCTb YCTOHtJHBOCTH.
TpeyrOJlbHHK, B,lI:OJIb KOToporo npl1 Ah = 0 KOpHH npHHHMalOT 3HII'IeHHSI
Ipe-AIII = Ip 1=1,
n epexo.n:uT fIpli Ah, He paBHOM HY JlIO, B cpliryPY HeCKOJlbKO 60Jlee CJlO)J{-
HylO llJlH P = - e AII fIOJIy'!aeTcS! JJHHetlHoe ypaBHeHue
a.a'j + al + 1 = 0, (15.3-12)
r.ae a., H 1- KOM6HHau,HH Ah, e Ah , e 2Ah 11 e Mh Ho .n:pyrlie .n:Be CTO-
p OHhl TveyrOJlbHHKa CTaHOBS!TCH fl-mep60JlaMH. Ha pl1C. 15.3-1 H 15.3-2
t5.5]
nporH03 no TPEM TO'1KAM
209
fIOKa3aHa ,1I)HI OTp«U3TeJlbHbIX 3H34eHI1tt Ah O.1l:Ha BeTBb fHrrepOOJIbI .11:0
rrepeCe4eHH5I C rrp5lMott (15.3-12). 3H34eHH5I BHYTPH .1I:Pyroti BeTBH .1I:OmKHbI
ObITb YCTOtl4I1Bbl, a B rrpOMeJKYTKe MeJK.1I:Y BeTB5IMI1 .11:0 JJJKHbI 6bITb Hey-
CToti4HBbIe 3Ha4eHH5I, TaK 4TO OHH, KpoMe, B03MOJKHO, CrreUHaJIbHbIX CJIY-
4aeB, He MorYT 6bITb rrOJIe3HbI. PHC. 15.3-1 H 15.3-2 rrOKa3bIBalOT 06JJa-
CTI-I YCTOt!4HBOCTH .1I:JIH KopHeti p, H P2' D.JIH 60JIbWHX 3Ha4eHHt! I Ah I
KopeHb Pa .1I:aJIeK OT 3Ha4eHHH e Ah H oWH6KOti OT oT6paCblBaHH5I 4JIeHOB
HeJIb3H rrpeHe6peraTb.
Ynpa}/{HeHHSI
15.3-1. Ha4cpmTb npRMYIO (15.3-12) lI.JlR Ah > 0 11 HCCJle1l.0BaTb 06J1aCTb
YCTOH'IHBOCTH. C
15.3-2. I-1C110Jlb3YR ynpaJKHeHHe 13.7-1, pa3pa60TaTb cooTBeTcTBYIOllJ.YIO
TeOp111O lIiIR ].114J4JepeHUHalibHbIx ypaBHeHHH. Hatim 06J1aCTb YCTOti4HBOCTI1
B 3aBHCI1MOCTI1 OT a 1 H Ah.
15.4. DOMexH oKpyrJIeHHH
TeopHH pacrrpoCTpaHeHI151 rrOMex oKpyrJIeHI151 .1I:JI5I .1I:I1cpcpepeHUHaJJbHOrO
ypaBHeHH5I 04eHb rrOXOJKa Ha COOTBeTCTBYIOIlI.YIO TeopHlO .1I:JI5I HeOrrpe.1l:e-
JIeHHblX HHTerpaJIOB, H3JIOJKeHHYIO B 13.6. D.JIH Ah = 0 rrp5lMa51
a
(13.6-3) a2 = - pa3.n:eJI5IeT 06JIaCTb pOCTa H 06JIacTb y6bIB3HI151 OWI16-
KH OKpyr JIeHI151. BHYTpI1 06JI3CTH YCTOt!'1I1BOCTH K03cpcpHuHeHT YCI1JJeHI151
WYM3 .n:aeTC5I CPOPMY JIot! (13.6-2)
I
N c = I +a 1 +2a 2 '
KOr.1l:3 Ah =1= 0, rp3HI1U3 Me)/{.1I:Y .1I:BYM5I 06JI3CT5IMI1 60JJee CJIOJKHa, HO
3.11:eCb He CTOI1T 3THM 3aHHMaTbC5I; rrpoC'ro HYJKHO rrOMHHTb, 4TO JIY4we
6bITb BbIwe JIHHHH, 4eM HHJKe ee.
4T06bI He 6blJI CJJHWKOM 60JJbWHM HeKOppeJIHpOBaHHbIt! WYM, XopOWO
COXpaH5ITb
N - ( a 2 + a2 + a2 ) 1/2
a- 0 , 2
.1I:OCTaT04HO MaJIbIM.
15.5. DporHo3 no TpeM TOllKaM
B COOTBeTCTBHH C CPOPMY JIot! KoppeKUHH (15.2-1) XOTeJIOCb 6bI HMeTb
rrporH03, HCrrOJJb3YIOIlI.Ht! HHcpOpM3UHIO 0 rrOCJIe.1l:HHX Tpex T04K3X, 3
HMeHHO:
yn+' = AoYn + A,Yn_' + A 2 Yn_2 +
, , , - h 5 y(5'
+ h (BoYn + B'Yn-l + B 2 Yn-2) + E5 51 +...
(15.5-1)
210
OBIllA51 TEOPH5! METOAOB I1POrH03A 11 KOPPEKUl1l1
rrn.15
1{09cpqmuileHTbl MO)l{HO Hat!Til ilnH 06WJHblM rrYTeM, I1JJil nOJIOJKilB 8
ypaBHeHill1 (15.2-2) b_ 1 = O. Pe3ynbTaTbl
A o = - 8-A 2 , B _ 17+A
0- 3 '
B _ 14+4A 2
B: - lt< !
E - _ 40-4A 2
5- 3 .
A I =9,
(15.5-2)
A 2 = A 2 ,
EOJJbIllOe 3HaIJeHHe AI' paBHoe .n:eB5ITil, 03HaIJaeT, 4TO .n:a)l{e B caMOM
JJY4UleM CJJY4ae rronYIJilTC5I 6onbilloe YCilneHHe lllYMa
Na=(A +A +A i/2 >9
B npOfH03e.
lln51 Toro 4T06b1 B npOrH03e no TpeM T04KaM il36eJKaTb 60JJblllOrO
YCilneHH5I lllYMa, MOJKHO c.n:enaTb cpOPMY ny T04HOt! TonbKO .n:o x 3 BKJlIO-
LIilTeJlbHO, OCTaBI1B OCTaT04Hblt! 4JIeH rrOp5l.n:Ka h 4 . 3TO rrpilBO.D:ilT K ce-
MeilcTBY rrpOfH030B:
Ao = 1 -- AI - A 2 ,
B _ -16+ 8A t + 16A 2
1- 12 '
A 1 = AI'
A 2 = A 2 ,
B _ 23+5At+4A2
0- 12
B _ 5- A l+4A2
2- 12 '
(15.5-3)
E4=9 -AI'
YTo6bl BOCIIOJIb30B3TbC5I p33HOCTbIO n-fO npOfH033 H IwppeKlum JJJHI
Oll,eHKH 01llI16KH (n + 1 )-fO npofH03a, CJle.n.yeT orrpe.QeJIilTb Pn+1 il3
mn+1 =Pn+1 - (Pn - cn) = C n + (Pn+1 - Pn)'
TaKa51 cxeMa ilHOf.n.a 6bIBaeT nOJle3Hot!, XOT5I MOJKeT Tpe60BaTb 04eHb
60JJbillOfO KOJlil4eCTB3 BbI4ilCJIeHilt!, oc06eHHo, KOf.Qa Tpe6yeTC5I TOIJHOe
peweHile .n.ilcpcpepeHll,ilaJIbHbIX ypaBHeHilt!. Mbl He 6y.n.eM 06CY)j{..QaTb ee
.QaJJbIlle; 51CHO, 4TO OHa ilMeeT OCTaT04Hblt! 4J1eH rr5lTOfO nOp5l..QKa.
15.6. npOl'H03bl TilDa MHJIHa
ECJJil Mbl XOTHM ilMeTb ceMb napaMeTpOB, 4T06bl ilCrrOJlb30BaTb HX B
rrOilCKax rro.n.xo..Q5IIllero rrpOrH03a, TO MOJKHO rrpl16aBilTb ilJlH O..QHO 60Jlee
CTapoe 3HaLIeHHe cpYH1([J.ilH, 3 ilMeHHO Yn-3' HJlH O..QHO CTapoe 3Ha4eHile
npOH3BO..QHOt!, a I1MeHHO Y -3' IlepBoe npe..QJJO)j{eHile BKJJI04aeT nporH03
MI1J1Ha, il Mbl Ha30BeM ero nporH030M «Tl1na MHJlHa», a BTopoe cO..Qep-
JKHT npOrH03 A..QaMca-53illcpopTa 11 Ha3bIBaeTCH COOTBerCTBeHHO.
15.61
npOfH03bl THnA MHJJHA
211
Oonpo6yeM CHa4ana Ha TH opMyny BH a
Yn+t = AoYn + AtYn_t + A Yn- + AaYn_a +
- h 5 y(5)
+ h(BoY +BtY -1 +B Y -2)+ E5 51+'" (15.6-1)
06bI4HbI MeTO aeT
Ao= -8 - A2+8Aa,
B _ 17 +A2 - 9A s
0- 3 '
B _ 14+4A2 -18A 3
t- 3 '
B _ -1+A2+9A3 j
2- 3 '
E - - 40-4A 2 +72A 3
5- 3 .
(15.6-2)
At = 9 - 9Aa>
.
A = A ,
Aa ==- Aa,
t.h06bl OCTaT04Hbl 4neH 9TO opMynbl HMen YKa3aHHbl BH,Q, He-
06XO HMO, 4T06bl YHKUIUI BnH5IHl:!51
Q(s)=(h - S)t - Ao(- S)t -- At(- h -S)t - A (-2h -S)t-
- Aa(-3h - S)t - 4h [Bo(-s) +Bt (- h -S) - B 2 (-2h - S) ]
6bIna nOCT05lHHOro 3HaKa n51 h s - 3h. B nnOCKOCTH (A , Aa) nl:!-
HH5I, B O.7Jb KOTOpO O(S) = 0, .lI.BH)J{eTC5I CHH3Y J{ OCH Aa = 0, KOr.ll.a S
MeH5IeTC5I OT h K -2h, H OCTaeTC5I Ha Aa = 0 n51 -2h S . -3h.
TaKHM 06pa30M, 4T06bI OCTaT04HbI 4neH HMe.7J HY)j{HbI BH.lJ., OrpaHl:!-
4HMC5I CnY4aeM Aa O.
ECJJH nOnp060BaTb YMeHblIIl:!Tb 01II116KY OT oT6paCblBaHH5I 4neHOB,
BbI6paB Aa = 0, TO Ha eM, 4TO At = 9, a ElTO ,n:OCTaT04HO Henpl151THo.
C pyro CTOpOHbl, ecnl:! 1I0lIp060BaTb MHHHMH3HpOBaTb YCHneHl1e IIIYMa
Na=(A + Ai +A +A iI2,
TO lIonY4l:!M
4
Ao=- m'
9 4
At=ffi' A 2 =-m,
113
Aa=m.
3IO 04eHb 6JlYI3KO K nporHo3Y MHJIHa
_ + 4h 2 ' . +2 ' ) + 14 h 5 (5) ( {J )
Yn+t- Yn-a 3 LYn - Yn-I Yn-2 45 Y v
c Ao = At = A 2 = 0 H Aa = I. BWlrpbl1II Ka)j{eTC5I HeCT05llll,HM.
TaKHM 06pa30M, l:!ccne Y51 9TOT MeTO.ll., Mbl npHXO HM K nporH03Y
MHJIHa. J{aK 6bIJJ0 3aMe4eHO B 14.4, ero MO)J{HO 06"be.ll.HHI1Tb B cxeMY
rn. 14 C mo60 1I0 XO 51Il1.e KOppeKlI,Hett J{oHe4Ho, B03MQ}j{eH 11 py-
ro Bbl60p napaMeTpoB A 2 l:! Aa.
212
OBUlA5I TEOPH5I METOJJ.OB IJPOfH03A H KOPPEKIJ,HH
[Tn. I
YnpamHeHHe 15.6-1. I1ccJle,aoBaTb npOrH03bJ BI1J1,a
h 4 (IV) (6)
Yn+l=AoYn+AIYn_l+AsYn_s+h(Boy +BIY -I)+Ei Y 41 '
HcnoJlb3YR A 2 KaK napaMeTp.
o T BeT: Ao=-4-5A2'
Bo=4+2A2'
ECJlH As = I, TO £4 =0 11 E5 = 12.
AI = 5 + 4A 2 , As = A2'
Bl =2+4A 2 , E\=4-4A2'
15.7. nporHo3bI nma AJJ:aMca-EarncpopTa
ECJlH BMeCTO TOro, 4T06w BOCnOJlb30BaTbC5I .lJ:OnOJIHHTeJlbHbIM 3Ha4e-
HHeM <PYHKIJ.HH, B03bMeM .lJ:OnOJlHlneJlbHOe 3Ha4eHHe npOH3BO.lJ:Hol1 Y;l-3J-
TO nOJlY4HM <POPMYJIY
Yn+I = AoYn + A 1 Yn_I + A 2 Yn_2 +
- h 5 y(5)
+ h (BoY + BIY -I + B2Y -2 + B3Y -3) + E. ----gr-, (15.7-1)
K03<p<PHIJ.HeHTbl KOTOpOti paBHbl.
Al = AI'
B _ -59+ 19A 1 +32A2
1- 24 '
B _ 37- 5A l+8A2
2- 24 '
B _ -9+AI
3- 24 '
E - _ 251-19A I -8A 2
.- 6
(15.7-2)
Ao = 1 - Al - A 2 ,
A 2 = A2'
B _ 55+9AI +8A
0- 24 '
B 9TOM CJlY4ae <PYHKIJ.H5I BJlH5IHH5I O(s) I1MeeT HYJIH B nJIOCKOCTH
(AI' A 2 ) B.lJ:OJJb JJHHHtI, nepBa51 H3 KOTOpblX - np5lMa51 Al = 9 - .lJ:.151
S = -2h nO.lJ:HHM3eTC5I H rrpH06peT3eT OTpHIJ.3reJIhHbJe T3HreHCbl yr.13
HaKJlOHa C YBeJIH4eHIIeM S, He HaKJla.lJ:blBa51 CYUleCTBeHHblX orpaHH4eHHti
H3 K03<p<pI-!IJ.HeHTbl.
P5I.lJ: CJIY4aeB, BKJJI04a51 nporH03 A.lJ:aMCa-6aw<popra, npl1Be.lJ:eH B
Ta6J1I1IJ.e 15.7-1. 3Ha4eHHe K09<p<pHIJ.HeHra YCHJleHH5I wYMa N c JI,.151 h -+ 0
3.n:eCb TO )j{e C3Moe, 4TO H .lJ:JI5I KOppeKIJ.HH, Hero MO)j{HO He nOBTop5lTb.
YnpamHeHHe 15.7-1. I-1cCJle,aoBaTb nporH03bl BI1.D:a
, , , _ h 4 y(lV) (6)
Yn+1 = AoYn + AIYn_1 +h (BoYn +BIYn_1 +B 2 Y n _2) +E. 41 '
HcnoJlb3YR AI KaK rrapaMeTp.
o T BeT:
23+5AI 5- Al
Ao=I- A t; Bo=- ; B 2 =12;
- 16+ 8A l -
Al=At; B I - 12 ; Ei=9-AI'
S 15.8J
OBlIHIE 3AME4AHI1ST 0 BbiBOPE METOJJ.A
213
T a 6 JI H U, a 15.7-1
nporHo3bl THna AJI,aMCa - 6auHpopTa
[3HaLfcHHJI E. -£5 BblLfHCJleHbl .D:JlR CJlYLfaR ai = Ai (i = 0, 1, 2)]
I A aMC -I Bblcom I TlpaBHJlo ( ,<1/3. I <1/2» I «2/3»
6awq,opT TO<H!OCTb «3i8»
Ao I 0 0 I 1 0
a- 2
AI 0 I 0 I 1 2
2 2 a-
A 2 0 0 1 1 1
a- a-
8 0 55 8 21 94 110 191
24 a- S 36 48 72
8 1 59 5 9 63 99 107
24 -a -S - 36 - 48 --"{?:
8 2 37 1 15 57 69 108
24 a- S 36 48 72
8 3 9 1 3 13 17 25
24 -a -S - 36 - 48 - 72
E. 251 110 243 121 161 707
6 a- -6 a- T 18
E.-E. 270 210 270 260 255 250
6 6 -6- 6 6 6
15.8. 06UI.ue 3aMeqaHHH 0 Bbl60pe MeTO.ll.a
rI pH 110CTpOeHHH KOHKpeTHbIX <POp MY JI npOfH03a H KOppeKll.HH npH-
XO.ll.l1TeH ypaBHOBeIllI1BaTb Tpl1 .n:0 HeKOTOpOt! CTeneHI1 HeeOBMeCTI1MbIe ll.eJII1:
1. YMeHbIlleHl1e OIlll16KU OT OT6paebIBaHI1SI 4JIeHOB.
2. 60JIbIllOt! 3anae YCTOt!4uBOeTI1.
3. 3auuna OT HenOJJa.n:OK e oKpyr JIeHI-1eM.
JlY4t1Je Beero Ha4aTb e BbI60pa KOppeKll.UU, TaK KaK OHa Hrpaer
60JIee Ba)l{HYIO POJlb. PHe. 13.5-1 nOKa3bIBaeT 06JIaeTb, U3 KOTOpofi
MO)l{HO BbI6paTb KOppeKll.I1IO. 5IeHo, 4TO KOppeKll.HH MI1JJHa (al = 1,
a 2 = 0) npilMepHo TaK )l{e TO<JHa, KaK HaHJIY<Jt1JaH, KaKaH MO)l{eT 6blTh
Hat!lleHa. PV\c. 15.3-1 nOKa3blBaeT Mepy YCT0t14ilBOCTl-I. ECJIil A =
nOJIO)j{HTeJlbHO Ha BeeM uHTepBaJIe UHTerpHpOBaHI1H, TO MeTO.n: MUJJHa
Xopow. Ho eCJIH A = : MO)l{eT 6bITb OTpUll.aTeJlbHbIM, TO KOppeKll.HSI
214
05UlA5! TEOPI151 METOD-OB npOrH03A 11 KOPPEKUI1I1
[rJl. 15
MI1J1Ha HeYCTOfi4HBa, H HY)/(HO Bbl6HpaTb KOMnpOMHCCHOe pemeHOe Me)/(.lI.Y
.lI.BYM5! H.eOpH5!THOCT5!MO: HeYCTOt!4HBOCTblO B OlllB6KOfi OT oT6paCblBaHH5!
4JJeHOB, H3Mep5!eMbIMH COOTBeTCTBeHHO BeJll1'-!I1HaMH
: h = Ah
H
E 5h'y(" (6)
5!
Me)/(.lI.Y 9THMH .lI.BYM5! BbIpa)/(eHH5!MH HeT onpe.ll.eJIeHHOrO COOTHOrnemUI,
XOT5! 5!CHO, tJTO 06a HMelOT TeH.n:eHUHlO 6bITb HJJH 60JlblllHMH, HJlH MaJleHb-
KHMH rro BeJlH4HHe O.ll.HOBpeMeHHo.
Mbl nOJJO)/(HM Ah = - 0,4, CtJHTa5!, 4TO npB pactJeTax c YMepeHHot1
TOtJHOCTblO TaKofi Bbl60p .lI.JI5! 60JlblllBHCTBa ypaBHeHl1fi orpa.n:HT Hac OT
6ecrrOKoficTB OTHOCIUeJlbHO HeYCTott4HBOCTH. CpaBHeHHe pl1c. 13.5-1 11
15.3-1 rrOKa3bIBaeT, tJTO B.lI.OJlb rrp5lMofi tJaCTH KOHTypa Ah = - 0,4
ollll15Ka OT oT6pacbIBaHfHl lIJIeHOB rrotJTH O.ll.HHaKOBa.
HerrOJla.ll.KI1 C oKpyrJJeHHeM HaBO.ll.5IT Ha MbICJlb rro.n:BHHYTbC5I BrrpaBo
2 I
BBepx H BbI5paTb .lI.Jl5! KoppeKUHH T04KY Gl = -3' G2 = 3' B HeKoTopoM
2 . M I.
CMblCJle 9TO Ha 3 MeTO.ll. HJJHa H Ha 3 «npaBHJJO Tpex BOCbMbIX».
MI1J1H H PefiHOJlb.ll.C *) rrpe.ll.JJaralOT HCrrOJlb30BaTb MeTO.ll. MHJlHa, rrepl1o-
J]:HlIeCKH rrpHMeml5l rrpaBI1JJO Tpex BOCbMbIX, HO rrpe.ll.JIO)/(eHHOe CMellleHHe
9TI1X MeTOJ]:OB Ha Ka)/(J]:OM lllary 50JIee Y.ll.OBJIeTBOpl1TeJJbHO o5ecnetJI1BaeT
YCTOlt4HBOCTb.
XOT5I, B351B Ah = - 0,4, MbI HaAe5!JlHCb Boo6Iue 1'!35aBHTbC5I OT
HeycTot!lIHBOCTI1, CJJeAyeT 3aMeTI1Tb, tJTO rrpH pellleHI1H ypaBHeHI1t! C 60Jlb-
IlIHM OTpHuaTeJJbHbIM HY}KHO BbI511paTb 60Jlee YCTOttllHBbIIt MeTOJ]:.
v13 pnc. 15.3-1 BHAHO, tJTO TOllKa (Gl = - 0,20, a2 = 0,15) .n:aer OlleHb
YCTOt!tJI1BbIt! MeTOA, HO 3aTO OH I1MeeT AOCTaTOllHO 50JIblilYlO (E5 = - )
OlllH5KY aT oT5paCblBaHH5! '-lJIeHOB.
Ynpa}l{HeH1-te 15.8-1. B ynpa)/{HeHI1H 15.3-2 HCCJIe.n:OBaTb BbI60p KoppeKTH-
pYlOIll.eH 4JOpMy.%1 4eTBepToro nopilJJ,Ka.
15.9. BbI60p nporHo3a
IlpH BbI50pe rpOPMYJlbI rrpOfH03a cYUlecTByer CHJlbHbII1 C05J1a3H
HCrrOJlb30BaTb B :;not! rpopMYJle J1,JJ5! Ai Te )/(e 3Ha4eHl!5I, KaKl1e 5YAYT
rrpl1H5ITbI AJl5! ai B rpopMYJle KOppeKUI1H. Tor.lI.a npl1 BblllHCJleHfIl1 TpaeK-
TOpl111 .lI.BI1)/(eHJiI5l JIYHbI HJlH nJIaHeT MO)/(HO BbItJHCJl5ITb JlHHel1HYlO KOM6H-
HaUHlO 3HatJeHHt! y (nOJlO)/(eHHt!) nporpaMMot!, ctJHTalOUlet! C ABOI1HOfi,
a Bblpa)/{eHI15! .lI.Jl5! y' - C o5wJHol1 T04HOCTblO II nplIToM J1,OCTII4b 50JIb-
moro YBeJlI14eHII5! T04HOCTH.
*) J. Assoc. Computing Machinery, vol. 6, pp. 196-203, anpeJlb 1959.
15.101
HEKOTOPbIE <1>OPMY JJbl
215
TaKoe COOTBeTCTBHe K09cpcpl1lJ.HeHTOB Ai H ai HCKJllOqaeT, KOHe4HO,
npOrH03bl THna MHJlHa, I1MelO!!J.He HeCKOJlbKO 66JlblllYlO TO'IHOCTb, 'IeM
MeTO.ll,bl A.lI,aMca. 3.n:ecb Mbl HCCJle.ll,yeM TOJlbKO Te CJJY4aH, B KOTOpblX
nO)l,06Hblt-l Bbl60p Ai H ai B03MomeH.
Pa3HocTb Mem.n:y nporHo3HpoBaHHbiM H cKOppeKTHpOBaHHbJM 3Ha4e-
HHS/MH Yn+1 OlJ.eHHBaeT nS/TYIO npOH3BO.ll,HYIO. TIpH paBeHcTBe Ai H ai
4JJeHbi C Y YHH4TomaIOTcS/ H Mbl HMeeM
Pn+1 - Cn+1 = h [BoY + B1Y -1 + BVJ -2 + B3Y -3 - (b_tP +1 + boY +
+ btY -1 + bVJ -2)]' (15.9-1)
'ITO Boo6!!J.e MomeT 6bITb Bbl'I11CJleHO cOBepllleHHo T04HO.
Ynpa)/{HeHHe 15.9-1. I1ccJleJJ,oBaTb coorBercrBYIOw.ylO reopl1lO MeTO)l,OB 'IeT-
Beproro nOpR)I,Ka.
15.10. HeKoTopble <lJOPMYJIbl
MomHO HCnOJlb30BaTb TOT cpaKT, 'ITO Pn - C n 06bl'IHO 60Jlee HJlH
MeHee nOCTOS/HHa OT llIara J{ wary, .lI,JlS/ YHH4TomeHHS/ OlllH6J{H B npor-
H03e, paBHott
Eh"y'''J
51
4To6bl YHH'ITomHTb 9TY oWH6KY, eCTeCTBeHHO HCIlOJlb30BaTb BeJlHQHHY
E E h"y' " ,
_ " (Pn-Cn)=- '
E" - E"
(Ha nepBOM llIary, KOr.ll,a OTCYTCTByeT Pn - cn' Mbl nOJlOmHM Pn - c n = 0.)
TeM me Cl10C060M MomHO rrOnblTaTbCS/ HcrrpaBHTb KOppeKlJ.HIO; nonpaBKY
eCTeCTBeHHO B3S/Tb paBHot!
E" E"h"y'''J
(Pn+l- Cn+1) 5!
E" - E"
TIpHMeHHM Bce 9TO, HanpHMep, K TaK Ha3bIBaeMOMY «npaBHJJY .n:BYX
TpeTbHX». <t>OPMYJlbl B 9TOM CJlY'Iae HMelOT cJle.ll,YIO!!J.Htt BH.n::
TIpOrH03:
- 2Yn_l+Yn_2 + h (191 ' 107 ' + 109 ' 5 ' I
Pn+1- 3 n Yn- Yn-I Yn-2- 2 Yn-3)-r
+ h" (0)
2160 Y .
113MeHeHHe:
707
m n + 1 =Pn+1 - 750 (Pn - cn)'
215
OSliI.MI TEOPI1)1 METOAOBnpOfH03A 11 KOPPBKL(1111
(fJI. 15
KoppeKUilIJ:
c - 2Yn_l + Yn-s +
n+l- 3
+ 7 (25m +t+91y +43Y _1+9Y _2)- 2 0 h 5 y(5).
OKOH'IaTeJIbHOe 3Ha'leHl-le:
43
Yn+l = Cn+l + 750 (Pn+l - C n +l)'
3Aecb nOJJY'IaeTCIJ TO'lHblft OTBer, eCJIJ1 y(5) (X) nOCTOIJHHa, TaK '1TO
,HOT MeTOA MO}\(HO Ha3Barb MeTOAOM meCToro nopIJAKa.
CJIeAyeT 3aMeTI-ITb, '1TO eCJJI-I Yn+l 6JII-I3KO K m n +1' TO MO}\(HO He Bbl-
'1ilCJIIJTb np0l13BOAHYJO y + 1, a HCnOJIb30BaTb 3Ha'leHile m + 1. ECJIll
€ = y + I -- m + h TO, npeAnOJJaraIJ, '1TO 1 Ah I = 0,4, B CJIeAYJOllleM
npOrH03HpOBaHHoM 3Ha'leHl-lil nOJIY'lilM omil6KY, paBHYJO
€h 1: 2 1 I \ € . 2,65 .1 A \. h 1,06€.
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HCrrOJIb30BaHHll Bh14HCJII1TeJIhHotl MaliJHHhI LlJl51 peliJeHH\! pasHhIMll MeTO-
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y'=y y (0) = 1 y=e x (15.12-1)
y'=-y Y (0) = 1 Y = e- x (15.12-2)
y'=- 2 xy 2 Y (0) = 1 1 (15.12-3)
Y = I +X2
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9TH MeTOltbl MO)\{HO npHMeH51Tb 11 Ha Ka)\{ltOM Illary, 9<p<peKTI1BHOCTb
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(16.2-1)
BW4HCJI5IeTC5I nOCJJe.n:OBareJIbHO
k l = hi (xn> Yn)'
k2=hf(Xn+ , Yn+ 1 ),
ka = hi (Xn + ' Yn + 2 ),
k4 = hl(x n + h, Yn + ka),
1
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(16.2-2)
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BbIlJHCJJ5IerC5I TaHreHC yr JIa HaKJJOHa (kl/h); HCTIOJJb3Y5I ero, MbI H.n:eM Ha
I10JIOBHHY wara BI1epe.n: H CMOTpHM TaHreHC yr JJa HaKJIOHa 3.n:eCb. J.1CI10JIb-
3Y5I HOBWt! TaHreHC yr JJa HaKJIOHa (k /k), MbI 0I151Tb Ha4HHaeM H3 (x n , Yn),
H.lJ.eM BI1epe.n: Ha rrOJIOBHHY wara H On5lTb 6epeM rrp06y TaHreHca yr JIa
HaKJIOHa. B351B 3TOT rrOCJIe.n:HI1t! TaHreHC yr JIa HaKJJOHa (ka/k), Mbl Orr5lTb
HalJHHaeM U3 (x n , Yn), HO .n:eJIaeM Terrepb rrOJIHblt! war Brrepe.n., r.n.e
CMOTpHM TaHreHC yrJIa HaKJIOHa (k4/h). 4eTblpe TaHreHca yrJIOB HaKJIOHa
ycpe.lJ.HHeM C BecaMH 1/ 6 , /6' 2/ 6 , 1/6 H, 6ep51 3TOT Cpe.llHHt! TaHreHC yrJJa
HaKJIOHa, .n:eJIaeM OKOH4aTeJIbHblt! war OT (Xn> Yn)K (X n +1> Yn+I)'
ECJIl-l I (X, y) Ha 3aBI1CHT OT y, TO ycpe.n:HeHl1e MO)KHO rrpOH3Bo.n:HTb
no q>opMYJIe CI1MI1COHa. 9roT MeTO.lJ. I1MeeT oCTaTo'JHblt! 4JIeH, nponop-
!J.l1oHaJIbHblt! h 5 .
Cyrn.eCTBYIOT Bapl1aHTbI MeTo.n:a, KacalOw.l1ecS! TOro, r.n:e CJIe.n:yeT 6paTb
Y3JIOBble T04KH H, CJIe.n:oBaTeJIbHO, KaKHe Beca npH.lJ.aBaTb TIM Ha pa3JII1'J-
HbIX war ax. Cyw.eCTBYIOT TaK)Ke MeTo.n:bl 60JIee HH3Koro nopH.n:Ka,
I1CnOJIb3YIOw.l1e MeHbwe Y3JIOBbIX T04eK Ha war.
04eBI1.n:Ho, B 3TOM MeTo.n:e I1pOna.n:aeT 3pS! BCS! npe.lJ.bI.n:yw.aS! HHq>Op-
Mall.H5I H Ka)K.n:blt! rrOJIHblt! war .n:eJIaeTCS! 3aHOBO, a CJIe.n:OBaTeJIbHO, 3TOT
MeTO.lJ. e.1I.Ba JII-I S!BJIS!eTCH TaKHM )Ke 3q>q>eKTHBHbIM, KaK MeTO.lJ.bJ, KOTopble
I1CI10JIb3YlOt cTapylO I-IHq>opMall.l-IlO. 04eBI-I.n:Ho TaK)Ke, 4TO He cyw.eCTByeT
222
CDEUI1AJJbHblE METOJJ,bI I1HTEfPI1POBAH!1$1
1 fJJ. 16-
I1pOBepKH TOro, SlBJIHeTCSI JIH war CJIHWKOM MaJIeHbKHM HJIH CJJHWKOM
60JIbWI1M, H XOTSI, B03MOlliHO, H3Y'leHI1e k j MOllieT ,/J,aTb KJIIO'l K 9TOMY,
ero 06bl'lHO He npHMeHSIIOT.
KallieTCSI, MeTO,/J, IlOJIY'leHHSI 9T11X ypaBHeHHI1 He I1CIlOJJb3yeTc5I B ,/J,PY-
rnx MeCTax 'l11CJJeHHoro aHaJJH3a, H Il09TOMY ero He CTOI1T ,/J,aBaTb 3,/J,eCb *).
06W.HI1 CMblCJI IloCTpOeHI1S1 HX B TOM, 'lTO CPYHKJ,J,HH I (x, y), KOTopble
HaXO.[(SlTCH CIlpaBa, Bce paCIlHCblBalOTC5I B p5l,/J,bl IlO CTeneH5IM h 11 COOT-
BerCTBYIOw.He np0113BO,/J,Hble npHpaBmmaIOTC5I, 'lT06bl YCTpaHi-ITb 60JIee
HH3KI1e CTeneHH h. B pe3YJIbTaTe ypaBHeHH5I (16.2-2) 51BJI5IIOTCH TO'lHblMH
,/J,JJ5I MHOr04JIeHOB CTeIleHH 4 (HJIH HHllie).
S 16.3. MeTOll.bl lI.JUI ypaBHeHHJI BToporo nOpJlll.Ka, KOrll.a
OTCYTcTByeT y'
MO)KHO I1CCJIe,/J,OBaTb MHoro qaCTHblX CJIyqaeB, HO B 9TOM naparpacpe
Mbl OrpaHI1'l11MCH 06w.etl cI1TyaJ,J,l1etl; paCCMOTpI1M ,/J,l1cpcpepeHUfIallbHOe
ypaBHeHl1e BToporo nOp5l.[(Ka C orcYTcTBYIOw.etl nepBotl np0!13BO,/J,Hotl.
YaCTo, Kor,/J,a npoH3Bo,/J,Ha51 np!1cYTcTByeT, npocToe npeo6pa30BaHHe Y,/J,a.
JIHeT ee. IVm HCClIe.[(OBaH!1H ,HOtl np0611eMbi np!1MeH!1M re llie MerO,/J,bI,
KaKHe I1cnOllb30BaJIHCb B r JI. 15.
Ha4HeM C KoppeKJ,J,I1H IlPH !1HrerpHpoBaHHH ,/J,H epeHQHallbHOro
ypaBHeHH5I
y"=f(x, y),
(16.3-1)
KoropylO Ilpe,/J,craBllM B BH,/J,e
Yn+l = aOYn + alYn_l + a2Yn_2 + h 2 (b_lY +1 + boY +
+ blY -l + b2Y -2)'
(16.3-2)
MeTO,/J, IlOCJJeJl,Hel1 r JJaBbl MOllieT HaBeCTI-I Ha MblCJJb 3KOHOMHTb ,/J,Ba Ila-
paMerpa, HO IlpOCTa51 IlpoBepKa IlOKa3b1BaeT, 'lTO HCnOJJb30BaHHe a 2 B Ka-
'leCTBe e,/J,HHCTBeHHoro napaMeTpa Il03BOJJI1T HaM HaJl,elliHO peWHTb Ilpo6-
JleMY YCTOt\qHBOCTH. TaKHM 06pa30M, MbI MO)\(eM C,/J,eJJaTb MeTO,/J, T0'lHbiM
,/J,JJ5I I, x,..., x 5 . Pe3YJIbTaTbI IlpHBeJl,eHbl B Ta6JJHI-Le 16.3-1.
MeTo,/J, B IlOCJJe,/J,Hetl KOJJOHKe HMeeT HYJIeBotl OCTaTo'lHbll1 'lJJeH; cJle-
LWBaTeJlbHO, OH HMeeT 60Jlee BblCOKYIO TO'lHOCTb, 'leM OCTaJlbHble. Ho
KpaTHb11i xapaKTepHCTl1qeCKHtI KopeHb 03Ha'laeT, 'lTO pacIlpOCTpaH5IIO"
w.a5lc5I oWH6Ka En 6Y.lI.eT BeCTH ce651 KaK
En = C, + C 2 n + C a n2,
'lTO HaBO,/J,HT Ha rpYCTHbie pa3MbIIllJleHHH. B ,/J,pyrnx CJIyqaJjx HMeeM
en = c, + C 2 n + C a (a2t.
*) CM. [20].
16.31
METOllbl llJJ5I YPABHEHl151 DE3 y'
223
T a 6 JI H U a 16.3-1
I G2 = - 1 I G2 = - 1/21 G2 =0 I G2 = 1/21 G2 = I
a O =2+a 2 1 3/2 2 5/2 3
a 1 = - (1 +2a 2 ) 1 0 -I -2 -3
a 2 = a 2 -1 -1/2 0 1/2 I
b_ 1 = 1/12 1/12 1/12 1/12 1/12 1/12
b o = (10 - a 2 ) : 12 11/12 21/24 10/12 19/24 9/12
b 1 = (1 - IOa 2 ) : 12 11/12 12/24 1/12 -8/24 -9/12
b 2 =-a 2 : 12 1/12 1/24 0 -1/24 -1/12
E. =- 3+3a 2 -6 -9/2 -3 -3/2 0
PI> P2' P3 = 1,1 ,a 2 1, I, -I 1,1, -1/2 I, 1, 0 I, 1, 1/2 1, 1, 1
KpaTHbII1 KopeHb p = I 51BJ151eTC5I HefI36e)l(HbIM, eCJ1H rjJopMYJIa 1leJIaeTC5I
T04HOil 1lJI5I I H x, a JIHHel1HbII1 pOCT OIlIH6KH npOHCX01lHT OT OIlIH6KH
fj j (KOTOpa51 He n05lBJJ5IeTC5I 5IBHO), pacnpOCTpaH5I5ICb JII1HellHo C n.
ITpel1MYllI,eCTBO 1lBYX HYJJeBbIX K03rjJrjJHlI.HeHTOB npl1 BbI60pe (CM. [26],
4JOPMY JJa (44.1» a2 = 0 1leJIaeT rjJOPMY JJY O'leHb npl1BJIeKaTeJIbHoll
h 2 h6y! 6)
Yn+l = 2Yn - Yn-l + 12 (Y +I + 10y + Y -I) - . (16.3-3)
Eho MO)l(HO TaK)I(e 3anHcaTb B BH1le
2 ( " il2Y _I ) h6y(6)
Yn+l =2Yn - Yn-l +h Yn+ 12 - 240' (16.3-4)
ITo CHMMeTpHH JIemo npOBepHTb, 'ITO O(s):=j:: 0 1lJI5I - h < s < h.
Y1l06HWIl nporH03 1lJJ5I 3TOil KOppeKlI.l1H TaKoll (npHB01lHM ero 31leCh
()e3 BbIB01la):
_ 2 _ + 4h2 ( " + " + " ) + 16 h 6 (6)
Yn+l - Yn-l Yn-3 3 Yn Yn-l Yn-2 240 Y , (16.3-5)
IIJIH
_ 2 + 4h2 ( " + il2Y _2 ) + 16 h 6 (6)
Yn+l - Yn-l - Yn-3 Yn-l 3 240 Y ,
( 16.3-6)
«ITPOrH03 MHHYC KOppeKlI.H5I» 1laeT
h6 (6)
240 Y ,
TaK 4TO oilln6Ka npH KOppeKlI.HH npn6JIH3HTeJIbHO paBHa
I
- I7 (Pn+l - CntI)'
224
CDEl\VlAJJbHbIE METOJ],b! I1HTE fPI1POBAHI151
[fJJ. 16
B TO BpeMS1 KaK olll1-l6Ka B npofH03e
16
17 (Pn - cn)'
choT pe3YJIbTaT MO)KeT 6b1Tb HcnOJIb30BaH, tJT06bl YJIYtJllll-ITb TOtJHOCTb.
XapaKTepl1CTI1tJeCKHe KOpHI-I npOfH03a CYTb 1, - 1, i, - i 11 He
C03,l1.a1OT TPY,l1.HOCTet!.
YnpamHeHHSI
16.3-1. Bb1BeCTlt paBeHcTBo (16.3-5).
16.3-2. rIoKa3aTb, I)TO cnpaBe.n.JUlBO paBcHcTBO (16.3-5), IICCJle.n.OBaB <PYHK-
1.],1110 O.
16.4. JlHHeHHble ypaBHeHIIH
JII-IHetJHbIe ,l1.l1cjJcpepeHI.l.HaJIbHbIe ypaBHeHI1S1 BCTpe1.IaIOTCS1 OtJeHb tJaCTO,
fI, HecMoTpS1 Ha 06111l-lpHYIO TeopHio, cBS13aHHYIO C HflMH, HHOr,l1.a npflxo-
,l1.flTCS1 peruaTb fiX tJHCJJeHHbIMH MeTO,l1.aMI1. CBOtlCTBO JJHHetiHocTH ,l1.eJIaeT
ox 60JIee JIerKHMH ,l1.JI5I peweHHS1, tJeM 06ll1.He ypaBHeHH5I. B KatJeCTBe
npHMepa paccMoTpHM ypaBHeHfle
y"=f(x)y+g(x), (16.4-1)
KOTopoe S1BJJS1eTCS1 tJaCTHbIM CJIY4aeM ypaBHeHHS1, paccMoTpeHHoro
B 16.3 (TaK KaK y' He CO,l1.ep)\(HTCS1 B HeM S1BHO). VlCnOJIb3YS1 KoppeR-
UHIO B BH,l1.e (16.3-4), HMeeM
2 h 2 A 2' _ 2' hOy(6)
LlYn-l- 12 i.1Yn-I-hYn-24(j'
I10,l1.CTaBJIS1S1 (16.4-1) BMeCTO y", HaXO,l1.I-IM
Ll2Yn_l - Ll2 [f(X n _l)Yn_l + g(X n _l)] = h 2 [f (x n ) Yn + g(x n )],
Ll2{[ 1 - ; f(Xn_l) JYn-l} = h 2 f(xn)Yn + h 2 (gn + Ll gn_I)'
Tenepb Ham1weM
[1- ; f(Xn)JYn= Y n ,
TaK '-ITD
A 2 Y n _ 1 = h 2 [f(Xn)Yn + gn + 1 1 2 A2gn_1 J.
TaKHM 06pa30M, OKOH4aTeJIbHO
Yn+l = 2 Y n - Y n - 1 + h 2 [f(Xn)Yn + gn + 11 2 Ll2gn_1J, (16.4-2)
f.n;e
Y n
Vn = h2
1 - 12 1 (X n )
(10.4-3)
16.5]
METO)]., KOTOPbIM I1CnOJJb3YET 3HA'-IEHH5I y, y' H y"
225
EhoT MeTO,lJ, lJaCTO npl1nHCbIBalOT HYMepOBY *); OH OlJeHb nOJJe3eH H
3cpcpeKTHBeH. OlJeBI1,lJ,HO, lJTO nO,lJ,06HbIe npe06pa30BaHI151, C,lJ,eJIaHHbIe
C u.eJIblO I1CKJIlOlJl1Tb npOm03, MOrYT 6bITb rrp0l13Be,lJ,eHbI BC5IKI1t1 pa3,
KO r,lJ, a ypaBHeHHe JII1HetlHoe, 11 MbJ He 6Y,lJ,eM paCCMaTpl1BaTb Bce B03-
MOJKHble CJIYlJal1.
Y npa>KHeHIfSi
16.4-1. Bbl'lHCJlHTb pellleHHe
y" + xy = 0, y (0) = I, y' (0) = 0
2
IJ.JlR 0",;; x"';; I, h = 10 ' HcnoJlb3Yil MeTOIJ. HYMepoBa.
16.4-2. nOKa3aTb, 'ITa He Tpe6yeTcR nporH03a D.JlR JlI1HeHHOro ypaBHeHHR
y' = p (x) y + Q (x).
16.5. MeTO));, KOTOpbIH HCnOJIb3yeT 3HaqeHHH y, y' H y"
MbJ OTMelJaJII1 HeCKOJJbKO pa3, lJTO KOr,lJ,a CPYHKll.H5I BbFII1CJJeHa, lJaCTO
6bIBaeT He,lJ,OpOro (B CMbICJJe MallIl1HHOrO BpeMeHlJ) BbJlJl1CJJI1Tb 11 np0l13-
BO,lJ,HYIO CPYHKll.1111. I1TaK, nYCTb 3a,lJ,aHO ypaBHeHl1e
y' = I (x, y);
(16.5-1)
Toma
,. _ of , + of .
y - oyY ax
(16.5-2)
YrrpaJKHeHl1e 13.8-1 ,lJ,aeT CPOPMYJJY (b_ 1 = 1/ 2 )
_ + h ( ' + ' ) + hi ( " + " ) f- h5 (5) ( 6 )
Yn+l - Yn '"2 Yn+I Yn 12 - Yn+l Yn - 720 Y .
(16.5-3)
IlO-BH,lJ,I.JMOMY, 3TO - npeBOCXOAHbJfi BbJ60p AJJ5I CPOPMYJJbJ KOppeKll.I.JH.
TaK KaK BCTpelJaeTC5I JJl1l1Ib tJJJeH C Yn' TO He CYllI.ecTByeT rrOCTOpOHHI.JX
KopHefi xapaKTepHCTI.JlJeCKOrO ypaBHeHI151, nOpOJK,lJ,aIOIlI.HX HeYCTotllJI1BOCTb,
>lCHO, lJTO OllIH6Ka OT oT6paCbIBaHI1H 'weHOB OtJeHb MaJJa 11 oKpyr JIeHl1e
TaKJKe HaXO,lJ,HTC5I rro,lJ, KOHTpOJJeM. MbI ,lJ,OJJJKHbI rr03TOMY HafiTI1 nporH03
B 061l1.eM BI1,lJ,e
Yn+l = AoYn + A 1 Yn_l + h (BoY + BIY -l) + h 2 (CoY + CIY -l).
06bIlJHbItI npou.ecc H3XOJK,lJ,eHI1H K03cpcpHll.l1eHTOB, ,lJ,eJJ31OIlI.HX npom03
TOlJHbJM ,lJ,JJH 1, X, x 2 , x 3 , x', ,lJ,aeT pe3YJJbTaTbI, npl1Be,lJ,eHHbIe B Ta6JJHu.e
16.5-1. B lJaCTHOCTH, npH AI = I rrOJJYlJalOTCH OlJeHb rrpl1BJJeKaTeJJbHbIe
K03CPCPHu.l1eHTbI, XOT5I Al = 0 ,lJ,aeT HeCKOJIbKO MeHbllIYIO OllII16KY.
*) CM, 112].
8 p, B. XeMMIIHr
226
cnEUI1AJJbHblE METOllbI I1HTErPI1POBAHI-151
IfJJ, 16
Ta6J1Hu,a 16.5-1
\ Al = 0\ Al ={! Al = 1 II I Al = 0\ Al =+, Al = I
I Ii 17 33 4
Ao = 1 - Al 1 2 0 ,Go=(l7-A I ):12 12 24 3
I
I
1 ; 7 15 2
AI=AI 0 2 1 IC t =O+A I ):12 12 24 3
i I
Bo = (-I + A I ):2 1 1 0 I E5 =(31+A I ):6 31 21 16
-2 -4 6 4 3
Bt=(3+AI):2 3 I 7 2 I I
2 "4 I
YnpaiKHeHHSI
16.5-1. Pa3pa6oTaTb Jl,eTaJiH MeTOJl,a nporH03a 11 KOppCKU,!HJ Jl,JlR CJlY'IaR
Al = I, BK. K)'1aR MOJl,HcpHKafl.HH rrpOrH03a H 0IWH'IaTeJlbHbIX 3Ha'leHHI!.
16.5-2. I-1cCJIeJl,OBaTb 4JYIIKfl.HIO BJIHHHHR JJ.JlR CJlY'IaR Al = I.
16.6. CJIY'Iai%, KOrJ1.a pemeHHe TpYJ1.HO annpoKcHMHpoBaTb
MHOrO'lJIeHOM
110 CBX nop BC!O.n:y npe.n:noJlaraJlOCb, '1ro paCCMarpBBaeMbIe £pYHKUBH
XOp0U10 npB6JJB)I{a!OTC5I MHOrO'lJIeHaMH Ha HHTepBaJIe pa3YMHotJ .QJIBHbI.
EJJ,BHCTBeHHoe BCKJJ!O'IeHBe 6b1J1o c.n:eJIaHO B 12.1, rJJ,e paccMaTpHBa-
JIHCb BHTeq)aJIbI, npe.n:CTaBBMbIe B BH.n:e
b b
f(x)dx= K(x)g(x)dx.
a
a
Mbl SaMeTflJIfI TaM, 'lTO eCJm HSBeCTHbl MOMeHTbI
b
mk= K(x)xkdx
a
iii £pYHKUI151 g(X) XOp0U10 annpOKcHMHpyeTC5I MHOrOI.JJleHOM, TO I1HTerpaJl
MOlKeT 6bITb JIerKO Bbl'-IHCJIeH, He3aBHCHMO OT TOrO, XOpOllIO JI!1 npH6J1I1-
)l{aeTC5I MHOrO'lJIeHOM CPYHKIJ.H5I f (X). TaKHM 06pa30M annpOKCHMHpOBa-
JlaCb JIHU1b lJaCTb rIOJJ,blHTerpaJlbHOtJ CPYHKIJ.HB.
TaKHe cHTyaIJ.HH B03HBKa!OT B rIp 11 pernemm .n:HcpcpepeHIJ.l1aJIbHbIX
ypaBHeHHtJ; lJaCTO peWeHH5I, KOTopbIe H1IJ.YTC5I, OlJeHb rIJIOXO npH6JII1-
)l{a!OTC5I MHOrO'-IJleHaMH pa3YMHotJ CTerIeHH rIpl1 pa3YMHotJ BeJlHlJHHe wara.
VI eCJlH H3BeCTHO, KaK BeJJ,eT ce651 KaKa5l-.Jm60 CPYHKUH51, BXO.Q511IJ.a51 B pe-
meHHe, TO MO)l{HO nOnblTaThCH BCnOJlb30BaTb 9TO, annpOKCHMHPY51 He
peWeHl1e, a JIH1IJ.b lJaCTb era.
16.61
PEilJEHI1E TPYJJ:HO AnnpOKCI1MI1POBATb MHOroYJJEHOM
227
D.JHI npHMep3 npe.lInOJJO)l{I1M, '1TO peilJ3eTC5I yp3BHeHI1e
ay" + by' + cy = ((x), y (XO) = Yo, y' (XO) = yo, (16.6-1)
rlle a, b 11 C - KOHCT3HTbl H ((x) J1erKO annpOKCUMUpOB3Tb; CKa)j{eM,
((X) Pi+qi(X-Xi_l) llJl51 Xi_l X Xi'
TaKI1M 06pa30M, f (x) annpOKCI1MupyeTc5I nOCJlellOBaTeJlbHOCTblO np5lMO-
JJ11Het!HbIX OTpe3KOB. Henocpe.ZJ:CTBeHHot1 nO..QCTaHoBKot1 JJerKO Hat1m
'1aCTHOe peweHHe (16.6-1)
. y(x)=Fi(x)=P i + Qi(X-Xi_l)'
Pellleime O..QHopollHoro lI.HCPCPepeHlI.U3JJbHOrO ypaBHeHI151 npUBOll.uT K xa-
paKTepHcTI1'1ecKoMY ypaBHtHulO
am +bm+c=O
C KOpH5IMU
- b + .1 b 2 - 4ac
ml = 2a '
- b - V b 2 - 4ac
m2= 2a
CJleJlOBaTeJlbHO, peweHue (16.6-1) B i-M uHTepBaJIe (Xi_l x Xi) Ha-
'1I1HaeTC5I TO'lKOt1
Y (Xi._l) = Yi .1'
y' (Xi_I) = y'i-l
11 lI.aeTC5I CPOPMY JJot!
y (x) = Clem I (X-X i _ l ) + C 2 e m2 (X-X i _ l ) + Pi (x).
LJTo6bl OnpelleJJ\1Tb C 1 U C 2 , UCnOJJb3yeM H3Y3JIbHble YCJIOBH5I llJJ5I UHTep-
BaJia
Yi..1 = C 1 + C 2 + Pi (Xi_I)' y'i-l = ml C I + m C2 + F'i (Xi_I).
Tenepb Bbl'lHCJIliM Ha'laJlbHble YCJlOBHH JJ:JlH cJle..QYIOUJ.ero UHTepBaJJa:
y (Xi) = Yi' y' (Xi) = y'i'
11 Mhl rOTOBbl K CJJell.YlOllI.eMY wary.
EUIll OJlHO UJIU 06a 3H3'1eHH5I mi OTpUlI.3Te.lIbHble U 60JlblllHe, TO'
pewelme nJloxo annpOKcHMHpyeTC5I MHOrO'lJleHOM U, HanUC3B pellleHHe-
B BHlI.e CYMMbl MHOrO'lJJeHa 11 9KCnOHeHlI.U3JJbHot1 '1aCTI1, Mbl H36erHell-J>
MHornx HenpU5ITHOCTet1. VICnOJlb30BaHHblt1 MeTOll HHKOUM 06pa30M He 3a-
BUCUT OT Toro, '1TO .ZJ:Jl5I annpOKCHMall.UI1 ((X) 6bl.1I11 B351Tbl JJI1Het1Hhle"
MHOrO'lJJeHbl, U, Bo06we rOBopH, MHOrO'lJJeHbl 60JJee BblCOKI1X nOpHlI.KOIt
6blJJH 6bJ 60JJee 9KOHOMH'IHbl B CMblCJJe M31l111HHOrO BpeMeHH. MeTOll He"
CB5I3aH 11 C Bbl6paHHbiM HaMU ypaBHeHl'leM BToporo nop5l.!J:Ka, nop5l.lWK
ypaBHeHH5I Mor 6bl 6bITb JlI06hIM.
VIcCJle.ZJ:yeM Tenepb llpyrot1 KJlaCC 33..Q3'1, B KOTOpblX 3ImpOKcI1MupyeM
MHOr04JleHOM He OKOH'I3TeJlbHblt1 OTBeT, 3 TOJJbKO HeKoTopylO Y3CTb ero.
S'
228
CnEUI1AJ1bHbIE METOllbJ I1HTErpI1POBAHI15!
[rJJ. \6
EOJIblmre OTpHlJ,aTeJIbHbJe xapaKTepl1CTI1'1eCKl1e KOpHI1 TI1ITI14HbJ ,LJ)HI
MHorHX 3aJJ:a4, B03HHKalOIJJ,HX B KOHTpOJIbHbIX CHCTeMax, ITPH nOJJY4eHHH
CKOpOCTH H3MeHeHI1H CHmaJIa, B 3aJJ:alJaX paCITpeJJ:eJleHlHI npOCTpaHCTBeH-
Horo 3ap5lJJ:a, B XHMHI1 rOpeHI1H. DpocrefiwHiJ BHJJ: ypaBHeHHiJ TaKoro pOJJ:a
ill
y'=!(x,y), rJJ:e ily A<O.
OJJ:HH 113 nOJJ:XOJJ:OB K 9TOMY ypaBHeHHIO OCHOBaH Ha ITpeJJ:ITOJlO)l(eHHI1,
4TO npOH3BOJJ:HaH % ' XOTH 60JIbWaH 11 OTpI1UaTeJlbHaH, 113MeHHeTCH MeJJ:-
lIeHHO. Mbl nHweM ( ITpH6J1I13HTeJlbHO paBHo A = const)
y'=!(x, y)+Ay - Ay, y' -Ay=j(x,y)-Ay = F(x,y),
fJJ:e, KOHe4HO,
aF al
ay = ay -A O.
PeIJla51 ypaBHelme npH nOMOIJJ,H 06bJ4HOrO HHTerpnpYIOIltero MHO>KHTeJI51
e- Ax , HMeeM
x
y (x) = CJe A (X-Xo\ + e A (X-Xo) e- A (S-xo\ F (B, y (B» dB.
Xo
Tenepb MbI CTaJlKHBaeMC5I C 3aJJ:a4eiJ nOJIY4eHI151 CPOPMYlI JJ:Jl5I HHTerpa-
lIOB BHJJ:a
h
I (h) = e- AS g(B) dB,
o
KOTOPYIO JlerKO peIJlHTb, cJleJJ:Y51 Y>Ke pa3BHTbJM MeTOJJ:aM. Dpom03 He
MO)l(eT I1CnOJlb30BaTb 3Ha4eHI1e g(h), HO MO>KeT I1CnOJlb30BaTb g(O), g( -h),
a TaK)j{e I (0), I (-h),. .., TorJJ:3 KaK B KoppeKnlPYlOlJJ,YIOCPOPMYJIY MO-
>KeT BXOJJ:HTb npomo3HpoBaHHoe 3Ha4eHHe g(h). IleTaJII1 OCTaBJlHeM 4HTa-
TeJlIO. 3JJ:ecb npeJJ:nOJIaraeTCH JJHIJlb, 4TO pa3HOCTb
F(x,y)=!(x,y) - Ay
[y-y(x)]
MO>KeT 6bITb annpOKCI1Ml1pOBaHa MHOr04J1eHOM C HCITOJlb30BaHI1eM JJ:OBOJlbHO
peJJ:Koi-l rpynnbl Y3J10BbIX T04eK, XOTH BO MHorHX cJlY4aHx 9TO BOBce He TaK.
B nepBOM H3 BbIIJlenpHBeJJ:eHHbIX npHMepOB HeannpOKCHMI1PyeMa51
4aCTb BXOJJ:I1J1a CJlaraeMbJM, BO BTOpOM OHa 6blJla MHO>KJ-JTeJleM. Ho B 06011X
CJIy t Ja5lX Mbl OTK333J1I1Cb OT ITp5lMOfi annpOKCI1M3UI111 peweHHH MHOr04JIe-
HaMH B nOJlb3Y annpoKCHMaUHI1 HeKoTopofi ero 43CTH.
YnpalKHeHHe 16.6-1. rIpe.anOJlO)/{HM, 'ITO I (x) MOllieT 6bITb almpoKCHMHpO-
BaHa KBaJJ,paTHbIM TpeX4J1eHOM B HI!TepBaJle 0 x k . 2n, H paccMoTpHM ypaB-
HeHHe y" + y = I (x). nOCTpOHTb Teopmo JJ,JlB '1HC. el!HOrO HHTerpHpoBafIH5I 3TOro
ypaBHeHI1H B J],aHHOM I1HTepBaJle. rIoKa3aTb, KaK pacnpocrpaHHTb ee nOCJlellOBa-
reJlbHO Ha HHTepBaJlbl 2kn x 4kn, 4kn x.s;; 6kn H 1. JJ,.
% 16.71
KPAEBbIE 3AAA'-IH
229
16.7. KpaeBbIe 3a!l:aIJ:H
D.o CI1X nop paCCMaTpl1BaJIHCb 3an.a'-Il1 C Ha'-IaJlbHbIMI1 YCJIOBIl5lMI1.
YaCTo 6blBalOT .a:aHbI YCJJOBI151 Ha peweIme B .a:BYX TO'lKax. Hanpl1Mep,
MO)!{eT 6bITb .a:aHO ypaBHeHHe
y"=!(x,y), y(O)=O, y(I)=O
11 Tpe6yeTC5I 3HaTb Y = Y (x) lIJl5I 0 X I.
i:ha cl1Tyalll151 MOlKeT 6bITb CBe.a:eHa K npelIbI.a:yw.eMY cJIY'lalO MeTO-
AOM npo6 11 oW116oK. MbJ Ha'-lHHaeM C ypaBHeHI151
y"=!(x, y), y(O)=O, y'(O)=A
11 npo6yeM Hat!TI1 ABa TaKHX 3Ha'leHH5I AI 11 A 2 , 'ITo6bI AJI5I
AI: y (l) < 0,
A 2 : y (l) > O.
T aI< KaK .n:JI5I 60JJbUll1HCTBa npaKTI1'1eCKHX 3a.n:a'l y (1) eCTb HenpepblBfIa51
CPYHl<UHff A, TO MO)!{HO, Aa)!{e HcnOJIb3Yff rpy6blt! MeTO.a: .n:eJJeHl1ff OTpe3Ka
nOnOJIaM, 3a 10 npo6 YMeHbwl1Tb .n:JJHHY oTpe3Ka I AI - A 2 1 B 210, T. e.
60Jlee '-IeM B 1000 pa3. BnpO'leM, .n:Jl5I HaXO)!{AeHI151 3Ha'leHI151 A, npH KO-
TOpOM Y (I) = 0, JlerKO Hat!TH ropa3.n:o 60JIee scpcpeKTI1BHble MeTo.n:bl.
O.IlJlaKO, BMeCTO TOro 'ITo6bI CBO.Ill1Tb KpaeBYlO 3a.n:a'lY K 3a.n:a'le C
3a.n:aHHbIMH Ha'-IaJlbHbIMI1 YCJJOBl1ffMH, Hepe.n:Ko BbJro.n:Hee pewaTb ee Heno-
CpelICTBeHHo. Ha STOT C'IeT I1MeeTCff 06WHpHaff Teopl151 ([11]), H MbJ
KpaTKo paccMoTpHM JII1Wb O.n:HH npHMep, 'ITo6bI nOKa3aTb HeKoTopble H3
co.n:ep)J{aw.Hxcff B Het! HAet!.
PaccMoTpl1M ypaBHeHl1e
y"=!(x)y+g(x), y(O)=A. y(1)=B.
Dpe)!{.n:e Bcero annpoKCI1MHpyeM y" BTOpOtt pa3HocTblO
h2y = 2Yn_I'
(16.7 -I)
(16.7-2)
C'IHTa5l, '-ITO HHTepBaJI (0 x 1) pa3.n:eJIeH Ha N HHTepBaJJOB BeJIH'IH-
I
HOt! h= N' TaKHM 06pa30M,
Yn+1 - 2Yn + Yn-I = h 2 [!(x n ) Yn + g(x n )]
HJIH
Y _ Yn =. 1 + Yn+1 - h 2 g (X n )
n - 2 + h 2 f (X n )
(n=I,2,...,N-l). (16.7-3)
DOJJY'-IHJlI1 N - 1 JII-IHet!Hoe ypaBHelIHe C N - 1 HeI13BeCTHbIM, KOTophle
MorYT 6bITb perneHbJ MHorl1MI1 pa3J1I1'1HbIMI1 cnoco6aMI1.
B Ka'leCTBe npl1Mepa paccMoTpl1M O.n:I1H 'IaCTHbltt cJIY'-Iat! ypaBHeHI151
(16.7-1). DOJIO)!{I1M A=B=O, f(x)=l, g(x)=x H N=4, T. e.
ypaBHeHHe
y" =Y+x, y(O)=O, y(l)=O,
230
CnEUVlAJlbHbIE METOllbl VlHTRfPI1POBAHI151
[f JJ. 16
Tpe6yeMoe peWeHHe H3BeCTHO;
sh x
Y = sh 1 - x.
Pa3HOCTHwe ypaBHeHI1H HMe T BH
Y9 - 2Yl = (Yt + + ) .
Ya- 2 Y9+ Yl= 1 1 6 (Y2+ ).
I ( 3 \
= 16 Ya + 4) ·
- 2Ya+ Y9
HJIH
I
16Y2 - 33Yl =4'
2
16Ya- 33Y2+ 16Yl=4'
- 33Ya+ 16Y2 - .
3TH ypaBHeHHH MO)f{HO peWHTb, HanpHMep, TaK. YMHO)f{HM nepBOe H3 ypaB-
HeHHA Ha l BTOpOe Ha 33 H TpeTbe Ha 16 H CJIO)f{HM. DOJIyqHM
(16 2 - 33 2 + 16 2 ) Y2 = (16 + 2 . 33 + 3 . 16).
311aH Y2' JJerKO HaiiTH Yt H Ya 113 nepBOrO H TpeTberO ypaBHeHHA. Ho
4TO eJIaTb, eC.1JH nO.1JY4eJ-IHOe peWeHl1e He OCTaT04HO T04HO? MO)j{JIo
11OJIO)f{HTb N = 8 H, HCnOJIb3YH BbI'iHCJIel-1HOe peWeHl1e JIH N = 4, npl1-
KHHYTb peWeHI1H JIH N = 8. 3aTeM, nO CTaBJIHH 91'0 peWeHHe B npaBWe
4aCTH BOCbMI1 ypaBHeHHA, cooTBeTcTBY W.HX (16.7-3), MO)f{HO Bbl4HCJIHTb
YJIyqWeHHble 3Ha4eHHH H nOBTopHTb npoll,ecc o Tex nop, nOKa He npe-
KpaTHTCH H3MeHeHHH. YHCJIO T04eK MO)f{HO YBeJJH4HBaTb KaK yro HO.
Ha06opOT, BMeCTO (16.7-2), HaBepllOe, JIY4we 6b1JIO 6bl B3HTb 60JIee
T04HY 4>OPMYJIY. YT06bl Hai1TH T8KY 4>OPMYJIY, HCRoJlb3yeM M8TPHUY
8 7 H3 10.3. MOMeHTbl y" (0) CYTb (0, 0, 2, 0, 0, 0, 0) H MOrJIH 6bl
6hITb 3anHcaHbl KaK BeKTOp-CTOJI6eu.. 3TO aeT BeKTOp-CTOJI6eu, COOTBeT-
cTBY W.Hn YllBoeHHoMY TpeTbeMY CTOJI6uy 8 7 :
4 2
- 54 - 27
540 270
1 -980 I -490
360 = 180
540 270
54 27
4 2
16.71
KPAEBb1E 3AllA'IVI
231
KOTOpWtl npHBOnHT K opMyne
h Y;' = 1 6 (2Yn+a - 27Yn+2 + 270Yn+l - 490Yn + 270Yn 1 -
- 27Yn_2 + 2Vn a) = 2 Yn-l - I A 4 V n _2 + 9 1 0 A 6 Yn-3'
! A'
6ep5J B Hetl TOJIbKO nepBble nBa '1neHa, Tal< Kal< BenH'II1Ha 95 "Y n -2' Be-
p05JTHO, .D.OmKHa 6bITb ManeHblwli, I-IalineM 3Ha'leHHe - !; Yn-l' YT06bl
BbI'IHCnHTb ero, HY)j{HhI HeCI<OnbKO 3Ha'leHIiJIi BHe HHTepBana (0 x I);
B '1aCTHOCTH, Hano 3HaTb Y (- ) H Y ( ). 9TH BeJ1H41iJHhI MorYT 6bITb
HatJlIeHbI 1iJ3 O'leB1iJlIHbIX COOTHoweHHI1:
V_I - 2yo + Y! = h 2 (yo + 0),
Y5 - 2Y4 + Va = h 2 (Y4 + I).
3aMeTHM, '1TO B neliCTBHTenbHOCTH HaM HY)j{Hbl JHIllIb 2 Y _I .1 A 2 Ya, KO-
Topble MorYT 6bITb HatJlIeHbI IiJ3 3Ha'leHIiJIi Yo H Y.: 2 Y-l = 0, A2 Y4 = h 2 .
I1cnonb3Y5J Ta6mIIJ,y 3Ha4eHIiJIi
6 62 I 63 I 64
Y-J
Yo
YI
Y2
y,
Y4
Y5
MO)j{HO BbI'IHCJJHTb 3Ha'leHH5J A. Yi' a HMeHHO 4 Y-l' A4 Yo, 4 y" Tpe6Ylo-
IlI.HeC5J, '-!T06bI ynY'IWI1Tb pe3ynbTaT.
9TH 3Ha'leHI15J 4 Yi' KOTopbIe He Y'lHTbIBanHCb B nepBOM BbI'IHCneHHH.
nO.D.CTaBHM Tenepb B KOppeKTl1pYJOIlI.l1e 'IneHbI B npaBoti 'I8CTH
2 Yn-I = h 2 (Yn + X n ) + /2 4 Yn3'
PeWHM 3ana4Y, OmITb He MeH5I5J [WeHOB 4 Yi' EcnH 3Ha'leHH5J . Yi HO-
Bora perueHH5J CYIlI.eCTBeHlIO OTJJIiJ'IaJOTC5J OT CTapbIX, TO 1l0BTOpllM IlpO-
liecc On5\Tb. <POKC [I I J, KOTOpbltl v.Cl10nb30Ban OI1HCaHHbItI npneM C oonb-
woIJ 3 eI<THBHOCTblD, Ha3BaJI ero MeTOnOM «pa3HOCTH H KoppeKUHH»
232
METOJ1 HAI1MEHbWI1X KBAJ1PATOB. TEOPI151
[fJI. [7
KaK Y)I{e 6hlJlO 3aMe4eHO, Mhl He 11hlTaeMC5I H3J1araTb TUJ,aTeJlhHO pa3-
pa6OTaHHYIO TeopHIO Jl.Jl5I KpaeBhlX 3aJl.a4, a JJHWh CTpeMI1:\IC5I Jl.aTh npo-
cretiwl1e npHeMhl, nOKa3hlBalOIIl.He, KaK C raKHMI1 3aJl.a4al\!I1 o6paIIl.anc!/.
Ynpa}l{HeHHe 16.7-1. BbInOJlHHTb Bbl'lHc. eHI1R npHMepa B 16.7, HcnoJlb3YII
MeTO,IJ. pa3HOCTI1 11 KoppeKu,HH, 11 cpaBHHTb pe3YJlbTaT C rrpaBI1J1bHblM OTBeTOM.
fJIABA 17
METO,U HAI1MEHbllUIX KBA,UPATOB. TEOPI1S!
17.1. BBeJI.eHlle
B 7.1 CTaBHJlI1Cb 4eThlpe OCHOBHhlX Bonpoca:
1. KaKl1e Y3J10Bhle T04KI1 Mhl 6YJl.eM I1CITOJJh30BaTh?
2. KaKoti KJJaCC annpOKcHMI1PYIOUJ,l1x CPYHKIJ.l1fi Mhl 6YJl.eM HCrrOJlh30BaTb?
3. KaKofi KpHTepl1fi COrJlaCI151 Mhl npl1MeM?
4. KaKYIO T04HOCTh Mhl XOTI1M I1MeTb?
D.o CI1X rrop 6hlJJO pacolOTpeHo MHoro pa3J1WlHhlX MeTOJl.OB .lI;Jl5I Bbl-
60pa Y3JJOB, HO Mhl BCer.lI;a rrOJlh30BaJJI1Ch MHOr04J1eHaMH 11 KpHTepHeM
T04Horo rrpOXO)l{.lI;eHH5I MHOr04J1eHa 4epe3 Y3J1oBhle T04KH. ITpoBe.lI;eM Te-
rreph I1CCJle.lI;OBaHl1e .lI;pyrl1x KpHTepHeB Jl,Jl5I Bw60pa KOHKpeTHoro MHO.O-
4J1eHa H3 o6IIl.ero KJlaCCa Bcex MHOr04J1eHOB crerreHI1 n.
OK 0 H 4 aTe JJ h H hI fi BhlOOp Kpl1TepH5I Jl,Jl5I KOHKpeTHofi 3a.lI;a4H 3a-
BI1CHT OT npeJl,blCTOpl1H .lI;aHHbIX ypaBHeHl1fi H n03TOMY He MO)l{eT 6hlTb
.lI;aH B Y4e6HHKe no MerO.lI;aM BhI4HCJleHl1fi. ITonp06yeM, OJl.HaKO, 06CY.lll1Tb
HeKOTopble KpHTepHH, pacCMoTpeB I1X .lI;OCTar04HO rro.rr.pOOHO.
Kor.lI;a XOpOill KpflTepl1ti T04Horo cooTBerCTBH5I? 113BecrHo, 4TO 3Ha-
4eHH5I CPYHKIJ.I1H B Bbl6paHHblx T04Kax I1CKa)[{eHhl WYMOM oKpyr JleHHH.
IToKa ypoBeHb wYMa CJlaOhlti, a BO MHOrHX pac4eTax Ha cOBpeMeHHblx
Bbl411CJlI1TeJlbHhlX MaWHHax C 8 - 12 .lI;eC5ITH4HbIMI1 pa3p5l.lI;aMI1 3TO TaK,
T04Hoe COOTBeTCTBHe 51BJlHeTC5I pa3YMHbIM MeTO.lI;OM.
ECJJI1 )l{e ypoBeHb wYMa BbICOK, 4ro MO)l{eT CJlY411TbC5I B HeKOTopbIX
BbI4HCJJeHH5IX - H rr04TH BCer.lI;a CJlY4aeTC5I, KOf.lI;a .lI;aHHbIe rro.rIY4eHbI H3
cpl13H4eCKI1X H3MepeHHti, - TO pa3YMHOCTb nOnbITKH I1CKaTb arrnpOKCHMH-
PYIOUJ,YIO CPYHKIJ.HIO no Kpl1TepHIO T04Horo COBrra.lI;eHHH B «ruYMHbIX» Y3-
Jlax He06xo.lI;I1MO HCCJleJl.OBaTb BHI1MaTeJlbHO. ITO-BH.lI;HMOMY, HaH60Jlee WI1-
pOKO I1CnOJlb3YIOIIl.l1tiCH MeTOJl. B «WYMHbIX» CI1TyaIJ.HHX - 3ro arrnpOKCH-
MaIJ.H5I rro HaHMeHbWI1M KBa.lI;paTaM.
17.2. MeTOJI. HallMeHbIllIlX KBaJI.paTOB
ITpe.LJ.TIOJlO)l{HM, 4TO rpe6yeTc5I H3MepHTb HeKoTopylO BeJlH4HHY H l\e-
JlaeTCH n 113MepeHHti, pe3YJlbTaTbI KOTOphlX paBHhI
Xi = X + 8i (i = 1, 2, . .., n),
17.2]
METOLl HAI1MEHhUH1X KBALlP A TOB
233
..lle si - ,no oum6KIl (!-!JIIl IlIYM) H3MepeHHt1, a x - 9TO «IlCTIlHHOe 3Ha-
qeIme».
MeTO,lI, HallMeHbIllHX KBa.n:paTOB YTBep)\{.n:aeT, '1TO HaHJIY'1Illee rrpl16JIIl-
iKeHHoe 3Ha'1eHl1e X eCTb T8Koe '1IlCJIO, .n:JHI KOToporo MHHHM8JIbH8 *)
CYMMa KBa.n:paTOB OTKJIOHeHHt1 OT Xi:
} n n
f(x) = l= (Xi - x)
i=1 i=1
B KOHe'1HOM C'1eTe, nOJIe3HOCTb ['noro MeTo.n:a onpe.n:eJIfleTCH TeM, HaCKOJIbKO
XOpOlliO 9Ta Mo.n:eJIb COOTBeTCTByeT OrIbITY Il KaK JJerKO MO)\{HO IlCrIOJIb-
30BaTb ee Ha rrp aKTI1Ke. CTOIlT 3aMeTIlTb, '1TO 9TOT MeTO.n: 9KBI-IBaJIeHTeH
npelIrrOJIO)\{eHIlIO, '-ITO HallJIY'1IllIlM rrpll6JIIl)\{eHlleM HBJIHeTCH cpe.n:Hee apHcp-
MeTl1'1eCKOe
n
Xa= !Xi'
i=\
YTo6bI .n:OKa3aTb 9KBIlBaJIeHTHOCTb 3TIlX orrpe.n:eJIeHllt1, rrOKa)\{eM CHa'1aJIa,
'1 fO MeTO,lI, HallMeHblllHX KBa.n:paTOB rrpllBO.n:llT K Cpe.n:HeMY apllcpMeTI1'-1e-
CKOMY. 3aMeTIlM, '-ITO
f(x)= (Xi-X)2
i
MO)\{HO paCCMaTpHBaTb KaK CPYHKIJ:1l1O OT X Il MIlHIIMll311pOBaTb 06bI'1HbIM
npHeMOM:
d d =-2 (Xi - x)=O,
X .
I
Xi - x=o,
i i
Xi-nx=O,
i
- I
X=- Xi=X a '
n .
,
TaKI1M 06pa30M, X = Xa MIIHIlMIl311pyeT 1: (Xi - x?, TaK KaK
d 2 f
dx 2 = 2n > 0.
3TO )\{e MO)\{HO rrOKa3an Il IIHa'1e. TIYCTb
f (x a ) = 1: (Xi - Xa)2.
TorlIa
f(xa) = 1: x l- 2x a 1: Xi + 1:x = Xl- 2xanxa+iix -
= 1: Xl - nx .
*) TeopeMa faycca - MapKosa yraep)/{.!(aeT, 'ITa eCJII1 CpeJ1Hee no BceM 0i
H KOppeJlRu,Hi/ no BceM 0i OJ paBHbI "YJllO, T8 an ou:eHKa HMeeT HaHMeHbIIIYIO
JJ,HcnepCHlO H3 Btex JlHHeHHblx OlleHOK.
234
METOll HAI1MEHbIlIHX KBAllP ATOB. TEOPI151
tW. 17
ECJm B3S1Tb JIIo60e 3Ha4eHI1e X b ' OTJIli4HOe OT Xa' TO
!(Xb) = (Xi - Xb)2 = x1 - 2nXaXb + nx .
B TaKOM CJIY4ae HMeeM
!(Xb) - !(Xa)=n(x - 2XaXb + xg)= n (X a - Xb)2 O.
TaKI1M 06pa30M, CYMMa KBaJlpaTOB Si MIiHIiMaJIbHa TOJIbKO npH X = Xa'
CJIeJlOBaTeJIbHO, JlOKa3aHO, 4TO «MeTOJl Hal1MeHbllJI1X KBaJlpaTOB» 11 «Bbl-
60p CpeJlHerO» 9KBI1BaJIeHTHW,
Ynpa>KHeHHe 17.2-1. HallTH npH6J1H)j{eHHoe 3Ha4eHHe B CMblCJle HaHMeHb-
IIIHX KBaJlpaTOB JlJlil 4HCeJl 2, 3, 2, 1, 2, 3 060HMH cnoc06aMI1.
o T BeT: X = !:-)j6.
17.3. .lI.pyrHe KpHTepHH
Bbl60p B Ka4eCTBe JIY4llJerO 3Ha4eHlH! CpeJlHerO 1i3 n H3MepeHHtt
cooTBeTcTByeT MeTOJlY Hal1MeHbllJl1X KBaJlpaToB. Ho B03MO)J(Hbl 11 n.Pyrl1e
MeTOJlbl, l1 4aCTO OHl1 60JIee YMecTHbl.
IlpeJlnOJIo)J(I1M, 4TO BMeCTO MI1Hl1MH3all.l1l1 CYMMbl KBaJlpaTOB Tpe6yeTcH
Ml1Hl1Ml13allHSI CYMMbl MOJlY JIett OTKJIOHeHl1fi
2:i X i- X \=!(X),
i
3TO Tpe60BaHI1e []pl1BOJll1T K Bbl60PY MeJll1aHHOrO (cpeJlI1HHoro) 3Ha'-le-
HI1S1 X m 113 Xi (eCJII1 KOJII14eCTBO Xi 4eTHO, TO []Pl1XOJll1M K Bbl60PY
JIlo60ro l13 JlBYX CpeJlI1HHbIX). l1.0Ka3aTeJIbCTBO 9TOro cpaKTa []pOCToe.
I1peJl[]OJIO)J(HM, '-ITO HMeeTCSI He4eTHOe '-II1CJIO 2k + 1 3Ha4eHl1fi Xi' Bbl-
6epeM B Ka4eCTBe X m CpeJlHee H3 HHX []O BeJIl1411I-Ie. TorJla JIIo60fi
cmmr OT X m no X, CKa)J(eM BBepx, 6YJJ.eT YBe.TlWH-IBaTb k '-IneHOB
I Xi - X I, JJ.JISI KOTOpblX Xi I-IH)J(e X m ' 11 YMeI-IbWaTb k 4JIeHOB I Xi - X [,
.QJJSI KOTOpblX Xi BblllJe X m Ka)J(.Qblfi I-Ia O.Ql1HaKOBYIO Bem'!4l1I-IY; HO '-IJIeH
I X m - X I 6Y.QeT TaK)J(e YBeJHJ4I1BaTbC5!, YBeJIl14I1Ba5!, TaKI1M 06pa30:l-J, BCIO
CYMMY OTKJIOl-Iel-Il1tJ.
l1.Pyrot! MeTO.Q COCTOl1T B MI1Hl1MH3allHI1 MaKCI1MaJIbHOrO OTKJIOHeHI151
BMeCTO MHHI1MH3alll1l1 CYMMW I1X KBa.QpaTOB. 3TO npl1BO.QHT K 3Ha4eHHIO
xmax+xmin
2 -x cp '
KOTopoe SlBJISleTCSI CpeJlHHOfi HHTepBaJIa Xi'
17.4. OWH6KH C HOpMaJIbHbIM pacnpe.QeJIeHHeM
Ilo-BH.QI1MOMY, llJHpOKO pacnpoCTpaHeHo MHeHHe, 4TO MeTO.Q HaHMeHb-
llJHX KBa.QpaToB []0.Qpa3YMeBaeT pacnpe.QeJIeHHe 01lm60K no HOpMaJIbHoMY
aaKoHY. Cor JIaCHO eMY, BepOSlTI-IOCTb TOro, '1TO ollJH6Ka Cj' HaXOJlI1TCSI
\\ 17.41
OUTI1BKI1 C HOPMAJJbHblM PAcnPE.QEJJEHHEM
235
B HHTepBaJle x, x + x, 3aJI.aeTC5I CPOPMY Jlofi
e - k2X2 D. x
V"1t .
3TO lHeHl1e OIll!1604HO, H 3JI.eCb CTOHT 06CYJI.HTb 9TY TeMY, '1T06bl npo-
HCHYJTb JI.eJlo. KpOMe Toro, HHOrJI.a BepSIT B TO, 4TO HOpMaJlbHblt! 3aKOH
eCTb 3aKOH np!1pOJI.bI. PacrrpoCTpaHeHa CPOPMYJI!1pOBKa: «MaTeMaTHK!1 no-
JlaralOT, 4TO HOpMaJlbHblt! 3aKOH JI.OJl)J(eH 6bITb cpH3H4eCKHM 3aKOHOM, B TO
Bpe 151 KaK cpH3HKH ,lJ,YMaIOT, 4TO OH .o:omKeH 6bITb MaTeMaTH4eCK!1M 3aKOHOM»,
O,lJ,Ha H3 xapaKTepHhlX cHTyall.Hfi, npHBo.o:51I1J.HX K HOpMaJlbHOMY 3aKOHY ,
([''jP WeJlb), Bblr Jl5I.o:I1T TaK: paCCMOTpHM MeTaHHe .o:pOTl1Ka C HeKOTopofi
BbICOTbI C lI.eJlblO nonaCTb B HeKOTO-
PYIO T04KY 0 Ha rOpI130HTaJlbHOM
nCJlY (pHC. 17.4-1). TIpe.o:nOJlo)J(HM
Tenepb, '1TO OW06KH He 3aBHC5IT OT
Bb!6opa CHCTeMbI KoopmmaT, a 3aBH-
CHI TOJlbKO OT paCCT05lHH5I .0:0 TO'lKH 0
11 4TO 60Jlbllme OW06KH MeHee Be-
POWfHbI, 4eM MeHbWl1e, 3TH npe.o:no-
JlO)!{el!H5I Ka)J(YTC5I BnOJlHe pa3YMHbIMH,
YaK KaK CHCTeMa KOOp,lJ,OHaT MOJKeT
6hITb BnOJlHe npOH3BOJlbHOt!.
TIYCTb Bep05lTHOCTb nona.o:aHH5I B nOJlocy x, x + D.x paBHa (npH-
6J1H)J(eHHO) f (x) D.x 1'1 COOTBeTCTBeHHO B nOJlocy y, y + D.y paBHa fey) D.y.
V!CXO,lJ,51 03 npe,lJ,nOJlO)!{eHH5I 0 He3aBHCI1MOCTI1 OWH60K no Ka)!{.o:ofi KOOp-
JI.I1HaTe, nOJlY4aeM, 4TO Bep05lTHOCTb nOnaJI.aHH5I B np5lMoyroJlbHHK nepe-
Ce'leHH5I JI.BYX nOJlOC paBHa
f(x) f(y) tlX D.y.
(17.4-1)
y
y+l1y
Y
()J
o
x+L1x
x
PHC. 17.4-1.
(17.4-2)
BBelI.eM Tenepb nOJl5lpHble KOOp,lJ,HHaTbI. Bep05lTHOCTb nOna,lJ,aHH5I B 9J1e-
MeHT nJlOlHa.o:H no npeJI.nOJlO)!{eHHIO He 3aBHCI1T OT HanpaBJIeHH5I 11 paBHa
g(r) D.x D.y.
(17.4-3)
TaK KaK (17.4-2) H (17.4-3) Bblpa)J(alOT O.o:HY 11 TY )!{e Bep05lTHOCTb, TO
g(r)=f(x)f(Y).
J1eBa51 4aCTb He 3aBHCI1T OT B, 11, .o:HcpcpepeHHHPY5I no B, HMeeM
ag (r) _ a - f( ) af (y) + f(y) af (x)
a8 - - x a8 a8 .
V!cnOJlb3YH COOTHOllleHH5I x = r cos B,
HaXOJI.I1M
(17.4-4)
(17.4-5)
y = r sin B,
af (x) _ df (x) ax - f ' ( ) . ( _ ) I
a8 - dx a8 - x y ,
af (y) _ df (Yl d)i - j " Lv)
a8 - dy d8 - x.
(17.4-6)
236
METon HAVlMEHhUlVlX KBAilPATOB, TEOPI151
[W. 17
rIollCTaBJISlSl (17.4-0) IJ (17.4-5), flOJIY4ae,ii
f(x) f' (y) x + ley) f' (x)( - y) = 0
BJ/11
I' (x) _I' (y)
xl (x) - yf (y)'
00 npelInOJIO)l{emno x H Y He3aBI1CI1MhI, a 3Ha4HT, o6e QaCTI1 9Toro
paIJeHCTBa paBHhI KOHCTaHTe, CKa)l{eM, K, 1. e.
I' (x) = K = I' (y)
xl (x) YI (y)
I1JII1, 4TO TO )l{e,
dl(x)
-=K.x.dx
fix)
H
d/(Y) = K. v. dy.
f(Y) .
HHTerpHPYSJ nepBoe 113 9TH X paBeHCTB, UOJIY4aeM
Kx 2
In l(x) = 2 + C 11J11i
I
Kx 2
l(x) = Ae 2 .
MbI upe.lWOJJO)I{I1JIil, 410 66JIbll1l1e 01ll116KIi MeHee BepOl/THbI, 4eM
MeHhWl1e; n09TOMY K ,QOJI)I{HO 6bITh 01plmaTeJIhHbIM, CKa)l{eM, K = - 2k .
OKOH4aTeJIbHO I1MeeM
f (x) = Ae- k 2 x",
fey) = Ae- k2y
H H3 (17.4-4)
g(r) = A 2 e- k2 (x 2 +y2).
Ho, TaK KaK KY,Qa-Hl1oy.n:b JI:pOTI1K .n:OJI)I{eH nonaCTb, TO
co
A 2 2'1t re- k2r2 dr = 1,
o
2 e - k2r2 1 "" A21t k
A 'It _k2 1 0 =7= 1, A= V- '
H OKOH4aTeJIbHO MbI nOJIY4aeM (rpopMYJIa (17.4-1»
f ( ) - - k"x"
X - V-; e ,
2" 00
g(r) r dr de = 1,
o 0
T, e, HOpMaJIbHOe (rayccoBo) pacnpe.n:eJleHHe.
3aMeTHM, 4TO 9TO He e,QHHCTBeHHbIt! cnocoo nOJJY411Tb HOpMaJJ['Hblt!
3aKOH Llpyrot! nOJI:XOlI, lI,aeT LI.eHTpaJIbHa51 flpel1,eJJbHa51 TeopeMa, KOTopaSJ
yTBep)l{llaeT, rpy60 rOBOp5l, 4TO CYMMa 60JIblllOrO '-lUCJIa MaJIeHbKI1X
OWH60K pacrrpe.neJIeHa HOpMaJIbHO.
Ha npaKTHKe HOp1\taJIbHblt! 3aKOH eCTb 06wmaSJ MOlI,eJIb BO MHorHX
flpI'!JIO)l{eHYI1JX. OTKJIOHeHl1e OT Hero npOHCXOlI,HT 06bI4HOOT HaJII14HSJ
17.51
llPOBElIEHME 1l01IX01I5IUlE ro MHOrolJ.nEHA
237
60JIbWerO, lJeM 9'1'0 npeJJ,CKa3bIBaeTC51 MOJJ,eJIblO, KOJIH4eCTBa nonaJJ,aHI151
B «XBOCT» paCllpe,lJ,eJIeHI151, r,lJ,e I X \ BeJII1K. TIpI14I1HOt! 9Toro 4aCTO
SlBJI51eTCSI He60JIbwoti «pa3Ma3aHHblt!» 9cj.Jcj.JeKT. B TaKI1X CJIY4a51X 06bIlJ!l3
CMeCb ,lJ,BYX HOpMaJIbHbIX Kpl1BbIX C pa3JIH4HbIMl1 3Ha4eHlHJMH k.
TeOpH5! Ka4eCTBeHHoro KOHTpOJISI, B 4aCTHOCTH, OCHOBaHa Ha B03MY-
ll1.eHH5!X, Ha6moJJ,aeMblX B «XBOCTax».
Ynpa>KHeHHe 17.4-1. OOKa3aTb, 'ITO (12 = ; , rJJ.e (1 eCTb JJ.HcrrepCHR pac-
k k2 2
npe.rreJ/eHHH f (x) = V"1t e- x.
17.5. npOBe}l,eHMe nO}l,XO}l,HUJ,ero MHOrO'lJleHa
OJJ.HH H3 HaH60JIee 06IJJ,HX CJIY4aeB npHMeHeHH51 MeTOll,a HaHMeHbWHX
KBa,lJ,paTOB COCTOHT B TOM, 4'1'0 I1MeeTC51 N Ha6JIlOlI,eHI1t! (Xi' Yi) (i = 1,
2, ..., N) H Tpe6yeTC51 npl16JIH3HTb 9TH ,lJ,aHHble MHOr04JIeHOM CTeneHl-I
M<N.
y(x)=ao+aIx+ ... +aMx M .
(17.5-1)
BbI4HCJieHHaSl KpHBa51 Y (x) B HeKOTopOM CMbICJie lI,aeT cr Jia)j{eHHoe
MHOJKeCTBO 3Ha4eHHt! Y (Xi)' KOTopble, Bo06Ill.e rOBOp51, OTJII14HbI 0'1' Ha-
6JIIOlI,eHHbIX Yi' MeTOll, HaHMeHbIllI1X KBall,paTOB YTBepJKlI,aeT, 4'1'0 CJie,lJ,yeT
BbI611paTb TaKott MHOr04JIeH, KOTOpbltt MI1HHMI1311pyeT cj.JYHKlll1lO
N
L: rYi - Y (Xi)]2 = !(ao, aI' ..., aM)'
;=1
(17.5-2)
JJo CI1X nop paCCMaTpI1BaJIC51 MeTO,lJ, Hal1MeHbWI1X KBa!1:paTOB TOJlbKO
,lI,Ji5J O,lI,HOro 113MepeHl:l5J; 04eBHlI,HO, 4'1'0 npl1Hlll1IUJaJIbHO H!-:l'IerO He Me-
H5JeTC51 npH nepexoll,e K (M + 1) H3MepeHHIO. MOJKHO paCCMaTpl1BaTb a o ,
a1> ..., aM KaK KOOpll,l1HaTbI Oll,HOt! T04KH B (M + 1)-MepHoM eBKJJl1l1,O-
BOM npoCTpaHcTBe.
JJJi51 HaXOJK,lJ,eHI151 MI1HHMYMa nOCTynaeM KaK B aHaJIl13e: lI,Hcj.Jcj.JepeH-
1ll1pyeM (17.5-2) no KaJKlI,ot! 113 HeH3BeCTHbIX a k
oj k
oak =-2 ","IYi-Y(Xi)]Xi=O
;
(k = 0, 1, 2, ..., M)
liJiH
"k "k + " k+l + + "k+M_
YiXi = ao -"" Xi ai.L.J Xi . .. aM LJ Xi -
i
M N
" "k+1
= .L.J a} .L.J Xi .
i=O L 1
( 17.5-3)
238
METOil HAVlMEHbnIVlX KBAilPATOB. TEOPI151
IfJI. 17
liJIS1 ynpomeHI1H BBe.lleM 0603Ha4eHl1H
IV
X7=Sk'
;=1
IV
y;X = h
;=1
YpaBHeHl.JS1 (17.5-3) npl1HHMalOT RI111
M
aiSk+j= Tk
i=O
(k = 0, 1, ..., M)
(17.5-4)
11 Ha3blBalOTCSJ «HopMaJIbHbIMYI ypaBHeHI1SJMI1».
OHYI 06pa3YIOT CI1CTeMY M + I mmef.1HblX ypaBHeHHfi, onpell.eJIl1'reJIb
KOTOpbIX eCTb t. = I Sk+j \,
IToKa)l{eM, '1TO t.::j: 0, EOIH 6bl t. = 0, TO 011Hop011HaSJ CHCTeMa,
cooTBeTcTBylOIl1.a5! (17.5-4),
M
1.: a;Sk+; = 0
;=0
flMeJIa 6bl HeHYJIeBOe pellJeHl1e. YMHo>KaSJ k-e H3 9Tl1X ypaBHeHl1i1 Ha ak
1-1 CyMMHPY5! no BceM k, nOJJY4aeM
M M N IV.'Ii .'Ii
o = ak 2: aj (xix:) = 2: ( ak x :) (. a;xf) =
k=O }=o ;= I ; = I k=O ;=0
N M 2 N
= 2: ( .L; ak x :) = 2: y (Xi) = 0,
;=1 k=O i=1
OTKY11a Bce Y (Xi) = 0, l = 1, 2, ..., N. TaK KaK N> M, 9TO HeBOS-
MO)l{HO BCJIe11CTBHe <PYH.llaMeHTaJIhHotl TeopeMbl aJIre6pbl: MHOrO'lJIeH CTe-
neHH M I1MeeT He 60JIbllJe M l<opHefi, eCJII1 He Bce ak = 0, TaKHM o6pa-
80M, He cymecTByeT HeHYJIeBOrO pellleHI151 11 t.::j: O.
B rrpl1Hlll1ne HallJa Sa11alJa pemeHa. Ha npaKTI1Ke )l{e pewaTb cn-
cTeMY (17.5-4) He TaK JIerKO, IlOTOMY 4TO onpe.n:eJJI1TeJIb t. '1aCTO 6bmaeT
BeCbMa 6JIHSOK K HYJIIO. LJT06bl YB!1J.1.en, KaK 9TO MO)l{eT CJIY'IHTbC51,
npe.n:IlOJIO)l{HM, '-ITO Xi 60JIee nJIH MeHee paBHoMepHo pacnpe.n:eJleHbl
B HHTepBaJIe 0 Xi';:;:;; 1. Tor11a
IV 1 I
Sk= S X: N X'''dX= k+1 N
;=1 0
OnpeJlemlTeJJb (c TO'lHOCTblO 110 MHO>KHTeJI5I NM + 1)
! i+)+1 1
(l, i=O, I, ..., M)
H3BeCTeH KaK onpeiJe.llUme/lb rll.llb6epma.
17.51
npOBEJJ.EHHE nonXOJJ,5IlUEro MHOroYJIEHA
239
Onpe.LJ.emneJIb rHJIb6epra nOp5l.LJ.Ka n HMeer BeJIH'IHHY
Kg;ropa51 6blCrpO CrpeMHrC5I K HYJlIO. B Ta6JIHu,e 17.5-1 npHBen:eHo He-
CKOJIbKO 3Ha'leHHtI Hn.
Ta6J1Hu,a 17.5-1
3HaqeHHH onpeAeJlHreJleA rHJlb6epra
n I H II n I Hn II n I Hn
n
I I
1 I \ 4 1,7.10- 7 I 7 4,8 . 10- 25
2 8,3 . 10- 2 5 3,7 . 10- 12 8 2,7. 10- 8 "
3 4,6 . 10- 4 6 5,4. 10- 16 \ 9 9,7 . 10- 43
I.ho6bl o6otirH rPY.LJ.HOCrH pellleHH5I CHcreMbl C O'leHb MaJIblM onpe-
.lI:eJlHreJleM, crOHr Bbl'lHCJl5ITb He Sk' a HeKOTopbltt HX 9KBHBaJleHT. 9ro
npHBO.LJ.I1T K opTOrOHaJlbHblM <PYHKU,H5IM H cneUl1aJIbHblM opTOrOHaJlbHblM
nOJlHHOMaM, KOTopble XOpOIllO l13Y'IeHbl 11 no KoropblM HMeerC5I 061llHP-
Ha51 JII1Teparypa.
C n:pyroti CTOpOHbl, 3aMeTHM, 'ITO ypaBHeHH5I (17.5-4) MorYT 6bITb
3anl1CaHbl B .lI:pyrnx 0603Ha4eHH5IX. 113 (17.5-1) HMeeM
aO+alXi +a2x + ... +aMxf! =Y(Xi)
(l = 1, 2, ..., N)
HJIH, B MaTpH'IHOM 0603Ha'leHHI1, Xa = y. TIYCTb XT - TpaHcno3HUHSI
MaTpHu,bl X. Tor.LJ.a (17.5-4) npl1HHMaeT BH.LJ.
XTXa=XTy.
9TOT flYTh Ha npaKTHKe He nprrrO.LJ.eH.
Ynpa}l{HeHHe t 7.5- 1. Il.aHo
x 1 0 1 1 2 1 3 I 4
Y \ 1 2 1 I 0 I 4
rIpoBecTH npRMYIO MeTon:OM HaHMeHbllil1X KBan:paToB.
2
o T BeT: Y = 5 (x + 2).
240
METOll HAI1MEHbWI1X KBAllP ATOB. TEOPI15J
IIJJ. 17
17.6. OpTOrOHaJIbHble 4>YHKlJ.HH
B CJIY4ae lI,BYX H3MepeHl1ti .LI.Be np51Mble Ha3bIBalO1'C51 «op1'orOHaJlb-
HblMI1» I1JII1 nepneH.LI.I1KYJI51pHbIMI1, eCJIH
1
tg8 1 =-TT'
g 2
91'0 YClIoBl1e MO)J(HO 3anl1CaTb B BI1.LI.e
sin 8 1 sin 8 2 + COS 8 1 COS 8 2 = O.
I1CnOJJb3Y51 .LI.OnOJIHI1TeJIbHbIe yrJIbI Cfl 11 Cf2 B nepBoM CJJaraeMOM, no-
JIY4aeM
COS Cfl cas Cf2 + COS 8 1 cos 8 2 = O.
B cJIyqae Tpex 113MepeHl-lil C yrJIaMI1 a, , 1 B Ka4eCTBe yCJJOBH51 nepneH-
.LI.HKYJIHpHOCTI1 HJJH opTOrOHaJIbHOCTI1 HMeeM
cos al cos a2 + cos I cas 2 + COS II COS 12 = O.
Y CJIOBHe op1'OrOHaJIbHOCTH JIerKO 0606ll1,HTb Ha CJIY4all n H3MepeHI11l
c yrJIaMI1 a(1), a(2), ..., a(n)
cos ail) cos a l) + . . . + cos ai n ) cos a n) = O.
Ha npaKTI1Ke OKaaaJIOCb, '11'0 HanpaBJI51lOllI,He KOCI1HYCbI
cosa)k) = Ai (k)
60Jlee y.u.06HbI, '-IeM caMH yrJIbI, TaK '11'0 06bI'-IHO yCJloBHe op1'oroHanb-
HOCTH yno1'pe6JIHe1'CH B <popMe
n
Al (k). A 2 (k) = O.
k=l
ECJlH Tenepb B HeKOTopOM CMbICJIe YCTpeMI1Tb k __ 00, TO MO)J(HO 3aMe-
l'HTb, '-11'0 paaYMHO HaaBaTb YCJIOBl1eM opTOrOHaJIbHOCTI1 paBeHCTBO
AI (k) A 2 (k) dk = O.
I1CXO.LI.51 113 9Toro, rOBOp51T, '-11'0 .LI.Be CPYHKLLI1H II (x) 11 /2 (X) opmo-
ZOftaAbJibl H.a UJimep8aAe a < x .:;;:; b, eClm
b
f, (X)f2 (x)dx=O.
a
17.61
OPTO rOHAJlbHbIE <1>YHKL\l111
241
c1>YHKIJ.1111 fl' 1'1>
... ,
1m 83aU.MftO Opm020fta/lbftbt, eCJJU
b
Ii (x)lj(x) dx= 0
(l ::/; j),
a
(17.6-1)
b
Vf(x)dx=Ai>O
a
(i = j)
(K CO)l{aJleHUIO, B 3TOM MeCTe 06WJHO yrroTpe6JISlIOT 0603Ha4eHHSI Ai; UX
He CJleJJ:yeT nYTDTb C HarrpaBJlSlIOIlJ,HMH KocHHycaMI1).
Mhl rrpeJJ:nOJ/araeM, 4TO Ii (x) JJ:etJCTBJ1TeJIhHbl, HenpephIBHbl J1 He paBHhi
rO)KJJ:eCTBeHHO HY J/IO.
TIYCTb .
Cf" _ Ii
oi - V A/
TorJJ:a
r { 0, [::/; j,
JgJx)gj(x)dx= I [_'
a ' -J.
(17.6-2)
<PYHK1UIH gi (x) Ha3blBalOTCSI opmoftOp.MUp08aftftbt.MU, a rrpoIJ.ecc
OT Ii K gi - ftOp.MUp08aftlle.M.
KJlaCCHqeCKotJ C!1CTeMotJ opTOrOHaJlbHblX CPYHKIJ.HtJ B
o X < 2'1t (J1J/H - 'It X 'It) SlBJI51eTCSI CI1CTeMa
rrepexoJJ:a
HHTepBaJIe
1, cos x,
sin x,
cos 2x,
sin 2x,
. ..,
cos Mx,
sin Mx.
. ..,
ECJ/I-J 3HaqeHl-JSI 3aJJ:aHhl B 2N paBHoOTCTOSlIlJ,J1X T04Kax, pSlJJ:hI KOCHHYCOB
1
H CHHyCOB 3aKaH4HBalOTCSI COOTBeTCTBeHHO 2" cos Nx J1 sin (N - 1) x
(CM. r J/. 6).
LIT06hI YBHJJ:eTh, KaK MeTO.n: opTOrOHaJIhHbIX CPYHKIJ.HtJ 06XO.n:I1T TPY.n:-
HOCTH, 110pO)l{JJ,eHHhle I1JIOXOti MaTpHIJ.etJ He!13BeCTHbIX K09cpcp!1IJ.l1eHTOB,
HaI10MHHM, KaK rrp0!13Bo.n:!1TCSI pa3JJO)KeH!1e, Tpe6yeTcSI rrpe.n:cTaBJ1Th Yi
B BH1J.e
a aN
Yi= 2. +aIcosxi+...+aN_Icos(N-1)Xi+"2CosNxi +
+b j sin Xi+'" +b N _ I sin (N-1)x i .
YMHO)KaSl Ha cos mXi HJ/J1
sin mXi J1
1tt
CYMMUpySI no BceM Xi = N '
1l0JlYQaeM (HcrrOJJb3YSI (6.2-3))
2N-l
a m = ! Yi cos mXi'
i=O
2N-1
b m = ! Yi sin mXl'
i=O
242
METOJl HAI1MEHbWI1X KBAJlPATOB. TEOPI151
I r n. \7
::ha C\1CTeMa ypaBHeH\1fi .1I.J15I He\13BeCTHbIX am H b m pernaeTC5I TPHBH-
aJlbHO - OHa Yllie perneHa.
Bce npOHCXO.lU1T TaK )l{e, eCJI\1 eCTb .n.pyroe MHO)){eCTBO opTOrOH3J1b-
HblX cjJYHKUHt! {Ii (x)}. DYCTb l(x)=ao!o(x)+...+aNIN(x). YMHO-
)l{aeM Ha Ij (x) \1 HHTerpHpyeM:
b
V(x)lj (x)dx = Ajaj. (17.6-3)
a
PerneHHe T3K)I{e TpnBH3JJbHO. K09cjJcjJnUHeHTbl a j' Hat!.n.eHHble T3KHM
crrOC060M, H3SbIB3KiTC5! KoarftifJlltJ,lleHma.AUl <Pypbe.
ECJI!-I cjJYHKUI15! \1SBeCTH3 TOJlbKO B OT.lI.eJlbHbIX TOqK3X x m ' TO I1HTe-
rpaJlbl 3aMeH5!KiTC5! CYMMaMH, a YCJlOB\15! opTOrOH3J1bHOCT\1 npHHHM3KiT
BH.n.
"{.Ii (xm)lj(x m ) = { i
npH i::/= i,
npH j=i.
3aMeT1-1M, 4TO Aj =1= O. ECJJn .lJ.aHHble I1MeKiT p33J\1-14HYKi CTeneHb Ba)l{HO-
CTH, B HHTerpaJl MOllieT 6b1Tb BBe.n.eHa BeCOB35! cjJYHKUH5! p (x) 0
b
P (x) Ii (x) Ij (x) dx, (17.6-4)
a
a K03cjJcjJ\1IJ.HeHTbI nOJJyqaKiTC5! H3 YP3BHeHHfi
b
P (x)/(x)l j (x) dx= Ajaj'
a
(17.6-5)
ECJII-I 33.lI.3H3 CMeWaHHa5! HHcjJopMaUHH, B HenpepblBHott H .lI.HCKpeTHot!
cjJOpMaX, TO MOlliHO HCnOJJb30BaTb IJHTerpan CTHJlbTbeCa.
P33BHTb 9TY TeOpmO JlerKO .lI.Jl5! HenpepbIBHOfi cjJOPMbl, XOT5J 60JlbWet!
qaCTbKi OHa 6Y.lI.eT HYlliHa .lI.Jl5! .lI.HCKpeTHoro cJlyqa5!.
Ynpa}l{HeHl-le 17.6-1. rIoKa3aTb, 4TO Po=l; P 1 =x; P i = (3x 2 -1);
P 8 = +(bX 8 - 3x) opTOrOHaJlbHbI ,IJ:JliI - 1 x.s; 1.
S 17.7. OOlll.He cBoiicTBa opTOrOHaJIbHbIX $YHKIJ.HH
Dpe)\{.n.e qeM nOJlb30BaTbC5I opTOrOHaJJbHblMH cjJYHKUH5!MH, pa35epeM
HeKoTopble 06il1.He TeopeMbl, lJT06bJ .lJ.aTb qHTaTeJJ!O npe.n.cTaBJleHHe 0 no-
Be.lI.eHllIl opTOrOHaJlbHbIX cjJYHKUHfi.
YI.lI.e5J JIImet1Hot! HesaBliCHMOCTI1 5JBJl5JeTC5! O.lI.HOt! H3 OCHOBHbIX H.n.et1
B MaTeMaTHKe. <f>YHKlI.HH Ii (X) Ha3bIBalOTC5J .tllll-LeUHO He3a8UCUMbl.Mll Ha
(a, b), eCJlH li3 paBeHCTBa ao!o (x) + ad1 (x) +... + amfm (x) = 0 BO BceM
17.71
OBliU1E CBOMCTBA OPTOroHAJ1hHbIX <fJYHKUI1M
243
HHTepBaJJe CJJe.n:yeT, 4TO Bce ai = a *); B npOTHBHOM CJIY4ae OHH JIHHet1Ho
3aBHCHMbJ.
IlpOCTbIM H Ba)J{HbIM npHMepOM MHO)J{eCTBa JJ!1Het1Ho He3aBHCHMblX
<jJYHKUI1H B npOH3BOJIbHOM HHTepBaJJe SlBJISleTCSI MHO)J{ecrBO <PYHKUHII
I, X, x 2 , x 3 , ..., xm,
TaK KaK no OCHOBHOt1 TeopeMe aJIre6pbI H3 TO)J{.n:eCTBa
a o + alx +. . . + amx m = 0
CJJe.n:yeT, 4TO Bce ai = O.
YHKUHI1 HenpepWBHwe 11 opTOrOHaJJbHWe B HHTepBaJIe JIHHeilHO He-
3aBHCHMbI. D.oKa3aTeJIbCTBO TpHBH3JIbHO: .n:onYCTUM, 4TO aolo + aUI + . . .
... + a"j m 0, H Bbl4HCJIHM K09<p<pvmHeHTbI YPbe (17.6-5)
b
aj= L O.p(x)lj(x)dx=O.
a
Hao60pOT, H3 CHCTeMbI JIHHet1Ho He3aBHCHMbIX <PYIIKUHt1 C nOMOIUbIO
npouecca WMH.n:Ta MO)J{HO nOJIY4HTb CHCTeMY opTOrOH3JJhHbIX <PYHKIlHil.
Ilpouecc COCTOHT !1 CJIe.n:Y!OJueM. IlycTb .n:aHO MHOiKeCTBO JIHHeilHO He-
3aBHCHMbIX <PYHKIlI1I1 fi (x). BbI4HCJIHM
b
P (x)f6 (x) dx = Ao> 0
a
[p (x);?; 0].
Tor.n:a paBeHCTBO
go (x) = [, V o (x)
1.0
onpe.n:eJISIeT nepBYIO OpTOHOpMHpOBaHHYIO <PYHKUHIO ga (x). IlpHMeHSISI
MeTO.n: MaTeMaTH4ecKol1 HH.n:YKUHH, npe.n:nOJImKHM, lITO Y}f{e nOCTpOeHbI
nepBbIe j OpTOI-IOplVmpOBaHHbIX <PYHKuuII gi (X) (i = 0, 1, 2, ..., j - I).
IloJIo)J{HM
Fj (x) = aogo+ algI +... + aj_lgj_l + fj(X).
(17.7-1)
YHKUIISi F j (x) OT JJH4Ha OT HY JIIJ, TaK KaK Ii (x) JJ!1HeilHO He3aBHCIIMbl,
a Ka)J{.n:aSl gi (x) eCTb JIuHeilHaSi KOM611HaIlHSI fk (x) .n:JIH k i. Mbl
D.OJ))J{HbI HMeTb
b
P (x) Fj(x) gi (x) dx = 0,
a
O i j-1.
*) <PYHKU,HIO f (X) = 0 31{eCb f1 Ilallbille HCKlllOtJaeM 113 paCCMOTpeHIIII.
244
METO!! HA!1MEHbmI1X KB}\JlPATOB. TEOP!151
[IJJ. 17
Ho no onpeJJ.eJIeHl-!\O F j (x) OTCIDJJ.a CJJeJJ.yeT
b
Aiai + p (x) gi (x)fj (x) dx = 0,
a
OTKYJJ.a H8XOJJ.I1TCH ai H TeM C8MbIM Fj(x). YT06b1 HOpMHpOB8rb Fj(x)
HYlKHO BbIl/I1CJII1Tb
b
p (x) F, (x) dX=Aj
a
[p (X) 0)
11 3aTeM nOJIO)!{HTb gj (X) = .
V Ai
T8KHM 06pa30M, mar I1HJJ.YKlI,I1H BimoJIHeH. ECJII1 eCTb TOJIbKO KOHel/HOe
'-IHCJIO N Y3JIOB X m ' TO cymeCTByeT no Kpai1Hei1 Mepe N mmei1Ho He-
38BHCHMbIX <PYHKlI,HI1 fj (x m ).
To, lITO I1X N, CJIeJJ.yeT H3 cymecTBoBaHHSI MHO)KeCTBa
{ 0 npH m '* j,
gj(x m )= 1 npH m=j,
(j= I,
N),
TaK K8K HHKaKoe MHO)!{eCTBO 9THX <PYHKlI,I1t! He MOlKeT 6bITb mmefiHO
38BI1CI1MbIM.
K CO)!{aJIeHHID, ceMeficTBO <PYHKUHi1, KOTopoe ITa JIY'-laeTCH B 9TOM
npouecce, onpeJJ.eJIHeTCH HeOJJ.H03Hal/HO. OHO 3aBI-ICI-IT OT BbI60pa V )'i > 0,
a TaK)!{e OT I-IHTepBaJIa, 110PHJJ.K8, B KOTOpOM MbI nepe611paeM <PYHKUHH,
11 BecoBol1 <PYHKlI,HI1 p (x).
Ynpa>KHeHHe 17.7-1. j],aHbI Po=l; P 1 =x; p(x)=1.
rIoCTpOHTb opToroHa.1bHble <PYHKU,HH P 2 (x) H P3 (x) Ha MH0JKeCTBe TOlfeK
- 2, -I, 0, 1, 2, cae P 2 H P 3 - MHorOlf.1eHbJ CTeneHH 2 H 3 COOTBeTCTBeHHO.
17.8. HepaBeHcTBo EecceJUI Ii nOJIHOTa
K09<p<p11Ul1eHTbI <pYPbe <PYHKlI. l1 11 F (x) OTHOCI1TeJIbHO OpTOHOpM11pO-
BaHHoro ceMeficTBa
b
aj= p (x)F (x) gj(x) dx
a
YJJ.OBJIeTBOp5lIDT HepaBeHCTBY 6eCCeJIH
b .M
P (x) P (x) dx aj
a ;=0
(p 0).
(17.8-1 )
{\ 17.91
METOll HAJ1MEHbUJJ1X KBAllPATOB 11 K03<!><!>J1l1J1EHTbI <l>YPbE
245
.1J.oKa3b1BaeTCJJ 3TO HenOCpel!CTBeHHO. Hanl1we,\1
b M
P (x) [F (x) - ajgj (x) r dx 0
a j=O
n paCKpOeM CKo6KH
b M
o P (x) p2 (x) dx - 2 P (x) F (X) ajgj (x) dx +
a a ;=0
b M 111
+ 2: ajajp (X)gjgj dx.
a j=Oj=O
l-1cnOJlb3YH onpel!eJleHl-le aj I1 opTOrOHaJlbHOCTb gj' HaXOl!ilM
b 111 M 111
p(X)F2(X)dX 2 al- al = af
a j=O j=O j=O
J(JlH Bcex M.
B HenpepblBHOM CJIY4ae, eCJII-I B (17.8-1) BbITIOJIH5IeTCH paBeHCTBO J(JlS?
JIIo6otJ <PYHKI1.I1I1 F (x), HenpepblBHotJ B (a, b), TO 6eCKOHe4Hoe MHO)J{e-
CTHO <PYHKI1.I1I1 gj (x) Ha3blBaeTC51 no//.ftbl.M *), a paBeHCTBO
b 00
P (x) F2 (x) dx = a'
a ;=0
(17.8-2)
Ha3blBaeTCH pa8eHCm80,N napCe8a/lfl.
B D.HCKpeTHOM CJIY4ae Y Hac He B03HHKaeT TPYJ(HOCTetJ rrpI1 nepeCTa-
HOBKe CYMMl1pOB3HI151, H JIIo6oe ceMetJCTBO N <PYHKUl1tJ, JIHHetJHO He3aBH-
CMMbIX H3 MHO)J{eCTBe N T04eK, 5JBJJ5IeTC5I nOJJHbIM H3 3TOM MHO)J{eCTBe
T04eK (cp. 6.3).
17.9. MeTOJl. HanMeHbwnx KBaJl.paTOB
Ii K09<p<pnll.HeHTbJ <l>ypbe
f{09<p<pHI1.HeHTbI CPYPbe aJ D.3IOT H311JIY4wee B CMbICJIe H3HMeHb-
IllHX KB3l!paTOH npH6JIM)J{eHMe, KOr.Il.3 F (x) pa3J13r3eTC5I no OpTO-
rOH3JlbHOMY MHo)J{ecTBY <PYHKI1.I1H gj (x). 4T06bl .QOK3a3Tb 91'0,
*) 3TO CBMICTBO qalU,e Ha3blBalOT 3aMKHYTOCTbIO CI1CTeMbI 4JYHKU,HH, a nOl!
IJO.IHOTOH MHOJKeCTBa nOHHMaeTCR OTCYTCTBlle 4JYHKlI.HII, opTOrOHaJlbHOH BceM
IPYHKlI.f1HM MflO)KeCTBa. BlJpolJeM, JlJlil 4JYHKlI.HH C HHTerpHpyeMblM KBal!paTOM
9TH CBoI1cTBa 9KBHBaJleHTHbI, (l7puM. peiJ.)
'246
METon HAYlMEHbWYlX KBAJlPATOB. TEOPYl5I
[IJJ. 17
MHHHMH3upyeM BWpallieHUe
b M
In = p(X) [P(X)- Cjgj(X) r dx= p (X) p2 (X) dx-
, a j=O
M M M
-2 Cj p(x)P(x)gj(x)dx+ CjCdp(x)gtgjdx=
;=0 i=O j=O
M M
P (X) p2 (X) dx - 2 Ct at + d =
t=o ;=0
M M
= p(x)p2(x)dx- al+ (at-ci)2.
i=O t=O
Ho nOCJIelJ,Hee BbIpa)j{eHHe MHHHMaJJbHO, eCJIH Ct = at, 4ro H rpe60-
sanOCb.
MbI nOJlY4HJII1 3aMe4areJJbHOe H 04eHb nOJJe3HOe CBOtJCTBO K03cjJepI1IlH-
€HTOB <pYPbe: Kalli.u.bItJ H3 K03cpcpHIlHeHToB at, .u.alOu HtJ JIY4wee npH6JIH-
)l{eHHe B CMbICJJe Hal1MeHbWI1X KBalJ,paTOB no CHCTeMe opTOrOHaJIbHbIX CPYH-
KIJ.utt, onpe.u.emleTcH He3aBHCllMO OT OCTaJIbHbIX, H eCJIH Tpe6yeTcH H3Me-
HI1Tb 4UCJIO I-lCITOJIb3yeMbIx CPYHKIJ.HtJ gt (x), TO He HYlliHO I-lCKaTb 3aHOBO
Y)j{e Hatt.u.eHHbIe K03cjJcjJl1IJ.HeHTbI. HCCJIe.u.yeM Tenepb 06paTHYlO 3ana4Y,
ECJII-l K03cjJqJHIll-leHTbI c t pa3JJOllieHH5I CPYHKIlYlH P (x) no MHollieCTBY
CPYHKIlHI1 P-t (x) B CMbICJIe HaHMeHbWI1X KBa.u.paTOB He 113MeHHIOTCH npl1
H3MeHeHr1fJ 4HCJIa HCnOJIb3yeMbIx P-i (x), TO P-t (x) .u.OJI)j{HbI 6bITb opro-
rOHaJIbHbI. TIOJIOlliHM
b M
g(CO,Cl,,,,,Cn)= P(x)[P(x) - CiP-t(X)}' dx.
a i=O.
TaK KaK g .u.OJI)j{HO 6bITb MUHlIMU3I1pOBaHO, TO
b M
,=O= - 2 p(x{P(X) -l: Ct!1j]p-j(X)dX
, a 0
'"JIll
b M b
P (x) P (x) P-j dx = cd p (x) P-t P-j dx.
a 0 a
(17.9-1)
ECJIH 9TO CBOttCTBO BepHO .lI.JI5I Bcex M, OHO .u.OJIlliHO 6WTb BepHO H
Jl.JIH (M + I)
fJ M+l b
P (x) P (x) P-j (x) dx = Ct P (x) P-tP-j dx.
a a a
(17.9-2)
17.101
OPTOroHAJlbHbIR 1\\HOfO"IJ1EHbl
247
Y13 (17.9-1) H (17.9-2) I1MeeM
b
C M+l P (x) fLM+l fL j dx = 0
a
.llJlfl JIIo6oro j, T. e. fL j opTOrOHaJlbHO fL 1\1+1 (a M 6wIO np0I13Bo.1bHO).
TaKHM 06pa30M, opTOrOHaJlbHbIe cjJYHKIJ.I1I1, HaXolK.lleHi-Ie K03cjJcjJl1IJ.l1-
eHToB <tJYPbe H H.lleIJ npH6JiH)!{eHHIJ B CMblCJle HaHMeHbUlHX KBa.llpaTOB:
TeCHO nepenJleTaIOTCIJ.
17.10. OpTOrOHaJIhHb!e MHOrOQJIeHbI
Ba)!{HbIM nO.llKJIaCCOM opTOrOHaJlbHbIX cjJYHKIJ.l1ti SlBJlSlerCSI nO.llKJIaCC
opTOrOHaJlbHbIX MHOr04JIeHOB, me k-ti MHOr04J1eH HMeeT creneHb k
(k=O, I, ...).
J1erKo nOKa3arb, 4TO k-t! opTOrOHaJlbHbIt! MHOr04J1eH Yk HMeer k .lleti-
CTBI1TeJlbHblX pa3J1H4HbIX KopHet! B HHTepBaJle HHTerpHpoBaHHSI. lI.eticTBI1-
TeJlbHO, npe.llnOJlO)!{HM, 'HO 4HCJlO pa3JH!4HbIX .lleticTBl1reJJbHbIX KopHeti
r < k. 06pa3yeM npOl-l3Be.lleHI1e
1t (x) = (x - Xl) (X - X2)" .(x - x r )
(r <k).
Tor.lla
b
P (X) 'it (X)Yk (X) dx = 0 (p 0),
a
TaK KaK 1t (X) MO}l{eT 6bITb pa3J10lKeHa no YOYl" .Yr' a Yk (x) opro-
rOHaJlbHO I:IM BceM. Ho 3TO HeB03MOIKHO, nOCKOJlbKY nO.llblHTerpaJlbHa5f
cjJYHKUHIJ He MeHSler 3HaK B I1HTepBaJle. CJIe.llOBaTeJlbHO, Ha (a, b) cy-
UleCTByeT k .lleticTBI1TeJlbHbIX pa3J1114HbIX KopHeti.
OpTOrOHaJIbHbIe MHOr04J1eHbI Yk (x) Y.llOBJJeTBOpSllOr Tpex4J1eHHoMY
peKyppeHTHoMY COOTHOJlleHHIO BH.lla
akYk+l (x) + (b k - X)Yk (x) + CkYk_l (x) = 0
(k 1). (17.10-1)
4T06bI nOKa3aTb 3TO, nOJJOlKl1M
Yi (x) = rJ.i Xi +. ..
(rJ.i> 0).
(BbI60p rJ.i - HeKoTopofO nOJIOIKHTeJlbHOrO llHCJIa - eCTb .lleJIO TOJIbKO
CQr JIallleHI1IJ.)
TOf.lla pa3HocTb
akYk+l - XYk
eCTb MHOr04JIeH CTeneHH k npl1 YCJlOBHI1, 4TO a k = . CJlen:oBa-
"k+1
TeJIbHO,
akY/l+l - xy" = ik YII + Ik-lYk-l +. . . -r-loYo'
(17.10-2)
'248 METOJl HAVlMEHbIllVIX KBAJlPATOB. TEOPI151 ITJJ.17
YMHOllil1M 9TO paBeHcTBo Ha p (X) Y m (X) 11 rIpOI1HTerpl1pyeM. I10JJY4aeM
b
P (X) (a k Yk+l - XYk) Ym (X) dx = 1m )'m' (I 7.1 0-3)
a
JLJJ5I m = 0, 1,..., k tpYHKI.LI151 Ym opTOrOHaJIbHa Yk+1> a .a:JI5I m = 0,
I, . . ., k - 2 rIp0l13BeJleHI1e xYm eCTb MHOr04JIeH CTerIeHI1, MeHbllIel-l k, 1-1,
JJe.n:OBaTeJIbHO, opTOrOHaJIbHO Yk- TaKI1M 06pa30M, 10 = II =...= Ik-g=O.
L(JJ5I m=k-l paBeHcTBo (17.10-3) rIpl-lHI1MaeT BI-I.a:
b b
P Yk (XYk_l) dx = - PYk ( a l Yk + Ck_lYk_l +. . . ) dx =
a a
ak_l , ,
= - ---a;; "k = Tk-l "k-l;
MCnOJIb3Y51 (17.1 0-1), rIOJJY4aeM
ak_l Ak
ck = Ik-l = - ----,-- =F O.
ak "k-l
JLJI5I m = k (17.10-3) rIpl1HHMaeT BI1.a: (I1CrIOJIb3Y51 (17.10-1»
b b
P (akYk+l - XYk)Yk dx = - p xY% dx = Ik = b k .
a a
TaKHM 06pa30M, paBeHCTBO (17.10-2) MOllieT 6bITb 3arII1CaHO B BI1.a:e
ak +
_ a Yk+l - XYk = IkYk Ik-lYk-l
k+l
{iJIH
ak+l('(k + x) ak_l ak+l Ak
Yk+l = a Yk - a2 Yk-l'
k k k-l
(17.10-4)
113 9TOI-I tpOPMYJIbI CJIe.a:yeT HeCKOJIbKO BalliHbIX pe3YJIbTaTOB. I1oKa-
)/(eM, rrpelli.a:e BCerO, '1TO k HYJIett MHOrO'lJIeHa Yk (X) pa3.a:eJIeHbI k - 1
HYJI5IMI1 MHOrO'lJJeHa Yk-l (X) rIpl1 YCJIOBI1I1, '1TO P (x);?: O. L(oKa3aTeJIb-
CTBO rIpOBe.a:eM rIO I1H.a:YKJJ.I1I1. Ha nepBOM IlIare HMeeM Yo (X) = (Xo > 0
II Yl (X) = (Xl (X + 10)' KaK 113BeCTHO, Yl (X) I1MeeT .a:el-lcTBIHeJJb-
HbItt KopeHb B E1HTepBaJIe I1HTerpHpOBaHl-I5l, TaK KaK
b
P (X) Yo (X)Yl (X) dx = O.
a
I1pe.a:nOJJOlliI1M, '1TO HYJIH Yk (X) pa3JleJIeHbI HYJI5IMH
Yk (Xi) (17.1 0-4) rIpllHI1MaeT BI1.a:
( ) ak_l ak+l Ak )
Yk+l Xi = - a 2 . Yk_l(Xi'
k k-l
Y k-l (X). B HY JISL{
(I 7.10-5)
B KOHJJ.e IIHTepBaJIa X = b KaK Yk+l (b), TaK II Yk-l (b) rIOJIOlliI1TeJIbHbI,
I10CKOJIbKy C'fapllIlle K09tptpl1JJ.I1eHTbI BbI6paHbI rIOJIOllillTeJIbHbIMII (Xi> 0,
17.111
KJJACCI14ECKI1E OPTO roHAJJbHblE MHO ro4JJEHbi
249
3 BCe HY"IH CPYHKIJ.HH JIe)J{aT BHYTpH HHTepB3JI3. B H31160JIbllleM HY JIe
Yk (Xi) = 0, Yk-l (X;) rrOJIO)J{I1TeJIbHO; C.lJe.u.O BaT e.lJbHO, Yk+l (Xi) OTpHIJ.a-
TeJIbHO B CH.lJY (17.10-5). B C.lJe.u.YIOUJ,eM rro Be.lJHqHHe HY.lJe Yk-l rro rrpeu-
rrOlIO)J{eHHIO I1H.u.YKIJ.HH HMeeT .n:Pyroti 3H3K, C.lJe.u.OBaTe.lJbHO, Yk+l (x) rro.lJO-
)J{[-ITeJIbHO, 11 T. .u.. TIo Mepe Toro KaK MbI nepexo.u.HM GT O.u.HOro HY lJ5<
Yk K CJIeLl,YIOUJ,eMY HY"IIO, MHOrOqJJeH Yk+l TQ)Ke MeH5"JeT 3HaK. TaKHM 06-
p330M, MbI rrOKa3aJII1, qTO HYJIH Yk (x) p33L1,eJJ5IIOT HYJIH Yk+l (x), 3a HC-
KJIIOqeHHeM HaHMeHblllero HYJI5I Yk+l (x). Ho TaK KaK MbI rrOKa3a.lJH, qTO'
Yk+l (x) HMeeT k + 1 pa3JJ11qHbIX Ll,etlcTBHTeJJbHbIX KopHeti Ha (a, b), rro-
ClleLl,HHtI HYJIb Yk+l (x) MeHbwe Bcex HY"leti Yk (x), H Ll,OKa3aTe.lJbCTBO
HHJ1,YKIJ.HH 3aKOHqeHO.
ECTb elll.e O.u.HO Ba)J{Hoe C.lJe.u.CTBl1e TpeXq.lJeHHblX peKyppeHTHbIx
cOOTHOWeHI1t1 (17.10-1) H.lII1 (17.1 0-2). 'KaK TO.lIbKO H3BeCTHbi KoacpcpH-
lJ,HeHTbl peKyppeHTHoro COOTHOWeHH5I KaK CPYHKlJ,HH k, a TaK)J{e Yo (x)
H Yl (x), TO MQ)KHO rrOC.lIeLl,OBaTe.lJbHO BblqHCJJ5ITb Yk (x) 6b1CTpee, 4eM
rpOM03L1,KI1M MeTo.u.OM WMI1L1,Ta (CM. 18.2).
17.11. KJlaCCMQeCKMe opTorOHaJlbHbIe MHOrOQJleHbI
CYllI.eCTBYIOT TPli no.u.p06HO flCC.lIeJ1,OBaHHbIX MHO)J{eCTBa opTOrOHaJIb-
HblX MHOrOq.lIeHOB: MHOrOq.lJeHbl JIe)J{aH.u.pa, Pn (x)
J f 0,
}1 P m (X)P n (X)dX=\ 2n 2 +1 '
MHOrO'lneHbI JIareppa, Ln (x)
m-::j::.n,
m=n,
r e- x Lm (x) Ln (x) dx = { 'I,
m-::j::.n,
m=n,
H MHOrOqneHbI 3pMHTa, Hn (x)
r 2 { 0,
J e- x Hm(x)Hn(x)dx= n V -
-00 2 nl 11:,
m-::j::.O,
m=n.
COOTBeTCTBYlOllI.He HM TpeX'lneHHble peKyppeHTHbIe COOTHOlllemm
(CM. (17.10-4» HMelOT BH.u.
(n+ 1) Pn+tCx) - (2n + l)xPn (x) + nPn_tCx) =0,
(n + I)L n + 1 (x) - (2n + 1 - x) Ln(x) + n 2 Ln_1(x) = 0, (n 1)
H n + 1 (x) - 2x Hn (x) + 2n Hn_l (x) = 0,
MHoro Ll,Pyrllx CooTHollleHl1t1, KaCalOllI.HXC5I 9THX CPYHKlJ,HtI, MO)J{HO
HatiTH B 06bl'lHblX pa60Tax. HYJJH MHOrOq.lIeHOB JIe)J{aHLI,pa, HanpHMep,
oKa3bIBalOTcH Y3.lIaMH rayccoBblX KBa.u.paTypHblX cpopMyn (rn. 10).
250
METon HAI1 MEHhUJI-1X K13AllP ATOB. TEOPI-151
[r JJ. \7
COOTBeTCTBeHHO 3TMM OpTOrOH3JJbHblM CPYHKl.U-IHM, Onpe.l1.eJ/eHHbtM .i1,JJH
I-IflTerpHp013aHM5I, cyw.eCTBYIOT OpTOrOHaJlbHble qJYHKlll1H .I1.Jl5I .I1.HCKpeTHhlX
MHO}!{eCT13. vb HHX HaH60Jlee pacnpoCTpaHeHo MHO}!{eCT130 CPYHKlll1t!, co.
DTBeTcT13YIOw.ee nOJlHHOMaM Jle}!{aH.I1.pa. VI3 co06pa}!{eHHt! Y.I1.06CTBa MHO.
uJ,a Bbl6HpalOT liHTepBaJl - 1 Xi 1, a HHor Aa 01' 0 .11.0 I npH paBHo-
OTCTOHIllHX Y3J1ax Xi' Ii 'H-ITaTeJlb ,lJ,OmKeH BHHM3TeJlbHO CJlemJTb 33 'reM,
J,3EOe 113 3THX MHO}!{eCTB DcnOJlb30B3HO B T36. Hlle.
Ynpa};{HeHl-le 17.11-1. ECJlI1 Po = 1, PI =x, nOKa3aTb, 'ITa l\JlR -1 x I
1 ( \ I ( ) I ( )
P2=2\3x 2 - 1), ,03="2 5x 3 -3x, ,04= 8\35x 4 -30x 2 +3 .
17.12. CpaBHeHHe MeTOJJ;a HaHMeHhmHX KBaJJ;paTOB
H pa3JIOmeHIUI B CTeneHHble PH.IJ;hl
Pa3J10}!{eHHe 13 CTeneHHot! p5I.ll CPYHKllHH Y (x)
co
'\1 ylnl (0) x n
y (x) = -'- nJ '
n=O
-6yn:y'm ycetteHHhlM AO
N
'\1 ytn! (0) x n
Y N(X) = ""' n! '
n=O
JIaeT OtteHb xopowee npU6JJfDKeHHe 136J1H3U X = 0, HO KOr,lJ,a X B03pa-
naeT, 91'0 npH6J1H}!{eHHe HMeeT TeHJJ;eHllHIO yxyn:maTbC5I. C n:Pyrofi CTO-
POHbl, MeTon: HaHMeHbWHX KBan:paTOB nblTaeTC5I Hat!TH 6o.1ee HJIH MeHee
p3BHoMepHoe (B 3aBHCHMOCTH 01' BeCOBot! CPYHKll11H p (X)) npH6J1H}!{emre
H3 HHTepB3J1e. VI 13 9TOM COCTOHT OAHO H3 OCH013HhlX p33J1H4HU: CTeneH-
Hble p5l.n:bl npH6J1u}!{aIOT 13 TOttKe, 13 TO BpeM5I K3K H3HMeHbWHe KBa.n:paTbl -
Ha HHTepBaJle.
Kor Aa AeJlO .n:OXO.n:HT .11.0 Henocpe.n:cmeHHblx BblttHCJleHlit!, TO He Bce.
f.n:a B03MO}!{HO TOttHO 13blttUCJlUTb npOU3Bo.n:HYIO B TOttKe, HcnOJlb3Y5I Y3J1bl,
p336poC3HHble no HHTepB3J1Y; B 3Tm! CJlY43e BMeCTO yce4eHHoro pH,lJ,3
lIaCTO HCfIOJlb3ylOT TO'JHO npU6J1H1KalOw.Ht! HHTepilomlllUOHHblt! nOJlHHOM.
B CJlY4ae HaHMeHbIllHX KBa.n:paTOB 'JaCTO B03HHKaeT B03MO}!{HOCTb 131,1-
60pa. MO}!{HO npe.n:CT3BHTb ce5e, liTO HMeIOTC5I HenpepbIBHble opTOrOHaJlb-
Hble IlOJJHHOMbl, K09cpcpHuueHThi <pYPbe (HHTerpa.1bl) no KOTOphlM MO}!{HO
BbIlIHCJ1HTb KaKHM-im60 4nCJleHHbiM MeTOJJ:OM. B .n:pyroM c. Y'Iae MO}!{HO
H3'13Tb C .I1.HCKpeTHblX Y3J10B H BeCT!i 13bI4HcneHHH, HCUOJlb3YH JlHCKpeT-
Hoe opTorOHaJlbHoe MHO}!{eCT130 CPYHKUHt!. TpY.D,HO cKa3aTb, K3KOt! U3
MeTo.n:OB AaeT JIY'JIllHe pe3YJlbT3Tbi 13 60JlblliHHCTBe CJIY4aeB. no Bcet!
£ep05lTHOCTH, Bbl60p 3aBHCHT OT o6CTOHTe.lbCTB, BKJlIO'la5I npe.D,1101lTeHHH
JlHua, 110J1b3YIOilI.erOc5I pe3YJIbTaT3MH.
17.131
METon HAVlMEHb[JJl1X KsALlPATOB C orPAHVl4EHf-HIMVI
25J
HYJKHO 3aMeTI1Tb, 4TO npl1 pa60Te C p5lLlaMI1 <t>YPbe B03MOJKHbi nBa
nOLlxOLla, npHBOJUIllJ,llX K COBepllleHHO OLlllHaKOBblM Bb14HCJIeHI151M. 3TH
nonXOnbI HanOMHHalOT pa3JII14HbIe B03MOJKHbIe B3r JI5ILlbI Ha npe06pa30Ba-
Hl1e CI1CTeMbI KOOpnHHaT B aHaJIHTI1'1eCKOt! reOMeTpl1l1, 0 KOTOpbIX lllJIa
pe'lb B 1.1. MOJKHO nOJIb30BaTbC5I JII06bIM H3 HI1X, B 3aBI1CHMOCTH OT
yn06CTBa, npl1 YCJIOBllH, 4TO MbI BCer.L/,a 6YLleM nOMHHTb 0 COLlepJKaTeJIb-
HOn CTOpOHe LleJIa.
17.13. MeTOJ1. HaUMeHbUlUX KBaJ1.paTOB C OrpaHHQeHUHMu:
npO)1.0JJIKeHHe npuMepa H3 1.9
B npl1Mepe, npHBeLleHHOM B 1.9, Mbl CTpOI1JII1 I1HTepnOJJ5IUHOHHbIfi'
MHorO'-!JleH no 11 TO'-!KaM Ha oTpe3Ke [0, 1] C lllarOM h = 0, I, HcnOJlb-
3Y51 I1HTepnoJI5IUHoHHYIO cj:>OPMYJIY HblOToHa. 6JIarOLlap51 pe3YJlbTaTaM 3THX
BbI'IHCJIeHl-Iti 6bIJIO C03LlaHO HOBoe JIa60paTopHoe 060PYLloBaHHe, n03BO-
JIHBlllee c06paTb 60JIee T04HbIe 11 60JIee 06llll1pHbIe LlaHHble.
BI13yaJIbHOe I1CCJIenOBaHlle J(aHHbIX nOKa3aJIO, 4TO pe3YJJbTaTbl an--
npoKcI1MI1PYlOTC5I MHOrO'lJIeHOM llleCTot! CTeneHI1, a rpacj:>HK 3Ha1.JeHI1t! B6JI113I1'
TO'lKI1 x = 1 I1MeJJ HaKJJOH OKOJIO 6, 4TO, Ka3aJIOCb, nOLlTBepJKLlaJIO 3TB
I1CCJIeLlOBaHH5I. «Ha paLlOCT5Ix» 6bJJl npoBeLleH MHOr04JIeH MeTOLlOM Hal1-
MeIIbllll1X KBanpaTOB, HO pe3YJIbTaTbl OKa3aJII1Cb Hl1Kyna He rOLlHbIMH.
B T01.JKe x = ° MHOr01.JJIeH I1MeJI nOJIOJKI1TeJIbHOe 3Ha4eHl1e, B TO BpeM5I'
KaK 3KCnepl1MeHT naBaJl 411CTblt! HYJIb, nOCJIe 'Iero MHOrO'lJIeH CTaHOBI1JIC5I'
OTpHuaTeJIbHbIM, 1.JTO cj:>11311'1eCKI1 HeB03MOJKHO.
HaH60JIee npocTblM OKa3aJIOCb OT6pOCI1Tb cB060nHbIIi 4JIeH MHoro-
4JIeHa. LIJIeH, conepJKallJ,111i x, OT6pOCI1Tb 6blJIO HeJIb35I, nOCKOJIbKY 113:
06CYJKneHI151 C 3aKa3'111KOM BbI5ICHI1JIOCb, 'ITO Kpl1Ba5I He LlOJIJKHa KaCaTbC5I
oel1 x B TO'lKe x = 0. BbI4I1CJIeHI15I, BbIflOJJHeHHble C Y4eTOM 3TI1X 3aMe-
1.JaHI1t!, naJII1 BnOJIHe npHer.!.TIeMblt! MHOrO'lJIeH, a OCTaJIbHa51 4aeTb. Bbl'lI1C-
JJeHHIi 6blJIa CTOJIb JKe npOCTot!, K3K 11 p3Hbllle.
LLpyroli o'-!eBHnHblt! enoe06 pellleHI151 3ana'll1 annpOKCHMa'U1111 3TI1X
.lI.3HHblX B TO'lKe x = 0 COCTOI1T B TOM, 4T06bI npl1naTb 3Ha4eHI11O B 3TOl1-
TO'lKe 60JJblllot! Bee, CKaJKeM 1000, a OeTaJIbHbIM 3Ha1.JeHH5IM LlaHHbl x
npl1.l1.aTb Beca, paBHbIe 1.
CY1ll.ecTByeT lI-1Horo Bapl1aUl11i Ha TeMY HaJIOJKeHHH nOnOJIHHTeJIbHblX
YCJJOBI1Ii H3 Me-ran Hal1MeHbllll1X KBanpaTOB. Hanpl1Mep, npennOJIOJKHlIl,
4TO B T04Kax a H b pellleHHe LlOJJJKHO npl1HHlIIaTb nBa 3anaHHbIX 3Ha4e-
HH5I 11 annpOKCHlIIaUI15I BeneTeH opTOrOHaJIbHbIMH nOJII1HOlllalllH. B 3TOM
eJJY'l3e MOJKHO HenOJJb30BaTb KJJaCel1'-!eCKHIi MeTOn MHOJKHTeJIet! J1arpaHJKa.
Ec. 11 6bI 3Ha'leHH5I B T04Kax,a 11 b 6blJJI1 paBHbI 0, TO MHOrO'lJIeH P n (x) MOJKHO
6wIO 6bI nOCTpOI1Tb, I1CnOJIb3YH BeeOBYIO cj:>YHKUI1IO (x-a) (x-b) . TOrLla
Qn (x) = (x -- a) (x - b) Pn (x)
6y nYT opTOrOHaJIbHblMI1 MHorOYJleHaMI1, npoXO.ll.5IllJ,I1MI1 4epe3 HY JIb B T04-
Kax x = a 11 .x = b.
252
METon HAHMEHbWI1X KBAnp A TOB. OP AKTV!KA
[fJJ. 18
Ynpa)KHeHl1e 17.13-1. rIoKa3aTb, 'ITa B MeTOJ],e opTOrOHaJlbHblX nOJlHHOMOB
113 Ha'laJlbHbIX llaHHblX MO)/{HO TaK BbI'IeCTb .1HHeHHYIO 4JYHKUIlIO, 'ITa YCJlOBI1R
l(a) = A H f (b) = B cBellYTcR K YCJlOBHIO , (a) =, (b) = O.
17.14. nOCJIenHlle 3aMeqaHHSI 0 MeTOil.e
ttaHMeHbllIHX KBail.paTOB
flOJIY'1IlB HeKoTopoe npe.n:cTaBJleHHe 0 npOBe.n:emIH MHOrO'1JleHOB Me-
ro.n:OM HaHMeHbllIHX KBa.n:paToB, 411TaTeJlb, B03MO)l{HO, O)l{11.n:aeT, 4TO aIHOp
Ilepetl.n:eT K paccMoTpeHHIO 3a.n:a'1 11HTerpHpOBaHM5J 11 peWeHH5J .n:Hq1CpepeH-
U11aJJbHbIX ypaBHeH11tt HeCMOTp5J Ha peaJIbHOCTb ElTOrO, MeTO.n: Ha11MeHbWI1X
KBa.n:paToB pe.n:Ko HcnOJlb3yeTC5J B Ka4eCTBe OCHOBbJ npl1 11HTerp!-lpOBaHI1I1.
HHTerp11poBaH11e HBJlHeTC5! npoueccoM cr Jla)l{HIJ3HI15J. ECJlI1 .n:HcpcpepeHUI13JIb-
Hoe ypaBl-JeHHe 11MeeT «WYMOBOt1 4JleH», TO 06bJt1ll0 ero cr JIa)l{111J31OT, 11
HOOle ElTOrO .n:JI5J I1HTerpHpOBaHH5J ypaBHemIt1 11CnOJIb3yIOT T04HO annpOK-
-CHM11PYIOlll.HtI nOJIHHOM.
.upyra5J H.n:e5J, KOTOpa5J MO)]{eT B03HHKHYTb y 4HTaTeJI5J, HO KOTOpa5J
'eIl(e He.n:OCT3T04HO H3Y4eHa, COCT011T B c. e.n:YlOlUeM: B l\WTo.n:e T04HO!!
3nnpOKCI1M3UH H MbI nOJJaramd Eo, E 1 , ..., Em_I = 0, 3 OCTaJIbHble Em'
Em+I' ... OCTaBJI5JJIH TeMH, KaKHe OHI1 eCTb. flpe.n:nOJIO)]{HM, 4TO Mbl nbI-
:raeMC5! MHHHM113HpOBaTb BbIpa)l{eHHe
00
m (a o , aI' ...) = 2:: akE'f,.
k=O
B npou:ecce TOlIHO!! annpOKCHMaIJIHf am = a mtl =. .. = 0, 110 MO)l{II0
-CJIerKa H3MeHHTb ak H TaKHM 06pa30M nOBblCI1Tb TOlIHOCTb annpOKCI1M
tiHH .n:JJ5J 60JJee BblCOKHX CTeneHe!! 3a ClIeT T04HOCTH 60JIee HH3KI1X. 9TO
MO)l{HO P accMaTpHBaTb KaK npe.n:JIO)l{emfe annpoKc11MHpoBaTb B npocTpaH-
nBe OWH60K, a He B npOCTpaHCTBe cpYHKUMt1. flo.n:06H35J TeOpH5J OCTaTO'1-
Horo 4JJeHa elUe B HaCT05llUee BpeM5I He pa3pa60TaHa.
fl pI1MepOM He3Ha'lHTeJIbHOrO YMeHbWeml5J TO'lHOCT11 annpOKC11MaU:HI1
IlJl5J HeKOToporo 3Ha'lem15J k, npI1Bo.n:5!lll.I1M K 60JIbWeMY nOBbfWeHI11O TO'l-
HOCTH npH 60JIee BbICOKOM 3Ha'1eHI1I1 k, 5JBJJ5JeTC5I H.n:e5J «oT6paCbfBaHH5I»,
KpaTKO paCCMOTpeHHaH B 9.5 B CB5J311 C HHTepnoJ!5JUHOHHOt! cpOPMYJIOt!
3BepeTTa. B )J.aHHOM cJJY'Iae MbI nocTynHJ!Hcb He60J!bwot! T04HOCTbIO B x 2
C TeM, '1T06bf HaMHoro YJ!Y4WHTb anrrpoKcHMaUl11O x 3 11 x .
r JJ A B A 18
METO.ll HAHMEHbllIHX KBA1I.PATOB. nPAKTHKA
18.1. OOllI.He 3aMeqaHHH 0 MHOrOQJIeHHOM cJIYQae
MeTo.n: HaHMeHbWI1X KBa.n:paToB I1cnOJJb3yeTc5I r JIaBHbIM o6pa30M B Tex
cHTyaUH5JX, r.n:e Ha.n:o onpe.n:eJJHTb KOElcpcprm.HeHTbI, Bxo.n:5!IUI1e mlHet!HO,
oc06eHHo KaK K09cpcpHUI1eHTbI MHOr04J1eHa. P3CCMOTpHM CHa'1aJJa I1MeHHO
T3KOtl cJ!Y'lat1, a 3aTeM nepet!.n:eM K APYrHM.
18.2]
TPEX4J1EHHOE PEKYPPEHTHOE COOTHOLUEHIiE
253
f{aK ObIJIO YKa3aHO B rrpe.QbI.QYllJ.etl r JIaBe, orrpe.QeJII1TeJIb HOpMaJIbHorO
ypaBHeHI1H (17.5-4) 06bllJHO 6bIBaeT OlJeHb MaJ/; rro3ToMY peweHHe onw-
CHTeJlbHO K03cpcpHll.l1eHTOB .QOJI)\{HO 6bITb .QOBOJlbHO Heonpe.QeJIeHHblM.
Heo6xo,IUIMO, o.n:HaKO, pa3JIHlJaTb .QBe BellJ.H: TOlJHOCTb K03cpcpHll.HeHTOB H
MaJIOCTb CYMMbI KBa.QpaToB oWH6oK. Ec.m yrroMHHYTbltI orrpe.n:emneJIb
MaJl, TO K03cpcplill.HeHTbI 6Y.QYT Hatl.QeHbI rrJIOXO; TeM He MeHee CYMMa KBa.Q-
paTOB OWH6oK MO)\{eT 6bITb 6JII13Ka K MHHHMYMY. Boo6llJ.e rOBopH, Kor.Qa
lJHCJlO onpe.n:eJJHeMbIX K09cpcpHll.l1eHTOB He rrpeBOCXO.QI1T rrHTI1-WeCTH, rrpH-
Moe pellJeHl1e HOpMaJIbHOrO ypaBHeHI1H 06bIlJHO rrpHeMJIeMO; HO rrpl1 60J/b-
weM I1X lJHCJIe CKopee Bcero BCTpeTHTC5I TPY.QHOCTH.
Do TeOpI1H paCllJHpeHHe CHCTeMbI opTOrOHaJIbHbIX MHOrOlJJ/eHOB 3aMe-
H5IeT npHMoe pellJeHHe. O.QHaKo orrbIT rrOKa3blBaeT, lJTO eCJII1 rrpoBo.QI1Tb
opTOrOHaJIH3aUHIO npH rrOMOllJ.H npoll.ecca WMIJ..rna, TO B03HHKHYT Te )\{e
TPY.QHOCTH, XOTH H B .QpyroM BI1.n:e. B rrpOll.eCce WMI1.QTa m-tl MHOrOlJJIeH
CTpOHTCH C rrOMOllJ.blO rrepBbIX p a3HoCTetl Bcex KOMrrOHeHT BeKTOp a x m ,
T. e. (x , x , ..., x'N), KOTOpbItI Jle)\{I1T B HarrpaBJIeHHI1 paHee orrpe.n:e-
JIeHHbIX MHOrOlJJIeHOB. Pe3YJIbTaT, KaK rrpaBUJIO, 6bIBaeT MaJI, eCJIlI
TOJlbKO m He.QOCTaT04HO BeJIHKO, H B KalJeCTBe rrOCJIe.QHero llJara ero ellJ.e
Tpe6yeTcH HOpMHpOBaTb. TaKoe HOpMl1pOBaHHe YBeJIHlJHBaeT OIlIH6KY, rro-
TO MY lJTO HOpMI1PYIOllJ.utI MHO)\{HTeJIb 06bIlJHO HMeeT BeJJH4UHY, MHoro
60JJbllJYIO e.QI1HI111.bI.
fpyoo rOBopH, 3aTpY.QHeHHe MO)\{HO BbIpa3HTb 3aMelJaHHeM, lJTO JJ;JIH
60JIbllJHX n BeKTop x n HarrpaBJIeH rrOlJTH Ty.n:a )\{e, KY.Qa H xn-I, x n - 2 H
T. .n:. TaKHM 06pa30M, ypaBHeHI1H, Be.n:yllJ.He K onpe.n:eJIHTeJJIO I'HJIhOepTa,
rrOlJTH JIHHetlHo 3aBHCHMbI; OTCIO.n:a 9TOT orrpe.n:eJIHTeJIb MaJJ.
18.2. TpeXQJIeHHOe peKyppeHTHoe cooTHorueHHe
ECJII1 BCe-TaKI1 rrbITanCH I1CrrOJIb30BaTb opTOrOHaJJbHbIe MHOr04JIeHbI,
TO, rrO-BI1.n:I1MOMY, MO)\{HO H36e)\{aTb 3aTpY.QHeHHtI, yrroMHHYThIX B rrpe.QbI-
.QYllJ.eM rraparpacpe, I1CrrOJIb3YH .QJIH 06pa30BaHHH opTOrOHaJIbHbIX MHoro-
4JIeHOB TpeX4JIeHHOe peKyppeHTHOe COOTHOllJeHHe (17.10-2).
3arrlIllJeM (17.10-2) B BH.Qe
po(x)=l, )
PI (x) = xPo (x) - IJ.IPO (x),
Pk+1 (x) = XPk (x) - ak+lPk (x) - k+IPk-1 (x), k> 1,
(18.2-1 )
r1I.e ak+1 11 k+1 rrO.QJIe)\{aT orrpe1I.eJIeHI1IO. Hatl.n:eM CHa4aJIa al' 113BecTHo
lJTO
b
p (x) po(x)') 1 (x) dx = O.
a
254
METOL\ HAI1MEHbllll1X KBAL\P A TOB. TIP AKTI1KA
IfJI. 18
Owo.n:a no (18.2-1)
b b
p(x)XdX=a.l p(x)dX.
a a
ITpe.n:nOJIm/(HM Tenepb, 'ITa 113BeCTHb1 Po (x), PI (x), ..., Pk (x) 11 'ITa OHii
B3aHMHO opTOrOHaJIbHbI. Tpe6yeTC!! BblqHCJ/I1Tb c.lle.n:YlOllI.Ht! MHOrOqJIeH
CI1CTeMbl Pk+l (x). TIOTpe6yeM CHaqaJla, qT06b1
P (x) PhI (x) Pk (x) dx = 0,
P (X)Pk+l (x)pk-J (x) dx = 0.
3TOro .QOCTaTOqHO, qT06bl onpe.QeJ/HTb a. n +l H n+l' YIcnOJIb3Y!! onpe.n:eJle-
HHe Pk+l (x) (paneHcTBa (18.2-1)), nOJlY1JaeM
p (x) XPk (x) dx = a.k+! P (x) Pk (x) dx + k+l P (x) Pk (x) Pk-l (x) dx,
) p (x) XPk (x) Pk-l (x) dx =
=.a.k+l P (x) Pk (x) Pk-l (x) dx + k+l P (x) Pk-l (x) dx.
TIOCKOJlbKY Pk (x) 11 Pk-l (x) opTOrOHaJIbHb1, I1MeeM
i p (x) xPT. (x) dx i p (x) XPk (x) Pk-l (x) dx
a. k +! = r ' k+l = \'
.\ p (x) Pi: (x) dx J P (x) pj. _ 1 (x) dx
3HaMeHaTe.1Jb Bblpa}/(eHI1!! .n:JI!! k+l 6bIJi y}/(e Bbl1JHCJleH Ha npe.n:bl.n:y-
III eM III are, Kor.n:a onpe.n:eJl5lJIOCb a. k (x); TaKHM 06pa30M, Ha Ka)/(.n:OM llIare
na.n:o Bb1qI1CJlI1Tb TpH HHTerpaJla.
OpTOrOHaJIbHOCTb nOJlyqeHHOrO MHOroqJleHa Pk+l (x) BCeM Pi (x)
(i<k-1) cJ1e.n:yeT 113 paBeHcTBa (18.2-1)
Pk+l (x) = XPk (x) - a.k+!Pk (x) - k+1Pk-l (x),
nOCKOJlbKY, YMHO}/(a51 ero Ha p (x) Pi (x) 11 I1HTerpl1pY5l, nOJlyqaeM
P (X)Pk (x) [XPi (x)] dx - Il k +l P (X)Pk (x) Pi (x) dx-
- k+! P (x) Pk-l (x) Pi (x) dx = 0,
llBa nOCJle.n:HHX HHTerpaJla paBHbl HYJl!O B CI1JIY opTOrOHaJlbHOCTI1, a B
l1epBOM .XPi (x) eCTb MHoroqJIeH CTeneHI1 MeHblllell, qeM k; n03TOMY
3TOT I1HTerpaJI TO}/(e p aBeH HY JIIO.
ECJII1 nOl1bITaTbC5I nOCTpOI1Tb MHOrOqJIeHbl Ha .Q!1CKpenIOM MHO}/(eCTBe
rOqeK xi U = 1, ..., N), TO I1HTerpaJIbl 3aMeH5ITC5I CYMMaMH. ECJIH
nblTaTbC!I CTpOHTb 60JIee N MHOrOqJleHOB (T. e. CBepx PN-l (x), TO ypaB-
HeHHe, onpe.n:eJI5IIOlll.ee Il N , BblpO.n:I1TC5I, KaK I1 .n:OJl)KHO 6bITb.
YnpamHeHHe 18.2-1. OpToroHaM13HpOBaTb CI1CTeMY MHoro". eHoB J, x, x 2 .
x 3 Ha HHTepBaJle O.s;; x.s;; 1, HcnoJlb3YH TpeX1.fJleHHblfi peKy'ppeHTHblA MeTOJJ,.
18.41
HEMHO rOlJJJEHHhIt't CJJYlJAt't
255
18.3. TIocTpoeHHe KBa3HopTOrOHaJlbHbIX MHorotJJleHOB
fIp5!Moti no.axo.a npHBOJJ}1T K HnpMaJIbHoti CHCTeMe ypaBHeHHti, Y KO-
TOpOti eCTb HeHY JleBbIe lJJIeHbl BHe r JIaBHoti J1.HarOHaJIH. YlcnOJIb30B3Hl1e
opTOrOHaJIbHbIX MHOrOlJJIeHOB npHBOJl.I1T K ypaBHeHI15!M, Y KOTOpbIX Bce
4JIeHbi BEe rJJaBHoti .aHarOHaJiH paBHbl HYJIIO; n03TOMY peweHHe CHCTeMbl
TpHBHa.1IbHO. ECJJH CTpOHTb opTorOHa. bHble MHOr04JIeHbl npOL\eCCOM illMHJl.Ta
(HJI!1 OpH nOMOIll.H TpeXlJJIeHHOrO peKyppeHTHoro COOTHOllJeHH5!), HO onpe-
Jlem-ITb HX HeT04HO, TO y OKOH4aTeJJbHblX ypaBHeHHtJ, KOTopble npH.aeTC5!
pew an, 6y.aYT 60JIbllJHe "WeHbl Ha rJIaBHotJ J1.l1arOHaJIH, Tor.aa K3K 4JleHbi
BHe .a,JarOHaJiH nponopUHOHaJIbHbl OTKJIOHeHHIO 01' opTOrOHaJIbHOCTH. ECJlJ1
OHO HeBeJII1KO, TO CHCTeMa ypaBHeHutJ COBceM JIerKO peWaeTC5!. 31'0 Ha-
BO.aHT Ha MbIC,Ib nonp06oBaTb nOCTpOI1Tb KBa311opTOrOHaJIbHbie MHoro-
4J!eHhl, He .aeJIa5! 60JIblliHX BbI4I1CJIeHHtJ. HaCKOJJbKO Mbl I1X c.aeJJaeM op-
TOrOHaJIbHbIMH, HaCTOJIbKO 4JIeHbi BHe r JIaBHotJ J1.l1arOHaJII1 CI1CTeMbl
HOpMaJIbHblxypaBHeHHtJ 6y.aYT HYJJ5!MH, H Hao6opOT, HaCKOJjbKO MbI OWI1-
6eMC5! B opTOrOHaJIbHOCTI1, HaCTOJIbKO OHI1 6YJl.YT J1.aJIeKI1 01' HYJIIJ.
Pa3MbIWJIeHI15! Ha 3TY TeMY npl1BOJl.IJT K MbICJII1, 41'0 OCHOBHoe CBOti-
CTBO, KOTopoe Ha.ao nOCTapaTbC5! COXpaHI1Tb, eCTb CBOtiCTBO 4epe.aOBaHH5!
KopHeti. BbJ6epeM nepBblti MHOrO'1JIeH paBHblM e.aHHI1Ue, Po (x) = 1. BTOpblM
Bb!6epeM npIJMYIO nOJl.xo.a5!lll.ero HaKJIOHa H, KpOMe Toro, C HYJIeM, 6 JIl-I 3-
KHM K cepe.aHHe Hawero MHO)l{eCTBa T04eK. D.aJIbllJe Bbl6HpaeM napa60JIY,
HYJIH KOTOpOti JIe)l{aT no pa3Hbie CTOpOHbl 01' HYJI5! np5!Moti, H T. .a., Ha
K3)1{J1.0M wary Bbl6Hpa5! HYJIH CJIe.aYlOlll.erO MHor04JIeHa TaK, 4T06bl OHH
pa3J1.eJI5!JI!-'Cb npeJl.WeCTBYlOlll.HM MHOr04JIeHOM, a TaK)I{e OCTaB. 5!JII1 MeCTO
.l1.J15! CJleJl.YlOlll.Hx MHOr04JIeHOB. fIoKa Mbl He nblTaeMC5! nOCTpOHTb MHoro-
4JIeH CTeneHH, 6JJH3Koti K CTeneHI1 MHOr04JIeHa, npOXOJl.5!lll.ero 4epe3 Bce
HaWH T04KH (a 31'0 6bIBaeT pe,lJ,KO, eCJII1 6blBaeT Bo06Ill.e), y Hac ecn
J1.0CTaT04HO CB060Jl.bl, 4T06bl BbI6pan KOpHH B YJl.06HbiX MeCTax, CJl.eJIaTb
y.a06HbIMH K03cpcpl1UHeHTbl MHOr04JJeHOB H TeM caMblM 06JIer411Tb 6peMIJ
npOMe)l{YT04HbiX BbI4HCJIeHHti.
OnblT pa60Tbl no 3TOMY MeTOJl.Y nOKa3bIBaeT, 4TO OH BeCbMa 3cpcpeK-
THBeH; 06bl4HO He.al1arOHaJIbHble 4JIeHbi Cl-ICTeMbI ypaBHeHHti 04eHb MaJIbl
11 ypaBHeHHIJ JIerKO pellJaIOTC5!.
Ynpa>EHeHHe 18.3-1. PaccMOTpHTe KBa31-1OpTOrOHaJlbHble MHOrO'lJleHbl
{-I:s;x:s;I), p=l; Po=l; Pl=X 1 ;
1
P2 =x 2 -4;
Pa = X (X2 - 1 9 6 )'
t 8.4. HeMHorotJJleHHblii cJlytJaii
I1HOr,lJ,a 6bIBaeT J1.aHa CHCTeMa cpYHKLI:Hti Ii (x) H Tpe6yeTC5! npH6JIH-
311Tb no MeTOJl.Y HaHMeHbWHX KBa.apaTOB HeKOTopble .aaHHbIe cpYHKLI:Heti I (x)
BI1J1.a
t (x) = aJ (x) +- aJ'2 (x) +-. .. +- antn (x).
256
METOJ.\ HAI1MEHbWI1X KBAJlP ATOB. OP AKTI1KA
[Tn. 18
Llet!CTByeM KaK H paHbWe. 3arIHCbIBaeM 0lllH6KY ci B T04Ke Xi H 06pa-
3yeM CYMMY KBa.n:paTOB. 3aTeM .lLH<p<pepeHUHpyeM rIO K09<P<PI1UHeHTaM
(KOTopble S!BJlS!IOTCS! rIepeMeHHblMH 3a.n:a4H). TIOJJYlJHBlllHecS! HopMa.1JbHble
yp aBHeHHS! <popMaJlbHO rIO.n:06Hbl ypaBHeHHS!M B MHOr04J1eHHOM CJJY4ae,
3a HCKJlI04eHHeM Toro, 4TO TerIepb onpe.n:eJl5IIOTCS!
Sk.j=Sj,k= L.Jj (xi)lk (Xi);
i
T k = 2: I(Xi)lk(XJ
i
ECJlH <PYHKltHH Ii (X) CHJlbHO JlI1Het!HO He3aBHCI1Mbl, T. e. HH o.n:Ha 113
HI1X C e.n:I1HYl'JHbIM K09<P<PI1Ul1eHTOM He MO)J{eT 6bITb BbIpa)J{eHa KaK J1f1Het1-
HaS! KOM6HHaUI1S! OCTaJlbHblX nJIIOC MaJlaS! nOrIpaBKa, TO M01KHO 01KH.n:aTb,
4TO y.n:aCTCH peWHTb HOpMaJlbHbie ypaBHeHHII 6e3 60JlbllHiX 3aTpYlLHeHI1t1.
EcmJ OHH 6J1113KH K Jll-JHet!HO 3aBI1CHMbIM, TO M01KHO I1CnOJlb30BaTb rIOXO-
1KYIO cxeMY opTOrOHaJIH3aUHI1, XOTH 3lLeCb, Bo06IUe rOBopH, He 6YlLer HI1-
KaKoro TpeX4J1eHHOro peKyppeHTHoro COOTHOllleHH5J H KBa3HopTOrOHaJlb-
Hblti MeTO.n: M01KeT OKa3aTbCH Tpy.n:HO 11306pa3HMb1M, eCJlH y Ii (X) MHoro
HY Jlet!.
18.5. HeJIHHeMHble napaMeTpbl
4aCTO CJlY4aeTC5J, 4TO TeOpH5J .n:aeT BI1.n: <POPMYJlbl, KOTOpOt! .lI.OJl}]{HbJ
ylJ,OBJleTBopHTb HeKoTopble .n:aHHbIe, H 9TH .n:aHHbIe .n:OJl}]{HbI HCrIOJlb30-
BaTbCS! .n:JlS! OrIpe.n:eJleHI1S! K09<p<PI1UHeHTOB <POPMYJlbI, npHMeH5J5J Kpl1TepHt!
HaHMeHblllHX KBa.lI.paTOB. Hanpl1Mep, npe.lI.nOJJO}]{I1M, 4TO .n:aHHaS! <popMYJla
TaKOBa:
y(x)=a+be cx ,
(18.5-1)
H CJle.lI.yeT onpe.n:eJlHTb a, b, C TaK, 4T06bl .lI.Jl5J .lI.aHHbIX (Xi' Yi) (i = 1, . ..
..., N)
N
'-, [ + b CXi ») 2 .
...:::.. Yi-(a e =mln=m.
i=I
(18.5-2)
ECJlI1 npo.n:H<p<pepeHuHpoBaTb no a, b, C COOTBeTCTBeHHO 11 npl1paBH5ITb
npoH3Bo.n:Hble HYJlIO, nOJlYlJaTCH ypaBHeHI15J, KOTopble Tpy.lI.HO peWI1Tb.
TIpe.n:nOJlo)J{HM, 4TO .n:aHHble HaHeceHbJ Ha rpa<pHJ{ H npHMepHo onpe-
.n:eJleHO, 4TO C = CI' Tenepb MO}]{HO onpe.n:eJlHTb a 11 b 06bl'IHbIM CIlOCO-
60M. 3aTeM MO}]{HO BbJ'II1CJlHTb m (CI) 113 (18.5-2) .n:Jl5I 9Toro 3H34eHH5I c.
LlaJlee, nonpo6yeM KaKoe-HH6Y.lI.b .lI.pyroe B03MO}]{HOe 3Ha4eHI1e C = Cg
H CHOBa Bbl4HCJlHM m = m (C2)' l1cCJleJJ:OB3HHe 9THX JJ:BYX 3Ha4eHI111
C I m (c)
c i 1 m (CI)
C s m (C 2 )
no.n:CK3)J(eT (no JIIlHet!HOCTH) HOBoe 3Ha C leHI1e C, CKa}]{eM ca'
\9.\]
BBEJJ.EH\1E
257
TaKliM cnOC060M MO)/(HO npH6JU-I3!1TbC5I K MAHAMaJIbHOMY 3Ha4e-
HHIO m (c). lleTaJI!1 CTpaTenJh ;nOHCKa 6y.n:YT 06cy)/(.n:eHbI n03)/(e. 04eBI1.n:HO,
O.n:HaKO, 4TO MbI npO.n:eJIaeM MHOrO 60JJbWe Bb14HCJIeHHtI no CpaBHeHl-IIO
C TeM CJJY4aeM, Kor.n:a Bce onpe.n:eJJ5IeMbIe napaMeTpbI BXO.n:HT JIrmetiHO.
YeM 60JIbWe napaMeTpOB BXO.n:I1T HeJIHHetlHO, TeM 60JIbWe BbI411CJJeHAtI
Tpe6yeTcH, 4T06bI HatlTH np!16JIA)/(eHHe cnoco60M Ha!1MeHbI1l!1X KBa.n:paTOB.
OnbIT nOKa3blBaeT, 4TO, Kor.n:a 411CJIO HeJIHHetlHbIx napaMeTpOB .n:OCTHraeT
4eTbIpex HJII1 I15nll, npou.ecc MO)/(er 6bITb KpatlHe MY4!1reJIbHbIM II Me.n:-
JJeHHbIM.
r JJ A B A 19
MHOr04JIEHbI 4E6bIllIEBA
19.1. BBeJl,eHHe
PH.n:bI cpYPbe I1MelOT MHoro 3aMe4areJJbHbIX cBoMcrB. OTMeTI1M CJIe-
.n:YIOI1l.!1e 113 HI1X:
1. }{a)/(.n:a5I CPYHKll.llH HBJIHeTCH paBHOKOJJe6JIIOI1l.etlCH, r. e. 4epe.n:yfO.
I1l.l1eC5I MaKC!1MYMbl 11 MHH!1MYMbI o.n:I1HaKOBbI.
2. O.n:H!1 H re )/(e aHamIT114eCKI1e BbIpmKeHHH (CHHYCbl !1 KOC!1HYC!>I)
opTOrOHaJIbHbl KaK Ha HenpepbIBHoM, TaK 11 Ha .n:l1cKpeTHoM MHO)/(eC'rH
paBHoOTCTOHI1l.HX T04eK (rJJ. 6).
Yr06bI YB!1.n:eTb, HaCKOJIbKO 3aMe4aTeJJbHO BTopoe CBoMcTBO, paccMoI'-
pl1M MHOr04JIeHbl JIe)/(aH.n:pa P n (x). ECJI!1 nepBbIe m 113 HAX, m = 0, I,...
. . ., n - 1 .n:OJI)/(Hbl 6bITb opTOrOHaJIbHbl Ha .n:!1CKpeTHoM MHO)/(eCTBe
T04eK, TO 4r06bl onpe.n:eJI!1Tb nOJIO)/(eHHe n Y3JIOBbiX T04eK (pa3pema51
n (n-I)
JJI060e AX pacnOJJO)/(eHl1e), nOHa.n:0651TcH 2 ypaBHeHHM
n
2.: Pj (Xi) P k (Xi)=O,
i=1
j =1= k.
He)/(eJIaTeJIbHOCTb :noro y)/(e .n:JIH He CJJI1WKOM 60JIbW!1X n 04eB!1.n:Ha.
8TO He 3 Ia4!1T, KOHe4HO, 4TO HeT CI1CreMbl MHOr04JIeHOB, opTOrOHaJJbHbIX
Ha .n:aHHOM MHO)/(eCTBe, HO 4JIeHbl 9TOtl CHCTeMbl 3aBHCHr, Bo06I1l.e rOBopH,
OT n - 4!1CJIa !1CnOJJb3yeMbIx T04eK. }{or.n:a n CTpeMATC5I K 6eCKOHe4HO-
crn, 4JIeHbi CI1CTeMbl, opTOrOHaJIbHotl Ha .n:I1CKpeTHOM MHO)/(eCTBe T04eK,
CTpeM5ITC5I K COOTBeTCTBYIOI1l.AM 4JJeHaM CI1CreMbI, opTOrOHaJIbHotl Ha He-
npepblBHoM I1HTepBaJIe.
B KJIaCCe opTOrOllaJIbHblX CPYHKll.Htt no.n:KJJaCC opTOrOHaJIbHblX MHoro-
4JIeHOB HMeeT pH.n: OC06bIX CBOttCTB:
1. OHH y.n:OBJIeTBOp5lIOT TpeX4JIeHHOMY peKyppeHTHOMY COOTHOWeHI1IO.
2. \.1x JIerKO BbI411cmHb 11 npeBparu.aTb B CTeneHHble p5!.n:bI.
3. \.1x HYJII1 pa3.n:eJI5IIOT .n:pyr .n:pyra.
9 P. B. XeMMHHr
258
MHO ro'-/JJEHbl '-/EBbIIlIEBA
l rn. 19
MHorO'-!J1eHbl LJe6blllIeBa 06J1aAa1OT BCeMI1 CBOtiCTBaMI1 KaK p5lAOB <pYPbe,
TaI{ 11 OpTOrOHaJ1bHbIX MHOr04J1eHOB, OHM H 51BJI5IIOTC5I, B CYllJ,HOCTM,
cjJYHKUI151MM <PYPbe cos nB, aaMaCKl1pOBaHHhlMH npOCThlM rrpeo6pa30BaHHeM
nepeMellHoti
B = arccos x.
(19.1-1)
Ta[{I1M o6pa30M, MHOr04JIeHhI LJe6bIIfleBa eCTeCTBeHHblM o6pa30M HrpmoT
ywmaJ1bHYIO pOJ1b cpeAI1 opTOrOHaJIbHbIX cjJYHKUl1ti. BbIpa/KeHHe <Pypbe
.lI.J151 opTOrOHaJ1bHOCTI1 CTaHOBHTC5I TaKI1M:
( 0
1t I
cos m B cos n B dB = { ;
o l
(m =j::. n) 1 I
(m = n =j::. 0) = (' T m (x) Tn (x) V dx ;
I J l-x 2
-I
(m=n=O»)
(19.1-2)
(0
N-I I
i ' cos mejCOSn9j=t
(m # n) ) 1 N-I
(m=n"* 0) = j O
(m=n=O)
T m (Xj) Tn (Xj).
3AeCb 0603Ha4eHO *)
Tn (x) = cas (n arccos x)
H I1cnOJlh30BaHa opTorOHaJIbHOCTh cos nx Ha I1HTepBaJ1e (00;;;;; x 0;;;;; 1t)
(CM. 6.7).
DOKa/KeM, 4TO Tn (x) - MHorO'-!JleH. no TeopeMe MyaBpa
cas nB + i sin nB = (cos B + i sin B)n.
Pa3JJara51 6I1HOM, B351B AeticTBHTeJ1bHbJe 4aCTI1 e 06eux CTOpOH U aaMeHUB
'-!eTlible CTeneHn sin B 113
(sin 2 W=(1- cos 2 8)k,
BI1AUM, 4TO cos nB eCTb MHOr04JIeH CTeneW-I n OT cos B. Ho cos (arccos x) = x,
OTCIOAa Tn (x) = cos (n arccos x) een MHorOtweH CTeneml 11 OT X
*) 0603HaQeHHe Tn (x) npOHCXO,lJ,HT OT (jJpaHI.\Y3CKoro HaUHcaHMII (jJaMMJlI1i1
4c6bI111eBa (Tschebycheff).
19.2]
HEKOTOPblE TO)j{J],ECTBA
259
I1pe06pa30Bam1e (1 G.I-I) MO)j{HO paCCMaTpl1BaTb KaK rrpOeKlI.I1IO ne-
peCelJeHI1t1 rrOJIYKpyra C MHO}l{eCTBOM rrp5lMbIX, HMelOIHHX paBHbIe YrJIbI
Me}l{JJ.Y c060tl [pl1c. (19.1-1)]. TaKI1M o6pa30M, MHO)j{eCTBO TOlJeK Xi' Ha
KOTOpOM CI1CTeMa lJe6bIWeBCKHX MHO-
rOlJJIeHOB Tn(x) opTOrOHaJIbHa,TaKOBO:
1t .
X j = cos N J
(j=O, 1, ..., N-l). (19.1-3)
9TO HepaBHoMepHoe pacrrOJIO}l{eHHe,
Y KOToporo Xj crYIHaIOTC5I K 060l1M :c
KOHuaM HHTe p BaJIa ( - 1 x 1 ) , P 19 1 1 4 6
HC. . -. e blIlleBCKHe Y3J10B ble
KOMrreHcl1pyeTc5I B HerrpepbIBHoM CJIY- TO'lKH.
lJae (CM. (19.1-2» BecOBotl cpYHKw1etl
I
f . TaKHM 06pa30M, rrOlJTl1 BbIfJOJJH5IeTC5I CBOt!CHW 2 p5lJJ.OB <PYPbe;
J l-x 2 '
npnWJIOCb TOJIbKO OTKa3aTbCH OT paBHoMepHoro pacnOJIO}l{eHlHI Y3JIOBhlX
TOlJeK. ITOCKOJJhKY MHOrOlJJleHbl Tn (X) eCTb, no cymecTBY, cos nO, TO OHB
TO}l{e 51BJJ5IIOTC5I paBHOKOJIe6JIIOIUl1MI1CSJ cpYHKIU-lSJMI1, n rrOCKOJIhKY OHH MHO-
rOlJJJeHbI, OBB 06JIaJJ.alOT BCeMI1 CBot!CTBaMI1 opTOrOHaJJhHhlX MHOrOlJJIeHOB.
19.2. HeKoTopble TOH<)1.eCTBa
MHorl1e CBotlcTBa MHOrOlJJIeHOB Ye6b1IUeBa CJIeJJ.YIOT fl3 cooTBeTcTBYIOIHI1X
TO)K.n.eCTB .n.JI5I rpHroHOMeTpl1lJeCKI1X CPYHKUI1t1. Hanpl1Mep, TO)K.n.eCTBO
cos (n + I) 0 + cos (n - 1) 0 = 2 cos 0 cos nO
CTaHOBI1TC5I, cor JIaCHO (19.1-3), paBeHcTBoM
T n + 1 (x) + Tn_l (x) = 2xT n (x)
(n 1),
(19.2-1)
KOTopoe 5IBJISJeTCSJ TpeXlJJIeHHbIM peKyppeHTHbIM COOTHoweI-lHeM, COOTBeT-
CTBYIOIHHM (17.1 0-1).
Vl3 TO}l{JJ.eCTB a
cos (m + n) 0 + cos em - n) 0 = 2 cos me cos nO,
nOJIYlJaeM
T m + n (x) + T m _ n (x)=2T m (x) Tn (X).
TICJIara51 m = n, HMeeM paBeHCTBO
T2n(x)=2T (x)-I,
(19.2-2)
KOTopoe HHor.n.a rrOJJe3HO JJ.JISJ rrOJIYlJem-l51 o.n.Horo MHOrO'lJIeHa BblCOKoro
nopHllK a.
g'
260
MHOro4J1EHbl 4E5bllIJEBA
(IJJ. 19
1-13 (19.2-1) JJerKO BbIBeCTH, '-ITO CTapUmt! '-IJleH MHOrOqJleHa Tn (x)
HMeeT BH.Il
Tn (x) = 2n lxn +...
To (x) = 1
(n ?;; 1),
(n=O)
(19.2-3)
H '-ITO 7 n (x) - MHOrOqJleH '-IeTHot! HJlU He'-leTHot! CTeneHH COOTBeTCTBeHHO
TOMY, '-IeTHO HJlI1 HeqeTHO n. DOCKOJlbKY
T n+1 = cas l(n + I) arccos x]
MMeeT npOH3BO.llHym
1 dTn+l _ - sin [(n + 1) arccos x]
n+l - -V I- x 2
TO
1 dT n +1 I dTn_l _ T +I T _I _
n+l - n-l - n+l - n-l -
sin(n+I)6-sin(n-l)6 2cosn6sin6 (19.2-4)
= sin 6 = sin 6 = 2 Tn'
ECJm nepem-rcaTb 9TO paBeHCTBO B BH.Ile
T' T'
-!::...=2T + n-2
n n-I n - 2
(n> 2),
TO npHXO.llHM K BWpameHHHM:
T 2n
2il = 2 (T2n_1 + T 2n _ a +. . . + T 1 ),
T2n+1 +
2n + 1 = 2 (T 2n T2n_2 + T2n_i +. . . + T 0 + 1
(19,2-5)
II.JIS! qeTHOrO H He4eTHoro CJJyqaS! COOTBerCTBeHHO.
YnpamHeHHH
]9.2-1. BblBecTH paBeHCTBO (19.2-5).
19.2-2. \ilcnoJlb3YH (19.2-1), .JJ,OKa3aTb
k T ( ) _ Tn+k +C (h, 2) Tn+k_s + C (k, 2) Tn+k_3 +... + Tn_k
X n X - 2 k .
] 9.2.3. .LI.oKa3aTb
TSk+l (x) - 2T2k-2Tsk_s+2Tsk_4- '" To'
X
19.3. KpHTepHH qeoblmeBa
4e6bIIIJeB rrOKa3aJI, 4TO H3 Bcex MHOrOlfJIeHOB P n (x) CTerreHl-J n co
T (X)
CTapWHM K09rpqmuHeHTOM 1 Y MHOrOlfJIeHa 2 -1' T04H351 nepXH5151 rpaHI>
a6COJl!OTHblX 3Ha4eHI1t! Ha HHTepBaJIe (- 1 X 1) HaHMeHblIJaa. ITo-
II 19.3]
KPI1TEPI1J':I '-IE5b1lliEBA
261
CKOJIbKY SepXH5m rpaHb I Tn (x) \ paSHa 1, YKa3aHHaSl BepXH5!5! rpaHb
I
paBHa 2 n - 1 .
D.oKa3aTeJIbCTBO ,noro 3aMe'-laTeJJbHOrO CBoltCTBa BbITeKaer H3 pac-
CMOTpeHHH pa3HOiTH
Tn (x)
Cfn-l (X) = 2 n - 1 - P n (x),
KOTopaH eCTb MHOrO'-lJIeH CTeneHl-l (n - 1) (TaK KaK lfJIeHbI x n YHI-IlfTO-
lKalOTCH [CM. (19.2-3)]). vb Toro CP aJU a, '-ITO Tn(x) eCTb cos nB, BI-I.a:HO,
'-ITO Tn(x) B I-IHTepSaJIe (-1 x 1) rrpHHHMaeT CBoe 9KCTpeMaJIbHOe
3HalfeHI-Ie n + I pa3, rro o'lepe.a:H rrOJIOIKHTeJIbHbIM I-I oTpHUareJJbHbIM.
ECJII-I 9KCTpeMaJIbHOe 3Ha4eHI-Ie Y P n (x) MeHbllie, lfeM Y tpn-I (x), TO
B 9TI-IX n + I 9KCTpeMaJIbHbJX TOlfKaX <Pn-l (X) ITO OlJepe.a:H nOJJOIKHTeJIbHO
(-j OTpI-IUaTeJIbHO; CJIen:OBaTeJJbHO, Y Hee .a:OJJIKHbl 6bITb n .a:etJCTBI-ITeJIbHbIX
IwpHejj MelK.n:y 9THMU TOlfKaMH. TaK KaK tpn-I (X) HMeeT CTeneHb n - 1,
_ Tn (x)
TO MO)/(HO 3aKJIlOlJl1Tb, 4TO tpn-.l (X) = 0 I-I Pn (X) = .
BTO CBOltCTBO npe.a:CTaBJJf!eT 60JIblliOlt HHTepeC B lJHCJIeHHOM aHaJII-I3e.
ECJJI-I KaKaH-JIv.60 OlliH6Ka MOlKeT 6bITb BblpaIKeH3 MHorOlfJJeHOM 4e6bI-
weBa CTeneHI-I n, TO JII060e n:Pyroe BbJpalKeHI-Ie .n:JJH OWH6KI-I B BH.n:e
MHorOlJJIeHa CTeneHl-I 11, HMelOIlLero TOT )/(e caMbllt CTapWI-I1t K03rpcpI-I-
IJ.UeHT, 6y.n:eT HMeTb Ha HHTepBaJIe (- I X l) 66JJbWYIO M a K C H -
M a JI b H Y 10 OlliH6KY, lJeM lJe6bIIlleBcKoe. B COOTBeTCTBI-II-I C 9THM,
«lJe6hIllieBCKI-IM rrpI-I6JII-IlKeHHeM» Ha3bIBalOT TaKoe *), rrpI-I KOTOpOM cTpe-
MHTCH CBeCTI-I K MI-IHI-IMYMY MaKCI-IMYM olliI-I6KI-I. !1Hor.n:a 9TO Ha3blBa!OT
«rrprlHurJnoM MUHHMaKca». DpH6JJ!1lKeHHe B CMbJCJIe HaUMeHbWHX KBa.a:pa-
TOB YMeHbwaeT cpe.a:HIOIO KBa.n:paTHlJHYIO ollm6KY, HO rrpu 9TOM .a:ony-
CKaeT OTAeJIbHbJe 60JIbllme ollJ!16KH; '-Ie6bJUIeBCKOe - YMeHbUIaeT 9KcTpe-
MaJIbHYIO OUII-I6KY, AorrycKaH 60JJbUIOe cpe.a:HeKBa.n:paTI-IlJHOe OTKJIOHeHue.
B Ka4eCTBe rrpocTolt I-IJIJIIOCTpaUlm paCCMOTpI-IM 33,LI.alJY I-IHTeprrOJl5!UHH
MHorOlfJIeHOM CTerreHI-I n Ha HHTepBaJIe (- 1 x 1). OCTaTOlJHbIfJ lJJIeH
(8.6-1) HMeeT BH.n:
(X-XI) (x-x 2 ) ... (X-Xn+l)y,n+1> (x)
(n+ 1)!
ECJIH Mbl XOTHM MHHI-IMH3I-IpOB3Tb MaKCI-IMaJIbHOe OTKJIOHeHI-Ie 3a clier
MHOIKHTeJIH (x - Xl) (x - x 2 ) ... (X - xn+d, TO HYIKHO Bbl6paTb Y3JJQ-
Bble TOlJKI-I B HYJIHX MHOrOlfJJeH3 Tn+1 (x). TeM CaMbJM MHOIKI-1TeJIb, KOTO-
pbJt! JIerKO KOHTpOJIl1pOBaTb, 6y.n:eT paBHOKOJIe6JIIOIlLI-IMCH MHOrOlJJIeHOM
C H3HMeHblliI-IM MaKCHMaJIbHblM OTKJIOHeHHeM.
*) 4e6bIIIJeBCKOe npH6J1H)j(eHHe He ClIe.n:yeT nYTaTb C Qe6b1meBCKHM HH-
rerpf1pOBaHHeM; cyw.ecrByer .lI.OBOJlbHO MHoro pa3Ho06pa3HblX H.lI.ett, HOCRW.HX
ero HMR.
262
MHOf04JlEHbi 4EEbiWEBA
[W. 19
t 9.4. 3KoHoMH3aUHSI
.apyrOti npocTotl, HO 04.eHb Ba)J(Hhltl npHMep HCrrOJIb30BaHHH 4.eObl-
weBCKHX MHOr04.JIeHOB - npoil,ecc «9KOHOMI13ail,Hl1» *), rrpI1Ha.lJ,Jle)J(auJ,Hfi
B OCHOBHOM JIaHLl.OllIY. ITYCTb JI.aH OTpe30K CTeneHHoro p5UJ,a cpYHKLl.HlI
{(x) = a o + a1x + a2x +... + aNx N (19.4-1)
B HHTepBaJIe (- 1 x 1). .aml CTeneHHoro pH.lI.a OllII10Ka OOWIHO
ohlBaeT "BeJlI1Ka Ha KOHLl.aX HHTepBaJJa H MaJIa B CepeJI.HHe. ITpoil,ecc
9KOHOMI133il,I1H H34.HHaeTC51 c Toro, qTO Mbl, HcnOJIb3Y5I Taom-Iuy
1 = To,
x = (3To+4T + T ),
x 5 = 116 (IOT 1 + 5T 3 + T 5 ),
(19.4-2)
X =T 1 ,
x 2 = ; (To+ T ),
x 3 = (3T 1 + Ta),
npeBpamaeM cTeneHHotl p5lJI. B pa3JIo)J(eHHe no MHOr04.JJeH3:.1 4eObJilleBa
f(x)=bo+blT1(x)+b2T (x)+ ... + b N TN (x). (19.4-3)
2:ho - p33JIO)J(eHHe no opTOrOHaJIbHblM MHOr04.JIeHaM. .aJI5I lUHpoKoro
KJJaCca <PYHKil,HtI pa3JIo)J(eHHe no 4.eOblIUeBcKHM MHOr04.JIeHaM CXO.!J.HTC5!
MHoro OblCTpee, 4.eM no JlIo6ofi JI.pyrofi CHCTeMe opTOrOHaJIbHhlX lHoro-
'IJJeHOB (OOOCHOBaHHe 9Toro CM. B 19.5). TaKHM 06pa30M, Mbl Ha.!J.eeMC5I,
4.TO b k B <popMYJIe (19.4-3) YOhlBalOT MHoro oblCTpee, 4.eM ak B (19.4-1).
.aJI5I HJIJIIOCTpail,HH paccMoTpHM npHMep, .!J.OCTaT04.HO npoCTofi .!J.JI5I
Toro, 4.T06b1 era JIerKO MO)J(HO OblJIO npOCIJHTaTb:
x 2 x 8 x' x
y=]n(l +x) x -2+3-4+5.
YlcrroJ1b3Y51 (19.4-2), rrOJIY4.aeM
I 1 1
y= T 1 -4(T o + T <0+ 12 (3T 1 + T 3 ) - 32 (3To+4T2 + T )+
t. II t 1 , 3
+ 80 (IOT I +5T3+ T 5 )=- 32 10+ '8 7 1 (x)- 8" T (x)+
7 t t
+ 48 T3 (x) - 32 T (x) + 80 T5 (x). (19.4-4)
B HHTepBaJIe (0 x 1) oT6pacbIB3HHe nOCJJe.!J.Hero 4JIeHa CTerreHHoro
p5l.!J.a (19.4-1) JI.aeT H3MeHeHHe Ha 1/5 (.!J.JJ5I x = 1), B TO >Ke BpeM5I
*) B PYCCKOM nepeBO,IJ,C KHHflI JJaH/.\ollla (DpaKTII'JeCKHC MeTO,IJ,bJ npHKJla)1-
HOfO aHa.1H3a, <PH3MarfH3, M., 1961) 9Tor MeTO,IJ, Ha3BaH ne.1eCKOnHlJeCKI1M
CABl1fOM nYTeM rrOCJle)lOBareJlbHOfO cOKpaU.\eHHR . (flpUM. pea.)
'Ii 19.51
MEXAHH31\WUI t1POU,ECCA 3KOHOMH3AU,HH
263
OT6paCbIBal1!1e TpeX 1I0CJJen:HI1X 4JJeHOB (19.4-4) n:aCT H3MeHeHHe MeHbwe
qeM Ha
7 I l 19 I
48 + 32 + 80 = 15.32 < 5,2 '
TaK KaK I Tk (X) [ 1 Ha I1HTepBaJle (-1 x 1). TaKI1M 06pa30M, MOJKHO
npen:CTaBHTh
II 11 3
Y = In (1 +x)R::; - 32 TO+8 TI(x) - 8 T 2 (x)
MHoro'weHOM BTOpOtl CTeneHH B 4e6hlweBcKotl lj:IopMe 60Jlee T04HO, '!eM
COOTBeTCTB YIOill.HM OTpe3KOM CTeneHHOrO pHn:a. Ye6blWeBCKOe pa3J10JKe-
Hue MOJKHO CHOBa npeBpaniTh B MHOr04JleH no Ta6JJ1JIJ,e
To = 1,
Tj=x,
T2=2x2-1,
T 3 = 4x 3 - 3x,
T4=8x - 8x 2 + I,
T = 16x" - 2 x :J + 5x,
(19.4-5)
\1TaK,
+ 11 + 11 3. 0
.JI = In (1 x) R::; - 32 8 x - 8 (2x" - 1) =
I II 3 2
=32+'8 X -T x
(0 x 1).
Boo6rn:e rOBopH, MO}J(HO O)j{l1n:3Th, 41'0 CTenellHot! pHn:, COCT05lrn:11t1 K3
60JlbWOrO '-H1 CJI a 4JJeHOB, npeBparn:eHHblti B 4e6b1weBCKOe pa3JJO}J(eHHe,
n:aeT npU6Jl1jJKeHHe MHOr04J1eHOM 3Ha411TeJJbHO MeHhwet! CTerreHH, TaK KaK
MOJKHO oT6paCbIBaTh MHoro nOc. en:HI1X 4J1eHOB 4e6blweBCKoro pa3J10JKe-
HH5I 6e3 60JlbWOrO yBeJlH4er1!15I OWH6KI1 no cpaBHeHI1IO C TeM, 4TO n:aBaJl
nepBOHa4aJJbHbIt! OTpe30K nenelmoro p5ln:a.
19.5. MexammaUHH npouecca 9KOHOMH3aUHH
Onl1C3HHbIti npouecc 9KOHOMH3aUl1H HCnOJlb3yeT Ta6J1HUbl (19.4-2) H
(19.4-5). ITepBa51 113 HHX - npoCTO XopOWO 113BeCTHoe Tpl1rOHOMeTp114e-
C!we TOJKn:eCTBO
,k ( ei6 + e- 16 ) ,,, I
(cos 0) = 2 = 2k-1 X
[ e'k6+r'k6 e'(k-2' 6 + e-i<k-2' 6 ei<k-4'&+e-i<k-4'& ]
X 2 + C1 2 + C 2 + ... .
r.n.e nOCJIe.n.HuJ:l 4J1eH
1 C7: cos 6,
l C m
'2 fl'
eCJlu k=2m+ 1.
eCJUi k = 2m.
264
MHO r04J1EHbl 4E6bILUESA
[fJJ. \9
MO)!{HO nOCTpOI1Tb 06e Ta61II1L1.bl nOClle,a:OBaTelIbHO, I1CnOlIb3YH paBeH-
CTBO (19.2-1). O,a:HaKo, BMeCTO Toro 'lT06bl HcnOJIb30BaTb Ta6mJlJ,bI,
nO-BH,a:I1MOMY, JIY4111e 3anporpaMMHpoBaTb 9TOT npouecc npocTblM cno-
C060M. O,a:I1H TaKoli cnoc06 OCHOBaH Ha 110,a:CTaHOBKe x = cas 6 B nep-
BOHa'laJIbHblli CTeneHHoli pH,a:
N N
Y= 2.: ak xk = 2.: a k coskO=
k=O k=O
= ao + cos 6 {al + cos6 [a2 +... + (a N _ l + cos6. aN)" .J}.
Ha4lma51 C Tpl1Bl1aJIbHOro pH.lI.a <t>YPbe B KpyrJIblX cK06Kax
a N _ l + aN cos 6
H YMHO)!{a51 Ha cas 6, 110JIY'lHM
I 1
"2 aN + a N _ 1 cos 6+"2 aN cas 26,
T. e. CHOBa p5l.ll. <t>YPbe.
Bo06w.e, eCJIH Ha k-M lllare 6blJI p5l.ll. <t>YPbe
a k) + a: k ) cos 6 + a k) cas 26 + . . . + ak k ) cas k6,
TO, YMHO)!{aH Ha cas 6, nOJIY4aeM
a(k) ( ark) ) ( a(k) a(k» )
-j- + a k) + -+ cos 6+ ++ + cas 26+
a(k)
++ cas (k+ 1)0,
T. e. CHOBa p5l,a: <t>YPbe, HO co,a:ep)]{aw.HIi O,a:HH 'lJIeH 60JIee BblCOKoro
nOp5l,a:Ka.
I1HblMH CJIOBaMIJ, HYJIeBOIi K09cpqJl1L1.l1eHT k-ro lllara BXO,a:H1; Ha
(k +. I )-M lllare KaK CJIaraeMOe B 1<09cpcpHuHeHT npl1 cos 0; Ka)]{,a:blt! K03CP-
cpHuHeHT (KpOMe HYJIeBoro), .lI.eJIeHHblft nOnOJJaM, BXO,a:HT KaK B CJIe,a:YID-
W.Ht!, TaK H B npe.ll.bl,a:YW.Ht! 4JIeH no OTHOllleHHID K TOMY MecTy, r,a:e
9'10'1 K09cp(!JI1UHeHT CT05lJI Ha k-M III are.
Ehot! npOCToft npOue.ll.ypbl ,a:OCTaT04HO, 4T06bl Bbl'lHCJIHTb K09cpcpH-
u,HeHTbl 4e6blllleBCKoro pa3JIQ)KeHH5I 113 pa3JIO)]{eHH5I BCTeneHHot! p5l.ll..
OTCID,a:a BI1,a:HO, 'lTO
aN
b N = 2 N - 1 .
rpy60 rOBOp5l, eCJIIJ paCCMaTpl1BaTb K09cpcpl1lI.HeHTbl KaK MaCCbl, TO
9TOT I1poueCC 06JIa,a:aeT CBOftCTBOM COXpaHeHI151 MaCC.
60JIee .lI.eTaJIbHOe paCCMOTpeHHe 9TorO npouecca nOKa3blBaeT, 4TO b k
,a:JI5I 60JIblllHX k CTpeMHTC5! K HYJIID 6blCTpee, 4eM ak' XOT5I 9TO H onpas-
,a:blBaeT YTBep)]{,a:eHHe, 'lTO 4e6blweBCKoe pa3JIO)]{eHHe 06b1'lHO CXO,a:HTC51
6b1CTpee, 'leM CTeneHHot! p5l.lI., HO He .lI.OKa3bIlJaer ero.
S 19.6J
CMEUlEHHbIE MHO roLfJJEHbI LfEB bIUlEBA
265
BeCb npoJJ,ecc MOiKeT 6bITb BblBeJ1.eH HenOCpeJ1.CTBeHHO 03 peKyppeHT-
BOrO COOTHOilleHI-I5I (19.2-1), 3anl-lCaHHOro B BI-IJ1.e
1 I
xT n ='[ Tn+1 +'[ Tn_I' n> 1,
xTo= TJ, n= 1,
HO Haill BbIBOJ1. nOKa3bIBaeT TaKiKe, KaK OT pa3JIOiKeHI-I5I B CTeneHHot! p5IJ1.
nepeltTH K COOTBeTCTBYIDw.eMY p5IJ1.Y <t>YPbe.
TIepeXOJ1.H Tenepb K npOll.eccy 9KOHOMI-I3aJJ,HH, paCCMOTpHM K09cpqJH-
JJ,HeHTbI b k tfe6bIIlleBCKoro pa3JIOiKemI5I 1-1 OT6poCHM Bce Te CYMMbI, KOTopbIe
MeHbille J1.0nYCTI1Moti Oml16KI1 (BMeCTe C OIllI16KOt! OTpe3Ka CTeneHHoro p5IJ1.a).
lho6bI CHona nOJJytfHTb MHOrOtfJJeH, 06paTI1M OnI1CaHHbIt! BbIIlle npo-
!Iecc. cho 06paw.eHl1e B03MOiKHO, TaK KaK CTapmHt! tfJIeH B03HI1KaeT
TOJJbKO B OJ1.HOM MeCTe npeJ1.b1J1.YIJJ,eti CTpOKH. OTIJpaBJI5I5ICb OT CTapillero
tfJJeHa (k + 1 )-ti CTpOKI-I, MbI MOiKeM BbItfl-lCJJI-ITb CTapml-lti tfJIeH k-t! CTpOKI-I.
3aMeIHM, 1JTO 9IOT nOCJIeJ1.HHt! tfJIeH BJIH5IeT Ha BTOpOti C KOHJJ,a tfJJeH
k-J! cmJ7{1/(a
C1:
(k+ 1)-i/ cmJlOJ(a
PHC. 19.5-1. flpou.ecc nepexo.aa K MHOrOIJJleH-
HOMY BH,IJ.Y.
(k + I )-ti CTpOKH. B03bMeM 3aTeM COCeJ1.1IHti tfJIeH C MeHbIllI-IM HOMepoM
H npOJ1.BHHeMC5I K HatfaJIY CTpOKI-I (pHC. 19.5-1). TIoCJIeJ1.HHt! mar onpe-
J1.eJII1I COOTBeTcIBYIOW.Ht! K09cpcpHJJ,l-IeHI CIeneHHoro p5IJ1.a a k). KaK Ha
np5IMOM, TaK 1-1 Ha 06paIHoM Illare cYMMa K09cpcpl-lJJ,l-IeHTOB COXpaH5IeTC5I
1-1 9IO l\lOiKeT CJIYiKI-ITb KOHTpOJIeM B KOHJJ,e, nOCKOJJbKY
N N
ak = b k .
k O k=O
TIoCJ1eJ1.Hee paBeHcIBo JIerKO nOJIYtfI-lTb, nOJIaraH x = 1 (T k (x) = 1).
vITal<, npouecc 9KOHOMI13aUI-IH JIerKO MeXaHI-I3HpOBaTb 6e3 1-ICf10JIb30-
BaHI15I 06Illl-lpHbIX Ta6J1l-1u.
YnpaJlmeHHe 19.5-1. HapHcYHTe 6J1oK-cxeMY npou,ecca 3KOHoMH3au,HH.
19.6. CMew.eHHbIe MHOrO'lJIeHbI qeobImeBa
YaCTO YJ1.06HO BMeCIO I-IHTepBaJIa - 1 :os;;; x:os;;; 1 I-ICnOJIb30BaTb muep-
Ba,'] O:os;;; x:os;;; 1. LlJJ5I 9TOt! ueJIH ynoTpe6JI5IIDICH CMew.eHHbIe MHoro'.weHbi
Ye6bIIlleBa
T (x) = Tn (2x - 1).
(19.0-1 )
266
MHOTO'IJ1EHbI 'IEBblIlJEBA
[fJJ. II}
V1MeeM
n(x)= 1,
T: (x)= 2x - 1,
n(x)=8x 2 - 8x+ 1,
n (x) = 32x 3 - 48x 2 + 18x - 1,
T: (x) = 128x 4 - 256x 3 + 160x 2 - 32x+ I,
Ii
= TJ,
- I (T * + T * )
X -2 0 l'
X'l = (3n+4Ti+ Ti),
X 3 = 3 (lOTJ'+ 151':+61':+ Tn
I
X4 = 128 (35n + 56Ti + 28T + 81': + Tn.
. . . . . . . .
MMeIOTC51 60JIee nO.lJ,p06Hble Ta6JIHUbi *).
LlJl51 cMew.eHHbiX MHOr01.JJleHOB 4e6hlllieBa CnpaBe.lJ,JlHBO peKyppeHT-
Hoe COOTHOllieHl1e
I1JiM
T +I (x) = (4x -2) r;: (x) - 1';_1 (x), n = 1,
XT;(X)={- 1';+1 (X)+-} T (x) + 1';_1 (x),
( 19.6-2)
r,l/,e
T (x) = cos [n arccos (2x - 1)] = 7 n (2x - 1).
( 19.6 3)
YnpamHeHH5I
19.6-1. npl1MeHHTe CMem.eHHble MHOrOTJJleHbl r K npHMepy 19.4. (j],JlR
SToro npl1J\eTcR npoJ\OJDKHTb Ta6J1HUbl ri; (x) H x n Ha OJ\HY CTPOKY.)
19.6-2. flpOCJlcJJ,HTe MexaHH3aUHIO (npou.ecca 9KoHOMH3au,HH) JJ,JlR CMem.eH-
HbIX illHOrOlJJleHOB ri; (x), KaK B 19.4 [HClIoJlb3yfiTe paBeHcTBo (19.6-2)].
19.6-3. HaHJJ,HTe lJ,JlR CJ\BHHYThIX MIIOrOlJJleHOB paBeHcTBa, 1I0J\06Hble (19.2-5).
19.7 T-npouecc J]aHlJ,owa
Ellie O.D.HH npneM nCnOJlb30B3HH51 MHOrOt!JleHOB l1e6hlllieB3 npe.D.JiO)l(HJJ
Jl31!UOIll. B OCHOBe 't-npouecc3 .D.JJ51 pellieH!:Hl JIllHetiHblx .lJ,HcpcpepeHI1HaJJb-
HhlX YP3BHeHHti C nOJinHOMn3JibHbiMn KoscpcpHuneln3Mn Jie)l(wr npoCT351,
*) Tables of Chebyshev Polynomials Sn (x) and C n (x), Nat!. Bur. Standards
(U. S), Appl. Math. Sl!ries 9, 1952.
19.7J
t-npOUECC J1AHUOillA
267
HO Ba)f{IIa51 H.lJ,e5l, 06hJlIHO 6bIBaeT TPY,lI,HO HatiTH OIlll'l6KY Bbl'-lI'lCJIeHIJOrO
peIlleHH5I 3a,ll,a'-lH, HO Ha 06paTHbiti BOnpOC: «,lI,Jl5I KaKOt! 6J1113KOti 3a,ll,a'-ll'l
TOJIbKO '-ITO Bbl'-lYlCJIeHHblt! OTBeT 6blJl 6bl TO'-lHbIM perneHl1eM?» - OTBe-
TI1Tb '-IaCTO HeTpY,lI,HO. Eho - O,ll,I1H 113 OTBeTOB Ha '-IerBepTblt! OCHOBHOt!
BO[]POC 7.1 «'imo' ecmb mO'i/iOCmb ?», H BO MHOl'HX CJIY'-I851X 9TO-
npaBH.'lbHblH OTBeT.
I.ho6bl npOI-I.'1.'1I0CTplIpOB8Tb '1:-Mero,ll" []pe,ll,[]OJIO)f{HM, '-ITO ,lI,aHO 06b1K-
HOBel-iHOe ,lI,l'Icj)(pepeHlJ.Ha.'lbHoe YP8BHeHlIe
y'+y=O;
y(O)= 1,
1I perneHHe 1Irn.eTC5I B BlI,lI,e MHOrO'-lJJeHa, T. e. Mbl H8,l1,eeMC5I 060PB8Tb
CTeneHHot! p5l,ll" KOTOpblM Bblp8)f{8eTC5I perneHYle. O'-leBH,lI,HO, HI1K8KOti
MHOrOlJ.JIeH He 6Y,lI,eT TO'-lHO Y,lI,OBJleTBOp5lTb 9TOMY ypaBHeHI1IO. P8CCMOTpl1M
M8, oe 113MeHeHlIe npaBoti 1:J8CTI'I, np1I68BHB J( Hetl MHoro'-lJJeH 4e6bIIlleBa,
<J603HalJ.851 '1epe3 't Be.'II11:JI'IHY M8KCI'IMaJIbHOrO H3MeHeHH5I
y' + y='tT (x).
4T06bl JIerKO 6blJlO npOCJJe,ll,I'ITb Bce Bbl1:JlICJleHI'I5I, n,pJIO)f{lIM n = 4 1I
nonpo6yeM HatiTH MHOrO'-lJleH
y = a + bx + cx 2 + dx 3 + ex!,
npl'lpaBHHBa51 K09cjJcjJHlJ,l'IeHTbl []PH O,ll,HHaKOBbiX CTe[]eH5IX x, []OJlY'-IaeM
b+a='1:,
2c+b=- 32't,
3d + c = 160'1:,
4e+d=- 256'1:,
e= 128'1:.
Ha'-l8J1bHOe YCJIOBl'le ,lI,aeT a = 1. OTCIO,lI,a []OCJIe,ll,OB8TeJIbHO nOJlY'-IaeM
b='t-l,
1-33'1:
c= ,
d - ( 160 + 33'1:-1 ) _ 353'1:-1
-3 '1: 2 - 6 '
1 ( 1 - 353'1: ) 1
e=T \ - 256'1: + 6 = 24 (1 -1889,,),
e= 128'1: = 2 1 4 (1 -1889'1:), '1:= 4 61 '
vhaK, o1lIlI6Ka, KOTOpa51 6blJIa BHeceHa B [JepBOHalJ.a, bHOe ypaBHeHHe,
paBHa
I n(X) 1 I
4961 4961 '
268
MHOrO'-lJJEHbJ '-IE6bJWEBA
[W. 19
M n HOBoro yp8BHeHHH HMeeM T04Hoe perneHHe
4960 2464 2 768 3 128 4-
Y = I - 4961 x + 4961 x - 4961 X + 4961 X .
CpaBHllM 91'0 C pemeHHeM B BJi1 e CTenellHoro pH a
x' X S x 4
y= I-x +2--6 + 24 '
y KOTopora M8KCHM8J1bH8H omHoK8 npl1MepHO
1 1
51= 120 '
B HameM cJlY'Iae JlerKO Oll.eHllTb OmI10KY, npol1cxo 5lI1I.YIO OT 06aBJIeHI1
lJeOblmeBcKoro 4J1eHa B l1cjJ<pepeHlI.HalJbHOe ypaBHeHl1e,
y' + Y= 'tTl(x),
x
y (x) = e- x . 1 + 'te- x r:- (6) e$ dB.
o
IlJJ5I 0 X ] cnpaBeJlJJllBO HepaBellCTBO
x x
\Y - e- x \ "s; I 'te- x Tt (6) e 8 d6 I 'te- x e$ d6 = 't(1 - e-.t)
o 0
0,665 =-- 3 X -4.
"s; 4961 """,1,4 .10.
1
nonYlJeHHOe 311a4eHHe Mlloro Mellbme 51'
Mbl npl1BeJII-J nHmb npOCTol-1 npl1Mep 't-MeTOJla. B KHHre J1aHlI.Oma [23)
cOJlep)KHTC5I rOpa3JlO oOJlee ml1pOKOe ero 113JIO)KeHl1e. O HaKo oeHOBHa51
H eH Ta )Ke: Mbl clJaoo MeH5IeM yelJOBl1e 3aJlalJH H nOJlYlJaeM TOlJHOe
pemeHl1e :nol-1 HOBOI-1 3a alJH. TIOCKOJlbKY B 9TOM clJYlJae MeHHIOTC!J
YCJIOBI1!J nepBOHalJaJIbHol-1 <pH3l1lJeCKotl 3aJla4H, HaM 91'0 OObI'IHO JlerlJe
HeTOJIKOBaTb, lJeM KaKoe-HHOYJlb npH6JJH)KeHHe, eJleJlaHHOe B xOJJ.e
4HCJleHHOrO pemeHI1H. 't-MeTOJl nOKa3blBaeT, KaK BalliHO aKKypaTHo OTBe-
THTb Ha 4eTBepTbltl oellOBllol-1 Bonpoe: «limo ecmb mO'i/iOctnb»?
Ynpa}/{HCHHe 19.7-1. OpuMeHHTb 't-MeTO,l{ K ypaBHeHHIO
xy" + y = 0; y (0) = 0, y' (0) = 1
JlMI (0 X I), nOJlaraJl n = 4.
19.8. BHIIon3McHcHHC T-MeTOJl,a
TIpe [10JlOlliHM, 41'0, BMeCTO Toro 4TOObl JletlCTBoBaTb KaK J1aHQOIll,
MI,l nblTaeMe51 pemJi1Tb 3aJlalJY B JlOO H npeJJ.CTaBJIHeM pemellHe Al1cpcpe-
peHlI.l1aJlbHOro ypaBlleHHH (e nOJII1110MHaJlbllblMI1 K09<PCPI111.I1eHTa H1) B BHAe
pH a no MIIOrO'-IJIeHaM 4eOblmeBa
ro
y (x) = 2: ak Tk (x).
k=o
(19.8-])
II 19.81
BVI,a0113MEHEHVlE <-METO,aA
269
Tor,aa
00
y' (x) = ak Tk (x),
k=O
Ii MO/KHO nCnOJJb30Ba1'b (19.2-5),
co
y' (x) = ".[:,/a k [2 Tk_l + 2 Tk_3 + ... + (2il) ]*) (19.8-2)
Ii TaKHe /Ke Bblpa/KeHn!! ,aJI!! c1'apwnx npon3Bo,aHbIX, '1T06bl HCKJIIO'IH1'b
Bce npOH3BO.ll.Hble.
3a1'eM HCnOJJb3YI01'Cf!
paBeHC1'Ba (19.2-1)
I
x Tn (x) = 2" [Tn+l (x) + Tn_l (x)]
(19.8-3)
11 1'aKne /Ke Bblpa/KeHI1f! ,aJI!! xkT (x) (CM. ynpa/KHeHHe 19.2-2), '1T06bl
HCKJIIO'IH1'b Bce CTerreHfJ x. TaK!1M 06P330M, Mbl fIp!1XO.ll.!1M K pH.lI.Y fIO
'1e6blWeBcKilM MHOrO'lJJeH3M Tn (x). MeTO.ll. np!1p3BHHBaHHH K03(pqJl1uneH-
TOB fIpH 0,aHH3KOBbIX CTeneHf!)C X B 06e!1x '1aCTHX 1'O/K,aeCTBa no x
OCHOBbIBaeTC5I H3 TOM, '11'0 p33Hbie CTerreHH x JJHHettHo He3aBHCHMbI.
Pa3 Y/K npOH3BO.ll.Hbie H '1JIeHbl C x HCKJIIO'1eHbl, T. e. I1MeeTC5J pa3JJO-
/KeHHe JJnlllb no Tn, TO MO/KHO TaIOKe npl1paBH5J1'b K03cjJqJl1UHeHTbi fIpn
O.ll.HHaKOBblX 4e6blllleBCKHX MHOr04JleHaX. CJIe.ll.OBaTeJIbHO, MO/KHO ,aett-
CTBOBaTb TaK, KaK eCJII1 6bl ilMeJJH .lI.eJIO C pa3JIO/KeHHeM peWeHH5I B CTe-
neHHot1 p5l.ll.. 060pBaB P5l,a, fIOJIY4HM oWil6KY, fIp!1MepHO paBHYIO fIepBOMY
OT6pollleHHOMY MHOr04JIeHY Ye6b1llIeB3.
Mbl MO/KeM C,aeJIaTb 31'01' 060PB3HHblf! p5J,a TO'1HbIM peWeHI1eM, npil-
6aBJI5JH K np3Bot1 4aCTil 1'aKHe MHOr04JIeHbi Ye6b1WeB3, 4T06bJ BCe '1JIeHbJ
YHH4TO/KHJIHCb. Pe3YJIbTaT 6Y.lI.e1' TOT /Ke, '11'0 H B OnHCaHHOM paHee
't-Me1'O.ll.e, HO 31'01' MeTO,a 60JIee rH60K B BOnpOCe 0 TOM, r,ae 060pBa1'b
peWeHl1e.
MeTO.ll. OnHCaH B 1'epMI1HaX CT3H.lI.apTI-IbIX MHOrO'lJIeHOB Ye6blllieBa;
TaK /Ke XOpOWO 31'0 MO/KHO C.lI.eJI3Tb .lI.JI5J cMew.eHHbiX MHOr04JIe-
HOB n (x).
.fLBe cjJOpMbl o,aHOrO H TOrO /Ke 't-MeTO.ll.a npHBe,aeHbI, lJT06bl nO.ll.4epK-
HYTb TO 06C1'OHTeJIbCTIJ0, 41'0 eCJIH HY/KHO fIOJIY4HTb OWH6KY B BH,ae
MHOr04JIeHa Ye6b1llIeBa, TO, nO-BH,aI1MOMY, JIY4We H a4HHaTb C npe,aCTa-
BJIeHH5I peWeHH5I B BH.lI.e p5J,aa no MHOr04JIeHaM Ye6bIweBa, 3 He CMeWH-
BaTh 06bIKHOBeHHbIe H 4e6bIWeBCKI1e MHOrOlJJIeHbI. HY/KHble TO/K.lI.eCTBa
Me/K.lI.Y MHOr04JIeHaMil JIerKO Hat1TH.
*) CI1MBOJI (2 1) B aHHOM cJlY'Iae 03Ha'laeT, '1TO nOCJle)J.HHM CJlaraeMblM
MOJKeT OKa3aTbCI! JlH60 2T 1 (npl1 '1erHOM k), JlH60 1 (npH He'leTHOM n).
(Dpu)oI. pea.)
270
MHO r04JJEHbl 'tEBblUJEBA
! L1I. 19
YnpamHeHl-l\1
19.8.1. vl3nO)!{HTe BH)l.OH3MeHeHH blfi 'MeTOJJ" HcnoJlb3Yil CMeIlleHHble M1!OrO-
'IJlCHbJ 4e6b11IleBa T (x).
19.8-2. OOKa3aTb, '1TO
i Tn (x) dx = -} [ T +(. ) - T -. .5 ) ] (n> 1),
To (x) dx = T( (x), Tl (x) dx = T 2 (x),
T. C. 'ITa H1UerpHpOBaHYle MOMeT 6bITb BbllIOJlHeHO HerrocpeJJ:CTBeHHO B &lHoro-
'1JIeHax 4c6blmeBa.
19.8-3. flpoBecTH Bbl'lHCJleHHJJ JJ:JlJJ rrpl1Mepa y' =y 19.7 rrpJJMbIM -, eTO-
HaM, KaK B 19.8.
19.8-4. OpHMeHHTb npRMofI -MeroJJ, K ynpa)/{HeHHIO 19.7-1.
19.9. HeCKOJIbKO 3aMeqaHlIM 0 qeObIWeBCKOM npllOJIIUKeHlIlI
Ye6blweBCKoe rrpl16JIH)j{emle BbI3BaJJO HeKoTopblt! rreperroJlox Cpe.lUI
Tex, KTO 3aHI1MaJJCH BbI'IHCJJeHHeM Ta6JJHu" r.n,e Ba)j{HO JJHillb TO'IHO 3HaTb
MaKCMMalIbHY!O OIJJI16KY. TaK, BMeCTO Taro 'ITo6bI I1CITOJIb30BaTb fJ,JIiJ MH.
TepnOJJHI.UHI MHor04JIeHbl, npOXO.ll.iJIlJ,He 'Iepe3 3a.lJ:aHHble T04KH, 6bI.ir0
npeJJ.JIo)j{eHO JJ.eJ!aTb Ta6JII1u,bl, I1CnOJlb3Y!O!lJ.l1e .lI.JlH HHTepnOJIiJUl111 4e6bl-
weBCKHC MHOrO'lJIeHbl.
Ye6blilleBCKHe npI16J1H)j{eHVIH VI npH6J111}KeHI1H no Hal1MeHbWHM KBaJJ.-
paTaM MorYT 6bITb 06'be.ll.HI-IeHbl B paMKax O.ll.HOt! HJJ.eH. Ilpe.n,IToJJO)l(!1M,
qTO Mbl MI1HHMI1311pyeM <PYHKUH!O
N I
mink = ( I ci Ik)T.
i=l
II.JJH k = 2 9TO - MeTOJJ: Hal1MeHbWI1X KBaJJ.paToB, JJ:JIH k = 00 - MeTO.ll. LIe-
6blweBa. choT <paKT npHBoJJ:HT K JJ:pyrot! T04Ke 3peHHH Ha 4e6blweBcKoe
npI16JHIJKeHHe. HatlJJ:eM CHa'IaJIa npH61IH)j{eHHe MeTO.lJ.OM HaHMeHbWHX
KBa.lJ:paToB. 3aTeM, HcnOJIb3Y5I KBaJJ:paT OIJJH6KI1 KaK BecoBYID <PYHKUI1!O,
CHOBa Hat!JJ:eM npI16JIH)j{eHI1e no HaI1MeHbWI1M KBaJJ:paTaM. LJeTBepTYIO
CTeneHb HOBOt! OWH6K11 HcnoJlb3yeM IWI{ BeCOBY!O CPYHKUI110 H nOBTO-
PUM Bce CHOBa. TaKuM cnoco6oM Mbl nOCTeneHHO npH6JIH3UMCH K moo'
Pa3JIo)j{eHl1e no MHOrO'IJIeHaM Ye6blweBa .n:aeT B3BeweHHoe npH6J1H-
)j{eHue no Hal1MeHbWHM KBa.ll.paTaM. ECJIH 9TO pa3JIO)j{eHHe CXO.ll.HTCH .n:o-
CTarO'IHO 6bICTpO, ro nepBbltl oT6poweHHblt! '1JIeH MO)j{HO paccMarpHBaTb
KaK OWI16KY 4e6blUleBcKoro npH6JIu)J(eHHH. B JJI060M cJlY4ae TaKoe pa3-
JIO)j{eHue .n:aeT npH6JJI1)j{eHHe B 4e6blweBcKoM CMblCJIe (CM. 19.3).
19.10. KpuTepHH COBIJaJI,eHHH MOMeHTOB
II.0 CI1X nop Mbl paccMaTpHBaJJH
MHor04JIeHOB KOHKpeTHot! <PYHKUUH
HOtl B aHaJIUTU4eCKHX onepaUHHx:
TpH KplHepH5I BbI60pa U3 KJlaCCa
.lI.JIH UcnOJIb30BaHUH ee BMeCTO AaH-
19.101
KPVlTEPVli1 COBTIA.!I.EHI15I MOMEHTOB
271
I) C03naap,HUe 8 Y3AOBblX mOI{fcax,
2) HatU,teH'bUlue lCBaapambt,
3) 'fe6blt/ieBClCOe HJlH .MUHll/iWlCCHoe.
Eu e OJU1H KpllTepHt!, KOTOpbIt! Y)I{e 6bIJI HCrrOJlb30BaH B 2.9 H KOTO-
pbIt! I1lHpOKO rrpHMerIHeTCH B CTaTHCTHKe, - STO rroJI.60p rrpU6J1H)I{3IOIllet!
cj:JYHKaHH no COsnaJI.eHHIO HeCKOJlbKI1X MOMeHTOB. B CTaTHCTHKe yrrOTpe6-
'JIHIOTCH 06bI'-IHO aeHTpaJlbHb!e MOMeHTbI (KpoMe nepBoro), HO eCJ!H Bce
MOMeHTbI .n:o k-ro COBna.n:aIOT, TO H aeHTpaJIbHbIe MOMeHTbI TO)l{e COBna-
.n:aIOT, TaK '-ITO MbI 110J1Y'1I1M O.n:I1H H TOT )l{e' pe3YJlbTaT, I1Cn0J1b3YIOTCH J1H
«MOMeHThI» HJIH «aeHTpaJlbHWe MOMeHTbI», XOTSI OIlH16KI1 oKpyrJleHHSI MO-
rYT 6bITb COBceM pa3HbIMH.
Eha KOHIJ.errmlH rrpe.n:CTaBJJHeT BeJlI1KOJlerrHwt! rrpHMep cnoco6a BbJ60-
pa KpvlTepl1SI. B CTaTHCTHKe MOMeHTbI HcnOJIb3YIOTCSI '-Iano; OTCIO.n:a HC-
HO, rrO'-leMY COBrraJI.eHue MOMeHTOB rrpH rrOJI.CTaHOBKe OJI.HOt! cj:JYHKUI1H
BMeCTO .n:pyrot! eCTb, B pOSITHO, XOpOIJlO BbI6paHHbIt! KpHTepHt! .n:JlSI CTa-
THCTH'-IeCKHX 3aJI.a'-l.
B Ka'-leCTBe rrpHMepa MeTOJI.a COBrra.n:eHHI1 MOMeHTOB paCCMOTpHM arr-
rrpOKCHMaUHIO cj:JYHKll.I1H
x 2
Y (x) = , e - 2a 2
a I" 211:
KBa.a.paTHWM rpeX'-IJleHOM. KaK rrpaBHJJO, y (x) rrpe.n:CTaBJIl1eT paCnpe.lle-
JIeHHe BepO!!THOCTeti H, KOHe'-lHO,
00
y (x) dx = 1.
-00
EcmI B3HTb .lI.Ba rrepBbIX 'WeHa pa3JJO)l{emUi B CTerreHHOt! pH.lI (TOJlbKO
TaM, r.n:e napa60JIa BhIIlJe OCI1 x), TO nOJJHa!! BepOSITHOCTb rrpH6J1H)I{eHHH
He 6bIJJa 6bI paBHa 1.
TaKI1M 06pa30M, MbI .n:OJ!)I{Hbl OTKa3aTbC!! OT rrpH6JUI)I{eHHI1 CTeneH-
HbIM pSIJI.OM 11 BbI6paTb rrapa60JIY BI1.n:a
Yo = A (1 - b 2 x 2 )
(-+ x +)
no COBrra.lIeHHIO MOMeHTOB.
MOMeHTbI rrepBOHa'-laJlbHOrO pacnpelIeJJeHHI1
+co x 2
1 \ k - 2a2
a V2n x e dx= mk
-co
HalOT
mo= 1, ml=O, m'1.=a'1..
272
P AlJ,110HAJ1bHbIE <!>YHKlJ,1111
[fn.2O
LlJI5I npH6J1H)I{eHH5I 110JIy t weM
Ink = A I Xii (l - b"x )dx=A ( : Il -b : ) I 'h =
-l/b
2A [I + (_I)k]
= b k + 1 (k+ I) (k+3)'
OTCI011a
4A
/no = 3b '
4A
m! = 0, m =l57J8;
CJlellOBaTeJlbHO,
4A
1 = 3b '
4A
Cj = J"5b3".
OTCI011a
I
b = ;Y5 '
3
A= 40-V5 '
vhaK, napa60Jla
Ji = 4cr} 5 r 1 - ( 5 : n
npH6J1H)I{aeT HOpMaJlbHOe pacrrpe.n.eJleHHe, cOxpaHSISI HYJleBOR, nepBbIR H
BTOpOti MOMeHTbI.
rJ1ABA 20
PAUHOHAJIbHbIE <l>YHKU m
20. t. BBe!l.eH \-Ie
11.0 CHX nop npe11I10JJaraJJOCb, '-1'1'0 Me)l{.!I.Y Y3J10BI.IMH TO'-lKaMH CPYHK-
IJ.\-ISI Be.!l.eT ce651 KaK MHOrO'-lJJeH. B 9TOti r JlaBe MI.I 6Y.!I.eM npe.!l.nOJlaraTb,
'-1'1'0 CPYHKIJ.\-ISI Be.!l.eT ce6SI KaK OTHOIlIeHHe .!I.BYx MHOrO'-lJleHOB, 06bI'-IHO
Ha3bIBaeMoe pal1,uoHallbHoii tjJYHltl1,ueii.
PaIJ.HOHaJJbHa51 CPYHKIJ.H5I 06J1a.!l.aeT cpYH11aMeHTaJlbHbIM CBOtlCTBOM oc-
TaBaTbC51 paUHOHaJlbHoi1 npH nepeHoce H paCT51)1{eHHI1 He3aBI1CI1Mot! nepe-
MeHHotl. PalJ.HOHaJlbHble CPYHKIJ.HH 06J1a.!l.a1OT pSI.!I.OM CBOtlCTB, '-IaCTO Tpe-
6yeMblx OT npH6J1I1)1{alOUJ,efi CPYHKIJ.I1H, KOTOpofi MbI co6l1paeMc5I 3aMe-
HS!Tb .!I.aHHYIO B aHaJlI1TH'-IeCKHX onepalJ.l151x. HaI160Jlee Ba)l{HO, 4TO pa-
!.I.HOHaJlbHbIMI1 CPYHKUJ151MI1 MO)l{HO rrpI16JII1)1{aTb TaKI1e, KOTopble I1pI1HI1-
MalOT 6eCKOHe'-lHbIe 3Ha4eHl-I51 11J151 KOHe4HbIX 3Ha4eHHtI aprYMeHTa.
Llpyroe CBOt! CTBO - TO, 4'1'0 I1MH MO)l{HO npH6J1I1)1{aTb np5lMble JlH-
HI1H, r JJaBHblM 06pa30M OCb Ox .!I.Jl51 60JlbIlIHX 3Ha'-lellHtI x, qero lIeJlb351
C.!I.eJlaTb HeTpHBHaJIbHblM MHorOTweHOM. PalJ.HOHaJlbHble CPYIIKIJ.HH JlerKO H
20.21
HErIOCPE.D,CTBEHHhIt% rIOJJ.XOJl
273
6bICTpO C1.Jt1'ralOTC5l BbJ1.JHCJII-ITeJJbHOt! MaWHHOt!. f103TOMY nx 1.JaCTO HC-
nOJIb3YIOT, t-IT06bI nOJIY4HTb npH6JIH)J{eHHe ,n:JJ5I BbI4HCJIeHI15I 60JIee CJIO)J{-
HOt! <PYHKI1.\,m. llJI5J 3THX I1.eJIet!, Bep05JTHO, HY)J{eH 4e6hIWeBCKHt! THn MH-
HHMaKCHbiX npH6JII1)J{eH\1t!.
Teopl151 annpOKCI1MaI1.I1H paI1.HOHaJJbHbIMH <PYHKI1.\151MH HaXO.n:HTC5I B 3a-
nYTallHOM, HO 6bICTpO pa3BHBaIOllJ,eMC5I COCT05IHHO; Mbi ,n:a.n:HM 0 Het! TOJIb-
KO C3MOe 06llJ,ee npe,n:CTaBJIeHHe. O,n:Ha H3 npH4HH, no KOTOpOt! Mbi He
6y.n:eM CJIHWKOM r JIy60KO 3axo.n:OTb B Hee, - Ta, 4TO cpe.n:HeMY BbI4I1CJIf1-
TeJIIO e,n:Ba JII1 npH.n:eTC5I c.n:eJIaTb B ro,n: MHoro paUHOHaJIbHbiX npH6JIVl-
)J{elJI1H, \1 60JIee 3<p<peKTHBHbIt! MeTO,n:, n03BOJJ5IlOI1I.Ht! C6epe1.Jb HeCKOJIbKO
ceKYH,n: pa60TbI MaWHHbI, nO-BH,n:HMOMY, He CTOHT B 9JIeMeHTapHbiX CJIY-
Q35IX 3aTpaTbi BpeMeH\1 BbJ4\1CJI\1TeJI5I.
20.2. Henocpen.cTBeHHblii non.xo)l.
f1YCTb ,n:aHa <PYHKUH5I, no KOTOpOtt MO)J{HO BbI4HCJl5ITb Y3JJbI (Xi, Yi)
1MH CaM\1 Y3JIbi H Tpe6yeTC5I no,n:06paTb no 3T\1M .n:aHHbiM paUHOHaJIb-
BYIO <PYHKLI.\1lO,
N(x)
y=/(x) D (x) ,
r.n:e N(x) 11 D (X) - MHOr04JIeHbi. CHa4aJIa peWHM, KaKoro BH,n:a MIIO-
r01.JJIeHbi CJJe.n:yeT HCKaTb. f1YCTb B 4aCTHOM CJIY4ae HaHeceHHe Ha rpa-
<pHK T04eK (log Xi> JOgYi) ,n:aJIO ,n:JI5I MaJIbIX Xi> 0 HaKJIOH, npHMepHO
paBHbIt! 3, OTKy.n:a MbI 3aKJJI04HJJH, 4TO Ha.n:o nonp060BaTb
_ /( ) GX 3 + '"
y- X 1+...
ILml 60JIbIllHX X T04KH (Xi' Yi) JJerJJH npHMepHO Ha np5IMYIO C nOJIO-
)J{HTeJJbHbIM HaKJJOHOM. Eho 3Ha411T, LITO CTeneHb N (X) ,n:OJI)KHa 6bITb Ha
1 60JIbWe CTeneHH D (x). 0603H34HM CTeneHb D (x) 4epe3 k. Tor.n.a
3 + 4 + + k-t-I
_/( )_ G 3 X G 4 X ... Gk+1 X
y- X - l+bJx+b2x2+...+bkXk'
(20.2-1 )
OCTaeTC5I BbI6paTb nOKa3aTeJIb CTeneHH k. Tpy.n:HO CKa3aTb, KaK HMeHHO
BbI6I1paTb k; eJUIHCTBeHHOe, 4TO MO)J{HO nOCOBeTOBaTb - npI1H5ITb BO BHH-
MaHlIe nOJIY4aIOlUeeC5I 4I1CJIO napaMeTpOB OTHOCHTeJIbHO «CJJO)J{HOCTH»
(cy6'beKTHBHO) .n:aHHbiX.
Kor,n:a k Bb16paHO, CTaHOBHTC5J H3BeCTHbiM 4HCJIO napaMeTpOB. 3aMe-
HIM, 4TO MO)J{HO <pHKCHpOBaTb O,n:HH K03<p<pHUHeHT, Tal< KaK BCTpe4aeTC5I
TOJlbKO OTHoweHHe N (X) H D (X), CJIe.n:OBaTeJIbHO, O.n:HH K03<p<pHUHeHT
(3aBe.n:OMO HeHY JIeBOt!) AOJI)J{eH 6bITb 3a<pHKCHpOBaH *). MbI BbI6HpaeM
*) npaKTH'IeCKH 06bJ'IHO JIY'Illle Bbl6paTb eJ],\iHHu,eti crapmllti K034J4JH-
lJ,f1eHT lfHcnHTe. R HJlH 3H3MeH3TCJlR: 9TO 9KOHOMHT O.D:HO YMHO)/{eHHe npH Bbf1lHC-
JIeHHf1 pau,HOHaJlbHOH 4JYHKQHH.
274
p AU!10HAJlbHbIE <l>YHKUYlI1
rfll. O
r-IJIH BbI4f!CJ15!eM COOTBeTCTBYlOllI,ee 4nCJ10 M Y31IOB, pacrr0J10JKeHHbIX TaM,
r.n:e 3TO KaJKeTC5I Han60Jlee B aJKHbIM. Tor.n:a I1MeeM
D(Xi)Yi=N(Xi)
(1=1,..., M),
CHCTeMY M ypaBHeHHfi, KOTOpylO HallO peIlIHTb. B npHMepe (paBeHcTBo
(20.2-1» HMeeM
Vi + b 1 X tYi + b 2 X 7Yi +.. . + b k X 7Yi = a3X + a4 X ' +... + ak+Jx:+ 1
(i = 1,2,..., 2k - 1). (20.2-2)
Eho - J1HHeltHf,Ie ypaBHemHI OTHOCHTeJ/bHO Hen3BeCTHblX at H bi> KOTopble
MorYT 6bITb pellleHbI npn nOMOlll.n 06bI4HbIX 6n6JJuOTe4Hblx nporpaMM.
l.JaCTo BCTpe4aeTCH MHeHne, 4TO 3Ta CI1CTeMa BblCOKoro rrOp5!lI.Ka. Ea-
CKOJ/bKO MO)j{HO CYlIHTb, 4HCJlO HeI!3BeCTHblX pelIKo rrpeBbIlllaeT 10-12,
TaK lJTO pelllemle Tpe6yeT He 60J/ee 1-2 TbIC5!lJ onepau:ntt; TaKa5! pa-
60Ta e.n:i3a JJI1 CJlO)j{Ha .n:Jl5I 60JlbIlIOfi MaIllnHbI, eCJlu 3TO Tpe6yeTcH ClIe-
JIaTb OlIHH pa3 Ha 3a.rra4Y.
npe)/{JJ,e tJeM npHHRTb annpoKCI1MHPYlOllI.YIO 4JYHKIJ,HIO, nO. e3HO Bbr'tJI1CJlHTb
,nocTaTotJHo MHoro ee 3Ha'leHHH. OJJ.Ha1KJJ,bl, B npaKTHKe aBTopa, 3HaMeHaTe. b
HMen KopeHb Me1K):\y ):\BYMR Y3J1aMH, TeM caMblM 06palll.aR B 6eCKoHetJHoCTb
npH6nH)/{alOllI.YIO 4JYHKU,H10. Me)/{):\y TeMH )Ke J],BYMR Y3J1aMH Y tJHCJlHTe. R TO)/(e
6b1n KopeRb, tJeM MaCKl1pOBaJlOCb nOBe):\eHHe <jJYHKi.l.HH.
ECJIn rrpn6JII1JKeHne HeYlIOBJIeTBOpf!TeJIbHO, TO Bce, 410 MOJKHO no-
np060BaTb, CBOlIHTCH K CJIe.n:YIOIlI.eMY:
1) C.rrBHHYTb Y3JIOBble T04KH;
2) H3MeHHTb BH.Q npH6JIHJKalOIlI.etl <PYHKu:nu;
3) n3MeHHTb k.
BbI60p OlIHoro 113 cnoc060B .Qel1CTBnfi 3aBHCHT OT TOro, n04eMY
f!MeHHO 6bwa HeYlIOBJleTBOpIlTeJlbHa anrrpOKcnMaU:I15!.
YnpamHeHHH
20.2-1. KaKoH BI1,n paU:HoHaJlbHOH 4JYHKU.HH cJle):\yeT Bbl6paTb, eCJlI1 y (x) ?: 0
H Y (x) 0 npH x co?
20.2-2. KaKoH BH):\ pau,HoHaJlbHOfi 4JYHKU.HII Bbl6paTb, eCJlI1 y (x) I1MeeT KO-
peHb npii x = a; HYJlb D (x) npH x = b; H TO H n:pyroet
20.3. IIeohImeBcI<oe npH6JIH)J{eHHe pau,HOHaJIbHbIMH <!JYHI<U,IHlMM
KaK YKa3aHo B 20.1, pall.HOHaJlbHaH <PYHKlI.HH 4aCTO HcnOJlb3yeTcH
KaK JlerKO BbI4HCJlHe We npH6J1H)j{eHHe TpaHcu:eH.rreHTHOti <PYHKlI.Hn. Kor.rra
TaKoe npn6J1H)!{eHHe .QOJJJKHO HCf10JIb30BaTbC5I MHOrO pa3, I1JIn B 4aCTO
nOBTopHlOu etlcH 3a.rra4e, nJlH B. cnell.naJlIl3HpOBaHHott MalllnHe, TO CKopee
BCero )!{eJIaTeJlbHO 4e6bIIlIeBCKoe paBHOKOJle6J1lOllI.eeCH (MHHIlMaKCHoe) npn-
6JJn)j{eHHe.
Ea'meM C BbJ60pa Ha60pa Y3. OB Xi H BbI4HCJlHM COOTBeTCTBYlOu.l.l1e
Yi' Do HHM HaXOlIHM paU:HoHaJlbHYIO CPYHKlI.HIO nO.Qxo.Q5IIlI.ero Bl1lIa (KaK
B 20.2).
20.41
OBPATHblE P!\3HOCTVI ICMMMETPW1HblE)
275
3aTeM pHcyeM KprmylO OlJm6KH, Bbl'lBCJI5I5I OIllH6KH BO MHOrHX TO'l-
Kax; TaM, r.lJ:e OIllH6Ka BeJIHKa, C.lJ:BHHeM TO'lKH n06JIB)I{e .lJ:pyr K .lJ:pyry,
TaM lKe, r.lJ:e JIOKaJIbHaH MaKCHMaJJbHaH onm6Ka MaJIa, pa3.lJ:BI1HeM TO'lKH
110JJ:aJIbille. C 9TI1MH BHOBb Bbl6paHHblMI1 y3JIaMH nOBTopHM BeCb npoQecc.
YI1CJIO nOBTopeHHt!, He06xo,UHMbIX JJ:JJH nOJIY'leHHH nO'lTI1 paBHOKOJJe6-
JIlOuLeticH Kpl1BOti OIllH6KI1, pe.lJ:KO ,UoxoJun .lJ:0 .lJ:eC5ITH. CKOpOCTb npI1-
DJIHIKeHH5I K paBHOKOJIe6JIIOmetiC5I Kpl'!BOti Oll1H6KH 3aBHCHT, cpe.lJ:H llpO'le-
1'0, OT cnoc06a nepeMemeHHH Y3.'IOB.
MbI He npe,UJIaraeM HHKaKHX .lJ:OKa3aTeJIbCTB TOro, '1TO 9TOT MeTO.lJ:
DYJJ:eT pa60TaTb, HO npaKTHKa IIOKa3blBaeT, '1TO p5l.lJ: npOCTbIX cxeM pa-
60TaeT npH,'II1'1HO. 4T06bl He nOTepHTb CJIHIllKOM MHoro 3HaKOB, 06bI'IHO
rpe6yeTcH HeCKOJIbKO «MaJIeHbKHX XHTpoCTeti» npH opraHH3aQI1I1 Bbl'lH-
CJIeyIHti.
. 20.4. 06paTHbIe pa3HocTH (cHMMeTpl{QHbIe)
B npe.lJ:blJLymeM naparpacpe I1CnO.'Jb30BaJIaCb HTepaTHBHa51 cxeMa, Tpe-
6YIOIlJ,a5I HeCKOJJbKI1X nOBTOpeHl1ti, KalKJLOe 113 KOTOpbIX COJLep)l{HT peIlle-
Hl1e CHCTeMbI JIHHetlHblx ypaBHeI-IHti (20.2-2). 31'0 3aCTaBJJ5Ier TIbITaTbC5I
HatlTH perym!pHbItl cooc06 perueHHH TaKoti CI1CTeMbI.
CJJeJLY51 MHJIHY *), paccMoTpHM '1aCTHbIt! cJIY'lat!
_ a o + a 1 x + a 2 x 2 + a s x 3
y- b o +b 1 x+b 2 x 2 ,
KOTOpbltl I1cnOJIb3yeT IlleCTb 3aJLaHHbIX TO'leK (Xi' Yi) (i = 1,..., 6).
fIpHBO.lJ:HM K 06meMY 3HaMeHaTeJIIO:
ao - boY -/- alx - b 1 xy + a2x2 - b 2 x2y + a3x3 - O.
IlpHMeH5I5I MeTO.lJ:, KOTOpbIt! 6bIJI I1cnOJIb30BaH .lJ:JI5I BbIBO.lJ:a CPOPMYJI (8.2-3),
JIerKO YBH.lJ:eTb, '1TO onpe.lJ:eJIl-ITe.'Ib
1 Y X xy X2 x2y x 3
1 Yl XI XlYl xi xiYl X8
I
1 Y2 X2 X2Y2 x X Y2 x =0 (20.4-1)
Y6 x 6 Xi\Y6 x: X Y6 x:
naeT Tpe6yeMoe peIlleHHe.
f1pI1BeJIeM CHa4aJIa BTOPYIO CTPOKY K BH.lJ:Y (1, 0, 0, 0, 0, 0, 0).
YMHQ)KaeM:
1. I1epBbIt! cToJI6ell Ha YI 11 BbI4HTaeM ero H3 BToporo croJI6Qa;
2. TpeTHt! croJI6eu. Ha YI H BfiI411TaeM ero H3 4erBepToro croJJ6u.a;
3. TI5ITblt! cTo.l!6eu. Ha YI 11 BbI4HTaeM ero H3 IlJeCTOrO cToJJ6u.a;
*) CM. [25]. MbI HCMlIoro 113MCHHJlH 0603Ha'leHHR.
276
P AUVlOHAJJnHbIE <l>YHKU11V1
rW.20
4. TI5ITbltl cToJl6eu. Ha Xl H Bbl'1HTae 1 ero H3 CelJ,bMOrO CTOJ!OU.3;
5. TpeTHt! cTo1l6en Ha Xl H Bbl'1HTaeM era H3 n5lTOrO CTOll 6u.a;
6. TIepBbItl CTOJI6eu. Ha Xl 11 Bbl'1HTaeM era 113 TpeTbero CTOJl6u.a:
Pa3J10)J{HM 3aTeM nOJlY'1HBlllHt!C5I onpelJ,eJlI1TeJlb no BTOpOtl CTpOKe; "
pa3lJ,eJlHM
7. fIepBYIO CTPOKY Ha Y -Yl;
8. BTOPYIO CTPOKY Ha Y2 - Yl;
TpeTblO CTPOKY Ha Ya - Yl;
9. CelJ,bMYIO C'fpOKY Ha Y6 - Yl'
TIpH lJ,eJleHHH MorYT B03HHKHYTb Henp!H!THOCTH, eCJlH KaKott-Hu6y,IU>
Yi = Yl (i #- 1) H JJ:a)J{e eCJlH OHI1 6J1H3KH. B JJ:ettCTBI1TeJlbHOCTH CTPOKY,
KOTOpa51 6Y1(eT npHB01(HTbC5I K BI11(Y (1, 0, 0, 0, 0, 0, 0), CJlelJ,yeT Bbl-
6HpaTb, HMe51 B BH1(Y 9TO 1(eJleHHe 11 CTapa5lCb !m6e)J{aTb 60JlblllOtl HH-
Tep<pepeHlJ,HI1 B pa3HocTH Yi - YI (11 TeM caMblM 1(eJJeHHiI Ha MaJJeHbKOe,
HeTO'1HOe '1HCJJO). 0603Ha'1HM
X-Xl
- = PI (X, Xl)'
Y-YI
Xi-Xl ( )
-=Pl Xi,XI'
Yi-YI
OnpelJ,eJJI1TeJlb (20.4-1) 3anHllleTC5I Tenepb TaK:
PI (x, Xl) X XPI (X, Xl) X2 X2Pl (X, Xl)
PI (X2' Xl) X2 X2Pl (X2' Xl) X X PI (X2' Xl) =0, (20.4-2)
1 PI (X 6 , Xl) X 6 X6Pl (X6' Xl) X X PI (X 6 , Xl)
T. e. OCTaHeTC5I Toro )J{e BH1(a, '1TO H paHbllle, HO TOJlbKO llleCToro no-
p5IlJ,Ka.
fIOBTOpHM npou.eJJ:YPY. YMHO)l{aeM:
1. fIepBbltl cToJl6eu. Ha PI (X2' Xl) H Bbl'lI1TaeM ero 113 BToporo CTOJlOu.a;
2. TpeTI1t! CToJl6eu. Ha PI (X2' Xl) 11 BbI'1HTaeM ero H3 '1eTBepToro
CTOJl6u.a;
3. fI5IThltl CToJl6eu. Ha PI (X 2 , Xl) H Bbl'1I1TaeM ero H3 llleCforo
CToJl6u.a;
4. TpeTHtl CToJl6eu. Ha X2 11 Bhl'1I1TaeM ero H3 mlToro cToJl6u.a;
5. IlepBbItl CToJl6eu. Ha X 2 11 Bbl'1I1TaeM ero H3 TpeTbero cToJJ6u.a;
Pa3J1araeM onpelJ,eJlI1TeJlb H lJ,eJlHM
nepBYIO CTPOKY Ha PI (x, Xl) - PI (X2' Xl)'
BTOPYIO CTPOKY WI PI (Xa, Xl) - PI (X2' Xl)'
3lJ,eCb pa3YMHbItl BbI60p CTpOKI1, npl1BOlJ,HMott K BHlJ,Y (1, 0, 0, 0, 0, 0),
MO)J{eT TIOMO'1b coxpaHHTb TO'1HOCTb.
ECJJI1 Mbl 0003Ha'1HM
X-X 2
u x = ) ( ) '
PI (X, Xl - PI X 2 , Xl
20.41
OBP A THblE P A3HOCTYI lCYlMMETPYI'-IHblE)
277
1"0 nOJlYllHM .lI.JlSJ (20.4-2) Bblpa)KeHl1e
1 U x x x U x x
1 ua xa xsua x
1 U4 X4 X4U4 x =0.
1 U5 X5 X5U5 xg
1 U6 x 6 X6U6 x
(20.4-3'
<1>YHKIJ.HSJ PI (Xi' X j) 6blJla, 01leBI1AHO, CI1MMeTpH4Ha OTHOCHTeJlbH<r
CBOHX nepeMeHHbIx. Ho 3TO HeBepHo AJlSJ cpYHKIJ.I1I1, KOTOpylO MbI 0603-
Ha411J111 Cli' 06bl4HO I1cnoJlb3YlOT CI1MMeTpl14HbIe cpYHKIJ.1111
X-X 2
(x - xd/(Y- YI) - (x 2 -xd/(Y2- YI) + YI =
x-x '
= (x -XI) I (Y-Yl)- (x 2 XI) / (Y2- YI) + YI =
_ X 2 -XI
- (x 2 -x)! (Y2 - y)- (X-XI) I (Y-YI)
+Y'
11 0603HallalOr HX P (X, Xl' X2)'
YT06bl npHBeCTH onpeAeJlI1TeJlb K HY)KHOMY BH.lI.Y, YMHO)KHM nepBbIfi;
cToJl6eu Ha YI H npH6aBI1M ero KO BTOPOMY CTOJJ6uy, rpeTHt! CTOJl-
6eIJ. - Ha YI 11 npl16aBI1M ero K 4eTBepTOMY CTOJl6IJ.Y. MMeeM
P2 (X, X2' Xl) X XP2 (X, X2' Xl) X2
P2 (x s ' X 2 ' Xl) Xa XSP2 (xs, X2' Xl) X
=0.
(20.4-4 )
1 P2 (X6' X2, Xl) X6 X6P2 (X 6 , X2' Xl) X
Bonpoc IJ.eHHOCTI1 3TOf! CI1MMerpl1H AJlSJ npaKTl1lleCKI1X BbIllI1CJJeHI1f! oc-
TaeTC5I OTKpbITbIM, HO TeopHSJ OT 3Toro Bblr JlSJ.lI.I1T npollLe 11 Mbl 6y.u.eM,
npl1,l1,ep)KI1BaTbCSI CI1MMeTpHllHoti cpOpMbl.
DOBTOpSlSJ npoIJ.ecc, nOJlY4HM, HaKOHen.,
1 1 P5 (x, X5' X4' Xa, X2' Xl) \
1 P5 (X6' x 5 ' X4' xS' X2' Xl) = O.
(20.4-5),
T04HO TaK )Ke, KaK 6blJlO YCTaHOBJleHO, 4TO cpopMYJIa HbIOTOHa eCTb
TO)K,lI,eCTBO, MO)KHO HanncaTb
I X-Xl X-Xl
Y=YIT ( ) - YI+ )+ . [
PI X, Xj pj (x 2 , Xl (X - X 2 )! P2 (X, X 2 , Xl) - Yd
= YI + (X - Xl) {PI (x 2 , Xl) + (X - X2)![P2 (x a , X2, Xl) - YI] +
+ (X - Xs)}[Pa (Xl> Xa, X2, Xl) - PI (X2' XI)]}-l =... (20.4-6)
278
PAUHOHAJJbHbIE tf>YHKUHH
I fJJ. 20
13 .QaHHOM 4aCTHOM CJlY4ae MbJ 3aKaHql-IBaeM IJJJeHOM
x-x.
P. (Xij, X., X 4 , X 3 , X 2 . X 1) - P3 (X.. X" X 2 , X 1)'
TaK KaK
P5 (X, X5' X., X3' X 2 ' X,) = P3 (X6' XB' X X3' X2' X,).
BeJll14HHbl Pi lJa3bIBalOTC5I 06paTHblMH pa3HOCT5IMH; y.Q06HO npe.QCTaBJJ5ITb
.ce6e ra6J1Hu.y 06paTHbiX pa3HOCret! B BH.lJ,e
X, y,
p, (X2' X,)
X2 J'2 P2 (X3' X2, X,)
PI (X 3 , X,) P3 (X.. X3' X2' X,)
X3 J'3 P2 (X4' X\I, X,)
PI (X4' X,)
X4 Y4
. 20.5. npHMep
PaCCMOTpllM 4aCTO BcrpeqalOru.YlOcR <PYHKU.UIO
I
Y= I +X2
H nocrpOHM Ta6J1H1J.Y ee 3Ha4eHI1tt f{aK I1 .lJ,JlH pa3.lJ,eJleHHblX pa3HocTet!.
BMecro TOro 4T06bl HCn0J1b30BaTb nepBYIO u k-IO crpOKH, MO)/{HO 6paTb
pa3HocTH CMe)/{HbIX CTpOK; nOJlb30BaTbC5I 6Y.QeM rOJ1bKO 3Ha4eBI151MI1 Bepx.
Bet! .QHarOBaJlH. \;haK. nOJIyqaeM Ta6J111L1.Y 20.5-1.
T a 6 JI H 11. a 20.5-1
x I y I
0 1
-2
1 1(2 -1 i
-10(3 0
2 1/5 -1/10 0 1
I
-50(5 40 01
3 Ij10 -1/25
-170/7 140
4 1/17 -1/46 0
-44 2 / 9 324
5 IJ26 -1)73
-962Jl1 I
6 1(37
20.51
TIPHMEP
279
CToJi6eu HYJJetl (HJJH nO'lTH HYJIet1) HrpaeT ry me pOJJb, '1TO YI B 06bJ'I-
HOtJ Ta6JJI1Ue p331-1OCTetJ; 011 nOK33bIBaeT, r.n:e HMO OCT3HOBI1TbCS1. Tenep&.
113 (20.4-6) np5JMotJ rrO.IICT3HOBKOtJ nOJJYlJaeM
y=l+
-2+
x-O
x-I
_?+x-2
- 2 --L x- 3'
:
lJTO np11 nepexo.n:e OT Henpep/,fBHot1 .n:p0611 K paUI1OHaJIbHot1 CPYHKUI1W
.llaeT
I
y= i+X2 '
Mhl He 6y.n:eM yrJJy6JJ5JTbCS1 B 9TY 06JJaCTb 11 npoCTO OTOlllJJeM 'iI1T3reJJ!1-
K 06P33UOBbJM PYKOBo.n:crBaM MHJJHa [25], MI1JJH-ToMcoHa [27} 11 XI1JJb--
.u.e6paH.u.a [14].
LJACTb III
ftEMHOr04JIEHHbiE npI16JII1}f{EHI151
I' JJ A B A 21
nEPHOAHQECKHE YHKUHH. AnnpOKCHMAUH YPbE
. 21.1. UeJlb 9TOH TeopHH
TIpouecc pemeHHH JIIo60t! npaKTH'JeCKot1 3a.D,alJH Ha MamHl e BKJJIO-
qaer TPH 3Tana: nJJaHHpOBaHHe, BbJnOJlHeHHe nJJaHa H HHTepnpeTaUl1Io pe-
3Y JIbTaTOB. 3TI1 TPH 3Tana MO)l{HO cpaBHHTb C TpeMH KJJaCCHlJeCKHMH cpa-
3a 1H llIaXMaTiwIJ Hrpbl: .D,e6IOT, MHTTeJlbllInHJlb H 3H.D,llInHJJb. B waXMaTax
Bce TPH cpa3bl HMelOT HCHO Bblpa)KeHHbllJ xapaKTep, XOTH Ka)l{.D,a51 H3 HI1X
He33MeTHO nepeXO.D,HT B cJJe.D,YIOW:YIO H onpe.D,eJIHeTCH npe.D,bl.D,yw:ett, H MO-
)KeT CJlyqlHbCH, lJTO Hrpa OKOHlJl1TCH B MI1TTeJlbWnHJle I1J1H .D,a)l{e B .D,e-
61OTe. TaK }f(e .D,eJlO 06CTOHT H B BbllJI1CJlHTeJlbHott np aKTI1Ke.
Ha nepBOM E1Tane Bce paClJeTbI HOCHT, KaK npaBHJlO, xapaKTep Ha-
6pOCKOB, B KOTOpblX orrpe.D,emJeTC5I 06"beM BbllJI1CJJeHI1t1, MalllI1HHOe BpeMH
I-I T. .D,. 06bllJHO Ha 3Tarre rrJJaHl1pOBaHH5I rrpHHI1MalOTCH pellleHH5I, nO.D,06-
Hble CJle.D,YIOULeMY: «CHalJaJla BbllJl1CJIHM q,YHKUHIO B HeKOTopott rPYIme
TOlJeK, 3aTeM 3aMeH!-IM I1HTerpaJi B HHTerpaJlbHOM ypaBHeHHH raYCCOBoIJ
KBMP aTypoIJ 110 CeMI1 TOlJKaM, pelllHM rrOJlYlJeHHYIO CHCTeMY ypaBHeHI1IJ
MeTO.D,OM I1CKJlIOlJeHI1H, rrO.D,CTaBHM nOJJYlJeHHble pemeHI1H B TaKI-Ie-TO H Ta-
Ime-TO <\JOPMYJIbl I-! HarrelJaTaeM HY)I{Hble pe3YJlbTaTbl». HalJaJIbHbItt 3Tal1
BKJlIOlJaeT B ce6H OueHKI1 BpeMeHI1 npOrpaMMI-!pOBaHHH, KO.D,I-IPOBKH, Ma-
WHHHoro BpeMeHI-!, OUeHKI-! Taro, Kor.D,a 6Y.D,YT nOJlYlJeHbI pe3YJlbTaTbI 11
KaK OHH 6Y.D,YT I1CTIOJlb30BaHbi.
06bI4HO B npouecce peWeHI-!H 3a.D,a4I-! B03HHKaeT 06paTHaH CBH3b Me)Kl.!Y
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3a.D,alJI1 B03MO)l{Ha JJljlllb rrOCJJe Ha4aJJa BbllJI-!CJJeHI1t1. TeM He MeHee BhIXO-
JJ.I1Tb Ha MaWI1HY HY)l{HO, JJHlllb Tw:aTeJlbHO BbHlOJJHHB 3Tan nJlaHHpOBaHH5I.
Ha 3Tal1e «BbHlOJlHeHH5I» I1JJaH lJaCTO MeHHeTCH. HanpHMep, nOCJIe Bbl-
lJHCJleHHH ynoMHHYTOIJ CPYHKUHH CTaHOBHTCH HCHWM, lJTO BMeCTO CeMI1TO-
lJelJHot! raYCCOBoIJ q,OPMYJlbl CJJe.D,yeT HCnOJlb30BaTb .D,eC5ITHTOlJelJHYIO, a
aTO MO)l{eT B.D,BOe YBeJIHlJHTb KOJlH'1eCTBO pa60Tbl .D,JlH pellleHH5I 110.7JY-
tJalOw:et!C5I CHCTeMbl ypaBHeHHtt. OWH6KH npH pellleHHH ypaBHeHHIJ MorYT
OKa3aTbCH 60JlbWHMH, H TOf.D,a .D,JI5J y JlYlJWeHI1H nOJlYlJeHHblX pemeHVIIJ
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3AMEHA TIEPEMEHHblX 11 BblBOP Y3nOB
28f
B cpa3e «HHTepnpeTaUHr1» He06xo.n:HMO He TOJ/bKO 06cy.n:HTb H 061.,HC-
HHTb pe3YJ/bTaTbl, HO TaK)I{e H npOBepl1Tb, COOTBeTCTBYIOT J/11 pe3YJ/bTaTbl
cpH3fjljeCKol1 Mo.n:eJ/H 11 He HBJ/HeTCH J/H 4aCTb H3 HHX CJle.n:CTBHeM <pop-
MaJ/bHO npOBe.n:eHHhlX BhI4HCJJeHHI1, a He cpH3H4eCKHX 3aKOHOMepHoC"rel1.
KpoMe Toro, HY)l{HO 061>51CHHTb Bce H3MeHeH!151 B TIJJaHe BbI4HCJ/e\-1111i.
TI04eMY OKa3aJ/aCh CHJ/bHO OTJ/H4aIOllJ,el1cH OT npe.n:nOJ/araeMol1 <PYHJ<lL!15f
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perneHHH? 3Ha411T J/11 9TO, 4TO pe3YJ/bTaThl 04eHh 4YBCTBHTeJ/bHhl K HC-
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Tpe6YlOT I1CCJJe.n:OBaHI1H.
06N4HO B MaTeMaTI14eCKHX Kpyrax npeHe6peralOT nepBol1 H TpeTbel1
cpa3aMI1, KaK He OTHOC5IllJ,I1MHCH K MaTeMaTI1Ke. Ho 9TI1 .n:Be cpa3bl, B OCO-
6eHHOCTl1 TpeTb5l, KOTOpOtt npeHe6peralOT 4allJ,e Bcero, I1MelOT pernalOULee-
3Ha4eHHe .n:J/5I ycnexa Bcett pa60TN.
TIOJJI1HOMl1aJ/bHhle MeTo.n:hl, KOTopble paCCMaTpHBaJII1Cb BO BTOpOI1 4aCTH
KHHrl1, He IIpl1cnoc06J/eHbl .n:JJH I1CCJ/e.n:OBaHl1l1 B cpa3ax nJ/aHHpOBaHHH 11
HHTepnpeTall.HH. OUW6KI1 B HHX Bblpa)l{alOTC5I 4epe3 npOH3Bo.n:Hble BbICO-
KHX nOp5l.n:KOB H, CJ/e.n:OBaTeJJbHO, Ha 9Tane nJ/aHl1pOBaHI1H pe.n:KO MoryY
6hITb Oll.eHeHbl T04HO; 60JJee n03.n:HH5I Oll.eHKa I1X Ha 9Tane I1HTepnpeTall.HW
TO)l{e He Bcer.n:a npUJ/l1BaeT CBeT Ha nepBOHalJaJ/hHYIO np06.'IeMY.
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MeHTOB TeOpl1l1 annpOKCI1Mall.lI11 cpYHKlI.115IMI1 C OrpaHI14eHHbiM cneKTpOM,
KOTOpa5I .n:J/H llwpOKoro KJ/acca 3a.n:alJ .n:aeT Mo.n:eJIH, n03BOJ/HlOllJ,l1e MaTe-
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pOBaHI151 11 I1HTepnpeTall.l1H. B pe3YJ/bTaTe B <pa3e nJIaHl1pOBaHI1H MO)l{HO.
c.n:eJ/aTb peaJ/HCTl1lJeCKYIO oueHKY pacxo.n:oB 11 nOTpe6Horo BpeMeHI1, a TO"
4TO npol13011.n:eT B npouecce BbI411CJ/eHHI1, MO)l{eT npOJ/I1Th HOBhl11 CBeT
Ha 113Y4aeMYIO cp113114eCKYIO cl1Tyall.HIO. TeOp115I H3J/O)l{eHa He nOJ/lW; OHa
OTBe4aeT He Ha Bce BonpOCbl 11 HHor.n:a Tpe6yeT HeCKOJ/bKO 60J/brne Bbl-
4HCJ/eHHI1, 4eM MHor04J/eHHaH Mo.n:eJ/b. O.n:HaKo OnblT pa60Tbl C Het! B Te-
4eHHe HeCKOJ/bKHX JJeT nOKa3bIBaeT, 'HO 4aCTO Mo.n:eJ/b <PYHKlI.I1H C orpa-
HHlJeHHblM cneKTpoM OKa3blBaeT 60J/brnylO nOMOllJ,b KaK BO BpeMH niJaHI1-
pOBaHlHI BbllJl1cJ/I1TeJlbHol1 pa60TbI, TaK 11 BO BpeMH HHTepnpeTall.l111
pe3YJ/bTaToB. TIoBTopHeM .n:eB113 9T0l1 KHl1rl1:
ll.e.llb paC"tem08 - nOHll.MaHUe, a He ttllC.Ila.
21.2. 3aMeHa nepeMeHHblX Ii Bbl60p Y3JIOB
B 60J/brnl1HcTBe BbllJl1CJ/eHI1I1 Y3J/oBble 3HalJeHI15I I1llJ,YTCH Ha MHO)l{eCTBe-
paBHoy.n:aJ/eHHhlx T04eK. TIYCTb paCCT05IHI1e Me)l{.n:y HHMI1 npHHHTO 3a e.n:H-
HHlI.Y, T. e. h = I::..t = I. HccJ/e.n:yeM CHa4aJ/a 9cpcpeKTI1BHOCTb TaKoro Bbl-
60pa .n:JJH CI1Hycol1.n:aJ/bHhlx cpYHKUI1I1.
lI.J/5I CI1HycoH.n:aJ/bHot! cpYHKlI.l1H BH.n:a cos ['It (n + 10) t + 'P J cYllJ,eCTByeT
.Qpyra5I CHHycoH.n:aJ/bHa5I cpYHKlI.H5I cos ['It (n - 10) t - rf], KOTopaH HMeer
o.n:HHaKOBhle C Het! Y3J/oBble 3HalJeHI151 B TOLfKax t = 0, 1/ 2/ ... 3TO.
'282
nEPHO H4ECKHE YHK HH. AnnpOKCHMA H YPbE
1m. 21
BH.liHO 113 rpHrOHOMeTpI1'!eCKoro TOIK.a:eCTBa
cos ['It (n + e) t + cp] - cos ['It (n - e) t - cp] =
= - 2 sin ('ltnt) sin ('ltd + cp) = a
(.a:m! ueJIblX t 11 n). TaKMM 06P330M, rrpil Bbl4HCJIeHYJIJ B 3TBX TO'1KaX '1aCTOTbJ
1t (n + e) H 'It (n - e) BeJl,YT ce6H TaK, KaK 6YJl,TO OHH OJl,HHaKOBb! npB YCJIO-
BrlH, '1TO cpa30Bble yr JIbl HaJl,JJelliaru.HM 06pa30M CB513aHbl Jl,Pyr C Jl,pyrOM.
choT 3cpcpeKT xopowo 113BeCTeH KHHO- 11 TeJIe3pWreJJ51M. B 060I1x
CJJY'1a51X Jl,BBlKyru.a51c51 KapTI1Ha cpl1Kcl1pyeTc5I 20 pa3 B ceKYHJI,Y. KorJl,a
!WJJeco BafOHa Ha'1lmaeT Bparu.aTbC5I, 3pHTeJIb cnepBa BHJI,HT, '1TO OHO
KPYTHTCH Bce 6blCTpee H 6blcTpee; HO 3aTeM Jl,BHlKeHI1e Ha'1I1HaeT 33MeJl,-
JIHTbCH, KOJIeCO OCTaHaBJIHBaeTCH H, HaKOHeLl., Ha'!I1HaeT Bparu.aTbC5I B 06-
paTHYlo CTOPOHY. TIpH Jl,aJIbHet!weM B03paCTaHHH CKOpOCTH Ka)l{eTCH,
6YJI,TO KOJleeo ODHTb HalJHHaeT Bparu.aTbC5I Bflepe.n:, 3aMe.lUI5JeTC5I, OCTaH3B-
JHWaeTC5I, KpynncH B o6paTHYIO CTOPOHY H T. .
choT XOpOllJO H3BeCTHblt! cTpo60CKOnH'1eCKHt! 3cpcpeKT lJaCfO HCnOJIb-
3yeTC5I JI,.1J5I aHaJII13a nepHOJl,H'!ecKHx 5lBJleHHtI. YCTpaHBalOT TaK, 4T06bl
BcnblWKH CBeTa nOHBJI5IJIHCb C lJaCTOTot! He IHOfO MeHbllJetl, '!eM nepHOJl,
3Toro 51BJIeHH5I. TaKHM 06pa30M, KalKJI,aH BCnblllJKa nOKa3blBaeT nepHOJl,H-
-1.JeCKOe 5lBJIeHlIe B '1yn 60JJee n03J1,Het! cpa3e ero lI,HKJIa. Cyru.eeTByeT
MHoro '-IaCTOT, KOTopble MorYT 6bITb HCnOJlb30BaHbl .a:Jl5I cTp060CKorrH-
1.JeCKorO OCBeru.eHH5I, Jl,alOru.ero OJl,HHaKOBbIe pe3YJJbTaTbI Jl,JI5I 3pHTeJI5I. TaK,
cum flpM O.LJ:Hofi '1aCTOTe Ha6J11oJl,aeTC51 Oflpe.LJ:eJJeHHa51 KalKyru.a5lCH CKO-
pocn Bpam:eHH51, TO rrpH lJaCTOTe, paBHot! flpH6JIH311TeJlbHO rrOJJOBHHe nott,
lleJIOBe'-leCKHt! r JIa3 6y.a:eT Ha6JIIO.a:aTb TY llie caMYIO CKopocn, O HaKO /Ca-
>lJeCTBO Kapnmbl HeCKOJJbKO yxy.a:maeTCH 113-3a HH3K0'1aCTOTHoro Mepuaml51.
B 3THX Jl,BYX npHMepax '1eJIOBe'-leCKI1t! rJIa3 BH.a:HT JJ,Be KOM110HeHTbl
JJ,BHlKeHH51 x H Y H, CJIeJJ,OBaTeJJbHO, MOlKeT OTJIH'1HTb I1p51MOe JJ,BHllieHl1e
T 06paTHoro. ECJIH OH BHJI,I1T TOJJbKO OJJ,Hy KOMrrOHeHTY, TO B HeKOTO-
p bIX CJJY'!3HX JJ,BHlKeHHe Brrepe.a: HeOT JIll '1 f!MO OT .aBIUKeHH5I Ha3a.a. 3.aeCb
npOHCXO)UIT CMellleHl1e '-IaCTOT, KOfJJ,a HeC!{OJIbKO pa3JIHlJHbiX '1aCTOT Jl,alOT
'O.LJ:r1H H TOT lKe pe3YJJbTaT BO Bcex paeCM<npHBaeMblX TO'1Kax. O'1eBI1.LJ:HO,
'1TO 31'0 - HeH36elliHoe CJJe.lI.CTl3YJe Bbl60pKI1 C paBHblMH HHTepBa,TlaMU;
9T1IM cpaKTOM He CJIellyeT rrpeHe6peraTb.
Ynpa}\{HeHl1e 21.2-1. HapHcYHTe rpa!jJHKI1 4JYHKU,HH CDS (I,lnt) II COS (O,91tt)
JIJliI O,s;; t,s;; 5 H pacc IOTpHTe nOBeJIeHHe I1X B Y3J10BbIX TO'lKaX t = 0, 1, 2, 3, 4, 5.
, 21.3. PHJJ.bl <l>ypbe; nepl'loJJ.HIIeCKHe HBJIeHlH!
ECJH1 Y = Y (t) - rrepHO.lI. H '-IeCKa51 cpYHKIJ,UH OT t C rrepHO.lI.OM 2N, TO
l'eOp!151 p5l.aOB $ypbe YTBeplKJJ,aeT (CM. rJJ. 6), QTO
ro
y(t)= -+! (akcos'Rkt+bksin ; kt)
k=l
(21.3-1 )
21.31
P51JJ.bI <l>YPbE; TIEPHOLlH4ECKHE 5IBJJEHH51
283'
npo YCJJOBOO, '-ITO Ha y (t) HaJJQ)l{eHbI LlOBOJIbHO CJla6bIe (C T04KO 3pemm
BbJllHCJIHTeJIbHoti M3TeM3THKO) OrpaHH4eHI15I. BeJI040HbI ak 0 b k onpeL\e-
JI51IOTC51 no CPOp IY JJaM:
2N
I \ 7t
ak=7[ yet) cos N ktdt,
I
I
2N }
b k = Y (t) sin kt dt. 1
(21.3-2)
B r JI. 6 6b1JIH npHBeL\eHbl KOHe4HbIe P51L\bI <Pypbe, nOCTpoeHHbIe no
:ma4eh!151M B 2N T04Kax. ECJIH BbI40CJIOTb KOHe4HbIti p5lL\ <Pypbe, no-
CTpoeHHbIti no 2N Y3JJOBbIM T04KaM 0, I, 2, ..., 2N - 1, TO nOJJY4HM,
N-I N-I
y(t)"" o + Ak COS l:r kt+ ! Bk sin l:r kt+ At cos 'ltt, (21.3-3)
k=l k= I
rLle
2N-1
1 7t
A k =7[ 1.. yet) cos 7[ kt,
/=0
2N-1
Bk = ! Y (t) sin ; kt. (21.3-4}
/=0
TIoK3)l{eM Tenepb, 4TO Me)l{LlY LlI1CKpeTHbIMIi II HenpepbIBHbIMH K09<P-
qmuHeHTaMH cYUJ.ecTBYIOT CJIeLlYIOUJ.lle COOTHOllIeHI151:
1
I
I
I
I
Bk =b k + (- b 2Nj _ k + b 2N . k)' I
i = I J+ J
co
Ao = ao + 2 2: a2NJ'
j=1
00
Ak = ak + (a2Nj_k + a2Nj+k)'
1=1
(21.3-5)
co
1J.Jl5J LlOl<a3aTeJIbCTBa npOCYMMnpyeM o6e 4aCTH (21.3-1) fIpn t = O
1, 2, ..., 2N - 1 II nOJIYlJIIM
2N - I 2N - I 00 2N - I co 2N - I
y(t)= :f- 1 + ak cas l:r kt+ 2: b k
/=0 /=0 k=l /=0 k=l /=0
. 7t
S m N kt.
ECJIH k =1= 0, 2N, 4N, ..., TO
2N- I
!
/=0
7t
COS N kt=O,
2N-l
2: sin ;;' kt = 0.
/=0
284
nEPMOlll14ECK ME <PYHK MI1. AnnpOKCMMA I191 <PYPbE
[VI. 21
ITpH k = 0, 2N, 4N, ..., nOJlara51 k = 2N j , nOJlYlIHM
2N - I 2N - I
! cos !ft(2Nj) t= ! 1 =2N,
J=O 1=0
OTClO.lI.a, HCnOJlb3Y5I (21.3-2) H (21.3-4), HMeeM
2N - I 2N - I
! sin (2Nj)t= ! 0=0.
1=0 1=0
00
2N o = 2N i + 2N ! a Njo
i-I
4TO COBnaAaeT c nepBblM H3 ypaBHeH!1Ii (21.3-5).
1tm
ECJlH YMHO)l{HTb (21.3-1) Ha cos Nt (0 < m < N) H npOCYM-
HlpOBaTb, TO aHaJlOrH4HO nOJlYlIHM
00 2N - [
NAm = ! ak !
k-I 1_0
1tk Tern
cos N t cas N t =
00 2N - I
'\'I I [ 1t (k + m) t + 1t (k - m) t ]
= ..:;. ak ..:;. 2" cos N cos N '
k=t 1=0
"0TKY.ll.a CJleAyeT BTopoe H3 ypaBHeHl-It! (21. 3-5). TpeTbe ypaBHeHHe nOJIY-
. 1tm t
'lfaeTC5I YMHO)l{eHl-!eM Ha stn N . U CYMMupoBaHueM.
CooTHoIlleHHe (21. 3-5) Me)l{.lI.Y KO.3q,ejmuUeHTaMH pa3J10)l{eHHtI Ha
HenpepbIBHoM H .lI.l1CKpeTHoM MHO)l{eCTBaX TOlfeK 5ICHO rrOKa3blBaeT IIBJIe-
HHe, KOTopoe MO)l{HO Ha3BaTb
«MI1MI1KpI151 4aCTOT». ECJII-J Mbl
rrpe.ll.CTaBI1M 4aCTOTbI B BI1Ae
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TO 5IBJJeHUe «MI1MUKPHU» 3a-
KJIlOlIaeTC5I B TOM, 4TO np5l-
Ma51 KaK 6bl CKJIaAbIBaeTC5I
rapMOIllKot!. [1epBa51 lJaCTO-
Ta, H8 KOTOpOt! npOHCXO.ll.UT
nepern6 np5lMoti, Ha3bIBaeTC5I
'tacmomoii lW/lO:HCeHllfl UJIH 4aCTOTOn HatiKBucTa H COOTBeTcTByeT 3Ha-
'-IeHHIO k = N. IlOCJIe.a:YIOUJ.He H8JIO)l{eHUR 6Y.lI.YT npOUCXOll.HTh 'Iepe3
TaKoti )l{e npOMe)l{YTOK. Bce T04KH KpHBOti (puc. 21.3-1), paCIIOJIO)l{eH-
HhIe Ha.ll. O.ll.HOti U TOti )l{e TOllKOt! ocu 4aCTOT, B pe3YJIbTaTe Bbl6opKI1
npOHBJI5IIOT ce6H KaK O.ll.Ha 48CTOTa. O.ll.Ha/KlI.bl np0I13Bell.5J BhI6oPKY, Mbl
Y)I{e He ClYJO}J{eM CHOBa pa3J1.emlTb 4aCTOTbl, KOTopble «MI1MI1KpUpOBaJm»
. 1 I I
:::=-=-=- : :=--------
J I "" I ....., "
k=IV k=ZIV k=JIV k=4IV lIocmoml1
(lfocmoma 1{{1110J/[[');Ufl)
PflC. 21.3-1. CMellleH!le '!aCTOT.
21.41
HHTEPnOJJ5IUI151 nEPI1011114ECKI1X <l>YHKIlI1A
285
Jl.pyr nOlI llpyra. Ha caMY ttaCTOTY HaJIOlKeHI151 Mbl !lOna.n.aeM OAHH pa3
B .D.E!a UHKJIa, KOr.D.a sin 'ltt = 0, COS'ltt =- (- 1/ (t - ueJJoe).
YnpamHeHHSI
21.3-1.TloJlY'lHTe TpeTbe ypaBHeHHe H3 21.3-5.
21.3-2. HaH,IJ.HTe ypaBHeHHe lJ,JlR An H3 21.3-5.
21.4. HHTepnOJISlIJ,HSI nepHO)1,HlfeCKHX rpYHKIJ,H"
Bo RTOpOtJ 4aCTH 6blJIa H3JJOlKeHa TeOpH5I MHOr04JIeHHotl annpOKCH-
MaUHH. 3atlMeMc5I Tenepb H3JJOlKeHHeM nO.Q06Hoi1 TeopHH .D.JI5I armpOKCI1-
MaUJ1J1 nepHO.QJ14eCKJ1X cpYHKUJ1t1 p5l.lJ,aMH <l>YPbe.
HeCMOTp51 Ha TO, 4TO 3a.Qal/a MOlKeT 6bITb nOCTaBJJeHa .QJI5I MHOlKeCTBa
HepaBHOY.D.aJJeHHbIX TOl/eK J1 J13BeCHIbI HeKOTOpbre pe3YJIbTaTbl .QJJ5I ,HOrO
CJJY4a5l, O.QHaKO H reopml J1 npaKTI1Ka 06WIHO HMelOT .n:eJIO C MHOlKeCT-
,BOM paBflOy.n:aJJeHHbIX TOl/eK. lLJHI CJIYl/a51 2N paBflOy.n:aJJeHHbIX Y3JIOBbiX
TOl/eK 3a.n:al/a HHTepnOJI5IUI1H pelliaeTC5I nYTeM npHMeHeHH5I KOHe4HOf0
p5l.n:a <l>YPbe (r JI. 6). 3Ha4eHHe cpYHKUHH B npOMe)KYTKax Me)J{.D.Y Y3JI8MH
Bbl4HCJJSleTC5I C nOMOllJ,blO p5l.n:a.
KaK H BO BTOpOtJ 4aCTH, nOCJJe nOJIYl/eHH5I cpOPMYJIbl Heo6xoQHMO
()/lpe.QeJIHTb OlllH6KY annpOKCHMaUHH. 2hy OlllH6KY MOlKHO pa3.n:eJII1Tb Ha
.D.Be l/aCTH: OlllH6KY, B03HI1KalOIlJ,YlO BCJIe.n:CTBHe HCnOJIb30BaHHSI KOHe4HO-
ro l/HCJJa l/JJeHOB pSl.D.a, H OlllI16KY BCJJe.QCTBl-le .QI1CKpeTHoro 3a,lJ,aHI151
<PYHKU I1H .
BJII1S1Hl1e .QI1CKpeTHOCTH 6blJIO Y)J{e HCCJIe.n:OBaHO; OCTaeTC5I BJJH5IHHe
.KOHel/HOCrH l/HCJJa l/JIeHOB p5l.n:a <l>YPbe.
ECJIl1 HenpepbIBHbItI p5l.n: <fJypbe HMeeT BH.n: (21.3-1), TO, nOllCraBJJ5I5I
cpOPMYJIbl IW9cpcpHUHeHTOB (21.3-2), nOJIYl/HM nepBble 2N 1.JJIeHOB, H3 KOTO-
PbIX, KaK H B CJIYl/ae .D.HCKpeTHoro p5l.n:a, B03bMeM nOJlOBI1HY KOCHHYCHOfO
'4iTeHa
2 N-I
I \' { l [ 1t 1t . 1t 1t ]
Y2N(t) = N ., y(s) 2+ "'" cos N kt cas N ks+ SIn N kt sin N ks +
b k=l
+ ; cas 'ltt cos 'Its} ds =
2N N-I
= y (sH + ! [cas k (t - s) ] + ; cas 'It (t - s)} ds.
o k= I
Y.T06bl npOCYMMHpOBaTb BeJJ!1'-II1HY *), CT05lllJ,Yro B cpnrypHblX cKo6Kax,
*) 3TOT pR,IJ. MomHO npOCYMMHpoBaTb 60Jlee 9l!eraHTHo, HcnOJlb3YR MeTO.llbJ
rn. 3. OlJ,HaKO 'IHTaTeJlb, BepoRTHo, YCflel! 3a6bITb lJ,eTaI!H, H f109TOMY Mbl 6y.n:eM
IIOJlb30BaTbCR 60Jlee rpy6bJMH, HO H 60Jlee 3HaKOMblMH TpHroHOMeTpl1'leCKHMll
.MeTo,aaMH.
286
nEPI10.IJ.I1lJECKI1E 4>YHKUI1I1. AnnpOKCI1MAUI151 4>YPbE
! rJJ. 2i
YMHO}/{HM ee H3 sin 2 (t - s); B pe3YJIbT3Te nOJIYlJHM
; {sin 2 (t - s) + [ sin N ( 1 + ) (t - s) - sin (1 - ) (t - s) ] +
+ [ sin N (2 + ) (t - s) - sin N (2 - ) (t - s)] +
+ [ sin (N - 1 + ; ) (t - s) - sin N (N - 1 ) (t - s)] +
+ sin 2 (t - s) cos 'It (t - s)}. (21.4-1)
Bee 'IJJeHbl, KpoMe .QBYX, yHHlJTO)Ka!OTC5I:
{sin [ ; (N- ; )(t-s)]+ sin 2N (t-s) eos 'It(t-s)}.
PacnHllIeM nepBbli1 H3 3TH X lJJJeHOB no cpopMYJle
sin [ ; (N - ) (t - S)] = sin 'It (t - s) cas 2 (t - s)-
- cas 'It (t - s) sin 2 (t - s).
B pe3YJlbTaTe BTOpOt! 'IJJeH nOCJle.QHet! cpOPMYJJbI YHl1lJTO}/{HTC5I co BTO-
pblM lJJleHOM npeJI.!JoCJle.QHet!, H Bblpa}/{eHHe B CKo6Kax npHMeT BI1,l1.
sin1t(t-s) eosiN (t-s)
2sin - (t - s)
2N
Tor.Qa
2N
Y'N (i) = \ y(s)
.'
o
1t
sin 1t (t- s) cos 2N (t- s)
ds.
. 1t
2sIil 2N (t-s)
(21.4-2)
nOCKOJlbKY Y (s) - nepHolU1lJeCKa51 cpYHKU H 5I, TO OTpe30K HHTerpnpOBaHWl
MO}/{HO ClI.BHHYTb B Jl!o6oi1 11m epBaJl lI.JlHHbI 2N; KpOMe Toro, nOJlO}/{HM
t- s=B:
N
Y N(t)= 2 y(t-B)
-N
. 1t
sm 1tO cos 2N 0
de.
, 11
sm 2N 6
(21.4-3)
21.41
VlHTEPnOJJ5ILU151 nEPI10JJ:I1'-IECKVlX 4>YHKL.\I1ti
287
ECJllI Y (t)= 1, TO
N . 6 1t6
I j Sin 1t cos 2N de.
1= 2N 1t6
sin 2N
-N
Ho, nOcKoJlbKy nHTerpnpoaaHl1e Be.l1.eTC5I no e, TO
N
1 \ ' sin 1t6 cos ,--
y(t)= 2N yet) ,1t6 ; de.
t, Sin 2N
-N
(21.4-4)
BbJ'Hna51 (21.4-3) li3 (21.4-4), nOJlY'H1.M onm6KY 01' HCrrOJlb30BaHH5I 2N
<IJJeHOB pSJ.Qa
N
\ . 6 1t6
I sm 1t cos 2N
eJN=y(t)-Y N(t)= 2N Lv(t)-y(t-e)] .1t6 dO.
. Sin 2N
-N
(21.4-5)
JIlo60nbITHO OTMenITb, 41'0 B 60JJbIllHHCTBe pa601' no pSJ.QaM <t>YPbe
CYMMupye1'CIJ He4e1'Hoe 'mOIO 4JJeHOB, u n091'OMY BMeCTO (21.4-5) nOJlY-
'J3eTC5I nOXO)J{35I, HO MeHee Y.lI.06Ha5I CPOPMYJI3 *). Ho npOC1'OTa CPOPMYJIbl
(21.4-5) 06MaH411Ba: lJeM 60JJbIlie Y3JIOB B3511'b, TeM .QJIHHHee 6Y.lJ.eT o1'pe-
30K I1HTerpl1pOBaH115I. O.QHaKo H3 np3K1'HKe oTpe30K I1Mee1' cpHKCl1pOB3H-
HYIO .lJ.JII1HY 11 H3MeH5Ier HH1'epBaJI Me)J{JJ:Y Y3JIOBhlMI1 1'04KaMI1. TI091'OMY
Mbl np0l13BO.ll.HM 33MeHY nepeMeHHblx 110603HaQaeM (41'0 MO)J{e1' npl1BeCTH
K HeKOTopoti nY1'3HI1LLe):
6 = 2N<p,
t= 2N r,
y (2Nx) = Y (x),
1/ 2
100 ,= [ ( ) _ ( _ )] { sin 2 1t'P cas 1t'P } d
.1\ . Y I Y 1 <P SIIJ 1t'P <p .
- 1/ 2
(21.4-6)
BeJlI1411HY B cpl1rYPHbIx cKo6Kax 4ac1'O Ha3bLBaI01' H.lJ.pOM K (<p). 04eBI1.LIHO,
41'0 K (0) = 2N. TIpl1 60nbIliHX N 4neH sin (2N7t<p) MeHHe1' 3H3K, a ctg'lt?
onpe.QenHe1' 3a1'yx3HI1e 9TI1X Kone63HI1tl TaK, 41'0 K ( + 1/ 2 ) = O. Om6a-
lOlli.a5! KOJIe6aHutl eCTb ctg 7t<p H
K(<p)=K(- <p).
*) 3HrMYH.LI. [44] Ha3b1BaeT nOJlYlJeHHYlO 4JoPMYJIY <MoJJ.Hqmu,HpoBaHHblM
-HlI.PUM>.
288
I1EPI10L\114ECKr1E <PYHKUI1I1. AI1I1POKCrlMAUI151 <PYPbE
[W. 21
ECJlH CPYHKLLI1I1 Y (X) r Jla,lJ:Ka5l, TO HaI160JlbWI1t1 BKJlaJl. B HHTerpaJl
1 I
.n.alOT 3Ha4eHI151 511!pa B6J111311 <p = 0, HarrpHMep rrpl:! - 2N <p :S; 2N
(pHC. 21.4-1). ITpl1 YBeJJH4eHUU
N rrUK CTaHOBHTC5I Bblwe Ii y)]{e.
ECJlI1, C Jl.pyrotl CTOPOHbI, CPYHK-
UI151 Y (x) CIiJlbHO MeH5IeTC5I, TO
TeHJl.eHUIiI1 K 33TyxaHI11O KOJle6a-
HUtt Ha6J1IO.aanCfl He 6y.aeT 11
npl1 y.aaJleHI1I1 OT T04eK PHTep-
rrOJl5lUI-IH B03MO)KHbI pe3KIie H3-
MeHeHH5I CPYHKIUJH Y (x), C03.!1,alO-
w,l1e 60JlbWI1e OWI16KI1.
3aMeTHM, 4TO rr01!I:JHTerpaJlb-
Hoe Bblpa)]{eHl1e C01!ep)]{HT rpYHK-
11,1-110, a He rrpOH3BO.!l,HYlO Bbl-
COKoro rrOp5l.aKa, KaK B CJlY43e
Pl1c. 21.4-1. 5!1I.PO pilJJ.a <Pypbe ,IJ.JlR N = 4. MHor04J1eHHot! arrrrpOKCHMa1l,I1H.
YnpamHeHHSI
21.4-1. Mbl ymc annpoKcHMHpOBaJlH TpHroHOMeTpH'IeCKHe 4JYHKU,HH nom1-
HOMaMH. flpH6J1H3bTC y (t) = (I - (2)2, (- I t I), pR1I.OM <Pypbe npH N = 3,
( 2 I 1 2 )
T. e. B Y3J10BbIX TOtJKax t= -3; -3; 0; 3; 3; J , Hart,nI1Te 4JYHKl\HIO
1
B TotJKe t='[,
21.4-2. B 3TOM naparpa4Jc BbIBeJJ,eH 3KBHBa. eHT HHTepnOJlHlJ.1111 J1arpaH}/(a.
I-1ccJle,nYHTc 3KBHBaJleHT HHTepnOJlill\HH 3pMI1Ta, KOTopaR npH6JI1J)/{aeT 4JYHKU,HIO
II npol13Bo,nHYIO B KamJJ,oit y3J10BOit TotJKe.
o T BeT: TaKoro He CYlllccTByer.
21.5. HHTerpHpoBaHHe
KaK Ii BO BTOpOtt 4aCTI1, I1CCJle1!OBaB Borrpoc 06 HHTeprrOJlHpOaaHl-IH
11 ero own6Kax, nepetl.!l,eM K 3a1!a4e HHTerpl1pOBaUH5I. ECJIl1 K09CPCPHUI1-
eHTbl Herr epbIBHoro p5I1!a <pYPbe y (t) YMeHbllJaIOTC5I rrpH N -+ co
.l1,OBOJlbHO 6bICTpO, TO K03cpcpl1uneHT Ao KOHe4Horo p5l1!3 <Pypbe 6Y1!er
2N
1!aB3Tb XOpOllJylO Ol1.eHKY HHTerpaJla Y (t) dt = Nao, nOCKOJJbKY 113
o
ypaBHeHl1tl (21.3-5) CJJe1!yeT
2N - I 00
Ao = ! y(t)=ao+2 ! aWJ'
t = 0 )= 1
TaKnM 06pa30M, OllJ116Ka, B03HHKalOw,aSl rrpl1 oueHKe liHTer-
paJJa Ha OCHOBe paBHOOTCTOSlUJ.llX II llMelOw,I1X paBHble aeca Y3JJOB,
21.51
HHTEIPHPOBAHHB
289
lIaeTC5I <POPMY JIOt!
ro
2N a2Nj,
1=1
B KOTOpOt! OTpa}j{aeTC5I Hen36e}j{HOe CMellHfBaHne 'laCTOT - pe3YJIbTaT
lInCKpeTHOCTH Y3J10B.
IlJI5I nepHOlIH'leCKnX <PYHKllHtJ <pOpMYJIa HHTerpHpOBaHH5I, f1CnOJIb-
aYlO111a5! 2N paBHOOTCT05l111HX H paBHOB3BellleHHbiX Y3JIOB, COOTBeTcTByer
<p O pMYJ1aM:
HblOTOHa - KOTeCa, nOTOMY 'lTO TO'lKH paBHOOTCT05lIIl.He;
4e6blllleBa, nOToMy 'lTO TO'lKn HMelOT paBHble BeCa;
{'aycca, nOTOMY 'lTO 2N y3JIOB TOqHO HHTerpHPYlor 4N <PYHKll.Htt
ITpOHJIJIlOCTpl1pyeM TO'lHOCTb oJTotJ <POPMYJIbl npH HHTerpHpOBaHHH
nepHOlIHqeCKHX <PYHKllH}!, PaCCMOTpl1M 9JIJIHnrHQeCKHe HHTerpaJIbl
1t/
II = VI - k 2 sin 2 6 d6,
o
1t/2
1 2 =
1 d6,
V I - k" sin" 6
rlIe I k I 1 - napaMeTp. KBalIpaTHbltJ KopeHb MO}j{HO npe.ll.CTaBnTb B BH.ll.e
611HoMa. ECJIn k 2 MaJIO, TO p5l1I CXOlIHTC5I 6bICTpO. Bblpa3HB sin 2n 6 'le-
pe3 sin 2p6 H cas 2p6 (r lIe p = 0, 1, 2, .. .), nOJJY'lHM 6blCTpO CXOlIll-
111l1tJC5I p5l1I <t>YPbe
( Cia - ria ) 2n _ I 2nia 2 (n-I) ia _
2j - 22nj2n Ie -C(2n,l)e +...]-
1
= 22n-li2n [cos2n6-C(2n, l)cos(2n-2)6+...J.
Ilpl1 MaJIblX 3Ha'leHHIlX k 2 pe3YJJbTaTbi 6YlIyr TO'lHbIMI1, HO npl1 k 2 , 6JIH3KOM
K I, HeJIb351 O)Kl1lIaTb BbICOKO}! TOQHOCTH.
IlJI51 9KCnepl1MeHTa BbI6epeM Y3Jlbi B T04Kax 0°, 30°, 60°, 90°, XOT5I
,D)I51 nOJIY4eHH5I nepv.O.ll.H4HOCTI1 CJle.ll.OBaJIO 6bl Bbl6paTb llleCTb T04eK Ha
Ta6J1Hu,a 21.5-1
9nJlHnTH4ecHH HHTerpan
k' I i, (Bb14.) 11 (Ta6.,.) I OWH6Ka
1 1,46746 1,4675
if
I 1,35064 1,3506
2
3 1,21099 1,2111 0,0001
4
10 p, B. XeMMHHr
T a 6 JI H II. a 21.5-2
k 2 I I" (BbJ4.) I 1 2 (Ta6.0.) \ OWH6Ka
I 1,68575 1,6858
"4
1 1,85410 1,8541
"2
3 2,15789 2,1565 0,0014
4
I
290
nEP OA qECK E <PYHKUI1I1. AnnpOKCI1MAUI151 <PYPbE
I rJJ. 21
nepUO.II 0 t 'It. B .lIef!CTBUTeJJbHOCTH MbJ BbllJUCJIUM 3HaIJeHUSJ .1HIllb
B 4eTbipeX Y3J1aX, O,lJ,l1H H3 KOTOpblX 0 = 0 TpUBHaJleH. Pe3YJIbTaTbI Bbl-
4UCJJeHHtl U Ta6JJU4Hble 3Ha4eHUSJ 9THX UHTerpaJIOB npl1Be.aeHbI B Ta6JII1-
u.ax 21.5-1, 21.5-2.
Vl3 paCCMOTpeHl1SJ STOra npl1Mepa MO)KHO B bIB eCTU CJIe.aYlOlllee:
BO-nepBbIX, K09<pqmu.neHTbI pSJ.aa <t>YPbe 4aCTO MO)KHO OUeHI1Tb C .aO-
BOJlbHO BblCOKOtl T04HOCTblO, n09TOMY MO)KHO OueH1Hb Onm6KY BbI4HC-
JIeHl1tl, He npncTyna51 K HnM;
BO-BTOpbIX, MeTO.a I1HTerpl1pOBaHH5I, HCnOJIb3YIOlli.Htl paBHOOTCTOSJllll1e
Y3JIbI C paBHblMl1 BeCaMl1, .a,aeT 04eHb T04HbIe pe3YJJbTaTbi .aJISJ nepHo.a,n-
4eCKl1X <PYHKIJ.Utl. EhoT MeTO.a MO)KHO HCnOJJb30BaTb .a,JI5I BbI4HCJIeHI151
<PYHKIJ.UU 5ecceJI5I J o (x) no <popMYJIe
1t
J o (x) = + cos (x sin cp) dcp.
o
YnpamHeHHe 21.5-1. Hafi,Il.I1Te 10 ( ) no IiHTerpaJlbHOfi <t>opMYJle, IIcnOJIb.
3YH IlleCTb Y3J10B.
21.6. MeTOJl. oomero onepaTopa
B r JI. IO 6bIJI ormcaH 06Ill.Hf! MeTO.II (TpeTut! MerO.II) HaXO)JcaeHHSJ
qJOpMYJJ, .lIaIOlli.I1X T04HbIe 3Ha4eHH5I ,lJ,JI5I 1, x, x 2 , ..., x n . AHaJIOf\1lJHbIM
nYTeM MO)KHO HaiJTH <POPMYJIbl, .llalOIlI.He T04Hble 3HalJeH1-151 .aJISJ
1t 21t 31t (N-I)1': t
1, cos N t, cas - N t, cas N t, ... , cas N ' cas rot,
. 1t t . 21t t . 31t t . (N - I) 1t t (21.6-1)
sm N ' SID N ' sm N' ..., sm N .
TaKHe YCJIOBn51 JIerKO nOJIYlJI1Tb, eCJII1 <PYHKIJ.I1SJ nepnO,lJ,[-jlJeCKaSJ [-j
flCnOJIb3yeTcH 2N paBHoOTCTOHIll.UX Y3JIOBbIX T04eK.
DycTb pe3YJJbTaT npl1MeHeHH5I JIUHetiHoro onepaTopa L K <PYHKIJ.UU
I (t) BbJpa)KaeTC51 4epe3 3Ha4eHU5I <PYHKIJ.UU B 2N paBHoOTCTOHIll.I1X T04-
Kax t=O, I, 2, "oJ 2N-1:
L [f(t)]=wJ(O)+wtf(l)+...+w2N_d(2N-1), (21.6-2)
Onpe.lleJJHlOllll1e ypaBHeHlul ,lJ,JISJ
1': 21': 1t(N-l)
1, CDS N tn, CDS N tn, ..., CDS N tn,
. l' , 1t(N-!)
Sin N tn, . . ., SID N tn
COS 'lttn,
21.61
6ylJ,YT HMeTb BUn
Mo
M I
= W o + WI
1t
=WO+WI CDS N
Mil
21t
=WO+WI CDS N
METOtl OBllJ,Ero OrIEPATOPA
+W
21':
+W CDS If
41':
+W'I. COS N
291
+...+WW_t.
+",+W N_I X
X (2N - I) 1':
CDS N .
+",+WW_l"X
X (2N - I) 21':
cas N .
M N - I
. " . " " . . . . . . . . . . . . . . . . . . . . . . " . .
, (N-l)1t (N-I)21t
=WO-r-WI CDS N +W cos N +",+WW_I X
X (2N - 1) (N - I) 1':
cas N .
MN =wo- WI
O + . 1t
JIll N+I = WI sm N
+W
+ . 21':
w sm N
- . . . - W'l.N _l'
+... +W N_l X
2N-l
X sin N
M\lN-1 = 0
" . . . " . . " . " . . . " . " , " . . " "
. N - 1 . 2 (N - I)
+WI sm -r'lt+W sm IV 'It+",+WW_l X
X . (2N-I)(N-I)
sm N 'If.
Or ClOlJ,a MaTpulla K09<pcp H llHeHTOB JJ,JISl HeH3BeCTHblX BeCOB W npHMeT
BIUl
-I 1
1t 21': 2N-I
CDS N cas N cos N 'It
N-l N-I (N - I) (2N -I)
COS 'It cos 2'1t cos N 'It
-1 -1 (21.6-3)
0 1': 21': 2N-I
sin N sin - sin N 'It
l N
N-I , N-l (2N - I) (N - I)
sin -r'lt sm 2'1t sm N 'It
to.
292
BCJIeJICTBlle
DEPVlOnW!ECKI1E 4>YHKlI,VlVI. ADDPOKCVlMAlI,VlS1 4>YPbE
[m.21
COOTHOWeHHtI opTOrOHaJlbHOCTll (6.2-3)
2N-I
m7t n7t
cos N k cos Ii k = N8mn
k=O
2N-I
. m7t n7t
Sill N k cos Ii k = 0 ,
o
2N-l
'\'1 . mlt k . n7t k
1.J Sill N Sill 7V = NUmn>
o
2N - I 2N _ \
1 = cos 1"(k=2N
o 0
(m, n =1= 0, N),
06paTHaS! MaTpl-lua 6YJI,eT paBHa TpaIlCflOHllpoBaHHot! M aTpHue (21.6-3),
KpoMe roro, '1TO nepBblt! 11 (N - 1 )-t! CTOJl6Ubl MaTpl-lllbl, COOTBeTcTBYIO-
1-1 cos tm, 6YJIYT pa3)leJJeHbl Ha 2N:
w.He cj:JYHKUHS!M 1
-I
2
1
( )
2
N-I
cas 1"(
N-!
cas 21"(
It
cas N
21t
cos7V
1 2N-I (N-I) (2N- 1)
2 cas N It cas N 1"(
1 0 0
2
1 It N-I
-2 sin N sin 1t
1 27t , N-12
2 sin N sm -- 7t
N
(21.6-4)
I , 2N-I
-2 S/IJ N
It
... sin
(N-l) (2N-l)
N It
npHMeHeHHe oHoro TpeTberO MerOJl,a 13 061l1.eM CJIYlJae MO)KHO pac-
CMOTpeTb Ha npl1Mepe. ITpe)lnOJIO)KHM, lJTO HY)KHO Hat1nl nepBYIO npOH3-
BO)lHYIO nepllOJl,l1lJeCKOfl cj:JYHKUI1H 13 TOlJKe t = O. 1J.JlH onpeJl,eJIeHHOCTll
nOJIO)KHM, lJTO HCCJIeJl,yeTcS! CJIYlJafl N = 3. Tor)la COCTaBJlS!IOIlI.He «MO-
l 7t t 27t t t . 7t t . 27t t 6
MeHTOB» ,lJ.JJS! . cos '8 ' cos 3 ' cos , Sill 3 ' sm 3 Y)lyr pa13Hbl
0,
a.
0,
0.
It
21t
3 .
3 I
21.7] HECKOJJbK 0 3AME4AHI1H OTHOCI1TEJJbHO OSrn:E fO METOj],A 293
YMHO)J{HB o6paTHYID MaTpl1lJ.Y H3 BeKTop MOMeHTOB, nOJJY411M
! 0
2 2 1 0 0 0 V31'
1 -1 -1 V3 V3 0
-1 -I I V3 - V3 0 1_ 1(3 1'
0 3
eh) e/a) -2 2 -1 0 0 j = e / 6) { 0
1 - V3 V3 1t
-I -1 3 V3
-1 -1 -V3-V3 I 3 'It
3 } -"Vi'lt}
(21.6-5)
Pe3YJIbT3T npe.n:cT3BJI5IeT co60t! BeKTop BeCOB; T3KHM o6pa30M, HCKOM351
OPMYJI3 npHMeT BH.n:
f' (0) = V 1t [3/(1) - 1(2) + 1(4) - 3/(5)].
Ynpa>KHeHHH
21.6-1. I-1cnoJlb3YR MaTpHu,y (21.6-5), HaHTI1 cpopMyny )J,JlR f" (0).
21.6-2. V1cnoJlb3YR (21.6-5), HaH)J,I1Te 4JopMyny j],JlR BbIQHCneHHiI
2Nu
f (t) dtdu.
o 0
. 21.7. HeCKOJIbKO 3aMeqaHHH OTHOCHTeJIbHO oomero MeTOJJ:a
"'"
B cJIY'lae 3nnpOKCllM31J.Hll MHOrO'lJJeHaMH M3TpH'lHb1t! MeTO.n: 6bIJI
.n:OCT3TO'lHO 9 eKTI1BeH. O.n:H3KO npH 3nnpOKCHMalJ.HH p5l.n:OM 4>YPbe .n:eJIO
06CT0l1T HH311e. P33JIH"IHe COCTOHT B TOM, "ITO p33JIO)J{eHHe YHKlJ.HH
B pH.D. 4>YPbe H3XO.D.HTCH JIer"Ie, "IeM onpe.D.eJIH1OTCH BeC3 npH nOMOIlLH
YMHO)J{eHI1H M3TpHlJ. (CM. 6.4 - 6.6). ECJIH y)J{e H3BeCTHbI K09 HlJ.HeHTbI
4>ypbe .D.JIH .n:aHHot! YHKlJ.HH, TO, YMHO)J{HB HX Ha «MOMeHTbI», HeTpY.D.HO nOJIY-
"IHTb OKOH1l3TeJIbHbIl-i pe3YJIbT3T. T3KHM 06p330M, eCJIH H3BeCTeH pH.D. 4>ypbe
N-I N-I
I(t)= a; + ! ak COS if kt+ at cas 'ltt+ ! b k sin N kt.
"=1 k= 1
TO, npHMeHHB onep3TOp L ( ), nOJJY"IHM
N-I
Llf(t)]= a L(l)+ ! akL(COS kt)+ Gt' L (cas 'ltt)+
"=1
N-I N-I N-l
+ ! bkL (Sin N kt) = Mo + ! a"M k + a: MN+ ! bkMN+k'
1<=1 k=1 k=I'
294
CXOJU1MOCTb PYlJl.OB <l>YPbE
[rJJ. 22
K03cpcpl1lJ,I1eHTbI p5l.1l.a Onpe.1l.eJI5IIOTC5I He3aBHCHMO .1I.pyr OT .1I.pyra H ,lJ,0-
BOJJbHO npOCTO, TaK KaK CI1HYCbI 11 KOCI1HYCbI 51BJI5IIOTC5I OpTOrOHaJIb-
HblMH ifJYHKII.H5IMH. TI09TOMY MeTO.1l. 06paTHblX MaTp1111. npl1 annpOKCHMa1l.1l1l
p5l.1l.aMli $YPbe, XOT5I H npe.1l.CTaBJJHeT TeOpeTH'leCKHt! llHTepec, He HMeeT
60JIbillOrO npaKTl1'leCKOro 3Ha'leH1l5l.
KaK 11 B 13.3, 3.11.eCb He06xo.1l.llMO YKa3aTb OC06blti cJIyqatt, KacalO-
llI.Ht!C5J HeOnpe.1l.eJIeHHOro I1HTerpHpOBaHI1H. 6Y.1l.eT JIll cpopMYJIa
b
Y= f(x)dx
a
TO'lHa .1I.JIH f (x) = 1, x, x 2 ,... llJIli .1I.JIH Y = 1, X, X2,... - He O,lJ,HO
II TO )l{e. KaK YKa3bIBaJIOCb B 1'3.3, npll ,lJ,aHHotl cpopMYJIe TO'lHOCTb
.1I.JIH f (x) = 1, x, x 2 9KBHBaJIeHTHa TO'lHOCTI1 ,1I,JI5I Y = x, x 2 , x 3 ,...,
nOCKOJJbKY I1HTerpHpoBaHHe cTeneHetl x YBeJJH'IHBaeT CTeneHb Ha 1. Tor.1l.a
Mbl HalllJIH Y,lJ,06HbIM .1I.06aBllTb YCJIOBlle TO'lHOCTH .n:JIiJ Y = I H TaKIlM
o6pa30M BOCCTaHOBHTb 9KBI1BaJIeHTHOCTb 060HX CJIY'IaeB.
B cJIY'Iae pH,lJ,OB <PYPbe, eCJIll CHaqaJIa annpOKCHMl1pOBaTb nO,lJ,blHTe-
rpaJIbHOe BbIpa)l{eHHe, a 3aTeM HHTerpaJI, TO '1JJeHbl COS kx 11 sin kx
nepetl,lJ,yT Apyr B ,lJ,pyra, a '1JIeH I nepet!.1I.eT B x. TaKHM 06pa30M,
B cJlY'lae pHAOB <Pypbe 9KBHBaJIeHTHOCTb npl16JlH)I{eHHiJ nO.1l.blllTerpaJIbHot!
ifJYHKlJ,HH 11 HHTerpaJIa 6y.n:eT nOTepHHa. TIo9TOMY C caMoro Ha'!aJIa He-
05XO.1l.HMO pelllllTb, KaKa51 '1aCTb aa.n:aql:l 6Y.1I.eT annpOXCI:IMl:lpOBaTbCSI
I1CITOJIb3yeMblM MHO)l{eCTBOM ifJYHKII.Ht1, TaK KaK annpOKCHMaUlll1 Ha pa3-
JIH'IHbIX 9Tanax pellleHI151 3a.1l.a'lH MorYT OKa3aTbC5I He3KBIlBaJleHTHbIMI:I.
Ha npaKTl1Ke 06bl'lHO 113BeCTHbi CBot!CTBa JIHlllb BXO.1l.HbIX HJlIl BblXO.1l.HblX
cpYHKlI.I1t1, 11 n09TOMY ,lJ,Jl51 annpOKCHMall.l111 BbI6HpalOT Te, cBoficTBa KOTO-
pblX I13BeCTHbl.
C 9ifJifJeKToM He3KBI1BaJIeHTHOCTH npn,lJ,eTC5I CHOBa BCTpeTI-ITbC5I B r JI. 26
npH p aCCMOTpeHl111 3KCnOHeHlI.HaJIbHbIX <PYHKlI.l1t1. TaK, npH HHTerpl1pOBa-
mlH cpYHKUHH e"i X nOCJle.1l.H5I5I nepeXO.1l.HT B caMY ce651 (c TO'lHOCTbIO
.11.0 nOCT05lHHOrO MHO)l{I1TeJl5I), 3a llCKJIIO'IeHHeM CJIY'Ia51 r:J.i = O.
fJJABA 22
CXO,LUfMOCTb P5UlOB <i>YPbE
22.1. CXO.1l.11MOCTb CTeneHHblX pHJ10B 1:1 pSlJl.OB <l>YPbe
CKOpOCTb CXO,lJ,HMOCTH CTeIIeHHblX P5l,lJ,OfJ H CKOpOCTb, C KOTOpOtl
CXO,lJ,HTC5I K CPYHKII.I1H nOCJIe.n:OBaTe.'lbHOCTb MHO)l{eCTB, COBlla,ILalOllI.llX C Het1
B 3a,ILaHHb1X TO'lKaX, Kor,ILa '1HC.'IO 9TI1X TO'leK B03pacraeT, 06e onpe.1l.e-
JI5lIOTC51 pacnOJIO)l{eHI1eM oc06eHHocretl ifJYHKIlHH B KOMnJleKCHot1 ilJlOCKOCTH.
STO He 3Ha'!I1T, '11'0, Bo0611 e rOBOp5l, nOBe.ll.eHHe <PYHKll.HH B.lI.OJlb .1I.et\CT-
22.2]
<l>YHKIlYlYI C npOCThIM PA3PbIBOM
295
BWreJlbHolJ OCH He orrpeJJ,eJl5JeT CKOpOCTH CXOJJ,HMOCTH, a CKopee 03Ha'1aeT,
qTO TPYJJ,HO rrepelJTI-I OT 3Ha'1eHHIJ Ha JJ,elJcTBHTeJlbHotl OCI1 K CKOpOCTli
CXOJJ,HMOCTH HH3'1e, K3K '1epe3 oc06eHHOCTH. Oc06eHHOCTH B KOMrrJleKcHolJ
rrJ10CKOCTH '1aCTO rrpaKTH'JeCKH HeB03MO)l{HO rroJlyqHTb; JJ,a)!(e rpy6bIe
oueHKH HMeIOTC5J peJJ,KO. HarrpHMep, paCC1I-IaTpHBaeTC5J 3aJJ,aqa BblqHCJJeHH5I
op6!1TbI JlYHbl. f{aK OueH!1Tb nOJlO)!(eH!1e 6. I1)1{at!lllHx oc06eHI-lonett
B KOMnJleKcHolJ nJ10CKoCTH?
HanpOT!1B, CXOJJ,HMOCTb P5JJJ,OB <PYPbe MO)!(eT 6bITb JlerKO orrpeJJ,eJJeHa
rro 3Ha'leH!15IM CPYHKUHH BJJ,OJlb .lJ,etlCTBWreJlbHotl OCH. B caMOM .lJ,eJle, Me)l{JJ,Y
H!1MH cyru.eCTBYIOT npOCTble cooTHorueHH5J. UeJlb HaCT05Jru.etl rJlaBbl Bbl-
BeCTH 9TH cooTHorueH!15J H nOKa3aTb, KaK HMH nOJlb30B3TbC5I. MO)l{HO
T3K)I{e OUeHl1Tb K09cpcpHl.I.HeHTbl P5J.lJ,OB <PYPbe JJ.O H3'13J13 Bbl'1HCJJeHHIJ,
qTO n03B0J1HT CJJ.eJl3Tb pa3YMHwe oueHKH Toro, CKOJlbKO qJ1eHOB nOHaJJ.o-
6HTC5I .lJ,JJ5J nOJJYQeHH5I 3a.lJ,aHHot! TO'1HOCTH (cp. C 21.5). TaKHM 06P330M,
B CT3.lJ,HH rrJlaHHpOB3HH5I Bhl'1I1CJleHHIJ Me-
TO.lJ, <pYPbe HMeeT Ba)l{Hhle npeHMyru.ecTBa
rrepeJJ. MHOrO'1JJeHHbIM MeTOJJ,OM.
22.2. <l>YHKUHH C npocTblM pa3pbIBoM
HatlHeM C np!1Mep3 (CM. pnc. 22.2-1)
CPYHKUIHI
t
Y= 21t (-1t<t<1t), (22.2-1)
'IC t
PH C. 22.2-1.
KOTOpa5J I1MeeT OLUm pa3pbIB BblCOTbl 1 B TO'1KaX t =:!: 1t (l1peJJ.noJla-
raeTC5J, '1TO y (t) nep!1oJJ.!1'1eCKa5J). Tal, KaK CPYHKUH5J HeqeTHa5J (T. e.
f (- t) = - f (t», TO Bce '1JleHbl C KocHHycaMH !1Ctfe3aIOT, B TOM qHCJIe
H rrOCTO>lHHbIIJ qJleH. f{09CPCPHllHeHTbl rrpH CHtlycax Ol1pe.lJ,eJJ5JIOTC5J TaK:
" "
bk= yet) sin ktdt= : sin ktdt=
-1t 0
1t
_ t -COSkt \ 1t + _l_ \ c osktdt= 1t k (-l)COsr.k= (-1t k-l .
- 1t 2 k 0 k1t 2 j
o
CJ1e.lJ,OBaTeJJbHO,
(t) _ 1 ( . t sin 2t + sin 3t sin 4t + )
Y --;:;- II Sill - --r - ....
8TOT p5J.lJ, CXOJJ,HTC5J K Y (t) npH Bcex 3HaqeHH5IX t, !CpoMe mo'l!Cu pa3-
pblBa (t = ::1:: 1t), r lI.e OH, OqeBl1l1.HO, CXOJJ.I1TC5I I{ HY JJIO (cpe.lJ,HeMY apHcp-
I 1
MeTHtlecKoMY JJ.BYX 3HaqeH!1t1"2 H - '[' KOTophle 5JBJI5IIOTC5I rrpeJJ.eJlb-
HhlMH 3HaqeHI15JMH rrpl1 t _ 1t HJlH -1t).
(22.2 2)
296
CXO,l1l1MOCTb P5IJJ.OB <t>YPbE
[In. 22
ECJIH nOJ/O)J(HTb
t = f - a + rc,
TO pa3pblB 6Y.lLeT nepeBe.lLeH B T04KY f = a. BMeCTo (22.2-2) HMeeM
(onycKaSi nnpHx Ha.lI t) JI,JlSI CPYHKlll1H Y (t) co CKa4KOM B T04Ke a:
co
I '\'1 (- I )k-I
y(t)=-:; '- -r- sin [(kt- ka)+ k'ltJ=
k=
co ro
I '\'1 ( I)k-I I sin (ka - kt)
= 1t "'" sin (kt - ka) cos krc = -; I.i k
k=l k=l
ro 00
'\1 sin ka kt I cas ka . kt
- 1t -'- -r COS - -;; .i.. Sin .
k=1 k=1
(22.2-3)
I
B STOM pSJJ1.e <I>YPbe KOscpcpHllHeHThI OUHTb y6bIBalOT KaK Ii"'
D.JJH CPYHKllHH C KOHe4HblM 4HcnOM M npOCTbIX p a3pbIBOB H nHHettHot!
Me)J{.lLY pa3pbIBaMH, y KOTOpOn CKa40K npH t = ai HMeeT BeJJH4HHY Yi'
MO)\{HO B3SJTb COOTBeTCTBYlOUJ.YIO JJHHettHYlO KOM6I1HaUHIO pSJJ1.0B BH.LI.a
(22.2-3) (BKJl104aSJ, 6blTb MO)\{eT, ew.e 4neH ao), 4TO .lLaeT
M 00 00
( t ) i '\1 ( ",sin kai kt cas kai . kt )
Y =-; Yi .i.. cos - Sill =
;=1 k=1 k=1
00 M 00 M
= ( Yi sin ka;) CO kt _ (! Yi cas ka;) Si kt .
k=1 i=1 1:=1 ;=1
Ole Mbl cpopMaJIbHO nepeMeHHJJI1 nOpH.llOK CYMMI1pOBaHHSJ.
TaKI1M 06pa30M, CPYHKllI1fJ C npOCTbIMH paspbIBaMH Ii rIpl!MOJlHHetlHbIMU
Y4acrK3Mu Me)\{J{Y paspbIBaMU .il.3lOT PSlJ{bI <I>YPbe, KOscpcpUllHeHThJ KOTOpbIX
1
y6bIBalOT KaK Ii; KpoMe Toro, MbI nOJ/Y'-IfjJIH npocTot! MeTOll .lLJIH rIO-
CTpOeHI1S1 t(1KI1X PSl.LI.OB.
IlJli! np (1KTH'-IeCKHX u.eJlet! BepHo H o6p aTHoe: eCJlH KOscPcpHUl1eHTbI
I
pi!.lla y6bIBalOr KaK Ii' TO CPYHKUHi! HMeeT npOCTble pa3pbIBbi. 4acro
BCTpe4aeTCi! CJIY4att HenpepbIBHott *) JIOMaHott, H nOToMY ero CTOHT
nCCJIeJ{OBarb OT.QeJIbHO, BbI'!HCJJIIM CHa4aJJa Koscj:JqmuneHTbI 4JIeHOB C KO.
") HanOMHHaeM, liTO AJiH HenpepblBHOCTH He06xo)lHMO TaKJKe {(1t) =/ (-1t).
\I 22.3J
<PYHKW1i1 C HErTPEPblBHbIMI1 BbICWI1MI1 rcPol13BolXHbIMI1
297
CI1HycaMI1, 0603Ha4a5I 4epe3 t i KOHUbl JIHHetlHbix Y4aCTKOa. /{03cpqmUl1eHw
flpM IWCI1Hye3x CYTb
n: 'i+1
'ltak = I (t) cas ktdt = I (t) cos ktdt =
-" i Ii
t i + 1
__ "'\1 I. . "\1 \ f' (t) sin kt dt
- '" Ii [f(ti+l) sm k (ti+l) - f(t i ) sm k (t i )] - .0;;;;. .) k .
i i Ii
f{or.Q3 eYMMHpOB3HI1e BbtnOJIH5IeTe5I Ha BeeM HHTepB3JIe, Bee 4JIeHbI YHH'lrO-
JKaIOrC51, TaK K3K I(t) npe.QnOJIar3JIaeb HenpepbIBHotl. B I1HTerpaJIb-
110M 4J1elle f' (t) 5IBJI5IeTC5I KOllcrallTotl B KaJK.QOM I1HrepBaJIe, CJle.QoBa-
reJIbllO, pe3YJIbr3T mITerpHpOB3HHH sin kt npHBollHT erne f{ O.QHOMY
1
MHO}KI1TemO Ii' B pe3YJIbT3re K03cpqJHUHeHTbI 'lJIeHOB e KOeH1-I}'eaMH
1
y6blBalOr KaK Ji.2' To JKe eaMoe npoHcxo.QHT .QJJ5I 4J1eHOB C CI-IHycaMH,
T3KHM 06P330M, .QJ/5! HenpepbIBHotl JloMaHotl Ko3cpcpHUI1eHTbI y5bIBalOT
I
KaK k2'
YnpamueHHe 22.2-1. HaHTH pRJJ, <Pypbe .1I.JlR nl1J1006pa3Hofi 4JYHK/.lf11f
{ I + t .1I.JlR (-I < t < 0),
f(t) =
t .1I.JlR (0 < t < I).
22.3. <l>YHKIJ.MH, MMeIOm.a!l HenpepblBHble npOH3BOlJ,Hbie 60JIee
BWCOKoro nopHAKa
ECJJH cpYHKUH5I HMeeT HenpepblBHYIO nepBYIO npoH3Bo.QHYIO 11 Henpe-
PbIBHYIO, KpoMe KOHe4Horo 4HeJIa T04eK, BTOPYIO npOH3BO.QHYIO, TO MOJKHO
noeTynl1Tb, KaK B 22.2, HHTerpl1p}'H .lI:BaJK.Qb1, 11 nOK33aTb, 4TO Ko3cpcp[1-
\
u,l1eHTbJ y5blB3lOT K3K Ji2 *)
" "
ak = - I (t) cos ktdt = - :k f' (t) sin ktdt,
-n -
TaK KaK 113-3a HenpepblBHoCTH npoHHTerpHpoBaHHbltl 4JIeH o6parnaeTC5I
B HYJ/b, Pa306beM oTpe30K HluerpHpOB3HH5! Herne pa3 npOI1HTerpl1pyeM
*) 60Jlee TO'lHOe ,IJ.OKa3areJlbCTBO n03BO.1Her YCTaIlOBI1Tb, 'ITa B :nOM c.1Y'Iae
1
K094JqJl1/.lHellTbJ <Pypbe cTpeMHTcR K llYJI!O KaK k3' (17PUM. ped.)
29R
CXOIlHMOCTb PHIlOB CPYPbB
[fn. 22
no lj3CTHM
M t j + J
ak = - f' (t) sin ktdt= - ni2
i= 0 Ij
M 'II:
i=O-1t
f" (t) cos ktdt,
I ak I ki 21t,
rIle M 2 ecn M3KCHMYM BTOpOti rrp0I13BOIlHOti. C b k IlOCTYIlHM 3H3J/OrHlJ-
HblM 06p330M.
1
ECJII1 K03cjJ<jmUl-leHTb! y6bIB31OT yme K3K k 2 , TO MO)j{HO 33KJIIO'IHTb, 'ITO
pHIl <t>YPbe CXO.ll.HTCH Be3Ile, T3K K3K
I ak cos kt + b k sin kt I 4 2 .
T3 me cxeM3, OlJeBI-I.lI.HO, rrpl-lMeHI1M3 K cjJYHKUI151M, I1MelOllJ:HM HeIlpe-
PbIBHYIO*) m-IO Ilp0I13BO.ll.HYIO; I-IX K03<jJ<jJI1Ul-leHTW <t>YPbe y6blB31OT K3K
1
k m + J (m - lieJIOe).
Ynpajf{HeHHSI
22.3-1. HafiTH pRJI <PYPbe JIJlR
{ t (n - t) JIJlR O.s; t.s; 1t,
{(t)=
- f (t) JIJlR 0> t? - 1t.
t 2
22.3-2. OYCTb f (t) = I - n 2 ( -1t E; t.s; 1t). KaK BeJIYT ceoR K094J4JflUHeHTbI?
22.4. Y J1Y'l1lleHHe CXOJl.HMOCTH pHJl.a 4>ypbe
P33JIO)l{eHI1e B pHIl <t>YPbe <jJYHKlil-lti C p33pbIB3MH HJII-I .lI.PYrHMH OCO-
6eHHOCTSIMI1 H3 IleticTBHTeJIbHoti OCI-I, oc06eHHO B !\oHU3X I1HTepB3J/3,
lJ3CTO BCTpelJ3eTCH B rrp3KTI-IKe. T3K K3K Y)l{e H3BeCTHO, lJTO oc06eHHOCTI1
H3 .lI.etlcTBHTeJJbHoti OCH orrpe.ll.emJIOT CKOpOCTb CXO.ll.I-IMOCTI-I, eCTeCTBeHHO
IletlCTBoB3Tb IlOlJTI-I TaK, K3K H B 4.2: 113 Il3HHoro pSIlJ.3 BbIlJeCTb COOT-
BeTCTBeHHO Bw6p3HHbIti, T3K)I{e MeIlJJeHHO CXOtl:SIUI,HtiC5I. T3KI1M 06P330M,
.lI.JI5I BbIlJHCJIeHl1ti 6YIlYT IlOJJYlJ3TbC5I 6bICTpee CXOIl5IUI,I1eC5I pH.lI.b!. M.lI.eR
.lI.OCT3T04HO OlJeBI1.l1.H3, 11 HeT He06xOIll1MOCTI1 OCT3H3BJII1B3TbC5I H3 HeJ1
60JIee IlOIlp06HO.
}(03<jJ<jJHI l1eHTbI ak 11 b k 3a.ll.31OTC5I HHTerp 3JIaMH, KOTopbIe 'I3CTO He
MorYT 6bITb BbIlJl1CJIeHbI 11 HX rrpl1XO.ll.I1TC5I H3XO.ll.HTb lJl1CJIeHHO 110 3H3'1eHI1HM
rrO.ll.bIHTerp3JIbHOil <jJYHKUHI-I B HeCKOJIhKI-IX Y3JIOBbIX TOlJK3X TaKHM 06p330M,
I
*) K094J4JHlJ.HeHTbJ <Pypbe flMeIOT nOpR,IJ.OK k mH TaK)j(e H llJIR 4JYHKU,HH,
Y KOTOpbIX m-R npOfl3BOJIHaR KYCO'lHO HenpepblBHa, T. e. MO)j(eT HMeTb KOHe'lHOe
IJHCJlO KOHellHblX CKaIJKOB. ([JPUM. pea.)
!> 22.51
cnEKTP MOiliHOCTVI
299
npHXO.lUHC5I CHOB3 CT3JIKHBaTbC5J C 3a.Q3'J3MH MHMHKpHH. O.QH3KO, npe!K.Qe
'JeM .L\eJI8Tb 9TO, BbJ'JTeM «J(aHOHH'IeCKHe» pa3JIO!KeHH5J OC06eHHocTeti,
Ta!(He K3K (22.2-3), a TaK!Ke HHTerpaJIbl OT HHX. 4T06bl cKoMneHcHpoBaTb
oc06eHHocTH cjJYHKUHH H ee npoH3Bo.QHbIX. Tor.Qa npH.QeM K 'JHCJIeHHoMY
HaXOiJWeHl11O KOscjJ<pHUHeHToB 6b1CTpO CXO.Q5IIlIerOC5I p5J.Qa .QJI5I 04eHb
rJI3.L\KoJ! cjJYHKUHH; MHMHKpH5J He 5JBJI5teTC5I Y!Ke )J,JI5I Hee cepbe3Hoti npo-
6J1eMoti. TaKHM 06pa30M, Bbl4liTaHHe oco6eHHocTeti He TOJlbKO YMeHbillaeT
BbJ411CmUeJlbHYIO pa60TY, HO TaK!Ke YJJY41llaeT TOtIHOCTb.
Ynpa>KHeHHe 22.4-1. BbI'!I1CJlHTb llepBble Tpn HHTerpana paBeHCTB (22.2-3)
)J.JlR l1oJlY4eHHR KaHOHl1'1eCKoi\ 4JOpMbI COOTBeTCTBYlOllllIX oc06eHHocTel1:.
22.5. CneKTp MOUJ.HOCTH
KaK B 4HCTOtl, TaK Ii B npHKJla.L\Hott MaTeMaTHKe, 06bJ1.1H0 HIlIYT HH-
BapHaHTbJ npe.L\CT3BJleHH5t - I1HuapHaHTbl no OTHOllieHHIO K KJlaccy npe06p a-
30BaHHtt. B KJlaCCe nepHO.IJ.114eCKI1X cjJYHKUl1tt nepeHOC ocett
t=t'+b
He .IJ.OJI!KeH MeH5JTb B rrpe.IJ.CTaBJleHIII1 cjJYHKlum TOro, 41'0 He 3aBHCHT
OT CHCTeMbl KOOpJl,HHaT. HerrOCpe.IJ.CTBeHHO BH.L\HO, 4TO KOscjJcjJHUHeHTbI
<!>YPbe ak H b k He 06JIa.IJ.afOT 9THM CBOt!CTBOM H MeH5tfOTC5t npH C.l(BHre
OCH, T. e. Kor.IJ.a H3MeH5JeTC5J Ha'I3JlO OTC4eTa BpeMeHH. f10JIara5t t = f + b
H HCnOJlb3YH rrepnO.lu-r4HOCTb I (t), 4T06bl C.IJ:BHHYTb npe.IJ.eJIbl B HHTerp aJIe.
110JIY4aeM
ak= I(t) cos ktdf= I(f + b) cos k (f + b) df =
- -
=[ f(f +b) cos kfdt]cos kb- [ I(f +b) sin Mdt] sin kb=
- -
= a'r, cos kb - b'r, sin kb.
AHaJIOrWIHO
b k = ak sin kb + b k cos kb.
XOT5t ak H b k He fiHBapHaHTHbI, BeJII1'IHHa
at + b% = (a;' cos kb - b k sin kb)2 + (ak sin kb + b'r, cos kb? =
= (aiY + (b;J2,
04eBH.lLHO, HHBapI-IaHTH3. BeJlH'IHHY ak + bj. Ha3bIBafOT .MOll1,flOCftlbFO '-Ia-
CTOTbI k 11 I1306pa!KafOT B BH.lLe .lLI1CKpeTHoro cneKmpa .Moll1,HOCfTln.
CymeCT ByeT n yraHHIla OTHOCHTeJlbHO Toro, 'ITO .lLaeT cneKtnp at + b;.
HJlH Vak + bk' HO Mbl 6Y.IJ.eM HCnOJlb30BaTb CJIOBa «cneKTp MomHOCTH»
H «cneKTp». I1Me5t B BH.IJ.Y O.IJ.Hy H TY )J(e BeJJH4HHY a + N.
Ynpa)l{HeHHe 22.5-1. BbI4HCIIHTb cneKTp 4JYHKU,HH H3 ynpaJKHeHHIJ 22.2-1.
300
CXOJUlMOCTb P5IJ10B <l>YPbE
[TJJ. 22
22.6. SIBJIeHlle rMooea
HallHeM C lIaCTHoro CJJY'Ia51 np5lMoyrOJJbHot! BOJJHbl H(t) c nepl10.1l.0M
271: (pHe. 22.6-1). ECJJl1 BblllHCJJHTb eYMMY nepBblx 2n '1JIeHOB, TO Bce
lIJJeHbI C KOCHHycaMH 6YJ1YT paBHbl Hymo H nOJIY'1aeM
n
H 2n (t)={+ : l: 2k 1 I sin (2k-l)t.
k=1
(22.6-1 )
rH66c OTMeTHJJ, liTO lIaCTHlIHa51 CYMMa H 2n npeBOCXO.1l.HT <PYHKI.I.HIO Ha
HeKOTOPYIO BeJJHlIHHY (pHC. 22.6-2). BOJJee TOllHO
H 2n ( 2: ) ---+ 1,08949 . .., Kor.1l.a n - 00 . (22.6-2)
)1ei-icTBllTeJJbHO, H 2n (t) He TOJJbKO npeBOCXO.1l.11T <PYHKI.I.HIO H (t), HO II
1
H(t)
--, 1
I I
I I 1/2
I I
I
-rr 0 rr 2rr Jrr t [1
rr t
PHC. 22.6-1. npRMoyroJlbHaR BOJIHa. PHC. 22.6-2. BJleHHe fH66ca.
HMeer TeHJ1eHll.H10 KOJJe6aTbC5I OKOJJO H(t), H KOJJe6aHIl51 YMeHbwaIOTC5I
Me.1l.JIeHHO, KOr.1l.a t Y.1l.aJI5IeTC5I or pa3pblBa.
4To6b1 06"bHCHHTb 51BJJeHHe, 3anHweM (22.6-1) KaK
n t
H 2n (t)= + l: cos (2k - l)xdx =
k=10
t n t
1 2 ,.., I + 1 sin 2nx
=- 2 +- ., c os(2k-1)xdx=- 2 - dx,
1t .iI..J 1t SIIJ X
o k=1 0
(22.6-3)
r.ae llcnOJIb30BaHa cjJopMYJIa
n
l: sin 2nx
cas (2k - 1) X = _ 2 ' .
smx
k=l
113 (22.6-3) HCHO, liTO MaKCHMYM II MHHHMYM .11J!H 0 t 'it ,aOeTH-
ralOTCH B TOllKax
dH 2n (t)
dt
sin 2nt _ 0
1t sint - ,
22.71
CH fMA-MHO}!{HTEJJH JJAHU,OWA
301
T. e. npH
t= m 1t
2n'
m= I, 2, ..., 2n-1,
(22.6-4)
H 4TO OHH 4epe.QYIOTC5I. I1x BeJII1411HbI 6blJIH BbI4f!CJIeHbl KapcJIoy [7J.
To, 4TO BepHo )J:JI5I 9TOtl Cl1eUl1aJIbHotl cjJYHKUHI1, 04eBH.QHO, BepHo
11 .QJI5I 60JIee 06Il!.l1x cjJYHKUHtl, TaK KaK pa3pbIB MO}!{HO paccMaTpHBaTh
KaK B03HI1KalOIl!.l1tl H3 np5lMoyrOJIbHotl BOJIHhI, npl16aBJIeHHotl K r JIaBHOfi
cjJYHKUHI1.
Ynpa}J{HeHHe 22.6-1. j],JliI 4JYHKIJ,HH B ynpa)/{HeHHII 22.2-1 HalJepTHTb CYMMY
nepBwx lJeTWpeX lJJleHOR
22.7. CHrMa-MHo}J{llTeJIll JIaHII.Orna
3aMeHI1M 6h1CTpO KOJIe6JIlOllI.YIOC5I cjJYHKII.111O Hm (t) cr Jla}!{eHHOfi
cjJYHKul1etl
t+
_ n \.2 n
H 2n (t) = J H 2n ('i.) d'i.,
t-
2n
(22.7-1)
r,lLe ycpe.QH5IeTc5I O,lLHO nOJIHOe KOJIe6aHHe H 2n (t), cOCpe,lLOT04eHHoe
OKOJlO t.
TIO,lLCTaBllB (22.6-1) B (22.7-1), nOJJY4aeM
"
t+2/i n
- n \. [ 1 2 '\1 I. ]
H 2n (t) = j :2 + -; 2k _ I sm (2k - 1) 'i. d't =
" k=1
t--
2n
n t+"
n [ 1t 2'\1 -I I 2n ]
=-; 2n +-; (2k_l)2 cos(2k-1)'t1 ,,=
k=1 t- 2n
. I 2 n I sin [ (2k - 1) 2 1t ]
= -:y + - '\l _ . n sin (2k - 1) t.
- 1t k 2k - I (2k - I) 2:
(22.7-2)
EeJIH epaSHI1Tb 9TO BbIpa}!{eHl1e C H 2n (t), TO rrOJIY4HM ,lLOnOJIHHTeJIb-
HbIt! MHO}!{llTeJIb
1t
sin (2k - 1) 21i
°2k_1 = 1t
(2k - I) -
2n
(22.7-3)
.lJ:JJ5I Ka}!{,lLOrO 4JIeHa npll CYMMllpoBaHllH.
302
CXOLlI1MOCTb P51LlOB <l>YPbE
[TJJ. 22
3<fJ<fJeKT 9TOrO MHO)l(HTeJHI Ok COCTOHT B TOM, 'iT06bJ YMeHhUlI1Tb MaK-
CIIMYM C 0,08949 LlO 0,01187 11 nepBbl!1 MHHHMYM C 0,04859 LlO 0,00473
II T. ll.. TaKHM 06pa30M, SIBJJeHHe rH66ca CHJIbHO YMeHbUlHJIOCb OT npH-
CYTCTBI1S1 O-MHO)!{HTeJJet!, KOTopble B03HHKJIH 113 cr JIa)l(HBaHHSI H Zn (t)
Ha KOpOTKOM IIHTepBaJIe .n:mlHo!1 1':/ n.
Mbl H3YQHllH lfacTHbl!1 CJIYlfat! npSlMoyrOJJbHOt! BOJIHbI; nOKa)l(eM Tenepb,
4TO BJJIISIHHe 0k-MHO)!OUeJJe!1 OC"raeTCSl TeM )I(e .n:JJSl JIIo60ro pSl.n:a <l>YPbe.
nYCTb I (t) (0::;::; t.;;;; 21':) HHTerpHpyeMaSl H nYCTb
91<
ak = I(t) cos kt dt,
o
k=O, 1, 2,....
21<
b k = I(t) sin kt dt,
o
k=1,2,....
- KOscp<pI'IIJ.HeHTbI <l>ypbe. Tor.n:a
n
fn (t) = a 2 + ! (ak cos kt + b k sin kt).
k=l
Tenepb BbllfHCJIHM
t+.2'...
2n n t
f, (t)= \' f, ( 't)d't= [ ao . + V ( ak Sink"t _bk cOSk't )1 2n ] =
n 1r./n J n 1r. 2 n i.. k k 1t
t-'::'" k=l t- 2n
2n
n
= : ( 2: aO+ ! { a; [Sin k(t+ 2 : )- sin k(t- ;n )J-
k=1
- b; [ cos k ( t + iz) - cos k ( t - i,;) J}) =
n
a o + n \1 ( 2ak . 1r.k kt + 2bk . rr.k . kt \
= :2 -; k.. T sm 2n COS -/i- SID 2n SID j =
k=1 '
. rr.k
n SJIJ-
a V 2n
= + i.. --.;;Jl (ak COS kt + b k sin kt)
k=1 2n
H On5ITb nOJIYQaeM O-MHO)l(HTeJIH (paBeHcTBo (22.7-3»
. rr.k
Stn 2n
Ok = --:;Jl'
2n
(22.7-4)
22.81
CPA.BHEHIifE METOnOB CXOllHMOCTH
303
BCTaBJleHHble B pa3.1H4Hble K09<jJ<jJI1IU1eHTbl p5l.1l.a <PYPbe. 3aMeTI1M, qTO
4J1eHbl sin kt H cos kt 06a I1MelOT MHO}[{I1TeJlb ak'
YnpamueUHe 22.7-1. flpHMeHHTb a-MHOiKHTeJlI1 B ypaBHeHIIH 22.6-l H Ha'lep-
THTb pe3YJlbTaT.
22.8. CpaBHeHHe MeTOJI.OB CXO.l1HMOCTH
f{poMe MeTO.1l.a a-MHO}[{I1TeJlet! J1aHlJ.OWa, cyw.eCTByeT xopowo H3BeeT-
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1,7
1,0
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:t:
L7,8
:::
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f'
0.5
0.5 0
0.2
0.4
0,6 0,8 1,0
BpeMJl, eel(
1,2
1,4
7,6
7,8
N
l 2 l
PHC, 22.8-1. 4acTHlfHble CYMMbl <Pypbe ="2 + "it "" 2n + I sin (2n + 1) t;
N n=O
u I I 2(N-n)+1 . .
<PeHepa = -'[ + 1t (N + I) 2n + 1 sm (2n + I) t, <Pypbe C a-MHO-
n=O
»J:IITeJleM
N . (2n + 1) 1t
I 2 1 sm 2N + 2 .
CXO.ll.HMOCTI1 ="2 + "it 2n + 1 (2n + I) 1t sm (2n + l)t.
n=O 2N+2
MeTOll JIaHlJ.owa TOJlbKO CHJlbHO raCHT ero. Prrc. 22.8-1 JJ;aeT COOTBeTCT-
BYlOw.ee cpaBHeHrre KpHBblX
pflaa C/Jypbe
CYAt.Mbt C/Jeiiepa
a-AtJiOJICUmellfl JlaftlJ,oma
304
cxomiMOCTh p nOB <l>YPbE
rrJ1. 22
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I I
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eel(
flpH6JIH)/{eHHe <Pypbe H npH6J1H)/{eHHe <Pypbe C MHOJKHTeJleM CXOJlI1MOCTH.
PHC. 22.8-2 fIOKa3bIBaeT CKOpOCTb, C KOTOpOft MeTO.u. cr-MHoIKHTeJI5I
fIpH6JIHIKaeT fIp5lMoyrOJIbHYIO BOJIHY KaK <jJYHKUHIO n. TaKHM o6pa30M,
JlalKe llJI5I YMepeHHoro n {(BpeM5I pOCTa» KpHBOft OqeHb HeBeJJHKO.
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STOM qaCTO fIOJIYQaIOTC5I HelKeJIaTeJJbHble B <jJH3HqeCKHX 3allaqaX 60JIbilme
KOJIe6aHH5I BblCOKOft qaCTOTbI.
JIaHUow [23] npellJIOIKHJI npaKTH'lecKHt1 fIpHeM, COCT05lUI.Hfi Ii HCfIOJJb-
aOBaHlUl
f(t+ )-f(t- )
21t
n
(22.9-1 )
B KaqeCTBe Oll.eHKH npoH3Bo.n:Hot1 t(t) B TOqKe t, BMeCTO npe.n:eJJbHOrO
npoll.ecca, KOTOpblfi HMeeT COMHl1TeJIbHOe <jJH3H'leCKOe 3HaQeHHe, llJ]H llH<jJ-
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305
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MbI He 6YlIeM BocrrpOH3BOlIHTb 31IeCb BbIB OlIbl; 3aMeTHM TOJIbKO, 4TO
me :JICe caMbte cr-MHO:JICU meAU (22.7-4), BBe.n:eHHbIe B cpopMaJIbHO lIHCP-
cpepeHuHpoBaHHblt! pH.Q, lJ.aIOT COOTBeTcTBYIOIll\1t! pe3YJIbTaT.
fJJABA 23
HEnEPHO HqECKHE YHKUHH. HHTErPAn YPbE
23.1. UeJIb rJIaBbI
B rJIaBaX 21, 22 rrepHOlIH4eCKHe CPYHKUI1H arrrrpoKcHMHpOBaJIHCb JIHHet!-
HOt! KOM6HHaUHet! rrepl101IH4eCKHX - CHHYCOB H KOCHHYCOB. 06paTHMCH
Terrepb K H3Y4eHHIO HenepHOlIH4eCKHX (PYHKUHtI 11 HX anrrpOKCHMall.HH
CHHycaMH 11 KocHHycaMH. BOT npHMep HerrepHo.n:H4eCKoti CPYHKll.HH, co-
CTaBJIeHHoti H3 lI.BYX rrepHOlI.IJT-IeCKHX,
y (t) = cos t + cos (1/2t)
(TaK KaK 1 H V 2 HeC0I13MepHMbI, TO y (t) HerrepI101IH4eCKaH). ITpH6JJH)\{eHl1e
HerrepHOLJ,114eCKoti CPYHKll.1111 nepHOlI.H4eCKI1MI1 HH4YTb He xy)\{e, 4eM rrpeLJ,-
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HerrepH01I114eCKHX 4JJeHOB
1, x, x ,...
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6bITb, Bce 4aCTOTbI 113 1IaHHOrO I1HTepBaJIa *). OCHOBHblM HHCTpYMeHTOM
rrpH 3TOM CJIY)\{HT HHTerpaJI <t>YPbe, KOTOpbIti 6YlI.eT cetl4ac KOpOTKO
pacnlOTpeH; 3a 60JIee CTporHM H3JIO)KeHHeM OTCbIJIaeM 4HTaTeJJ5I K CTaH-
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nOMoralOT BbHlEI1Tb npeHMyw.eCTBa HaCT05lw.erO MeTO.lIa Ha 3Tallax IlJIa-
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23.2. 0603HaQeHHH H KpaTKoe H3JIO}l{eHMe pe3YJIbTaTOB
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.lIOKa3aTeJlbCTE 60JIbWHHCTBa C.lIeJIaHHbIX YTsep)l{.lIeH11 t!.
P5I.lI <t>YPbe Ha HHTepBaJJe - N t N MO)l{HO 3alll1CaTb Tal, (CM.
(21.3-1) 11 (21.3-2»:
00
co
f(t)= a 2 + ak cos kt+ !b k sin Hkt,
k=1 k=1
r.lIe
N
ak= t(t) cos };ktdt
-N
N
bk= j(t) sin };ktdt
-N
ECJJn nCIlOJIb30SaTb KOMf1JIeKCHOe
rpYHKIlHtt
e,a + e-,ij
cos 6 = 2
TO'f10JJYlIllM
00 ":kt
I(t)= CkeN ,
k=-oo
r.lIe
(23.2-1 )
(k = 0, 1, 2,.. .),
(23.2-2)
(k = 1, 2,. ..).
npe.lICTaBJJeHl1e Tpl1rOHOMeTpl14eCKlIX
e,ij _e-,a
sin 6=
21
(23.2-3)
N i1t
I \' --kl
Ck= 2N f(t)e N dt.
-N
JIerKO BM.lIeTb, tITO
f ak - ib k
I 2 '
Ck= I 1,
ak+1bk
2 '
(23.2-4)
k>O,
k=O,
k<O.
(23.2-5)
23.21
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307
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nOJJO)KI1TeJJbHY!O, TaK H OTpl1lJ,aTeJJbHYIO lJaCTOTY.
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00
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f.ll.e
00
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(23.2-7)
-00
<I>YHKlIJiSl F (cr), rpy60 rOBOp5l, cooTBeTCTByer K03cpqmlll1eHTaM c k B p5l.ll.e
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-&1<0<&1
IlIHpl1HbI 2&1 H eCJJI1 HHTepBaJI Me)j{.lI.Y Y3JlaMI1 paBeH t, TO, lJT06bI H36e-
)j{aTb HaJIOJKeHH5I, He06xo.ll.HMO HMen
2&1M< 1.
(23.2-8)
308
HEf1EPHOJ],114ECKHE 4>YHKUHH. HHTE fP AJJ 4>YPbE
[rJJ. 23
\1Ha4e rOBOp5l, 'lT06bl H36e)J(aTb HaJJO)J(eHH5I, Mbl .aOJJ)J(Hbl I1MeTb rro Kpan-
Hen Mepe .aBa Y3JJa Ha CaMOn KOpOTKOt! BOJJHe, rrpHCYTCTBYIOUl.et! B pa3-
JJO)J(eHHH.
ECTeCTBeHHO B03HI1KaeT BOTIpOC: «MO)J(HO JJI1 BOCCTaHOBI1Tb cpYHKlJ.HIO
C OrpaHI1'leHHbiM crreKTpOM rro ee Y3JIaM, eom B caMOM .aeJIe CJJe.aOBaTb
(23.2-8)>>? I1HTYI1THBHblM apryMeHToM 3a TO, 'lTO 9TO B03MO)J(HO, 51BJJ5IeTC51
aHaJJOrl151 C I1HTeprrOJJ5IlJ.liOHHOt! cxeMot! J1arpaH)J(a ( 8.3). <t>YHKlJ.I151
sin ret
ret
06JJa.n:aeT TeM cBotieTBoM, 'lTO ee 3Ha'leHHe paBHo 1 IIpl1
npH t= + 1, + 2, + 3,... TaKHM 06pa30M, cpYHKlI.H5I
sinre(t-k)
re (t -k)
x=O H HYJIIO
(23.2-9)
HrpaeT TY )J(e pOJJb, 'lTO 11 Lk B HHTeprrOJJ5IlJ.I1OHHOt! cxeMe JIarpaH)J(a,
TaK KaK OHa paBHa 1 rrpH t = k H HYJJIO BO Beex OeTaJJbHbJX Y3JJax.
naJJee,
f(k) sin re (t - k)
7C (t-k)
rrpHHHMaeT 3Ha'IeHl1e cpYHKlI.I1li f(k) rrpn t = k H HYJJb BO Bcex OCTaJJb-
HbJX Y3JJaX; CJJe.aOBaTeJlbHO, cpopMa.'1bHOe pa3JJo)J(elme
00
f(k) sin re (t-k)
'-' re(t-k)
k=-oo
rrpoxo.aI1T '1epe3 Bee Y3JJbJ. J1erKo Bn.aeTb, 'lTO
1
sin ret \ I i1ttcr d
----:;t = j 2 e a.
-1
(23.2-10)
sin ret
CJJe.n:OBaTeJlbHO, cor JJaCHO (23.2-6) '1aCTOTbI, npncYTcTBYIOUl.He B ---;r-.
orpaHH'leHbJ. 3.n:ecb a MeH5IeTC5I OT - 1 .ao 1; 4T06bl rrpe.acTaBHTh
9KerrOHeHTY B cpoPMe (23.2-6), CJJe.ayeT B351Tb
1/
sin ret \ Q 7ticr't d '
---;;z- = e- a,
-1/
IIOCJJe '1ero IIOJlY'laeM HHTepBaJi - < 0' < . Eho COOTBeTCTByeT
o)J(l1.n:am151M; B351B HHTepBaJJ Bbl60pKI1 !::J. t = 1, MbJ rrOJlY'lHJ1li, 'lTO 'laCTOT3
1 I
npo6eraeT liHTepBaJJ OT -2 .n:o 2' ITo cyrn.eCTBY, 110JlY'leHHoe pa3JJO-
II 23.21
OB03HA4EHH H KPATKOE 113flO EHHE PE3VflbTATOB
309'
eHHe npe,lJ,CTaBJI5IeT c06011 meope.MY 8 bl 60plC 11 TeOpHH MH<j.JOpMalI.MM*);
eCJIM 3a,lJ,aHbl He06xo,lJ,MMble paBHOOTCT05lIll.Me Y3JJbl ,lJ,JI5I <j.JYHKlI.MI1 C orpa-
H!14eHHbIM crreKTpoM, TO no 9THM Y3JJaM MO HO BOCCTaHOBI1Tb <j.JYHKlI.MIO.
TIo aHaJlOr!1!1 C 22.5 BeJII14!1Ha
IF (cr) 1 2
(23.2-11 )
4aCTO Ha3b1BaeTC5I cneKTpoM MOlljHOCT!1; CJIOBO «MOIll.HOCTb» np!1wJJo
!13 HH eHepHblx np!1MeHeHMI1, 11 Mbl !1CnOJlb3yeM ero npocTo no Tpa,lJ,M-
liMM. CneKTp MOlliHOCT!1, !1JJI1 npocTo cneKTp, nOXQ}K Ha 06bl4Hblti
onH14ecKMI;j: KaK BXO,lJ,51Il1.MI1 JIYt.J CBeTa pa3JJaraeTC5I npI13Mol1 Ha OT,lJ,e.'Jb-
Hble lJ,SeTa (4aCTOTbl), TaK !1HTerpaJl <PYPbe pa3JIaraeT <j.JYHKliHIO Bpe-
MeHI1 t(t) Ha rap IOHI1KI1 aMfIJII1TY,lJ,bl F (cr).
B 23.7 6Y,lJ,eT nOKa3aHO, t.JTO ,lJ,JI5I <j.JYHKliI1!1, 3a,lJ,aHHol1 Ha 6ecKo-
He4Hol1 llp5lMOll (- 00 < t < 00), Bbl60p Y3JIOB Ha orpaHH4eHHoM oTpe3Ke
,lJ,OJI eH paCWI1pI1Tb cneKTp, !1 t.JeM Kop04e ,lJ,JJ!1Ha Y4acTKa, Ha KOTOpOM
. -6epyrc5I ,lJ,aHHble, TeM 60JIbWe paCWI1peHHe CfIeI<Tpa.
CKa3amloe Bblwe pacKpbIBaeT cYTb 9TOll rJIaBbI M nOKa3bIBaeT ee
3H34eHI1e ,lJ,JI5I Bblt.JI1CJIHTeJIbHOll npaKTI1KI1. 5lCHO, 4TO H 9<j.J<j.JeKT paccTa-
HOBKI1 Y3JIOB, 11 9<j.J<j.JeKT Bbl60pa KOHet.JHoro t.JMCJIa Y3JIOB MorYT 6bITh
XOT5I 6bI OT4aCTH nOH5ITbl B paMKax TeopHH <j.JYHKlI.Hll C orpaHM4eHHblM
cneKTpoM, HO HX HeJIh351 nOH5ITb B paMKax KJIaCCl1t.JeCKOll TeopHM nOJlM-
HOMHaJIbHblX llpH6JIH eHMll.
B OCT3BWellC5I t.J3CTH rJI3Bbl p33BHB3eTC5I TeOpH5I MHTerpaJl3 <PYPhe
B TOll Mepe, B K3KOli 9TO He06xo,lJ,MMO, 4T06hl HeMHoro n05lCHHTb CKa33H-
Hoe Bblwe. Mbl He XOTMM OK33aTbC51 BOBJle4eHHblMH B BonpOCbl M3TeMa-
TH4eCKoli CTporOCTH, TaK KaK KJIaCC <j.JYHKlI.l1Ii, KOTopble ,lJ,eIiCTBHTeJlbHO
np06YIOT afInpOKCHMI1pOB3Tb Ha Bbl4MCJlMTeJIbHblX MaWMHax, orpaHH4eH
«<j.JYHKliH5IMH C XOPOWHM nOBe,lJ,eHHeM» 11 He BKJIIOt.JaeT naTOJIOrH4eCKHe
CJIY43H, KOTOpblMM TaK t.JaCTO 33HHMaIOTC5I MaTeMaTMKH.
IlJI5I Tex, KOMY TPY,lJ,HO MaHHnYJIHpOBaTb C KOMllJleKCHblMH 4HCJIaMH,
MO HO npe,lJ,JIO HTb CJIe,lJ,YIOIlI.He npOCTble Yllpa HeHH5I ,lJ,JI5I npaKTHKM.
ECJIH OHH Bbl30BYT 3aTPY,lJ,HeHH5I, 6Y.lJeT JIYt.Jwe, npe .lJe 4eM .lJBHHYTbC
.n.aJIbWe, eIll.e pa3 nOBTopMTb 9TY TeMY.
YnpaiKHeHHSI
23.2-1. j],OKa3aTb, 'ITa I e'x I = I (JJ,JlR JJ,el1cTBHTeJlbHblX x).
23.2-2. j],oK!l3aTb, ,!TO TpHroHOMeTpH'IeCKl1e 4JOPMYJlbl CJlO)KeHHR CJleJJ,YIOT
H3 paBeHcTBa e!ae! = e' (a+ ).
23.2-3. a ib = U + iv; Bblpa3HTb U, v 4epe3 a H b.
*) B HameH .1HTepaType no TeopHH HH4JopMau,HH l(JlR 9TOH TeopeMbl npHHHTo,
Ha3BaHHe «reopeMa KOTeJlbHHKoBa». CooTBeTcTBYlOlltee pa3i10)j{eHHe Ha3blBalOT
P5liJOM f{orne.2bHUKOBa. CM. B. A. K 0 T e JI b H H K 0 B, <0 rrponYCKHOH cnoc06-
HOCTf1 «94J!1pa» H rrpOBOJlOKH». MaTepHa.1bl K flepBOMY BcecolO3HoMY C"he3)1Y
rro BorrpocaM peKoHcTpYKUl111 JJ,eJla CBR3H, M., 1933. (DpUM. peiJ.)
:itO
HErIEPI10LUi<{I!CKI1E <PYHKUH"I1. H"HTI! rp AJT <PYPbE
1m. 23
t
23.2-4. u + Iv = (a + ib) 2, HafiTI1 U, v (06paTHTb BHHMaHHe Ha 3fJaK u).
00
23.2-5. j],OKa3aTb, <{TO eCJII1 Ck = C_k, TO CYMMa pR.11a Cke ikt JJ,efi-
k=-oo
<:TBHTeJlbHa.
23.2-6. j],oKa3aTb, qTO eCJlH m II k - u,eJlble, TO
e/mxe-ikx dx = { 0,
-1t 21t,
m*k,
m=k.
23.2-7. nYCTb f (t) = t (-1t < t < 1t). Hai1T11 K094J4JI1I.lHeHTbI <PYPbe 11 pa3-
JlO)/{eHIIII
00
f (t) = Ckeikt.
k=-oo
00
23.2-S. flYCTb f (t) = F (cr) e21ticrt dcr.
- 00
HailTH npe06pa30BaHHe <Pypbe 1J.JlR f' (t), r (t), tIkI (t).
23.2-9. flYCTb
{ t (-1t<t<1t),
f (t) =
o B OCTa.1bHbIX TO'lKax.
HafiTH F (cr).
23.2-10. flYCTb
{ cr (-1t <cr <1t),
F (cr) = 0
B OCTaJlbHblX TOqKax.
Hal1TH F (t).
23.3. I1HTerpaJI «t>ypbe
4T06hI «BbIBeCTIU I1HTerpaJI <t>YPbe 1-\3 pH.D.a <t>YPhe, I1CKJlIO<{HM K09<p-
4mUHeHTbI Ck' no.n:CTaBJJiHl (23.2-4) B (23.2-3):
00 N /1t k i1t kt
t(f) = ! [ t(f) e - IV t df] eN 2 '
k=-oo - N
npeJJJlO.'IaraerCH, '-!TO <PYHKIJWI f (f) nepHOJl.HqeCKaH B HHTepBaJIe - N
f N. IIJIH npI16JJH}[{eHI15! HenepHOJI.H<{ecKotl <PYHKUI1H 3acraBHM N
CTpeMI1TbC5! K 6ecKoHe<{HocTI1, JI.JI5! <{ero nOJJO}[{I1M
1
2N =Acr,
II 3aMeTI1M, <{TO B 9Tofi CYMMe paCCTOHHfJe Me}[{Jl.Y TO<{Ka:.1fJ, B KOTOpbIX
6epYTCH COCeJl.Hne 9KcnOHeHTbI, Be.n:eT ce6H KaK Acr, KoropaH npH N -J> 00
6y.n:eT crpeMHTbCH K Hymo. VlMeeM
00 N
t(f) = [ t(f) e-21tlkf'\°dfj e2"ikf O .:h
k=-.oo-N ."
2Hl
nPEOEPA30BAHVlR <l>YPbE HEKOTOPbIX <l>YHKUVlA
all
Ho kt:.o - a H I1PH N - co, rr0J1Y4aeM HHTerpaJl
00 00
t(t) = [ t(t) e-2 ito dt] e2 iot do.
-co -00
nOJIO)KHB Terrepb
00
F(a)= 1(t)e-21tita dt,
-00
(23.3-1)
IlOJIYlJaeM
00
t(t) = F (0) e21tiat do.
-00
(23.3-2)
fOBOp5IT, '-ITO <PYHKIJ,H5I F (0) 5IBJI5ieTC5I npeo6pa30BaHHeM <l>YPbe <PYHK-
UHH t(t). COOTHOllleHfIl1 Me)KJJ:Y 9TfIMH <PYHKUI15IMH nO'-lTH CHMMeTpfI4Hbl,
pa3HI1lJ,a TOJIbKO B 3HaKe nOKa3aTeJI5I 0 - - o. 06e <PYHKUI1I1 HecYT OJJ:HY
11 TY )Ke HH<popMaumo, TaK KaK Ka)KJJ:a5I MO)KeT 6bITb HattIleHa 113 .n:PY-
fOU, HO TOJIbKO B pa3HblX <popMax: f (t) B 06J1aCTI1 BpeMeHH, a F (0)
B 06JIaCTH '-IaCTOT. B03MO)KHOCTblO H3YlJeHH5I I1H<pO pM aUf!l1 B JII060M HS
JJ:BYX BHJJ:OB 11 06"b5ICHl1eTC5I B OCI-IOBHOM IJ,eHHOCTb npeJJ:CTaBJIeHfI5I <PYHK-
IJ,I1H B BH.n:e HHTerpaJIa <l>YPbe.
BbIweCKa3aHlIOe He 5IBJI5IeTCI1, KOHelJHO, CTpOrl1M JJ:OKa3aTeJIbCTBOM,
9TO Bcero JIHWb JJ:OBOJJ: B nOJJb3Y Taro, 4TO BblCKa3aHHbIe YTBep)KJJ:e-
HI15I BepHbJ. YCTpeMJI5I5I N K 6eCKOHelJHOCTI1, Mbl OTKa3bIBaeMC5I OT npe.n:-
nOJJO)KeHHU OTHOCI1TeJIbHO nepl10.n:fI4HOCTfI <PYHKUfIH.
23.4. npeoopa30BaHHe <l>ypbe HeKoTopblX <f!YHKU,HH
CooTHOWeHHI1 (23.3-1) fI (23.3-2) nOKa3blBalOT, 4TO Ka)K.n:oU <PYHK-
UHH I (t) cooTBeTCTByeT npeo6pa30BaHI1e F (0) 11 Hao6opoT. CYllI.eCTBYIOT
06WI1pHble Ta6JII1lJ,bl ([6J, [8]) <P0PMYJI, CB5I3blBalOIlI,I1X I(t) 11 F (0). Bbl-
BeJJ:eM TOJIbKO HeKOTopble 113 HHX - Te, KOTopbIe IlOHaJJ:065ITC5I .n:JI5I JJ:aJIb-
HetimeU pa60Tbl,
B Ka4eCTBe IlepBoro npl1Mepa paCCMOTpHM CPYHKUI1IO C OrpaHfIlJeH-
HbIM cneKTpOM
00
t(t)= F(0)e21tiot do,
-co
(23.4-1 )
r.n:e (pfIC. 23.4-1)
(
F(o)= { 2Q
l 0
llJI5I
io!<Q,
(23.4-2)
.n:JII1
lal>Q.
312
HEf1EPI101H14ECKI1E q,YHKUI1I1. I1HTEfP AJI q,YPbE
[fJI. 23
T3KI1M 06p330M, F o) OrpaHl-l4HB3eT ell.I-lHl14HYIO OllOUJ.3l1.b, 3 I(t) HMeeT
Bce lJ3CTOTbI B HHTepB3J1e - Q < 0< Q H HHK3KHX BHe ero. 1,1coOllb-
3Y51 (23.4-2), 330HU1eM (23.4-1) B BH.Qe
-.Q
r 5 )
Ij(?SI) :
I-
S}
&1
1 \'
I(t) = 2Q j e 'd'i d:J =
-2
_ J.- . e 2 ,,/crl \ &1 _
- 2Q 21tit - :< -
e 2 "19.1 _ e- 2,,121 I sin 21tQt
2i . 2Q1tt = 21tQt '
PI1C. 23.4-1. flpHMoyroJlbHbIH I1M-
lIYJlbC.
(23.4-3)
4ro coorBercrByer (23.2-10) (pHC. 23.4-2). I1cnoJlb3YH (23.3-1), 3aMe4aeM,
4ro rrpeo6pa30BaHHe HMeer BH.Q
00
F (0) =
sin 2nQt e- "icrt dt,
27tQt
(23.4-4 )
-co
T.Qe F(o) lI.aHa B (23.4-2).
B Ka4ecrBe BropOn HJIJJlOCrpal1.HH COOrHOllJeHHH Me)l(.rr.y npeo6pa30Ba-
l-HlHMH <1JYPbe npell.nOJlO)l(HM, "Ira 3a.rr.aHa napa CPYHKl1.Ht! I(t), F (0), npHtJeM
ro
I(t) = F(cr)e "icrtdcr.
(23.4-5)
-OJ
l{aKolt CPYHKIJ.Hfj 11 (t) coorBercrByeT npeo6pa30BaHHe <1JYPbe F (0) e "icry?
f{t)
_1.. .L t
m ZSJ
PHC. 23.4-2. <PYHKU,I1H C OrpaHlllJeHHbIM cneKTpoM
( Sin 21tQt )
21tQt .
HMeeM
00 co
11 (t) = F (0) e "icry e "icrl dcr = F (0) e 1ticr (y+t) da.
-00 -00
{10JIO>KlIM Y + t = z; rOr.ll.a !-I3 (23.4-5) cJle.ll.yer
00
It (t)= fez) =fCY + t)= [F (0) e 2roicry ] e2TCicrt do.
-00
(23.4-6)
23.51
<l>YHKlum C OTPAHI-14EHHblM CnEKTPOM 1-1 TEOPEMA BblEOPKI-1
313
TaKI1M 06pa30M, B pe3YJIbTaTe YMHO)({eHH5J cjJYHKll.HtI Ha 3KCnOHeHTY ap-
rYMeHT npe06pa30BaHHH C.lI.BHraeTC5J. YpaBHeHHe (23.4-6) 11 HO r.ll.a Ha3bI-
BalOT meope.Mou ciJouca.
23.5. <l>YHKU,HH C OrpaHHQeHHblM cneKTpoM
H reopeMa BM60pKH
B 23.2 6blJI .lI.aH HarJI5J.lI.Hblt!, HO He 04eHb cTporHtI BblBO.ll. meo-
pe.!.lbt Ob16opKU. BCJJe.ll.CTBHe ee Ba)({HOCTH .lI.JI5J BbI4HCJIHTeJIbHot! npaKTHKI-I
.lI.a.ll.HM .lI.pyrot! ee B bIB O.ll.,
UeHTpaJJbHaH l-I.lI.e5J TeopeMbI Bbl60pKll COCTOHT B TOM, 4T06bl 6paTb
Y3JIOBble T04KI1 cjJYHKll.HH C orpaHH4eHHblM cneKTpOM B.lI.BOe 4aULe HaHBbICIlleti
4aCTOTbI. IToHHTHe cjJYHKll.HI1 C orpaHH4eHHblM cneKTpOM IllHpOKO HCnOJIb-
3yeTcH BO MHornx 06J1aCT5JX HaYKH. Ha npaKTHKe 4aCTO BCTpe4aeTcH CJIY-
4at!, Kor.ll.a BHe HeKOToporo HHTepBaJla F (0) 04eHb MaJla, 11 MO.ll.eJlb"
cjJYHKIl.HH C orpaHWieHHb!M cneKTpoM HBJl5JeTCH nOJIe3Hott O.ll.HaKO He06-
XOJUIMO CKa3aTb HeCKOJlbKO CJ10B npe.ll.OCTepe)({eHfUl. ECJII-I cjJYHKUHR
c .orpamt4eHHblM cneKTpoM, TO MaTeMaTI-I'JeCKaR MO.ll.e.7Jb rOBOpl.JT, !JTO OHa
He MO)({eT 6bITb «orp3HH4eH3 BO BpeMeHI1», T. e. He MO)({eT 6bITb p3BH3
HY JJlO .lI.JIH Bcex I t I > to HH npH K3KOM to. B 43CTHOCTH, eCJIH I (t) npe.ll.-
CT3BJIHeT 9J1eKTpH4eCKHt! TOK, TO npH.lI.eTC5J .lI.OnYCTHTb, 4TO OH IlleJI
Bcer.n.3 II 6Y.ll.eT H.lI.TI1 Be4HO. COOTBeTCTBeHHO eom I(t) «orp3HH4eH3 BO
BpeMeHI1», TO OH3 He MO)({eT 6bITb <PYHKUHetl C orpaHH4eHHblM cneKTpOM.
O!JeBI1J1.HO, He HY)({HO C411T3Tb M3TeMaTI-I'JeCKY!O MO.ll.eJlb pe3JIbHO cY!lI.e-
CTBY!O!lI.et!; OHa 5JBJI5JeTCH nOJle3Hot! 3nnpOKCI1M3UHet! <pH3H4eCKliX 5JBJIe-
HHtI, HO He 06H33TeJIbHO T04HO COOTBeTcTByeT 11M.
4aCTOT3 HaJIO)({eW15J 5JCHO CBH33H3 co CMeIlleHl1eM 4aCTOT li3-3a Toro,"
4TO Y3J1bI Bbl6paHbl P3BHOOTCTOH!lI.HMH. 4T06b! Bb!8eCTH TeopeMY BbI60PKH"
H3lJHeM C <POPMYJI
00
F (0) = I (t) e- "iat dt,
-00
00
I(t)= F(o)e "iat do. (23.5-1}
-00
ECJlI1 FI (0) rrepHOJUI4eCK3H (pl1c. 23.5-1, a) c nepl10Jl.OM - Q < 0 < Q,
TO F 1 (0) HMeeT pa3JJO)({eHHe B p5J.lI. <t>YPbe
00 ;" k
F 1 (0) = 2: Ck e "2 a,
k=-oo
(23.5-2)
rn.e
2 in
C - "2 1<0 1 ( k ) 1 ( k \
Ck= 2Q J F 1 (0)e do = 2Q fl 2Q = 2Q f 2Q )-
-1/
(23.5-3}
314
I-lEITEPV!OllW-IECKV!E <l>YHKUVlI1. VlHTE fP AJJ <l>YPbR
[fJL 23
Tenepl> paCCMOrpl1M np5lMoyrOJJbHblfl HMnYJIbC P (0) (pHC. 23.5-1, 6)
P('J=! i Q
1lJJ5I
10\<Q,
10\>Q.
1lJJ5I
Do (23.4-3) ero npeo6pa30BaHHe eCTb
( t ) _ sin 21tQt
P - LnQt .
(23.5-4)
HaKOHeu, rrOCKOJJbKY npe1lnOJJaraJIOCb, 'no F (0) I1MeeT cneKrp, orpa-
HHl.JeHHbJtt HHTepBaJIOM - Q < (j < Q, paCCMOTpHM
F (a)=F 1 (a)P (a) 2Q
:H HCIIOJIb3yeM (23.5-2) H (23.5-3):
k ( k ) ';jka
F(o)= ! Cke P(a)2Q= ! II 2Q P(a)e .
k=-co k=-oo
Tenepb np0l13Be1leM o6paTHoe rrpeo6pa30BaHI1e II npllMeHllM reopeMY
<:1lBl1ra (paBeHcTBo (23.4-6»:
co . ( k ) 00
k ) sm 2nQ t - 2Q '\1
I(t) = co/I ( 2Q 27tQ(t- ) = I( :Q )
sin 1t (2[/t - k)
n (2Qt - k)
(25.5-5)
TaKHM 06pa30M, nOJJYIJHJIaCb TeOpeMa Bb16opKH.
.il.JISI caMOn lJaCTOTbI HaJIO)KeHH5I HeB03MO)/{HO BOCCTaHOBHTb CPYHKUHfO,
IJ:crrOJIb3Y5I TeopeMY BbI6oPKH, TaK KaK eCJIH tit = Slit (t) = sin 7tt.
F,(6)
a)
dJ
-Q 0 Q 6
PliC. 23.5-1. a) PI (a) - nepHOIU1'1CCKaH; 6) P (0)-
npHMoyroJlbHwA HMUYJlb
TO Mbl nOJIYlJl1M Bce 3Ha4eHHSI B Y3JIax paBHblMH HYJIfO H no reopeMe
BbI60pKIJ t (t) = O.
23.61
TEOPEMA CBEPTKI1
315
23.6. TeopeMa cBepTKH
.IT.pyrHM nOJJe3HbIM COOTHOllleHHeM, CB5I3aHHbIM C npe06pa30BaHHeM
<PYPbe, HBJJSleTCSI meope.ua CBeplnlCU. TI pe.LJJlOJIOmIJM, '-ITO I1MeIOTC5I .n.Be
<PYHKllHIJ f (t) 11 get). CBepTKa f (t) C get) onpe.n.eJJ5IeTC5I KaK
00
h(t)= f(s)g(t-s) ds.
-00
(23.6-1)
3aMeTIJM, '1TO eCJIH .n.JJ5I <pHKcHpoBaHHoro t HanHcaTb t - s - s', TO nOJIY'-IHM
00
h(t)= f(t-s')g(s')ds',
-00
(23.6-2)
a CJIe.n.OBaTeJIbHO, CBepTKa 1 C g Ta me, '1TO H CBepTKa g C f.
TIoCTaBHM Tenepb Bonpoc: «4TO 51BJI5IeTC5I npe06pa30BaHHeM <PYPbe
.lI.JI5I CBepTKH h (t)>>? TIo onpe.n.eJIeHI1IO HMeeM
00
H (a) = h (t) e-21tita dt.
-00
HCnOJIb3Y5I (23.6-1), nOJIY'lHM
00 00
H(a)= l(s)g(t-s)dse- 21tita dt=
-co -00
00 00
= f(s)e-21tisa[ g(t_s)e- 21ti (t-s)adt]ds=
-co -0:)
00
= I(s) e-21tiSaO(a) ds=F (cr) o (cr).
-00
(23.6-3);
TaKHM 06pa30M, npe06pa30BaHllC c6cpmlcll iJ6YX r;jJYH1C11,llii ecmb npo-
ll38eiJclillc llX npeo6pa306aHllu. BblwenpHBe.n.eHHbII1 <popMaJIbHblt! BbIBOD.
.n.OKa3bIBaeT YTBepm.n.eHMe JJ,JI5I D.BYX <PYHKlI.Ht! BpeMeHIJ; B ynpamHeHlJ1J
23.6-1 paCClYIaTpHBaeTC5I c. Y4at! CBepTK!1 D.BYX <PYHKlI.!1t! OT 4aCTOTbI cr.
VlHTepecHbl11 pe3Y JIbTaT, KOTOpbltI CJle.n.yeT !13 TeopeMbI CBepTl{l1, Ka-
CaeTC5J CBepTKI1 f (t) C f (- t)
00
h(t)= f(s)f(-t+s)ds.
-00
8TO paBHO npe06pa30BaHHIO npOH3 B e.n.eHI151 npe06pa30BaH!1t!
00
h(t)= F(a)F(_a)e21tiatda.
-00
'316
HEIIEPI10Jll1'-IECKI1E <t>YHKUI1I1. I1HTEfPAJJ <l>YPbE
IfJJ. 23
DOJlaraSi t = 0, HMeeM
0::> 00
h(O)= / (s)ds= F(o)F(-o)do.
-C}) -co
(23.6-4)
Ynpa>KHeHHSI
23.6-1. .L!.oKa3aTb, '1TO eCJlI1
0::>
H(a)= 5 F('i:) O(a-'i:)d'l:,
-0::>
1:0
h (t) = f (t) g (t).
23.6-2. .L!.oKa3aTb 4JOPMYJlY, cooTBeTcTBYlOllIYIO (23.6-4), HCrrOJIb3YR ynpa)jwe-
HHe 23.6-1.
23.7. 3<P<PeKT KOHeQHorO CYl\Il\IMpOBaHlul
Mbl MO)!{eM paccMaTpHBaTb KOHelJHOe lJHCJJO Y3J10BbIX TOlJeK, Bbl6paH-
-Hoe 113 6eCKOHelJHot! nOCJJe.JlOBaTeJJbHOCTI1 Y3JJOBblX TOlJeK, KaK np0l13Se-
.JleHl1e I1CTHHHOt! YHK 1111 BpeMeHI1 H npHMoyrOJJbHoro I1MnYJlbca
P(t)={
Itl<T,
Itl>T.
TaKMM 06pa30M, I1MeeM
/1 (t)=p{t)/{t)
H no Teoper.1e CBepT1(H ( 2 .6)
0::>
F ( ) - \' F( ) Sin 21tT("t-a) d
1 0 - 't 21tT("t-a) 'to
-0::>
(23.7-1)
lJT06bl nOHSlTb CMblCJI 9Toro BblpmKeHHH, paCCMOTpi-lM <PYHK 111O / (t) =
.= e21ticrjt. CneKTp f (t) COCTOHT H3 e.JlHHCTseHHot! lJaCTOTbI 13=131 H csepTKa
-llpe.JlCT aBJlHeT co6ot! «pa3Ma3blSaHl1e» 9TOt! lJaCTOTbl s cneKTp
F 1 (0) = sin 21tT (a 1 -a)
2nT(a1- a )
(23.7 -2)
LJeM 60JlblIIe T, TeM y)!{e eHTpaJIbHblt! ml[{ (pHC. 23.4-2). B HeKOTopOM
-cMblCJle aTO pa3Ma3bIBaHHe 11 ITpe.JlCTaBJIHeT 9 cpeKT KOHelJHotl BbI60PKf1,
Kor.Jla Mbl ITblTaeMCSI onpe.Jlem-ITb lJaCTOTY ITO KOHe'lHOMY KYCKY lJf1CToti
CHHYCOH.Jlbl. ECJlH Tenepb npe.JlCTaBI1Tb cneKTp COCTOH1JJ.HM 113 MHOrl1X
4aCTOT, TO Ka)!{.JlaSi H3 HHX 6Y.Jler pa3Ma3aHa TeM )Ke CaMbIM MHO)!{f1TeJleM
(23.7-2); .JlJlSl npe.lLeJIbHOro HenpepblBHoro pacnpeJI,eJleHHSI lIOJlYlJHM (23.7-1).
B OnTl1Ke 9TO pa3Ma3bIBaHHe COOTBeTCTByeT pa3peWalOmet! cnoco6HOCTH:
4eM .JlJlHHHee no speMeHu CUrBaJl, TeM JlylJwe MO)!{HO pa3JI,eJlUTb JI,Be
6J1U3KUe lJaCTOTbl.
\I 24.1]
BBEJ1EH!1E
317
fJJABA 24
JIHHEtiHbIE «1>HJIbTPbI. CrJIA)l{HBAHHE
H AH«1>«1>EPEHUHPOBAHHE
24.1. BBeJ{eHMe
CJlOBO «cpHJlbTp», KaK H BblpaJKeHHe «crreKTp MOI!(HOCTH», B03HHKlm
fl3 cpH3H'leCKHX HBJleHHti, H3YIJaeMblX TeopHeti 3JIeKTpHIJeCKHX u,eneti, HO
nOJlYIJHJlH 3Ha'lI1TeJlbHO 60Jlee umpoKoe rrpHMeHeHHe. <t>HJlbTpbl, C KOTO-
pblMH MI,l 6YJ1eM HMen lI.eJlO, YlI.aJlHIOT H3 CPYHKU,HH HeKOTopble 4aCTOTbl,
rrpHMepHO TaK, KaK MaCJlHHblt! cpHJJbTp YJ1aJlHeT H3 JKHlI.KOCTH 4aCTl-HlbI
orrpeJ1eJleHHblX pa3MepOB. TeopHH rrOCTpOeHH5I cpHJIbTpOB 04eHb pa3BHTa,
HO MbI MOJKeM 3l1.eCb JlHWb KOCHYTbCH 3TOt! TeMbI; rrpHJ1eTCH YJ10BJIeTBO-
pHTbC5I J1eMOHCTpaIJ.Het! OCHOBHOti HJ1eH H crroc060B paC4eTa HeKOTopblX
(jJHJlbTpOB. Tex, KTO HHTepecyeTcH nOCTpoeHHeM (jJHJIbTpOB J1JI5I KOHKpeT-
HblX I1HJKeHepHbIX cHTyaIJ.Ht!, a He Bb1tJI1CJlHTeJIeti, MbI OTCbIJlaeM K CTaH-
.LI.apTHbIM KypcaM TeopHH 9J1eKTpH4eCKHX IJ.erret! H rrpOeKTl1pOBaHHH
(jJI1JIbTpOB.
B 3TOt! r JIaBe 6y J1eT 113YtJeHa pOJlb cpl1JIbTpOB B J1BYX THrrl1tJHblX CI1-
Tyau,HHX: C ueJIblO cr JIaJKI1BaHI151 H J1l1cpcpepeHIJ.l1pOBaHH5I J1aHHbIX. MbI
CTapaTeJJbHO H36eraJJI1 9THX TeM BO BTOpOt! tJaCTI1, B Kpat!HeM CJIYlJae
KOpOTKO yrroMHH8JIH 0 HHX; 06e OHH HBJlHIOTCH TPYJ1HbIMH H J1eJII1K8T-
HbIMI1, H rrpH6JIHJKeHHe MHOr04JJeH8MH He lI.8eT 0 HHX HHK8Koro rrpell.-
CTaBJIeHH5I. .ilaJIbHetiwHe rrpl1Mepbl 6YJ1YT lI.aHbI B r.ll. 25.
THrrHtJHblt! crroc06 HCrrOJIb30B8HH5I MHorOtJJleHHblX rrpH6JIHJKeHHt! lI.JI5I
lI.l1cpcpepeHIJ.l1pOB8HI1H COCTOI1T B TOM, tJT06bl rrpOBeCTH HHTeprrOJl5lIJ.HOHHblti
MHOrOtJJleH IJepe3 Y3JIOBble T04KH, npOll.l1cpcpepeHIJ.HpOBaTb 9TOT MHoro-
4JIeH, a 3aTeM BbI411C.'1I1Tb np0l13BOll.HYlO B Y3lIOBbIX T04Kax. .iloCTaT04HO
nOCMOTpeTb Ha OCTaTOtJHblti llJIeH HHTepnO.llHIJ.110HlJOrO MHOr04JIeHa (CM.
(8.6-1 ))
(x- xI! (x -x 2 )... (x- Xn+l) fW+1! (8)
(n+ 1)1
4T06bI YBHJ1eTb, IJTO B Y3JIOBblX T04Kax 8nnpOKCHMHPYlOIlI,HtI MHOrOIJJleH
nOIJTH HaBepHHKa nepeceK8eT CPYHKIJ.HIO; CJJell.OBaTeJIbHO, Y3J1oBa51 T04K8
eCTb n01JTI1 H8HxYlI.Wee B03MOJKHOe MeCTO J1JIH BbI4HCJIemIH rrpoH3-
BOll.HOtl.
l1HOrll.a, npeJKlI.e 4eM lI.HcpcpepeHIJ.HpOB8Tb, J1J1H rrOJIY'leHYIH r.ll3l1.KOrO
MHoro'weHa HCrrOJIb3YlOT MeTOll. HaHMeHbWHX KB8J1paToB. BTOT rrpoIJ.ecc
MHHI1MH3HpyeT CYMMY KB8l1.p8TOB OTKJIOHelIHtI OT rrepBOHaIJaJJbHbIX J13HHbIX
B Y3J10BbIX T04Kax, HO HWlero He rOBopHT 0 rrOBeJ1eHHH MeJKJ1Y Y3J10-
BhlMH T04KaMH, TaKHM 06pa30M, XOT5I MOJKHO Hall.eHTbCH, 4TO npOH3-
BOJ1HaH OIJ.eHeHa rrpaBI1JlbHO, HeJIb3H 6bITb cnHWKOM YBepeH1-1bIM, 4TO
MHor04.11eH, rrpOBeJ1eHHblti rro CflOco6y HaHMeHbWHX KBall.paTOB, He HMeeT
318 JJHHEAHblE <l>HJJbTPbl. CfJJA)!{HBAHI1E H J1H<I><I>EPEHUI1POBAHI1E [fn. z.t
Me)!{,ay Y3JI0BblMI1 TOLJKaMH KOJle6aHHtI, KOTopb1e 6YAYT CHJlbHO BJlHHT1>
Ha olJ,eHKY np0I13Bo,aHOti.
l-1cnOJJb30BaHHe AJIH :JTot! lJ,eJJH tPYHKIJ,HtI C OrpaHHqeHl-IblM cneKTpoM,
XOTH H He OTBeqaeT Ha Bce BonpOCbI. Bce )!{e AaeT HeKOTopoe 1l0HHlIJa-
Hue CUTY3IJ,UfI.
24.2. llpMMep npocToro CrJIamMl:lalQm.ero i»MJIbTpa
DYCTb A3Ha tPYHKll.IUI t (t), 3aBUCHluaH OT BpeMeHl1, H no TOn HJm
fIHOI1 npUqHHe Tpe6yeTcH cr R3AI1Tb paBHOOTCTOHll(He Y3RbI tk no JJf1Hetl-
not! tPOPMY Jle
fk+fk+l +fk+2 _ g
3 k
(t k = hk).
(24.2-1 )
4TO CAeJJaeTCH npu 3TOM co cneKTpoM tPYHKIJ,Uli?
DpeA110JlO)!{I1M CHaqaJla, qTO t (t) - npoCTaH CI1HycoHAa t (t) = e2"i '
(0' - .n.eticTBUTeJlbHOe 4!1CJlO). Y.n.06HO BBeCTH 0603H3qeHI1e
21t0' = OJ.
3J1eCb 0' u3MepHeTCH B IJ,U!<J1ax B eJ1I1HI1IJ.Y BpeMeHI1, a OJ - B pa.n.113HaX
tI eAHHHIJ,Y BpeMemJ. T3KI1M 06pa30M,
t (t) = ei(J)t.
B03bMeM J1J1H y,a06CTBa h = At = 1; Tor,a3, I1CnOJlb3YH (24.2-1), no-
Jlyql1M
f k = ei(J)k,
ei(J) (kH). ,ei(J) (k+t,
gk = -----;r--(e-1(J) + 1 + e ' ''') = (1 + 2 cos OJ).
CJleAOBaTeJlbHO, 3Ha'lemlH f YMHO)!{I1JJI1Cb H3 BeJJH'II1HY, He 3aBI1CHIlJ.YIO
OT fl,
= ! 1+2COS"' 1
OJ 3.
Ba)!{HO 3aMeTHTb, qTO MHO)!{HTeJJb nOJJyql1JJCH He 33BHCHIlJ.HM OT k BCJleA-
CTBHe JJHHet!HOCTH gk' ECJlH 6bJ gk 6bJJJH HeJlHHetlHbJMH tPYHKIJ.I1HMH aT Ii-.
10 3TOT pe3YJlbTaT He 6bJJJ 6bl cnpaBe,aJlHB.
B cneKTpe MHO)J{WreJJb CTaHOBHTCH paBHblM
( 1 + 2 cos "' ) 2
3 I
(24.2-2)
211: I
'ITa paHHO Hymo, KorAa OJ = 3"" IWI1 0' ="3' ECJJH HaqepTliTb 3TY KPH-
BYKJ (pHC. 24,2-1) K3K CPYHKlJ,HKJ qaCTOTbl 0, TO MO)!{HO y6eAHTbC5I, '1TO
24.31
npVlMEP nOCTPOEHVl5I <t>YIJJbTPA
319
<jJOPMY JJa cr JJa)KHBaHH5J (24.2-1) ,lI,aeT CHJIbHbl1i 9cpcpeKT
WI1HCTBa 'IaCTOT B BepxHet! nOJlOBI1He cneKTpa. TaKHM
<jJYHKlJ.I-]I-! C OrpaHl1'1eHHbIM cneKTpoM nOKa3blBaeT,'ITO
Ha'laJJbHWM CHrHaJlO npocTot!
JJl1Het!HbJt! cpl1JJbTp \; (ypaBHe-
Ime (24.2-1 »: OH a'3BewIlBaeT
'IaCTOTbl cor JJaCHO KpHBOt! Ha
pHC. 24.2-1.
Mbl 3a.1(aJJH Bonpoc 0 .1(et!-
nBHH JlHHet!Horo cpHJlbTpa
Ha e,ll,HHCTBeHHym 'IaCToTy,Ho
Ha caMOM .1(eJle YBI1,l1,eJlI-I
ero ,lI,et!CTBHe Ha Ka)K.1(ym
'lJaCToTY; CJle.llOBaTeJJbHO, HCnOJlb3Y5I npe.llCTaBJJeHHe npe06pa30BaHHeM
$YPbe, Mbl MO)KeM pa60TaTb c cpYHI IJ.I151MH, COCTaBJIeHHWMH H3 Bcex
'IaCTOT.
nO.llaBJleHH5I 60Jlb-
06pa30M, MeTO.ll
.lleJlaeT c nepBo-
f,O
0,8
0,5
0,4
0,2
0,1
o 1/8 1/4
J/8 1/2= 6
= '10C/17omll IIOI/OJl{t'IIUJI
Pf1C. 24.2-1. <P11J1bTpbl 1/3 (!k + fk+1 + fk+2).
24.3. npIlMep nocTpoeHIIH <tJIIJIbTpa
KaK H3BeCTHO, paCCT05JHHe Me)K.llY Y3J1aMH HenocpellcTBeHHo CB5J3aHO
c 'IaCTOTot! HaJIO)KeHH5I B cneKTpe. ITpe.llnOJlO)KI1M, 'ITO cpYHKIJ.H5J (Cl1fHaJI)
HMeeT OrpaHl1'1eHHbltl cneKTp H MaKCHMaJJbHym 'IaCTOTY 10 IJ.I1KJJOB B ce-
KYH.llY. 3TO BbIHY)K.llaeT CTaBHTb 20 TO'leK B ceKYH.llY, 'IT06bl H36e)KaTb
CMeWHBaHH5I 'IaCTOT.
D.onycTHM, 'ITO HHTepec npe.llCTaBJl5ImT JIHWb 'IaCTOTbl, MeHbWHe
.5 UHKJJOB B ceKYHllY. Mbl He pHCKHeM B35JTb 10 TO'leK B ceKYHn:y, Tal{
KaK TonIa 'IaCTb cneKTpa C 'IaCTOT3MI1 Bblwe 5 UI1KJlOB B ceKYH,lI,Y HaJlO-
)I(\1J1aCb 6b] Ha HH'repecymll l1t! Hac HHTepBaJI.
<PHJlbTp 24.2 .llaeT B03MO)KHOCTb YCTpaHI1Tb BepxHmm nOJlOBHHY
cneKTpa. KaK TOJIbKO cpHJJbTp YCTaHOBJJeH, OH YMeHbwaeT 9TH 'IaCTOTbl
no KpatiHet! Mepe B ,lI,eB5ITb pa3.
Mbl 1l0CTymIM eIlI.e JlY'Iwe, nOCTaBHB BCJIe,ll, 3a rrepBWM BTOpOt!
crJla)KI1BamIlI.Ht! cpl1J1bTp
gk + gk+1 + gk+2 + gkN h
4 = k'
(24.3-1 )
Ha npocTym CHHYCOH.llY e21Ciat = ei(J)[ 9TOT cpl1J1bTp .llet!CTByeT TaK:
h _ e iw ,' ( 1 + iw + 2i(J) + 3i(J) _ e iwk . I _e4<w _
k - 4 e e e - 4 I _ eiw
e i (J)(k+2) e2i(J) _e- 2iw
= -- 4e iw / 2 . e iw / 2 _ e iw/2-
ei (k + 8/ 2 )
4
'"
sin :2
sin 2",
320
JJYlHEHHbIE <l>YlJJbTPbl. C fJJAlliYlBAHYlE YI n.YI<I><I>EPEHlJ,YlPOBAHYlE
I fJ1. 2
CJIen.OBaTeJIbHO, CneKTp YMHOlliaeTC5J Ha BeJIWumy
( sin 20) ) 2 = ( Si 41ta ) ,
4' 0) 4 Slll1ta
Slll "2
IWTOpa5J MMeeT HYJJM npM cr = ]/4' 1/ 2 , ...
Pe3YJIbTaT nOCJIen.OBaTeJIbHOrO n.ettcTBfl5J n.BYX <pfWbTpOB (24.2-1) I}
(24.3-1) npen.CTaBJJ5JeT c060tt <pflJIbTp
(24.3-2)
h _ fk + 2tk+l + 3fk+2 + 3fk+B + 2fk+4 + fk+5
k- 12 .
KOTOpbltt OqeHb XOpOllIO
non.aBJI5JeT BepxHlO1O nOJJOBflHY cneKTpa (CM.
pHC. 24.3-1, rn.e HapHcoBaH JIOrapM<pM
CfIeKTpa B 3aBHCMMOCTH OT cr).
TIoCJIe TaKOrO cr JJalliMBaHM5J MOlliHO
cnoKottHo onYCTHTb qepe3 On.HH 'we-
Hbl h k , COXpaH5J5J, CKallieM, h 2k , H nOJIb-
30BaTbC51 pe3Y JIbTaTOM 6e3 cepbe3HbiX
onaceHHI1 OTHOCHTeJIbHO CMemeHI-151 (qa-
CTOTa HaJlOllieHH5J paBHa Tenepb 5 cefC- 1 ),
eCJIU TOJIbKO MOill.HOCTb BepxHett qaCTI1'
cneKTpa He 6blJIa CJ1HllIKOM BeJIHKa no
cpaBHeHMIO C HHlliHel1. TIpH OKOHQa-
TeJlbHOM MCnOJlb30BaHHH pe3YJlbTaTOB
Heo6xon.MMO yqHTblBaTb 9<p<peKT, no-
JIyqeHHbl11 BCJIe.n.CTBI1e <pUJIbTpaLJ,flU Ha
qaCTOTaX, 60JlbmHX qeM 5 LJ,UKJlOB
B ceKYHn.y. TaKHM o6pa30M, Ll.ett-
CTBue <pI-JJlbTpa (24.3-3) n03BOJl51eT
BbI6HpaTb qaCTb 3HaqeHI111 <PYHKl!.J1I],
He HMeS! npfl 9TOM Henpfl5lTHOcTel1 CMe-
meHUS!. TOqHOCTb IJ:JI5J qaCTOT Melli.n.y
1/3 fI 1/2 nepBOHaqaJIbHott QaCTOTbI
HaJIOllieHH5J 6yn.er He OqeHb xopomeJj,
2ln (F)
IJ
1/8 1/4 1/.3 1/2 IJ
'1{Jcmoma }f{J/lOJffCHUJI
fJll/J1t1l 1/z
-1
-2
-3
-4
P!1C. 24.3-1. 4acTOTHaR xapaKTepu-
CTUKa 4JHJlbTpa
1
h k = 12 (fk + 2fk+l + 3fk+2 +
+ 3fk+3 +2fk+4 + fk+.)'
(24.3-3)
Ynpa}/{HeHHe 24.3-1. PaCo,!OTpeTb BJI)UJHHe CrJla)KHBaHHR Ha crreKTp ,HfHa,1a
rrpH ycpe.l\HeHI1H m nOCJle.l\OB3TeJlbHblX 'IJlenOB.
24.4. <l>HJIbTpbI Booom.e
O ( sin 1tma \2
T BeT: . I
m Sill 1ta )
VI3 CKa3aHHoro Bblwe BblTeKaeT, qTO KaKYIO-HH6YLl.h JIIJHettHYIO KOM-
CHHaUHIO paBHOOTCT05lill.HX Y3JIOB MOlliHO paCCMaTpl1BaTb KaK HeKOTOpylO
«qJ!JJlbTpal!.HIO» CPYHKIlHH. B npOlllJlOM, nOKa COOTBeTCTBYIOw.a51 TeOpU51 He
6bJJIa n.OCTaTOqHO 5JCHa, Hepen.KO CJIyqaJlOCb, 4TO HeKOTopble HeBI1HHO
24.5]
AHAJJI13 flPOCTbIX tPOPMYJI .D.JJ5I .D.l1tPtPEPEHL\I1POBAHI151
321
BbIr JJ5I.lIeBllme npe06pa30BaHH5I, BbInOJIHeHHbIe Ha paHHett CTa.lIHH BbJtIHCJIe-
HHtt, cYllJ,eCTBeHHO BJII151JIH Ha nOJIYIJeHHbItt pe3YJIbTaT, KOTOpbItt 3aTeM
HHTepnpeTHpOBaJJC5I KaK cpH3HIJeCKHtt 9cpcpeKT, a He KaI< 9cpcpeKT o6pa-
60TKI1 .lIaHHbIX.
B KaIJeCTBe rrpI1Mepa npe.lICTaBI1M ce6e 6eJIbJt! WYM (B KOTOpOM, KaK
H B 6eJJOM CBeTe, rrpHCYTCTBYIOT Bce IJaCTOTbI C O.lII1HaKOBot! aMrrJJI1TY-
.lIoU, HO CJIYIJatJHott cpa3otJ) 11 rrpH o6pa6oTKe ero Y.lIaJIHM rrOIJTH Bce
IJaCTOTbI, KpoMe He6oiIbilloro HHTepBaJIa; Tor.n:a rrOCJIe.n:yIOIlJ,Ht! aHaJIH3
rrOKa3aJJ 6bI HaJII1IJI1e rrpeo6JIa.lIalOIlJ,etJ IJaCTOTbI, 11 O.lIHa 11 Ta )j(e IJaCTOTa
rr05lBJI5IJIaCb 6bI He3aBI1CI1MO OT HCTOIJHHKa 6eJIoro wYMa, 6y.n:b TO pbI-
HOIJHbIe lleHbI HJII1 KOJII1IJeCTBO n5lTeH Ha COJI1UI.e. TaKHe .lIaHHbIe MorYT
HMeTb crreKTp, OTJJI1IJHbltJ OT rJJa.D.KOrO; HO aHaJIH3 .D.aHHbIX, HcrrOpIJeHHblX
IlIYMOM, Tpe6yeT 3HaIJHTeJIbHbIX 06CY)K.D.eHHt!.
C .D.pyrotJ CTOpOHbI, KaK 6bI.lIO rrOKa3aHO B 24.3, pa3YMHoe HCnOJIb-
30BaHlJe cpHJIbTpOB MO)j(eT 6bITb OIJeHb rrO.lIe3HbIM. BTa 06JIaCTb 06Il111pHa
11 06blIJHble KYPCbI TeOpHI1 llerretJ, B03MO)j(HO, 51BJI5IIOTC5I .TIYIJWI1M I1CTOIJ-
HHKOM .D.a.lIhHet1Il1HX CBe.D.eHHtJ, TaK KaK 9TI1 H.D.eH pe.D.KO rrpOCaIJHBaIOTC5I
B BbIIJHCJII1TeJIbHbIe KpyrH. LlaJIbHet!IlIHe peKOMeH.D.aII.HI1, aBTopHTeTHble,
HO TpY.D.HbIe .D.JI5I IJTeHH5I, MO)j(HO Hat!TH y 6JIeKMaHa H TblOKH *).
24.5. AHaJIu3 npocTbIX <POPMYJI JJ:JIH JJ:lI<p<PepeHlJ,lIpOBaHUH
B03MO)j(HO, caMot! rrpOCTot! H HaH60JJee IlIHpOKO HCnOJIb3yeMot! cpop-
MYJJOt! .D.JI5I BbIIJHCJIeHH5I np0I13BO.D.HOt! IJepe3 paBHOOTCT05lllJ,He Y3ilbI
51BJI5IeTC5I cpOPMY JIa
t ' _ ik+t-ik-l
k- 2h
(h - IlIar).
(24.5-1 )
ilpOaHaJJH311PyeM, IJTO OHa .lIeJIaeT C pa3.TIUIJHbJMI1 IJaCTOTaMH. ilpe.lInoJIo-
iKI1M, IJTO
I(t) = e21tiat = e ioot .
(24.5-2)
ilYCTb B k-t! Y3JIOBOt! TOIJKe t k = kh. Tor.D.a
Ik = e21tiakh = e iookh .
.D.ncpcpepeHlI.HpOBaHne f (t) .D.aeT
df (t) I = dfk = [we iookh .
dt t=kh dt
(24.5-3)
C .D.pyrot! CTOpOHbJ, (24.5-1) .D.aeT olleHKY
, iwkh e iwh - e- iooh . iookh sin IJJh
Ik = e 2h = (we --;;;-jz .
(24.5-4)
*) CM. 121.
11 P. B. XeMMHHr,
322 JIMHEHHblE <!>I1JJbTPbJ. CfJ1A)I<I1BAHME !-1 IlI1<!><!>EPEHUMPOBAHME [TJJ. '24
OTHOllleHHe Bbl4HCJJeHHOrO OTBeTa (24.5-4) K BepHoMY (24.5-3) OmlTb He
3aHHCHT OT k H paBHO
sin ooh
-;;;j/
(CM. pHC. 23.4-2). T81<HM 06pa30M, (jJopMYJla (24.5-1) HBHO YMeHbillaeT
aMrrJJHTY.1lY Bcex 4aO'OT, KpoMe (j) = O. KaK MO)J{HO 6bIJlO 6bI O)J{H.1laTb,
1t I
npH (j) = Ii' T. e. (J = 2h' .1lJlH npO!-I3BO.1lHotl fIOJlY4aeTcH HYJleBOe 3Ha-
4eHHe ((J = 2 1 h eCTb 4aCTOTa HaJIO)J{eHliH).
DO.1l06HbiR acpcpeKT HMeeT MeCTO 1-/ .1lJJH BTOpOR np01-/3BO.1lHott
f " ik+1-2ik+ik_l
k = h2
Ylerro.%3Y5I (24.5-2), nOJlY4aeM olieHKY
. ,ooh
iw'z 2 + -iO)h . 2 (I h) sm'"')
{ I.'_ iwkh e '- e _ _ ;J Iwkk -cosoo _ _ 2 iwhk
," - e h2 - (j) e w 2 h 2 (j) e ( W;l r .
Dp aBHJlbHblt\ OTBel eelb
f " - . (j) 2 e i w hll
h-- ,
TaK 4TO OTHOllleHHe pe3YJlbTaTOB paBHO
, 2 00h
8m 2 002 h2
( hr rv 1- 12 +""
KOTopoe OrrHTb YMeHbwael Bee BeJlH4I1HbI, KpOMe (j) = 0, l1.Jl5I 4aeTOTI>I
4 I "
h(j) = 'it H8XO,lJ,HM 3H8'JeHHe npOH3BO.1lHOR, paBHoe - fi2 e .
24.6. KaK 1-I30elliaTb BbI'U!CJleHRH npOR3BO!l.HbIX?
Llano MO)J{HO H36e)J{aTb 3a.1la4H oueHKH rrpOH3BO.1lHott l1.J!SI rrpl1Mepa
[]pe.1lnO. O)J{HM, 4TO HMe!OTC5I 3Ha4eHH5I fk H H3BeeTHO, 4TO TeOpeT!JlleCKR
f (t) Y.1l0BJJenWp5ler .1lH(jJcpepeHUH8J1bHOMY ypaBHeHHlo HToporo rrOpH.1lKa
HH.1la
r = H(j, t).
Ylcno.1b3Y5I ero, MO)J{HO nepettTH OT 3H84eHHt\ f (t) K 3Ha'leHIHIM r (t) Ii
npOHHTerpHpOBaTb nOCJJe.1lHIO!O CPYHKUI110, tHo6bl rIOJJY'IHTb f' (t). l{acTo
HllTerpHpOBaHHe rrpe.1lnO'lHTaIOT .1lHcpcpepeHuHpoBaHH!O I1PH YCJJOBI1H, 4TO
OTpe30K, Ha KOTOpOM llMeIOTC5I .1l8HHble, He HaeTOJlbKO BeJlHK, 4T06bI He-
24,71
lIIETOlt <!>\1JJOHA
323
60J!bwne cnCTeMaTfiI-/eCKHe nOrpeWHOCTH HHTerpnpOBaHIH! lIaJ!1l B pe3YJIb-
raTe 60JJbWYIO onm6KY.
IlJJH lIaCTHblX CJIYlIaeB n3BeCTHbl MHOrOllHCJIeHHble .n:pyme XfITpOCTI1,
HO, nO-BI1.n:HMOMY, o6rnet! TeopnH, Kor.n:a H KaK MOJKHO H36eJKaTh ltHcpcpe-
peHuHpoBaHHH, nOKa HeT.
24.7. MeTOlt <PHJIOHa
Te JKe caMble npOCTble H.n:eI-l, KOTOpblMH Mbl nOJJb30BaJIHCb Bblwe,
MorYT .n:aTh npe.n:CTaBJleHHe 0 TOM, liTO B .n:etJCTBHTeJIbHOCTI1 .n:eJIaeT
cpopMYJJa, He B.n:aBaHCb B .n:eTaJJbHblt! aHaJIH3, KOTOpbltl 6blJJ npon3Be.n:eH
.n:JIH cr JIa)KHBalOIlLero H .n:HcpcpepeHunpYlOrnero CpHJJbTpOB. B TO JKe BpeMH
6y.n:YT BbIB e.rreHbJ HeKOTopbfe nOJJe3HbJe (POPMYJJb!.
4aCTO BC"rpellaeTC51 3a.n:alla Bbl'-lI1CJJeHI-IH muerpaJJOB BH.n:a
I (k) = Vet) cas kt dt.
a
(24.7-1)
<PH.nOH *) npe.rr.ryO)i{H.n MeTO.n:, KOTOpblt!, cpOpMa.nbHO rOBopH, OTHOCI-ITCH
KO BTOpOtJ '-IaCTI-I ,notJ KHHfH, TaK K3K OCHOBbIBaeTCH H3 MHOfO'-lJIeHHotJ
annpOKCI-IM3Ul111 I(t). TOllHee, HHTepBaJJ (a, b) .rre.nI1TCH Ha 2N fJHTepBa-
JIOB, H B KaJK.n:OM .n:BO!1HOM I1HTepBaJJe I (t) annpoKCHMHpyeTcH KBa.n:paT-
HblM TpeX'-IJJeHOM. TaJ(fIM 06pa30M, 9TO HanOMI1HaeT cOCTaBHYIO cpOPMYJIY
CHMncoHa, KpOMe Toro, liTO nolt HHTerpaJJOM HMeeTCH erne MHOJKHTeJIb
cas kt.
Vl.n:el1 MeTOlla OlleHb npOCThI, HO Tpe6YIOT IlJII1HHbJX BbJKJla.n:OK **), H
Mbt npHBelteM JIHWb pe3Y JJbTaTbI. B03bMeM war
b-a
h=2N'
(24.7-2)
H nycTb C n - cYMMa Bcex lIeTHblX Op1IHHaT I(t) cos kt, 3a HCKJJlOtJeHHeM
Toro, '-ITO OT nepBo!1 H nOCJJe.n:Het! op.n:mJaT 6epeTCH nOJJOBI1Ha
1
C 2 /1 =2 I(a) cos ka + I(a + 2h) cos k (a + 2h) +
I
+ I(a + 4h) cos k (a + 4h) +.. '+2 I(b) cas kb. (24.7-3)
nYCTb TaK)J(e C n_l - cYMMa Bcex HeqeTHbIX op.n:HHaT
C n_1 = I(a + h) cas k (a + h) + I (a + 3h) cas k (a + 3h) +...
...+ r(b - h) cos k (b - h). (24.7-4)
*) L. N. G. FiJon, Proc. Roy. Soc. Edm., XLIX (1928-1929).
**) CM., Hal1pHMep, [40].
11.
324 JlI1HEHHblE <l>I1JlbTPbI. CrJIAml1BAHI1E 11 Jj,I1<1><I>EPEHLl.I1POBAHI1E [rJI.24
TOf.n.a
u
V (t) cos kt dt = h {IX [I (b) sin kb- I(a) sin kaJ + C2n+IC2n_l}' (24.7-5)
a
f.n.e
_ 6" + 6 sin 6 cos 6 - 2 sin" 6 _ 26 3 _ 28 5 + +
IX- 83 45 315 4725 ...,
p_ 2[6(1+cos"6)-2sin8cos6] _2 + 26 2 + + 266 +
1"'- , 8 3 -3 15 105 567 .."
4 (sin 6 - 8 cos 8) 4 28 2 6 4 8 6
,= 8 3 = 3" -15 + 210 - 11340 +... (24.7-6)
IJ
fI - kh - k (b-a)
- - 2N .
(24.7-7)
TaKa5! me cpopMYJla OTHOCHTC5! K HHTerpaJIY
/)
t(t) sin ktdt= h {- IX [I (h) cos kh - tea) cos kaJ + S2n + "[S2n_d,
a
(24.7 -8)
r.n.e S2n H S2n_l - COOTBeTCTBYIDill.He CYMMbI .n.Jl5! I(t) sin kt.
I1orIp06yeM rIOH5!Tb 9TI1 CPOPMYJlbl rIPH rIOMOlll.H rrpocToro rIpHMeHe-
HH5! u.n.eit, 0 KOTOpblX IllJla pe'ib BbIIlle.
BeJlIjYUHbI, KOTopble 3.n.eCb BbIYHCJ])!IDTC5!, 5!BJl5!IDTC5! K09cpcpHl.ll1eHT8MH
<PYPbe, T8K YTO HmeTC5! 8MrIJIHTy.n.8 rapMoHHKH t(t) C 'laCToToit k (pa-
.n.H8H). BeJll1'iHH8 h ecn p8CCT05!I-me Mem.n.y Y3JlaMH, 8 BeJJH'iUHa
fI=kh
- ero rIpOH3Be.1leHHe Ha HCCJJe.n.yeMYIQ 'iaCTOTY. TeopeMa BbI60pKH rIOKa-
3blBaeT, 'lTO He.'!b35! Ha.n.e5!TbC5! Ha.n.emHO BbI'lHCJJHTb 3MrIJII1TY.n.y, eCJJH He
2n
BbI6paHbJ no KpaitHeit Mepe .n.Ba Y3JJa Ha Y'laCTKe .1lJJHHOit Ii; HO .lI.ame
TOr.ll.a nOTpe60B8JJOCb 6bI 6eCKOHeYHOe MHomeCTBO Y3J10B. TaK KaK HHTep-
BaJJ (a, b) KOHeyeH, TO MomHO OmH.lI.aTb, 'iTO fI I\IH0ro MeHbwe, 'leM 'It
(9TOT BOrIpOC 6y.n.eT eme 06cym.n.aTbC5! B cJle.n.YlOl.lJ,eit rJJaBe).
Ta6JJHlibl K09cpcpHliHeHToB !J., , 'I .n.aHbI Tp8HTepoM [40] (M5! fj B pa-
LIHaH8x) .lI.O fI= 1,50 H KOrIaJJOM ([20J, CTp. 539) (.lI.Jl5J fj B rpa.ll.ycax)
LIO 45°. B 060HX cJlY'la5!x H B rIepBOHa'iaJlbHoit pa60Te <PHJlOHa HlJyerO
He CK838HO, rrO'leMY Ta6J1HlibJ 06pbIB8IQTC5! HMeHHO B 9TOM MeCTe. ITo-
4eMY, eCJJH t (t) Be.n.eT ce651 KaK MHOrO'lJJeH, HeJIb351 rrpHMeH5!Tb CPOPMY JIY
T3K .lI.aJleKO, KaK XOTeJIOCb 6bI, H .lI.eJJ8Tb k, 8 CJle.n.OBaTeJJbHO H fI, co-
BceM 60JIbIllI1M? Pa3 c.n.eJJaHa «aHaJIHTHlJeCKa5! 3aMeHa» <j:>YHKliHH MHoro-
1jJleHOM, TO HHTerpHpOBaHHe .n.eJI8eTCH 8HaJIHTl1yeCKI1. 4TO OCTaHaBJlIJB8eT
II 24.81
3AKJlI04I1TEJJbHbIE 3AME4AHI15!
325
Hac? 04eBH.n:HO, B OCHOBe ,:noro Jle)l{HT TeopeMa BbI6oPKH, orrpe.n:eJlHIO-
w.aH BeJIH4HHY 6, KOTOpOti ew.e MO)l{HO rrOJJb30BaTbCH.
Mbl TaK}Ke O)l{H.n:aeM, 4TO rrpH 6 -+ 0 rrOJJY4aeTcH <popMYJla CI1MrrCoHa.
YpaBHeHHH (24.7-5) C (24.7-6) H (24.7-7) S1CHO rrOKa3blBa!OT 3TO, rrpl14eM
OTI<J10HeI!HH 6y.n:YT rropH.n:Ka 6 2 .
B npHHll.Hrre MO)l{HO rrp0l13BeCTH .QeTaJlbHblti aHaJll13 11 orrpe.n:eJlHTb
T04HO, 4TO .n:eJlaeT MeTO.n: <PIIJlOHa C Ka)l{.n:otl 4acToTotl, HO Mbl 3THM 3a-
HHMaTbCH He 6y.n:eM. MeTo.Q rr03BOJlHeT c.n:eJlaTb 3TO, HO BbIKJla.n:KH CJJlIlll-
KOM 3 arrYTaHbl, 4T06bl BocrrpOH3Bo.n:I1Tb HX B Y4e6HHKe.
YripaHmeHHH
24.7-1. BblBecTH paBeHcTBo (24.7-5).
24.7-2. BblBecTH paBeH<):W (24.7-8).
Mbl He H3J10)l{HJIIl (H He c06l1paeMC5I .n:eJJaTb 3Toro) 3a.n:a4H CrJla)l{H-
BaHHS1 H .n:H<p<pepeHll.HpOBaHHH *) BO Bcex .n:eTa Jl5IX. f{aK Y)I{e 6blJJO CKa-
3aHO, 3TO 04eHb .n:emJKaTHble BOrrpOCbI. O.QHaKO OCHOBHble HI1TH COOTBeT-
CTBY!OIlI.I1X paCCY)I{.n:eHHtI .QOJI)I{Hbl 6bITb S1CHbl. HaM .n:aHa <PYHKll.I1S1
(CHmaJI), HCKa)l{eHHaS1 lllYMOM, HarrpHMep CHmaJl OT.n:aJIeHHoti IlJlaHeTbl,
lIJ1H <PYHKll.HS1, Bbl.n:aHHaS1 3J1eKTpOHHoti MalllllHoti C 01llH6KaMlI oKPyr JleHHS1.
I1YCTb ,aa)l{e Y Hac eCTb TeopeTH4eCKHe Oll.eHKlI cneKTpa <PYHKll.lIl1 H
lllYMa. Tor.n:a Mbl CTaJlKHBaeMCS1 C Tpy.n:HblM BorrpocoM C03.n:aHH5J <pHJIbTpa,
KOTOpbltl c.n:eJIaJI 6bl C ClIrHaJIOM TO,
4TO HaM HY)I{HO, a TaK)I{e, eCJIH MO)l{-
HO, y.n:aJI5IJI 6bl HeHY}J{Hblti WYM.
Ha rrpaKTIIKe 4aCTO rrpHxo.n:HTC5I
CTaJIKIIBaTbCS1 C 3a.n:a4etl Oll.eHKII aMnJIII-
TY.n: HeCKOJIbKHX 4aCTOT 1I3 CYMMap-
Horo cneKTpa, KOTOpbltl 3a.n:aH. 6 bIB aeT,
4TO 113MepeHHwtI crreKTp BWrJl5l.n:HT
rrpllMepHO KaK Ha pllC. 24.8-1.
STOT PIlCYHOK HaBo.n:IIT Ha MWCJIb
o TOM, 4TO Mbl BH.n:I1M 3.n:eCb rrp5l-
MoyrOJIbHIIK **) 6eJJoro lllYMa (HaBepHoe, BMeCTe C 4aCTOTaMII, HaJIO}J{HB-
lllIlMHCS1 Ha IIHTepBaJJ, .n:orrycKaeMblti .n:aHHoti Bb160pKOtl) ITiI!OC ClImaJI Ha
HH3KHX 4aCTOTax. Mbl MOrJII1 6bl rr03ToMY crrJJaI-JHpOBaTb 06pe3a1OIlI.I1t1
<pHJlbTp, KOTOpbltl y.n:aJIHJI 6bl BepXHlO1O 4aCTb crreKTpa (BMecTe C HeKO-
TOpOti 4aCTblO CHmaJIa), H 06"be.n:HHHTb ero co BTOpblM <pHJIbTpOM, KOTO-
pbltJ .n:eJlaJI 6bl C CHrHaJJOM TO, 4TO HaM HY}J{I-IO.
24.8. 3aKJIlOllllTeJIbHbIe 3aMellaHHH
l!llomoma
o
l/Ilcmoma
1I1l110J/{f1/fl/fl
Pl1c. 24.8-1. THnl1'!Hblt"r cneKTp.
*) HarroMHHaeM qHTaTeJllO a-MeTO,a JJaHu,oma H3 22.9.
**) B rJl. 32 .n:aeTCR cneKTpanbHblH aHaJlH3 IllYMa, nOJlY'IalOllI,erOCiI OT 4JJlYK-
ryaL\I1H BbJ60PI<H; qTO )/{e KacaeTCiI cooTBercTBYlOU(ero H3yqeHHiI cneKTpa ruYMa
OKp)'L1eHI1R, TO ero ell\e HY)/{HO c,aellaTb.
326
I1HTE fP AlIbi J1 J1Y14>4>EPEH!lYlAlIhHblE YP ABHEHI151
rln, 21;
CJleAyeT nOHHMaTb, '1TO, npe)l{Ae '1eM CTpOHTb qmJlbTp, HY)I{HO npeJl,-
CTaBJJ5ITb ce6e, 113 KaKHX 4aCTOT COCTaBJleH AaHHbltl CHrRaJJ. f{ CO)l{aJleHI1!0,
Bonpoc 06 H3MepeHHH cneKTpa MOIll.HOCTH ( 24.4) Jle)l{HT 3a npe.ue.13MH
Ha4aJJbHorO Kypca, H Mbl .uOJl)l{Hbl TI03TOMY OCTaBI1Tb 3.ueCb 9TH ABe
Ba)KHble TeMbl, JlHWb CJJerKa KOCHYBWHCb HX.
fJ1ABA25
HHTErPA W H AM EPEHUMAnbHWE YPABHEHM"
25.1. COJl,ep>KaHl1e rJIaBbl
B 9TOt! r JlaBe 6YAYT paccMoTpeHbl HeKoTopble <jJOPMYJlbl rmTerpnpo-
BaHH5I C T04KH 3peHI151 HX BJJI151HI151 Ha pa3J1H4Hble 4aCTOTbl, a IaIOKe
cjJOPMYJlbl AJl5I HHTerpHpOBaHH5I ,lI,HcjJ<jJepeHIlHaJlbHb]X ypaBHeHHtI. B KOHe4-
HaM C'IeTe, pe4b 6y.ueT H,lI,TH 0 MeTO,ll,aX HHIerpHpOBaHJ15I, OCHOBaHHblX
Ha npH6J1H)I{eHHH <jJYHKIlH5IMH C orpaHH4eHHb]M cneKTpOM. B 4aCTHOCTl1,
6YIIJT paCCMOTpeHhl MeTO,ll,bJ '--le6h]WeBa.
Mbl BHAeJJH, 4IO HCIIOJJb30Barme 0603Ha4eHH5I e21tiat nOJIe3HO 11PH
pa3J1H4Hb1X npeo6pa30BaHI15JX; OHO YJJ:06HO TaK)I{e npH OIleHKe IOlJHOCTI1
ljJOPMYJI. 9TO nOCJIeAHee 06CT05lTeJlbCTBO y)l{e He5lBHO HCIIOJJh30BaJJOCb;
Hcc,1eAyeM ero 60Jlee BHHMaTe.llbHO.
IlpeJJ:t1o.ll0)l{HM, lJTO paCCMaTpHBaeTC5I CHHycOHJ1.a A sin 21tcrt. ECJlH
H3Mep5lTb Ka4eCTBO <jJOPMY JJbI BeJJI1411HotJ 01l1H6KI1, TO, 4T06b] YCTaHOB!1Tb,
HaCKOJlbKO ,lI,aHHa51 OWI16Ka cepbe3Hee, HY)l{HO pa3J1H4aTb CJlY4aH
A = 1 H A = 1000.
Donee p a3YMHo OIleHI1BaTb oTHoclneJlbHY!0 0IllH6KY. OJl,HaKO JJ:Jl5I 3Ha-
4eHl1t1 B6J1H3H HatJaJla KOOp,ll,HHaT OTHOCI1TeJIbHa51 oWH6Ka MO)l{eT 6bITb
04eHb 60Jlb1llOtJ, H 3,l1,eCb cJleAyeT I1peJl,n04eCTb nCIlOJIb30BaHI1e a6COll!0T-
HOt! 0IllH6KH.
IlpH paCCMOTpeH!111 KOMIIJleKCHotl CI1HYCOHJJ:b] Ae21tiat Jl,Jl5I AeJ1cTBH-
TeJJbHbJX a H t 1\Ib] BHJJ:I1M, '-ITO ee MO.n:YJJb 5IBJJ51eTC5I ljJI1KCl1pOBaHHblM no
BeJJl1lJlme
I Ae21tiat I = I A I
11 OTHOCHTeJJhHa51 01llI16Ka AaeT YAOBJleTBOpHTeJJbHb]t1 KpHTeplltJ T04-
HOCTfL
KpoMe Toro, npH BbllJI1CJJeHlJ!1 MOJ1.YJl51 BeJJ!1tJI1Hbl Ae21tiat+i'f .ljJa30Bh/tJ
yroJJ cp IIClJe3aeT. TaKIIM 06pa30M, npH 113YlJeHIIH <jJYHKIlHH Ae21tlal lJaCTO
MO}[{HO npeHe6pelJb <jJa30Bb/M yr JJOM cp HJJH BBeC1'a <jJa30Bbltt yroJI B I<09ljJ-
<jJHIlHeHT A (KoTopbrt! MO)l{eT 6bfTb KOMlJlfeKCHblM lJaClfoM). Ho, K3K 6Y,ll,eT
BH.uHO B A3J1hHer.llIeM, OTHOCWre.llbHotJ cjJa30tJ pa3JJH4HbiX <jJYHKIlHt! npe-
He6peraTb HCJlb351.
25.21
MErOll nEPEllA T04HOH <l>YHKl\i111 n:JJ I1HTE rpI1POBAHI1
327
25.2. MeToa nepeaaTOqffOH C\JYHKIUHI JJ;JISl IfHTerpHpoBaHlIH
B r J1. 13 paccMaTpHBaJIC51 KJIaCC CPOPMYJI llJI5I Bbl'U'lCJIeHli51 Heollpelle-
neHHblX HHTerpanoB. OCHoBHoe Bblpa1l<eHHe (13.3-1) HMeJIO BHll
Yn+! = aOYn + aIYn_l + a2Yn_2 +
+ h (b_ !Y +l +boY +blY -! +b2Y -2)'
(25.2-1 )
rlle Y;' - 3Ha'lemJJl nOllb!HTerpanbHot! CPYHKll.Hn, a Yk - Bbl'lHcneHHble
3Ha'leHn51
t
yet) =y (0) + y' (x) dx.
o
PaCCMOTpnM STn cpopMynbl eme pa3, B CBeTe TerrepeU!Hero rrOllXOlla.
DpellrrOJI01l<I1M CHa'1aJIa, '11'0 nOllbIHTerpaJIbHa51 CPYHKll.115I (BXOll) eCTb
CI1HYCO l1 lla
y' (t) = Ale21ticrt = Ale iCIJr
(ill = 21'tcr).
(25.2-2)
(HHlleKC I OTHOCI1TC51 KO BXOllY.) HCTI1HHOe 3HalJeHI1e TaKOBO:
iA I . iA I
y(t)=- _e lCIJt +c. C=y(O)+-.
w w
(25.2-3)
TaK KaK OCHOBHa51 rpopMYJJa (25.2-1) JIHHet!Ha, TO M01l<HO flOJIaraTb, '1TO
Bbl'1HCJIeHHble 3Ha'leHl15I (BbIXOll) HMe!OT TY 1I<e '1aCTOTY, XOTH, B03M01l<HO,
llPyrl1e cpa3Y H aMlIJIHTYLT.Y. DYCTb Ao - aMIIJII1TYlla BblXOlla, IIpW!eM Be-
nl1lJl1HY Ao MO)l{HO ClJl1T aTb KOMIIJIeKCHot!, '1T06bl BKJI!O'1I1Tb B Hee cpaao-
BWti yrOJI. TaKHM 06pa30M, rrpellnonaraeTC51, '1TO BWlJilCJIeHHa51 y (t)
C rO'1HOCTb!O LT.O 01lm60K oKPyr JIeHil51 HMeeT BilLT.
y (t) = Aoe iCIJt . (25.2-4)
Do,UcTaBI1M Tenepb 3TI1 CPYHKlll1H (25.2-2) 11 (25.2-4) B OCHOBHY!O cpop-
Ao
Myny (25.2-1) il pa3peWI1M ee OTHOCI1TeJlbHO OTHOWeHI1H Ii;
[
Ao h (b_ 1 + boe- iCIJh + b 1 r 2iCIJ I! + b2e-3iwh)
"'II;- = 1 - aoe- iCIJh - a 1 e- 2i ",h - a2r3iCIJh
(25.2-5)
3TO OTHOllleHile, Ha3hIBa!OI11.eeCH nepe8amolfrwii cjJYHlcll,ueii, RBnHeTCH
MHO)lUneJIeM, Ha KOTOpbIt! YMHo)l{aeTcH aMTIJIl1TYLT.a llaHHoii '1aCTOTbI, LT.JJ51
nOJIY'1eHHH BbIXOllHOt! aMlIJIHTYLT.bl Tofi )l{e '1aCTOTbl. HLT.eH nepeLT.aTO'1Hott
CPYHKIlHH eCTb npOCTo cpopMaJJbHOe YTBep)l{'ueHl1e Toro, '1TO '1aCTO llena-
JIOCb B rJI. 24 I1pl1 pacc IOTpeHHH nOBeLT.eHI151 OTlleJJbHOii '1aCTOThl.
PaccMoTpilM 60JJee LT.eTaJIbHO npOCToii cnY'1atl npaBI1JIa TpaUeQHfi
(12.2-1)
+ h ' + .
Yn+ 1 =Yn .2 (Yn +-1 y,,).
328
!1HTE IP AJJbJ 11 JlI1<P<PEPEHUI1AJJbHbJE YP ABHEHI151
[IJJ. 25
O'JeBI1JlHO, B (25.2-1)
a o = 1,
I
b_ 1 =b o =2'
II nepe aTo'JHaH YHKUHH I1MeeT BI1
a 1 = a 9 = 0,
b 1 =b 9 =0,
04 0 _ h ] +e- iwh
-2' l_e-iw!l
(iJh
h cos 2
2 . . (iJh
tS1n-:2
. ( h t (iJ h )
= - l ,2 C g -:2 .
(25.2-6)
CpaBimBaH a30BbIe yrJJbI 9Toro pe3YJIbTaTa C I1CTI1HHbIM 3Ha'Je-
HI1eM (25.2-3), BH HM, 'JTO OHI1 'JI1CTO MH!1MbIe, H MhI I1Mee:\1 npa-
BI1JJbHYIO a3Y JIH Bcex 00. ECJJ[1 cpaBHHTb aMf1JIHTy.a:bI, OCTaBJIHH B CIO-
pOHe Ha'JaJIbHbIe yCJJOB!1H KaK Hecyw.eCTBeHHbIe, TO CJJe.a:yeT CpaBHI1Tb
h (iJh I
"2 ctg"2 C w' LlJJH MaJIeHbKI1X ooh HMeeM
3- t (iJh -3- [ _ (iJh _ (U)h}3 _ ] -1.._ (iJh2 _ <u 3 h 4 _
2 C g 2 - 2 (iJh 6 360 . " - (iJ 12 720 . ..
CJJe.a:OBaTeJIbHO, OTHOWeHl-le BbJ'JI-lCJIeHHOrO OTBen K TO'JHOMY, paBHoe
ro 2 h 2 (iJ 4 h 4
1 - ----r2 - 720 -. H,
ro 2 h 2
OTJJH'WeTCfI OT 1 Ha BeJIH'JI1HY nOpflD-Ka - ---r2'
M nepe.a:aTo'JHafl YHKUHfI, H OTHOWeHI1e BbI'JnCJJeHHOrO OTBeTa K TOq-
HOMY HBJIflJOTCH YHKUHflMH qaCTOTbJ 00 H wara h. OTHoweHHe COLLep)l{I1T
TOJJbKO npoH3BeLLeHHe ooh H nOKa3bIBaeT, KaK BbI60p illara BJII1f1eT Ha
qaCToTY IIPH OLLHOt! 11 TOt! }/(e oilll16Ke.
Tenepb paCCMOTpHM OPMYJJY CI1MIICOHa (12.2-2)
Yn+ 1 = Yn-I + (y +1 + 4y + Y -l).
.IJ.JJfI (25.2-1) 3.a:eCb I1MeeM
ao=O,
I
b -1 = 3" '
al=l,
4
b O =3"'
a.a=O,
I
b 1 =3'
b 2 =O.
ITepeLLaT041'lafl YHKUHfI HMeeT BH.a:
04 0 _ h (1 + 4ie- iwh + e- 2iwh ) _ ih cos <uh + 2
A;- - 3 (1- e- 2iwh ) - - 3' sin ooh
n onflTb JJ.aeT IIpaBHJJbHYIO a3Y LLJIfI Beex 00 (MHO)l{I1TeJIb - i) 11 JJfI
I
MaJIeHbKI1X ooh aMnJIHTYJJ.Y rropflJJ.Ka .
(25.2-7)
25.2J
METO)l DEPE)lATO'-lHOt% (I>YHKl\1111 )lJJ5! I1HTEfPI1POBAHI151
329
HaKOHell" paCCMOTpl1M npaBHJIO TpeX BOCbMbIX (12.2-3). IlepeJl,aTo'-l-
HaH <PYHKll,I1H 3J1,eCb I3bIr JlHJI,I1T TaK:
3h
8'
1 + 3e- iwf + 3e- 2iwh + e- siwh
1- e- 3iwh
3wh wh
. 3h cas '---2 + 3 cos 2"
- (-l) -8 3wh ' (25.2-8)
sinT
4T06bI CTaHJI,apTI130BaTb npl1BeJl,eHHbIe <P0PMYJJbI, B03bMeM OTHoweIme
BbI4Hc.1eHlioro no HHM oneTa K T04HOMY 11 BbI6epeM MacwTa6 w TaK,
4T06bI h CTaJlO paBHbIM eJl,I1HI111,e..
TorJl,a fIOJlY4I1M:
Jl,JIH fIpaBI1J1a TpafIell,I1fi
COOTBeTCTBYIOw.He rpa<p HKI1
npeJl,CTaBJleHbI Ha pI1c. 25.2-1.
TIo OCI1 OpJl,I1HaT I1CfIOJlb30BaHa
JlOrapJ1(pl\H14eCKa51 WKaJla, 1160
eCJlI1 6Y.lI,eT Jl,Onyw.eHa 01l1I16Ka
B OfIpe.ll,eJleHl-ll1 9Toro OTHOWeHI1H,
CKa)l{eM, BJI.BOe, TO He HMeeT
3Ha4eHI1H - B KaKYIO CTOPOHY.
TIo OCYI a6Cll,HCC OT JlO)KeHbI KaK
2"
3Ha4eHYlH 4aCTOTbI W, TaK 11 3Ha4eHH5I CKOpOCTH BbI60pKH - (OCYI Ha-
w
npaBJleHbI B npOTlmOnOJlO)l{HbIe CTOpOHbI). J1erKo BH.lI,eTb, 4TO, eCJlI1 CY-
Jl,I1Tb no BeJII14HHe 01llH6KH, <popMYJla CI1MnCOHa - JlY41IIa51 H3 Tpex (JI,JIH
Jl,0C'raT04HO 60JlbIllOfi 4aCTOTbJ BbI60pKI1).
Ha pl1c. 25.2-1 nOKa3aHa TaK)I{e KpHBa51 Jl,Jl5I <POPMYJlbi
w w
HI (w) ='[ ctg"2;
)lJIH <POPMYJIbI CHMfIcoIIa
w 2 + cas w
H 2 (w) = ' 3 '- . ; (25.2-9)
sm w
Jl,JI5I npasl1J1a Tpex BOCbMbIX
aw w
cos--+3cos-
aw 2 2
H 3 (w)=-S' . 3w
sm 2
2.5
2.0.
1,5
,
I ,
:I
Cl/M/7COHJ!
fljll/BLI/IO J/8--:,; ,
I )- IlP081J/I
TIJt(lI
/ ' I
.,':;; ::-To'liloe l/Ilme
,""c \ o?PIJPiBlIHre
'-
----.
.. fljlllBIJ/10
mpll/7e«11
\ \
\ \
\ \
\ i
I 1,0.
'q
0,5
"i
.", '" 0.
'" -0.5
" '
::;: -10
:j,
';2-1,5
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3-20.
<'\J'
I
o
6
Ii
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-3,5[/
fl,2!t fl, 4ft UI fl,5rc fl,8ft
1 5 1 Z/08;; 4
I/LiCIlO YJIl08 Hll «II/(/I
7t
,
2
PIIC. 25.2-1. 4acrOTHaH xapaKTepHcrHKa
HeKoropblx 4JOPMYJl HHrerpHpOBaHI1R.
Yn+ I =Yn-l + h (o,3584Y +I + 1,2832y + O,3584Y _I)'
1{09<P<PHlll1eHTbI ee 6blJlH onpe.1l.eJleHbI 9KCnepHMeHTaJlbHO J1ea THKOM
1t
TaK, 4T06bI B I1lUepBaJle 0 W '[ OIllH6Ka I1MeJla 4e6bIWeBCKYlO <POPMY
(CM. rJJ. 19), a TaK)I{e 4T06bl <popMYJla 6hIJIa T04HOfi )J.JI5I 00=0. HCHe,
330
I1HTEfP AJJb! 11 !lI1<1><I>EPEHl\I1AJIbHbIE YP ABHEHI1S1
frJ1. 25
liTO CKOpOCTb BbI60pKH 6b1JIa 3J1.eCb YJ1.BOeHHOIt no cpaBHeHlHO C HaItK-
BHCTOM.
TOT cpaKT, !-ITO rpacpm< IUHl npaBHJla TpaneuHIt H/leT BHH3, TOfJ1.a KaK
LlJI5I cpOpMY, bl CllMnCOHa - BBepx, Tpe6yeT HeKOToporo n05lCHeHI15L <POp-
MYJIa CHMf1COHa nOBblJl1aeT 3MnJlHTYJ1.Y BbICOIWX 'IaCTOT, TOrJ1.a l<aJ< npa-
BHJIO TpaneUHIt CTpeMHTC5I CrJIaLlI1Tb HX, XOT5I B LletlCTBl-neJIhHOCTH MaJIO
liTO H3BeCrHO 06 9cpcpeKTax oJ<pyr JleHH5!, HeO)!{HJ1.aHHbIe CKaliKH CPYHKIUlH,
06YCJJOBJleHHbIe oKpyr JIeHHeM, nOpO)[{JLaIOT BhICOKHe '-laCrOTb!; CJJeJLOBa-
TeJIbHO, cpopMYJJa CHMncoHa YBeJlH'-lHBaeT HX, Tor J1.a KaJ< npaBHJJO Tpane-
UHtI CTpeMI1TC5J HX CrJJaJ1.HTb. ECJlH paccMaTpHB3rb CPOPMYJIY CHMncoHa
KaJ< CPYHJ<UHIO CJ<OpOCTH BbJ60pKH, TO npH I15ITI1 Y3J1aX Ha UHJ<JI 01lJH6J<a
Ha war COCTaBJI5IeT OKOJlO 1,5%, 4ro nO'lTH BCerJ1.a CJlHWKOM MHora.
DpH ceMH Y3J1aX Ha UHKJI nOJlY'-laeTC5! rpy6oe, HO I1HOrJ1.a nOJle3HOe
npH6J111)[{eHHe, a npH ,n,eC5JTH cOBepWaeTC5I oWH6Ka, MeHbWa51 0,001, 4TO
BO MHorHX CJlY'-laHX J1.0CT3rO'-lHO xopowo.
Ilo cox nop paccMarpHBaJlaCb rJJaBHbJM 06pa30M eJ1.HHCTBeHHa5J '-la-
CTOTa (J) = 2'1to. DpeJ1.nOJlO)[{HM, 'ITO 6epeTCH npOH3BOJJbHa5! CPYHKIU1H
(BXOJ1.)
00
fI(t)= F(o)e 1ti.tdo.
-00
f{a)[{J1.bIJ1 'weH npeo6pa3yeTC5J npn nOMomH nepe,n,aT04HoJ1 CPYHKUHn
0(00) = o (2'1tcr) = 0 1 (0)
H Ha BblxO.!l.e nOJIY'-laeTC5I
00
fo(t)= F(cr) OJ (0)e 21ti . t dcr.
-00
CrrpaBa HMeeM npeo6pa30BaHHe npOH3Be.!l.eHHH H, npHMeH5I5J TeopeMY
CBepTKM ( 23.6), nOJlY'IaeM
00
fo(t)= fr('t)gJCt-'t)d't.
- 00
(25.2-10)
TaKHM o6pa30M, CBepTKa npeo6pa30BaHHR gj (t) nepeJ1.aTO'-lHOIi CPYHKUHH
gj (0) C BXO.!l.HOt! CPYHKunetl II (t) ,n,aer BbIXOJl,HYIO CPYHKUHIO 10 (t).
Depe.lI.aTOliHaJJ CPYHI<UH5J OJ (0) COJl,ep)[{HT BCIO HHcpopMall}IIO 0 cpop-
MYJJe llHTerpnpOBaHllH H B HeKOTopOM CMbICJJe 9KBllBaJIeHTHa npHMe-
HJJBwetlCJJ cpopMYJle.
Ynpa>KHeHHe 25.2-1. PaccMoTpeTb YCJJOBI1I!, Heo6xoL1HMbJe L1JIJI Toro, 'lr06bJ
.,q>opMYJla HHTerpl1pOBaHI1H HMeJla I1paBHJlbHYIO 4Ja3Y (- i).
20.11
OBillYiE <l>OPMYJlLI YlHTEfPYlPOBAHYl5I
331
25.3. 06m:He 4IOPMYJIbI HHTerpHpoBaHHH
f1pHMeHeHl1e nepeJl.aTOlJHoti CPYHKll.HH Jl.aeT TOlJHY!O cpa3Y D:JJ5I Tpex
CPOPMYJI, KOTopLle I1CCJIeJl.OBaJIHCb B 25.2, a TaK)j(e Jl.JI5I CPOPMYJI,
I1CCJIeLl.OBaHHbIX B rJI. 24. KorJl.a I1cCJIeJl.yeTC5I 06illa5! cpopMYJIa (25.2-1)
Jl.Jl5! CJIYlJaeB, nepelJl1CJIeHHbIX B Ta6JII111.e 13.7-1, Jl.eJIO 6YLl.eT 06CT05!Tb
He TaK rJIaJl.KO. Hanpl1Mep, npl1 I1COOJIb30BaHI1H MeTOJl.a AJl.aMCa - Baw-
cpopra nOJIYlJaeM (h = 1)
iw
3iw
DiU)
iw
9+ 1ge-iw_5r2iw-!-e-siw [ge 2 -!- 1ge
. --l
24 (1- e- iw ) -
-2 _5e-T +e-T]
. w
48slIJ 2
11 lJHCJIHTeJIb He 5!BJI5!eTCSI lJl1CTO MHHMbIM, TaK KaK MHI1Ma5! lJaCTb BbIpa-
)KeHH5I B CK06Kax Jl.JISI Ma-
JleHbKI1X 0) eCTb
TaK lJTO OTKJJOHeHl1e nepeJl.a-
TOlJHOrO lJl1CJIa aT lJl1CTO
MHI1MOrO OlJeHb MaJIO Jl.JISI Ma-
JJeHbKI1X 0).
To, lJTO cpa3a HenpaBI1JIb-
Ha, He CJJeJl.yeT npl1HI1MaTb
CJIHWKOM Bcepbe3. XOT5I 11
aMnm1TYJl.Y, H cpa3Y MO)j(HO
113YlJaTb OTLl.eJJbHO, 3J1.eCb I1H-
TepeCHO JII1Wb, HaCKOJIbKO no-
JlYlleHHbltI OTBeT OTJIl1lJaeTCR aT I1CHlHHoro. PaCClIlOTpHM
[ 9 sin - 19 sin ; +
. 3w . 5w ] 5
+ 5 sm 2 - sm 2 r--..J 0) .
D.ef1CTBI1TeJIbHa5! llaCTb inoro
BbIpa)/{eHI1S1 I1MeeT BI1J1.
[ W W
9 cos 2 + 19 cas 2 -
3w 5w ]
- 5 cos 2: + cas 2: r--J 24,
2,5
0)
2,17
1,5
l,IJ
0)
0,5
I' 17
j -.2,5
-2,17
7,5
1,17
0,5
--- ...--
17
o,Zn:
i17 8 54
lfucllo yJIl08110 l/Uf(/l
PI1c. 25.3-1.
,
::
7'l
n03TOMY Be/H1lJI1HY
I i o - 1\, (25.3-1)
KOTOpa5! Jl.aeT xo powY!O eJl.I1HY!O Mepy OWI16KH *). OTHOWeHl1e \ "':\0 I)
*) B HCKoropblX 3aJ].a'laX, HanpHMep BO MHorHX 3a,IJ.a'laX aKYCTI1KI1, 01lIH6Ka
B 4Ja3e !1MeCT BTopOCTcnCHHOC 3Ha4eJlHe; HO B JJ,pyrHx 3aJJ,a4ax, raKI1X, KaK onpe-
J].C:lellHe B3al1MHblX cnCKrpOB, OHH Moryr HMcrb nepBocTencHHYIO BaJKHOcrb.
332
I-1HTE IP AJlbl M J],I1<1><1>EPEHIJ.I-1AJJbHblE YP ABHEHI151
{IJJ.25
KOTopoe HCnOJJb30BaJ10Cb paHbllle, MO}/{eT 6bITb 6JI1-I3KO K I H Bee Me
HMeTb 60JJblllYIO OlllH6KY B cpa3e; Mbl He BCTpeTHJJHCb C aTHM .ao enx
nap, taK KaK B H3yqaBWHXC5I CJJyqa5lX cpa3a 6wla B TOQHOCTH npa-
BHJlbHott.
Pne. 25.3-1 nOKa3b1BaeT :JTy BeJIl-I4HHY KaK cpYHKIlHIO QanOTbl
T a 6 JI 11 U a 25.3-1
I lroAo II
A;' - AJ\SI pa3JJIIQUblX MeTOAOB HHTerpHpoBaHHSI
l.-lacToTa I I I AJ\aMca - I I rpex
BbiOOPhH Yron Tpanenni! 531I1<jJopT3 CHMnCOH3 I BOC'bMb!X
I
25,00 0,08n 5,27 X 10- 2 1,05 X 10- 4 2,23 X 10- 5 5,06 X 10- 5
16,67 0,12n 1,19 X 10- 2 5,29 X 10- 4 I,J4X 10- 4 2,61 X 1O- t
12,50 0,1611: 2,11 X 10- 2 1,66 X 10- 3 3,66 X \0-4 8,49 X \0-4
10,00 0,2011: 3,31 X 10- 2 4,02 X 10- 3 9,08 X 10- 4 2,15 X \0-3
8,33 0,2411: 4,78 X 10- 2 8,25 X 10- 3 1,92 X 10- 8 4,67 X 10- 3
7,14 0,281t 6,53 X 10- 2 1,51 X 10- 2 3,66 X 10- 3 9,16 X /0-3
6,25 0,321t 8,57 X 10- 2 2,54 X 10- 2 6,44 X 10- 3 1,68 X 10- 2
---
4aCTOT3 I YrOA I OJ\l!oi\ I nO.IJOBHHbJ I JJ,nyx I ,5>
Bbl60PKB TpeT" rpeTei\
25,00 O,081t 5,05 X /0-5 5,00 X 10- 5 3,44 X /0-5 1,28 X 10- 4
16,67 0,121t 2,60 X /0-4 2,55 X /0-1 1,76 X 10- 3 6,24 X 10- 4
12,.")0 0,167t 8,39 X /0-4 8,10 X 10- 4 5,66 X /0-4 1,88 X 10- 3
10,00 0,207t 2,1 J X 10- 3 2,00 X 10- 3 1,41 X 10- 3 4,34 X \0-3
8,33 0,247t 4,53 X 10- 3 4,18 X \0-3 3,02 X 10- 8 8,51 X \0-3
7,14 O,287t 8,75 X \0-3 7,85 X 10- 3 5,78 X 10- 3 1,49 X \0-2
6,25 0,321t 1,57 X /0-2 1,36 X 10- 8 1,03 X 10- 2 2,41 X \0-2
T a6JU-ma 25.3-1 n.aer 3HalJeHH5I aeJlHQHHbl (25.3-1) n.Jl51 ynOTpe6rr-
TeJIbHbIX CI<opoCTeli abI6opKH. VICCJJe.n:oaaHHe ra6JIHQbI nOI<a3bIBaeT, !Jro
(13 YCTOt!4HBbIX MerOn.Oa (KpOMe Mero.n:a CHMTICOHa) HaHMeHbWY!O OlllH6KY
.llJ151 HH3KHX n YMepeHHbIX cl<opocret! BbI60pKH HMeer npaBHJ10 .llBYX
TpeTet!.
25.4. lbHP<IJepeHIJ.HaJIbHhle ypasHeH0!1
VIMeerC!I cyw.ecrBeHHaR pa3HHIJ.a Me}/{py Bbl4HCJIemleM HeOnpe.lleJleH-
Horo HHrerpaJIa H perneHHeM 06bIKHOBeHHoro .aucjJcjJepeHIJ.HaJlbHOrO ypaa-
HeHH5I. lho6bI 9ro craJIO flCHO, npe.llITOJIOIKHM, 4TO eCTb .llHcpcpepeHQliaJIb-
Hoe ypaBHeHHe
v'=Ay+f(t)
2S.41_
LIVI<1>1>EPEHUVlA.nfJHfJm YPAfJHEHl1jI
333
II HeOnpeJIeJIeHHbIft I1HTerpaJI
y' = t(f).
JInH HHTerpaJIa 6nOK-cxeMa BbI4HCJIeHHft I1MeeT BH
t(f)-- I 1 -- y (f),
a JInH JIH epeHUHanbHoro ypaBHeHHH
l(t) -- m -- y (f).
L \ A\ J
B nOCJleJIHel! HMeeTCH neT JJH o6paTHott CBH3H.
9TO pa3J1H4He JIenaeT 60nee nOJIXOJIHruHM lI.pyrol! nOll.X01I K tHlTe-
rpHpoBaHHIO ypaBHeHHH, XOTH npH )l{eJIaHHH MO)l{eT 6bITb I1cnonb30BaH
MeTOJI nepell.aT04Hol! YHKUHH. Mbl 6Yll.eM nOJJb30BaTbCH MeT01l.0M Hccne-
1IOBaHHH BenH4HHbI OIlIl16KH Ha Ka)l{1I.0M wary BMeCTO OTHoweHHH Bbl-
xOll.a K BXOll.Y, TaK KaK 06bl4HO 9TO 60nee cyrueCTBeHHo lI.JJH lI.H e-
peHIll1anbHblx ypaBHeHHtt. B lI.ettcTBHTeJJbHOCTH Mbl JJHUlb JIaeM pa3HhIe
OTBeTbl Ha Bonpoc «KaKOBa T04HOCTb ?», TaK KaK pa3Hble cnY4aH Tpe-
6YlOT npl1MeHeHHH pa3JJH4HbIX KpHTepHeB T04HOCTH.
fIpl1MeM o6ruylO OPMYlIY (15.2-1) (KoTopaH Bblr lIH1IHT TaK )l{e, KaK 11
B 25.2 H 25.3, HO HBnHeTCH JIpyrott no CMbIClIY, TaK KaK 3HaqeHHH y'
Tenepb nOJJyqalOTCH H3 3HaqeHHI! y lJepe3 lI.H epeHuHalIbHoe ypaBHeHHe)
Yn+l = aoYn + aV'n-1 + aV'n_ + h (b -1Y + l + boY + b1Y - 1+
+ b2Y -2)' (25.4-1)
"{Ta6bl H3YlJHTb lJaCTOTHoe nOBell.eHHe (25.4-1), nOll.CTaBHM YHKUI1IO
y (f) = e2TtIat = e lwt ((0 = 21t"0)
Ii BblQI1CJJHM OUlH6KY KaK YHKUHIO lJaCTOTbl
0(0) = 0 1 (w) = elw(t+h) _ aoe lwt _ a 1 e lw (t-h) _ a eIW(t-2h) _
- hiw [b _leiw(t+h) + boe iwt + b 1 e iw1 (t-h) + b2elw(t- h)1,
10 1 (w) \ = 11 - aoe- Iwh - ale-2lwh - a2e-3lwt -
- ihw (b_ 1 + boe- Iwh + ble- Iwt + b e-3,wh) I. (25.4-2)
EClIH B3HTb Tenepb o6ruyIO CPYHKUHIO
co
y(f)=l(f)= P(0)e2Ttlatda,
- co
334
I1HTEfPAJJhl 11 JJ.11<l><l>EPEHUI1AJlhHhlE YPABHEHWI
If JJ. 25
TO olUI16Ka 6Y.1I.eT
(X)
R[j(t)]= F(a)0(a)e21tiatda.
- oJ
(25.4-3)
I1crrOJlb3Y5I TeopeMY CBepTKH, I10JJY4aeM
oJ oJ
R [jet)] = j (s) get - s) ds = get - s) y (s) ds (25.4-4)
-Cf) -00
B 3TOM BH.1I.e OUlI16KY MO)!{HO cpaBHHTb C olUH6KOll H3 r JI. 1l.
B qaCTHOCTH (CM. paBeHcTBo (11.3-8»,
B
R [jet)] = pm) (s) O(s) ds,
A
rJl.e O(S) 6hIJla Ha3BaHa CPYHKll.l1en BJlH5IHI151 11 COCTaBJJeHa 113 4JJeHOB BHJl.a
(X k - s)J + ..,
3n:eCb Orr5lTb BbICTYI1aeT Ba)!{HOe pa3J111411e Me)!{n:y JJ.BYM5I TeOpH5IMI1: MHO-
r04J1emlblll MeTOJJ. I1pI1BOJJ.I1T K BbIpa)!{eHI1IO OlllH6KH B BHJJ.e HHTerpaJla
OT m-t! np0I13BOJJ.HOtt CPYHKll.1H1, YMHO)!{eHHOt! Ha CPYHKll.111O BJlI151HH5I, Tor.Qa
KaK MeTO.1I. HHTerpaJJa <pYPbe rrpl1BO.1l.I1T K I1CnOJlb30BaHHIO caMOl] CPYHK1J.I1H
HJlI1 ee npe06pa30BaHI1H, a He m-tt I1p0I13BOAHOtl 11, KOHe4HO, ApyrOtl
CPYHKIU1I1 BJlHHHHH (CM. TaK)!{e (25.2-10». OJJ.HaKo BI1.11. CPOPMYJlbl (25.4-4)
o6MaH411B; eCJlI1 cpYHKU,HIO get - s) I1CCJIeJJ.OBaTb TIlI.aTeJJbHee, TO MO)!{HO
YBH.QeTb, 4TO STO npoCTO aaMaCKHpOBaHHa51 cpopMYJla (25.4-1).
25.5. nOCTpoeHHe iPHJ1bTpOB no MeToJJ.Y qeoblmeea
l{ I1epBotl CTaJJ.HH nOCTpOeHI15I cpnJJbTpa I1pnBoJJ.I1T Y)I{e rrOHI1MaHHe
Ht(eH MeTopa. Ha npaKTI1Ke B03HI1KaeT TaK MHoro pa3J1WUlblX aan:a'l, 4TO
HeJlb351 HaAe5ITbC51 OXBan1Tb HX Bce. MbI YAoBJleTBOpI1MC5I nosToMY TOJJbKO
6er JJbIM o4epKoM, I1pOHCH51IOIlI.HM 4e6bIIlleBcKutI I10AXOA K KOHCTPYKll.HU
cpHJIbTpOB. OAHa U3 ll.eJletl STOt! KHurH - I1pHBeCTH 4HTaTeJJ5I K TOMY,
'1To6h1 OH Mor caM I1pnMeHHTb npePJlO)!{eHHble I1pnHW-Inbi n CTpOnTb cpop-
MY JIb!. MbI HaAeeMCH, 4TO JJ.OCTur JlH STOt! ll.eJlH.
DpeJJ.nOJlO}KI-IM, qTO Tpe6yeTcH nOCTpOHTb TaKot! MeTOJl. HHTerpl-lpOBa-
HUH AHcpcpepeHu,113J1bHbIX ypaBHeHHt!, 4T06bl olUI16Ka B 06J1aCTn '1aCTOT
HMe,1a 4e6blweBCKYIO CPOPMY, a He CPOPMY, HMeBWYIOC5I paHee - 04eHb
XOPOWYIO BHaqaJle 11 Bce YXY.QliIalOIlI.YIOC5I, Kor.Qa qaCTOTa BoapaCTaeT. D.JlR
HeKOTopblX u.eJletl qe6bIWeBcKI1t1 BH.Q OlllH6KH HBHO I1pe.QI10qTHTeJlbHee
own6KH B BH.1I.e CTeneHI-lOrO pH.Qa.
HatlHeM C npeo6pa30BaHI1H Hawet! 06J1aCTI1 qaCTOT, HO TaK, qTo6bl OHa
6blJla 3aKJlI04eHa Me}f(,a:y - 1 H + I; nocJle 3Toro MO)!{HO nCI10Jlb30BaTb
CTaH,LI.apTHbJe Qe6bJlUeBCKI1e MHOr04J1eHhI. ECJlI1 npHH5ITb o61l1.YlO q,OPMY JIY
25.51
nOCTPOEHI1E <l>I1JJbTPOB no MET01J.Y YEEblllIEBA
335
npe1J.bl.n:ymero naparpacpa (paBeHcTBo (25.4-1 », TO npl1.lJ,eM .lJ,JI5I 0llIH6KI1
B '-IaCTOTe 0) K cpYHKlllm 0 (paBeHcTBo (25.4-2»
Q( r:;) = eiw(l+h) _ aoe iwt _ a1eiW(t-h) _ a"eiw(t-"h) _
_ ihO) [b_teiw(t+h) + boe iwt + bteiOJ(t-h) + b"eiW(Hh)].
(25.5-1 )
BCnOMHI1M Tenepb r JI. 19. TaM 6blJIO 3aMe'leHO, '-ITO eCJII1 HY)I{HO !-IMeTb
OWI16KY II BI1.lJ,e MHoro ' weHa Ye6b1IlIeBa, TO CJle.n:yeT paCCMaTp!-lI33Tb Bce
Bblpa)l{eHHe 8351TbIM B BH.n:e MHOrO'-lJleHOB Ye6h1111eBa. .il.JI5I 9Toro HY)I{HO
YMeTb Bblpa311Tb e iwz '-Iepe3 MHOrO'-lJleHbl Ye6b1111eBa. B H3BecTHolt KHHre
B3TCOH3 «TeOpH5I 6eCCeJleBblX CPYHKI1.I1J'h [41] npHBo.n:!-ITC5I npoH3Bo.n:51!l(a51
CPYHKJU 15I
z/2 ( t - \
e t) =
00
t n I n (z).
n=-oo
fIOJlO)l{!-IM t = ie i6 11, fiCnOJIb3Y5I paBeHcTBo
J_n(z)=(-l)nJn(z),
fiJI!-I, '-ITO TO )l{e,
(i)-n J_ n (z) = (i)n J. (Z),
nOJlY'-IHM
00
eizcosO =J o (z)+2 CitJn(z)cosne.
n=1
[JOJIO)l{HM cos B = 0); Torn.a
OJ
eiZw=J O (z) + 2 (itJn(z) Tn (0)
n=1
(-- I :S; 0) :S; 1).
(25,5-2)
B (25.5-1) BblHeceM 33 CK06KI1 '-I.1JeH e iwt fi nOJlY'JHM
I OCr:;) I = le iwh - aO - at e - iwh -' G2 e -"iwh - thO) (b_ l e iWh + b o +
+ bte- iwh + b"e- 2iwh ) I.
l1cnOJlb3Y51 (25.5-2) (z = h, 0, - h, - 2h no o'-lepe.lJ,l-l) !-I COOTBeTCT-
BylO!l(He TO)l{n.eCTsa .n:Jl5I O)T/I (0), MO)l{HO npeBpaTI1Tb Bce B p33J10)l{eHHe
no '-Ie6bl1lIeBCKHM MHOrO'-lJleH3M c KOMnJIeKCHbJMH K09cpcpl111.I1eHT3MI1 (BKJJIO-
QalO!l(HM!1 6eCCeJleBbI CPYHKUHH).
ECJII1 npHpaBHI1BaTb Bce K09cjJcjJI1Ul1eHTbl HYJIIO, TO nOJlYtJHM O.1l.HO ypaB-
HeHHe .lJ,JI5I Ka)l{.lJ,Oro KOMnJIeKCHOro K09cjJcjJHlI.HeHTa. MO)l{HO HcnOJIb30B3Tb
Bce 9TH ypaBHeHH51, HO JIYtJIlIe COXpaH!-ITb CB060.lJ,HbIMH O.lJ,I1H HJlH .lJ,Ba
napaMeTpa .lJ,Jl5I YCTolttJHBOCTH !-I .lJ,PyrHx )KeJI3eMbIX CBOtlCTB. Pe3YJJbTaTOM
6b1JJa 6bl Kp!-lBa51 Oll1H60K, KOTOpa51 He paBHa HYJlIO npH HYJIeBolt '1aCTOTe,
'-ITO, Bep05ITHO, npHBO.lJ,HT K 33TPyn.HeHH51M; npOCToe ypaBHeHHe
y' =0, V(O)=A # 0
336
I1HTETPAJJbl 1-1 JJ,1-1<1><I>EPEHLlI-1AJJbHblE YPABHEHI-151
[TJJ.25
HMeJ/O 6bl peWeHHe, OTJ/H'lHOe OT nOCT05lHHOt!. 4T06bJ I1cnpaBHTb 9TO,
MO}l{HO Ha'laTb C Tpe60BaHI151, 'lTO HYJ/eaa51 'laCTOTa YJJ.OBJJeTBOpSJeTCSJ TO'lHO,
B pe3YJJbTare 'lerO
1 = ao + al + a2'
11 TOJ/bKO 3aTeM Ha'laTb npHpaBHI1BaTb K03cpcpHlI,I1eHTbl HYJ/IO. TaKHM o6pa-
30M, MOJKHO .a06I1TbC5I YBepeHHOCTH, 'lTO B npeJJ.eJJax oKpyr JJeHH5I HY J/e-
BaSJ 'laCTOTa Bbl'lHCJ/ReTC5I TO'lHO. 3aMeTI1M, liTO eCJ/H coxpaHHTb TOT JKe
caMblt! I1HTepBaJ/ 'laCTOT H nSMeHHTb CKOpOCTb Bbl60pKH sa C'leT nSMe-
HeHHR h, TO MeH5IIOTCSJ Bce K03cpcpHlI,HeHTbl, a He TOJ/bKO MHOJKHTeJlI:, h,
KaK B cJ/Y'lae MHOrO'lJleHa.
vhaK, Mbl B OCHOBHOM 06pHCOBaJ/H MeTOJJ. 4e6blweBa. VIs CKaaaHHoro
BI1LLHO, KaK HatiTH JKeJlaeMYIO CPOPMYJJY npH YCJ/OBHI1, 'lTO MbJ corJ/aCHbl
HeCKOJIbKO nopa6oTaTb, 'lTo6bl lIOJ/Y'lHTb ee. VI 3ro BepHo .n:J/51 WHpOKOfO
!(JJaCca CPOPMYJ/ - 06UJ,Ht! MeToJJ. lIaCTO onpaB.n:blBaeT ce651 npl1 YCJJOBI1I1,
'lTO Mbl c.n.eJJaeM IleKoToPYIO TPy.aHYIO pa60TY 11 HMeeM HeMHoro ao06pa-
JKeHI151, 'lr06bI npeo.n:oJJeTb HeCKOJ/bKO 3anyraHHbIx MeCT. VlCKYCCTBO Ha-
XOJK.n:eHIiSJ CPOPMY. npaKTI1'leClO1 CBO.n:I1TC5I K HaYKe; Mero.n.bl, pa3BHTbIe.a0
CI1X nop, JJ.ocTarO'lHbl, 'lTo6bl o6eCne'lHTb 60JIbWe CPOPMYJ/, 'leM cyw.ecT-
ByeT MeCTa, lIT06bl nepe'lHcJ/HTb HX B KHHre BO MHoro pas 60JIbweti, 'leM
3ra, TaK KaK cyw.eC'myeT MHoro KOM611Hall,Hti n.n.eti, KOTopble Moryr 6bITb
HcnOJIb30BaHbl rrpn pas. n'lHbIX 06Cr05lreJlbCTBax. ECTb Ha.n.eJK.LLa, 'lTO
'1I1TaTeJ/b Tenepb 6y.aeT lIYBcrBoBaTb, 'lro OH MOJKer C HeKoTopblM
Tpy.n.OM BbIBO.n.I1Tb CPOPMYJ/bl, COOTBeTcTBYlOllLl1e cl1Tyamm, BMeCTO Toro
4T06bl rro.n.rOH5ITb cl1ryalJ,111O rro.n: KJ/aCCI14eCKl1e CPOPMYJJbl; Ha.n.eeMC5I, 4TO
OH TaK 11 6y.n.eT nocTynaTb.
25.6. HeKoTopble .u.eTaJUI MeTo.u.a qe6blmeBa
TaK KaK MeTO.LLbI 4e6blmeBa He lIaCTO BcrpellalOTCSJ B JII1TepaType,
paCOIOTpl1M JJ.eTaJlbHee CPOPMYJJbl FlHTerpl1pOBaHFl5I. TIYCTb cpopMYJla FlHTe-
rpHpOBaHI151 I1MeeT CTaH.n.apTHYIO CPOPMY
Yn+! =aoYn +atVn_1 + aV'n-2 + h (b_tY +1 +boy +
+btV -I+bV' -2). (25.6-1)
TI pe.n:rrOJJOJKI1M, KaK 11 B MeTo.ae <t>YPbe, liTO
y (t) = e ioot ;
I1CnOJJb3Y5I ypaBHeHl1e (25.2-2), rrOJlY'lHM
OJ
y(t)=J o (t)+2 L (i)nJn(t).Tn(w).
n=J
(25.6-2)
npOn3I!o.n.Ha51 paBHa
00
y'(t)=J (t)+2 2: (i)nJ (t)Tn(w),
n=!
(25.6-3)
25.61
HEKOTOPbIE lJ,ETAfll1 METOlJ,A 4EBblf11EBA
337
TIOl1CTaBHM Tenepb 9TH JJ.Ba Bblpa)j{eHH5J B (25.6-1), fJCnOJlb3Y5J COOTBeT-
CTBYK>Ull1e 3Ha4eHI15J I, a HMeHHO 1= h, 0, - h, - 2h, 11 BblHeceM 3a
cK06KY ei<J?nh:
00
J o (h)=2 (ilJn(h) Tn (w)=
n=1
OJ
= aoJ o (0) + ad J o (- h) + 2 .L; (i)nJ n (- h) T" (w)]+
n=1
00
+a2 [J o (- 2h)+2 (i)nJ n (- 2h) Tn (W)]+
n=!
00
+ b_Ih r J u (h) + 2 L: (itJ (h) Tn (w)]+
n 1
00
+boI112iJ;(0)T](w)J+blh[J (-h)+ 2 (i)nJ (-h)Tn(w)] +
n=\
00
+ b 2 h [Jo( -2h) + 2 (it J (- 2h) Tn (W)].
n=!
TIpeo6p33yeM 9TO Bblpa)l(eHI1e TaK, 4T06bl nOJJY4!1Tb pa3J1o)j{eHl1e no MHO-
rO'iJleHaM l{e6blweBa, H npI1paBH5IeM K09cpcpHIl.I1eHTbl npY! To (w):
J o (11) = ao + alJ o (h) + a 2 J O (2h) + h lb_I10 (h) - blJ (h) - b2J (2h)].
TIpH T12 <iJ) :
J 1 (h) = - all] (h) - a211 (2h) + h lb _11; (h) + b o l; (0) +
+ b i l; (h) + b 2 1; (2h)],
(25.6-4)
Tk (<iJ) .
TIp I] 2 (if '-'
J k (h) - (- II [alJ k (h) + a2Jk (2h)] + h [b_IJ;, (h)-
- (-I)kbIJ;' (11) - (-llb 2 J;' (2h)].
TIepBblt! (Bblpo)j{l1eHHblt!) CJlY4at!, KOTOpblt! HY)I\HO pacCMOTpeTb, aKBH-
B3J1eHTeH npaBI1JJY Tpanell.l1t!; nOJJO)KI1M a l = a 2 = b] = b 2 = O. TIoTpe6yeM
TaK)Ke BbI1l0JJHeHI151 YCJJOBH5I, 4T06bl (25.6-1) 6b1JIO T04HblM .D:Jl5I Y = 1;
aTO lJ,OCTHraeTC5I npl1
ao = 1.
Tenepb B03bMeM Koacpcpl1ll.l1eHTbI npl1 To (w) 11 TI (w):
Jo(h)=1 +hb_IJ (n), JtCh)=h[b_IJ (n)+boJ (O)].
338 I1HTErP AlIbI VI D.VI <1> <1>EPEHl1.l1 AJJbHblE YP ABHEHVl5I
(TJJ. 25
VIs HHX nOJlY4aeM J (h) = - JiCh),
b - 1- J o (h) b _ 2 J 1 (h) - 2hh_ 1 J; (h)
_1 - hJ! (h)' 0 - h .
(25.6-5)
D.JlH Toro 4TOObl nOHHTb H npOBepHTb 9TH paBeHCTBa, HCCJleD.yeM, 'ITO
npOIKXO.lJ.lH npH h - 0. }{3K HSBeCTHO,
1 I Z r [ (Zj2)2 I (z/2)'
In(z)=nr \2; 1 - n+ 1 T 2\ (n+ I) (n+2)
(zj2)6 ]
- :i!(n+l)(n+2)(n+3) +....
T aK 4TO
H
ft2 ft'
Jo(h)= 1-- 4 +64- ...,
h h 3 h"
J I (h)=2- 16 + 384- ...,
h' h' hO
J 2 (h)=8- 96 + 3072 - ...
ft' ft' h'
4-64+'" 1- 16 I
b_I= - ( h' ) -2'
2-16+ ... 2 1- 8
h 3 ( I 3h' )
h-S+'" -2b_ 1 h 2-16+'" 1
b o = h - 2.
CHa4a.7la MO)KeT nOK3saTbCH y D.I1BHTe.llbHbIM, 'JTO K09cpqmll.HeHTbJ b_ 1
H b o paS.lIH4HbJ, HO He60.llbllIOe paSMblill.lleHl-le nOK3sblBaeT, 4TO B 9TOM
I.leT HI14erO HeO)KHD.3HHoro. Bce D.e.110 B TOM, 4TO Mbl npOBO.IlI1,1H KPHBYIO
OW1-l6KH 4epe3 HYJlb npl1 00 = 0, a nOTOM JlbIT3JJHCb CD.eJlaTb r,1aBHbltI
4J1eH B pa3JJO)KeHHH OWH6KH npOnOpll.HOHa.llbHblM T 2 (00), KOTopoe, 04e-
BI1D.HO, He HMeeT HYJl51 npH 00 = O. B D.etlcTBJ1Te.llbHOCTH 9TOT cJlY'Jat!
BnOJIHe TpHBHaJleH.
Tenepb paCCMOTpHM C.lIY43t1, KOrD.a J1CnOJlb3YIOTCH D.Ba CTapblX 3Ha'Je-
HHH CPYHKll.1111 J1 npOH3BOD.H3H, nOJJO)KI1B .rJI-IWb a2 = b 2 = O. B 9TOYI CJJyqae
COXpaHI1M D..lI5I YCTOtl411BOCTI1 OD.I1H nap3MeTp. TaKHM 06pa30M, I1MeeM
ypaBHeHHSI:
To (00):
TI (00):
T 2 (00):
1 = a o + at>
J o (h) = ao + al10 (h) + h [b _Il (h) - bl1 (h)],
1 1 (h) = - al11 (h) + h [b _11; (h) + b o l; (0) + b 1 1; (h)J,
1 2 (h) = alJ2 (h) + h [b_l1 (h) - bll (h)].
CB. 'JJJ.:
i 25.61
HEKOTOPbIE .D.ETAJJI1 METOJJ.A 4EBbIWEBA
339
Bropoe ypaBHemle MO)\{HO 3al1HCarb, I1CKJIIO'II1B ao, B BHJJ.e
0=(1 - J o ) (1 - al) + hJ (b_ 1 - b 1 ),
3 '-IerBeproe ypaBHeHl1e KaK
0=-J2(1-al)+hJ (b_l-bl)'
()TKYJJ.a BI1JJ.HO, '-Iro
al= I,
b_ 1 = b 1 .
MbI HaXOJJ.I1MCSI Ha rpaHl1ue ycrofi l lI1BocrH (CM. ynpa)\{HeHl1e 13.7-1). Vl3
nepBoro ypaBHeHI1S1 nOJIY'-laeM
Tenepb rpeTbe
ao=O.
1
ypaBHeHl1e I1Meer BI1JJ. J 1 (0) =:2'
2J 1 = 2hb IJ + hg o .
(25.6-6)
l{orJJ.a h 0, OHO nepexoJJ.l1r B ypaBHeHI1e
2b_ 1 +b o =2,
(25.6-7)
i{aK H JJ.OJJ)\{HO 6bITb.
3TH pe3YJJbrarbI HanOMHHalOr MeroJJ. THKa (CM. 25.2), KoropbIfi
1t
1-ICnOJIb30BaJI 4erblpe Y3JJa Ha UHKJJ h =:2' B CBoeM MerOJJ.e OH npl16aBI1JI
YCJJOBl1e (25.6-7), 03Ha'laIOUJ.ee, 'ITO MeroJJ. HHTerpl1pOBaHH5I T04eH, Ko!'.a,a
y (t) ecrb np5lMa51 JIHHH5I. VlcnoJlb3Y51 MerO.ll. THKa, MO)\{HO pelllHTb ypaBHe-
HI151 (25.6-6) H (25.6-7), KOTopble .lI.aJJ.yr
2Jdh) - h _ 7
b_ 1 = 21; (h) _I = 0,3D 85
(h = ; ),
"-ITO O'leHb XOpOlllO COBna.ll.3eT co 3H34eHI1eM 0,3584, Hat1.lJ.eHHbIM THKOM.
nOJJY4eHHoe pellleHI1e npHMeHHMo npH Jl1060t1 llaCTOTe r Bb16opKH. B 9TOM
npl1Mepe 'IJJeHbl 60JJee BblCOKoro nOp5l.lJ.Ka B pa3JJO}KeHI1I1 4e6blllleBa, o'le-
flH.lI.HO, .lI.alOT 04eHb MaJJeHbKHtt BKJla.lJ..
TIpI1 JJ.pyroM MeTO.lJ.e MO)\{HO CJJ.eJlaTb CPOPMYJlY T04HOtt .lJ.JJ5I 4aCTOT,
i{OTOpble cooTBeTcTBYIOT HYJl5lM T3 ((0), a I1MeHHO (0 = 0 11 + 1123 ; TaKI1M
()6pa30M, KpHBa51 OlllH6KI1 I1MeJla 6bl BHJJ. T3 ((0).
3uI npHMepbl nOKa3bIBalOT HeKoTopble npHHUHl1bI 4e6b1llleBCKOro Me-
TO.lJ.a. B KOHKpeTHbIx CJIY4a5lX, OJJ.HaKO, 4aCTO I1MeIOTC5I oco6bIe o6cr05l'reJJb-
CTBa, KOTophIe BJII1S1IOT Ha Bhl60p CPOPMY JIbI, cooTBeTcTBYlOuJ,et1 JJ.aHHot1
cHTyaUHI1. 50JlbIlIOe KOJIH4eCTBO T3KI1X JJ.eTaJlefi MelllaeT JJ.aJIbHefillieMY
l-I3JJO)\{eHHIO 9Toro MeTO.lJ.a B 9JIeMeHTapHoM Kypce.
340
eKcnOHEHUKAJJbHA AnnpOKCHMAUH
[TJJ. 2&
I' J1 A B A 26
3KcnOHEHUHA bHA AnnpOKCHMAUH
26.1. BBe)J.eHHe
Y!3 Tpex KJJaCCOB CPYHKlJ,l1ft, KOTopbIe 6bIJJI1 paccMoTpeHbI B 7.4 I-f
KOTopbIe HHBapl1aHTHbI OTHOCI1TeJJbHO cn:Bl1ra He3aBI1CI1Mot! nepeMeHHoft -
MHOrOYJIeHOB, CI1HYCOB 11 KOCI1HYCOB 11 9KCOOHeHT, - Mbr H3YYHJJI1 MHoro-
YJIeHbt B r JIaBaX C 7 no 20, CHHyCbI 11 KOCHHYCbI B r JIaBaX C 21 no 25,
H Tenepb B On:HOft KOpOTKOft rJJaBe noroBopllM 06 9KcnOHeHTax.
llJIH 06'bf!CHeHI1f! TaKoro HeCOOTBeTCTBI11I B I131I0)!{emm MO)!{HO npl1BeCnr
HeCKOJIbKO n:OBOn:OB. Bo-nepBbrx, KlIacc MHOrOYlIeHOB IIHBapl1aHTeH TaK)!{e
OTHOCI1TeJIbHO 1!3MeHeHI151 Macwra6a I!, CJIen:OBaTelIbHO, B HeKOTopbIX OTHO-
weHHHX 60JJee Ba)!{eH. BO-BTOpbIX, 3HaYIUeJIbHa51 '1aCTb r naB 0 MHorO'llIe-
Hax 6bIJIa rrOCB5Iru:eHa pa3BI1TI1IO TeXHI1KU, KOTOpaf! MO)!{eT 6bITb I1CITOlIb-
30BaHa 11 n:JI5I n:BYX n:pyrl1x KJIaCCOB. B-rpeTbI1X, MHOrO'llIeH yaCTO I!Meer
MeHbwe napaMeTpOB, yeM 9KCnOHeHlJ,IIaJJbHOe BbIpa)!{eHlJe, '1TO n:eJJaeT 9TI!
rrapaMe-rpbr 60JJee yn:06HbIMI1 n:1I5I IJCCJJeJWBaHI1II, HO MeHee 9cpcpeKTI1BHbIMI1
npl! HCITOJIb30Bam'll! BO MHOfHX cI1TyaUI!51X.
BepHo TaK)!{e I! TO, YTO 9KcrrOHeHTbI MorYT 6bITb npen:craBJJeHbI KaK
CHHYCbI H KOCI!HyCbI 4HCTO MHHMoro aprYMeHTa, H MbI Onf!Tb Haxon:I1M, 4TO
HeKOTopbIe n:eTaJIH y)!{e HCCJJen:0l!aHbI. Eru:e on:Ha rrpI1YI1Ha COCTOHT B TOM,
YTO JJI1repaTypa 11 I1CTOpHf! MaTeMaTI1KH OTBOn:f!T KJIaccy 9KcrrOHeHT BTO-
pOCTeneHHYIO pOlIb no CpaElleHHIO C n:BYMf! n:PYfHMI1 KJJaCCaMI1. HaKOHeu,
nepexon: OT '1aCTHOCTet! K 06!l(I1M nn:e5lM eCTb '1aCTb nlIaHa 51T0t! KHnrH,
H n09TOMY MbI YMbIWJIeHHO onYCKaeM cet!yac MHoro ,lLeTaJIeft.
3an:a'lH 9KcnOHeHUHaJIbHoft annpOKCI1MaUHI1 MO)!{HO rpy60 pa3.QeJII1Tb
Ha n:Ba KJIdCca: Te 3 an:a'l 11, rn:e nOKa3aTeJII1 CTeneHeft 3an:aHbI, n Te, rn:e
OHH HeI13BeCTHbI. PaCCMOTpHM CHal/ana CJJY'Iat!, KOr,lLa OHH 3a.QaHbI.
26.2. 0 HaXO}J{)J.eHHH CPOPMYJI, HCnOJIb3YlOIlI.HX 9KcnOHeHTbI, Kor)J.a
nOKa3areJ1H 9KcnOHeHT H3BeCTHM
B r 1I, 10 6bIJI pa3BHT .QOCTaTO'lHO 06!l(Hft MeTO.Q HaXO)!{.QeHI1f! cpOPMY JI.
TaM )!{e OH 6bIJJ npl1MeHeH K CJIY'IaIO MHOrO'lJIeHOB; llCITOJIb3yeM 9TOT
MeTon: CHOBa.
IlpeJuIOJIO)!{HM, YTO HMeeTCf! JIHHet!Hblt! onepaTOp L (/), Hanpl!Mep,
HHTerpHpoBaHHH, n:HcpcpepeHUl1pOBaHH5!, I1HTepnOJIHl.I.llH H T. n:. (CM. 10.3),
11 Tpe6yeTcH npen:CTaBHTb OTBeT IWK JU1HeftHYIO KOM611HaIJ.lllO Y31I0BbIX
3Ha4eHHt! C BecaMH (npoH3Bon:Hbre I1CKJIIO'-IaIOTC5! TOJJbKO ,lLJIf! yn:06cTBa)
L(!) = wo/(xo) + Wt/(Xt) + ... + wn_J(xn_t). (26.2-1)
3n:eCb HMeeTC5! n napaMeTpoB wo, Wt, ..., Wn_t 11, CJJe.a:OBaTeJIbHO, MOMHO
(Mbl Han:eeMclI) cn:eJIaTb cpoPI'.qJJY TOYHOt! .LJ:JJ5I n CPYHKIJ.Hft . eaoX, ea"x,...
26.21
9KcnOHEHTbI C 113BECTHblMI1 nOKA3ATEJ151MI1
341
. . ., e aa - IX. l-JaCTo aO = 0, TaK 4TO B 411CJlO TaKI1X CPYHKUnff BXO.ll.I1T
KOHCTaHTa.
Y paBHeHI151, COOTBeTCTBYIOIlI.l1e (10.3-1), rrpHHHMalOT BH.lI.
akx + akx + + akXa_1
mk=WOe 0 Wje _ 1 ... wn_je
(k=O, 1, ...,' n-1),
(26.2-2)
f.ll.e, eCTeCTBeHHO,
mk = L (e akX ).
(26.2-3)
B03HI1KaeT Borrpoc, MQ)KHO JlH 06paTHTb 9TY CI1CTeMY ypaBHeHHtI; 9TO
saBHCHT OT 3Ha4eHH5I orrpe.ll.eJlHTeJlH
I e akXj \
(k = 0, 1, ..., n - 1;
1=0, 1,..., n -1).
(26.2-4)
ITpe.ll.IJOJlO)l(HM, 4TO ak paclJOJlO)l(eHbI paBHoMepHo 11, 60Jlee Toro, 4TO
ak = k. Tor.ll.a 1J0JJO)l(I1M
eX} = Uj
(26.2-5)
H orrpe.ll.eJlI1TeJlb IJpHHI1MaeT BHA
\ uJ\,
(26.2-6)
a 9TO - orrpeJJ:eJII1TeJIb BaHJJ:epMoHJJ:a (CM. 8.2), He paBHbItJ HYJlIO, KpOMe
CJlY4a51 Xi = Xj AJl5I HeKOToporo i -::j::. j. cho H He YJJ:HBI1TeJlbHO, TaK KaK
saMeHa rrepeMeHHotl (26.2-5) IJpl1BOAI1T Hac K CJlYlIalO MHOrOllJleIlOB.
PaCCMOTpHM Tenepb 06IlI.11t1 cJlYlIatl paBHoMepHo paclJOJlO)l(eHllblX 3Ha-
4eHI1t1 ak 11 rrOJIO)l(HM
ak=a+bk.
TorJJ:a (26.2-2) rrpl1HHMaeT BHA
a-I a-I
mk= 22 Wje akx ;= 22 (wjeaxj)(eoXii'.
/=0 }=o
ITOJlO)l(I1B
axj_-
Wje - Wj'
e oxi = Uj,
HMeeM mk = W j uJ, liTO orr5lTb eCTb CJlY4atl MHOrOllJJeHOB. TaKHM 06pa30M,
CJJYlIatl paBHOMepHO paCnOJlO)l(eHHbIX rrOKa3aTeJletl eCTb IJpOCTO 3aMaCKH-
pOBaHHWR CJlYlIatl MHOr04J1eHOR
ECJlH pelIIl1Tb CI1CTeMY OTHOCI1TeJlbHO Wk (HJIH W k), TO HatlJJ:eM IlCKOMYIO
CPOPMY JJY.
KorJJ:a IJOKa3aTeJIH H3BeCTHbI, HO HepaBHOMepHO paCIJOJlO)l(eHhI, .lWKa-
SaTeJlbCTBO TOro, "ITO orrpeLI:eJlHTeJIb (26.2-4) OTJJHlIeH OT HYJIH He TaK
npOCTO, 11 MbI He 6YLI:eM 3LI:eCb AaJIblIIe paCCMaTpHBaTb 9TOT CJlY4atl. Bo
Bcex CJlY4a5lX pelIIeHlle ypaBHeHl-It! (26.2-2) .n.aeT HCKOMYIO CPOPMYJIY.
:342
9KcnOHEHUl1AJJbHAQ AnnpOKCHMAUI1
frfl. 26
YnpaH<HeHHe 26.2-1. HaHJI,HTe )J.BYMH pa3JIfIlfHbIMH McrOJl,aMH 4JOPMYJlY
I
! (x) dx = a o ! (0) + al/ 2 ! (1/2) + ad (1),
'j{oTOpaf! rO<iHa Jl,JlH ! (x) = I, e- X H e- 2X .
26.3. HeH3BecTHbIe nOKa3aTeJIH
CJlYtJati HHTepnOJl5ul,HI1 HJlH npe.ll.CTaBJleHHSI CPYHKIJ.HH B BI-I.lI.e CYMMbI
9KcnOHeHT C HeH3BeCTHblMH nOKa3aTeJlSlMH Ba)l{eH, TaK KaK OH SlBJlSleTC5I
DCHOBOti aHaJlHTl1tJeCKI1X 3aMeH. ITpOHP, .lI.aJJ npoCToti MeTO.lI. HaXO)l{.lI.eml51
flOKa3aTeJlet! CTeneHH .lI.Jl5I paBHOOTCTOHIlJ,HX JJ:aHHLIX. ITYCTb
I(x) = Aoe aox + A t e a1x +...+ All_teak-IX (26.3-1)
JJ,JlS! HeKOToporo MHO)l{eCTBa paBHOOTCTOmIJ.HX 3Ha4eHHt! X = Xf U = I, ..,
..., n). He 6Y.lI.eT orpaHI14eHHeM C4HTaTb, 4TO Xj = j.
ITpOHH 3aMeTHJI, 4TO eCJJH Bce '-!JIeHbl e aix (l = 0, 1, .. ., k - 1) Y.lI.OB-
JleTBOpmoT HeKoTopoMY pa3HocTHOMY ypaBHeH!1IO k-ro nopS!JJ:Ka C noCTO-
S!HHbJMH K09<p<pHIJ.HeHTaMH, TO xapaKTepHCTH4eCKHe KOpHH 9Toro ypaBHe-
HHS! paBHLI p = e ai . CJleJJ:OBaTeJJbHO, I(x) TaK)I{e Y.lI.OBJJeTBOp5leT 9TOMY
pa3HOCTHOMY ypaBHeHH . ITYCTb 9TO pa3HOCTHoe ypaBHeHHe eCTb
rW+cJ(j+ 1)+. ..+ckfU+k)=O
(j = 0, I, 2, .. .). (26.3-2)
B03MO)l{HLI .lI.Ba CJJY4aH. ECJlH Mhl HMeeM pOBHO CTOJlbKO )J{e ypaBHe-
HHti (26.3-2), CIWJlbKO HeH3BeCTHblX C m (m = 1, 2, ..., k), TO CJle.n:yeT
paCCMOTpeTb «([]epCI1MMeTpH4Hhlti» onpeJI.eJJ!1'reJlb (CM. 10.6)
IfU+n)l.
ECJJJj OH He paBeH HYJl , TO MO)l{HO pellll1Tb ypaBHeHI1S! .n:JJS! Cj' 3HaS!
.ef' Hati.n:eM xapaKTepHCTI14eClWe ypaBHeHHe
pk +Ctpk-t +...+ Ck=O
11 H3 ero KopHet! Haxo.n:I1M ai' 3aMeTbTe, KaK CI1JJbHO 9TO HanO 1!1HaeT
MeTOJJ: pellleHHS! CHCTeMbl 11,JJS! raYCCOBOti KBa.ll.paTypbl ( 10.6). Tenepb,
Kor.n:a 113BeCTHbl ai' MO)l{HO pellIHTb nepBble k ypaBHeHI1t! .n:JJS! Ai' TaKHM
06pa30M, 2k paBHoMepHo pacnOJJo)J{eHHhlx Y3J10B rex) onpe.ll.eJlS! T 2k
HeH3BeCTHblX Gi H Ai'
ECJII! HMeeTCS! 60JJbllIe 'leM 2k Y3J10B, TO MO)J{HO HCnOJlb30BaTb MeTO.n:
HaI-lMeHbllIl1X KBan:paTOB (r JJaBbl 17 11 18) 11 nOJlY4HTb HOpMaJJbHbIe ypaB-
HeHHS!, COOTBeTCTBy IlJ,l1e (26.3-2), 113 KOTOpblX Hal1TH no 04epe.n:H Gi 11 Ai'
KaK TOJlbKO eCTb HHTepnOJlHpY IlJ,a51 CPYHlW.HS!, Mbl MO)l{eM npOH3BeCfI1
aH3!lI1TI1'!ecKY 33MeHY H HatlTI1 JI 6y <POPMYJlY, KOTopaS! HaM HY)J{Ha.
ECJJ!1 Mhl XOTI1M npl1tiT!1 K OTBeTY, He npOXO.lJ51 4epe3 !1HTepTIOJJI1pylO-
!lLYIO <PYHKIJ.I1IO, JI.OCTaT04HO TIpOCTO TIOJJ:CTaBHTb MOMeHTbI Ha MeCTO 3Ha-
26.41
f1PEJ!.Yf1PE}!{J!.EHYI5I
343
4eH!1t! rjJYHKUl1!1 (26.3-1); eCTeC"l'Be!-1H0, 41'0 MbI HC110.1Jb3yeM CTO.1JbKO
MOMeHTOB, CKOJIbKO HeH3BeCTHbIX, a HMeHHO 2k. OmITb }!{e 06UUitJ MeTOJ1.
.n:OCTaT04eH .n:JIH HaXmKJ1.eH!1H rjJOPMYJIbI, KOTOPYIO MbI XOTHM HMeTb, o.n:HaKO
C.1Je.n:yer 6bITb OCTopmKHbIM. OTBeT MO}!{eT 3aBHCeTb OT Toro, B KaKOM
MeCTe 3a.n:a4!1 npOH3Be.n:eHa annpOKCHMall!1H rjJYHKUHH CYMMOti 9KcnOHeHT.
Ynpa}j{HeHMe 26.3-1. nOJI06paTb <jJ)"HKIUfJO YI = Ale- ajX + A 2 e- a2 x 110 clIe-
JIYIOLU,I1M JJ,aHHbIM:
o
2
3
5,5
5,0
5,1
4,:3
26.4. npeJJ:ynpem.ll.eHIHI
XOTH B npI1Hwme !3bIIlIe 6b!Jl0 rrOKa3aHO, KaK HaXO./U1Tb rrOKa3aTe.1JH r
Ha npaKTHKe He Bcer.n:a Bce H.n:eT TaK XOpOlllO: lJIlOr.n:a CJIY4aeTCH, tiTO
4HCJIO 4JIeHOB B rrpe.n:CTaB.1Jem'llJ (26.3-1) He H3BeCTHO, a .n:O.1J}!{HO 6bITh
Hatl.n:eHO. I1.1JJIIOCTpallHet! 9Toro HBJIHeTCH pa.lI,HOaKTHBHbItI pacna.Q, r.n:e 4JIeHb!
COOTBeTCTBYIOT pa3.'!H4Hb!M rrepHo.n:aM rrOJIypacnaJ1.a B llenH pacna.n:a, KOTO-
pblt1 HCCJJe.n:yeTCH.
PaCCMOTpHM npocTotl CJIY4atl rronbITKH pa3JII-I4HTb
A A ' [ -sf + '1 ]
A -at __ -(a+')t + _ -(a-E)t _ A -at e e
e H 2 e 2 e -e 2 '
BhIpa)!{eHHe B cK06Kax paBHO
E 2 t 2 E 4 t 4
1+ 2 +2"4+'"
Pa3HOCTb 3aBHCHT 01' 8 2 , H TOJIbKO .n:JIH 60JIbIlIHX t MO)!{HO Ha.n:eHTbCH 3a-
MeTHTb ee Ha rjJOHe lllYMOB H3MepeHHt!; HO .n:JI5I 60JJblllHX t BeJIlI'JI1Ha e- at
MaJJa! AHaJIOrH4H351 CHTY3UH5I B03HHK3eT B npe06p330BaHHH flanJJaca
co
t(t) = F (0) e-. t do.
D
ECJIH .n:aHO F (a), JIerKO BbI4HCJJHTh Jet), HO eCJIIi eCTb Jet), 3a.Q3HHaH
no T04KaM, TO 3a.lI,34a H3XO)!{.n:eHHH P (a) 60JIee Tpy.n:Ha. .naHllolli [23] ro-
nopHT: «... 3TO nOKa3hIBaeT, 4TO qJH3H4eCKHe Ha6JJ1o.n:eHH5I npe06pa30B3-
HHH flanJJaca HHKOrJ1.a He MorYT nplJBeCTH K pellIeHlJlO 3a.n:a4H BOCCT3HOB-
JIeHI151 OpHflmaJIa C .lI,OCT3T04HOt! CTeneHblO T04HOCTI1». 0./U-13 H3 TPYJ1.1lOCTetf
COCTOHT B TOM, 4TO 3Ha4eHH5I F (0) ,lJ..'!H 60JIbllIHX 0 onpe.n:eJ!HIOT 3Ha4eHH5I
f (t) ,lJ.JI5I MaJIeHbKHX t H Hao60pOT. ECJJH j (t) Y,lJ.OBJJeTBOpHeT HeKOTOpb!M
344
OCOBEHHOCTI1
[IJJ. 27
)LOII0JlHlneJJbHbIM YCJJOBIHIM HJJH eCJlH Mbl paCnOJlaraeM .n:OnOJJHHTe,1lbHotl
I1HcjJOpMalJ,Het! 0 P (0), TO 06paTHOe npe06pa30BaHHe MO)j{HO I1HOr.n:a Hat!TH.
YHTaTeJlb He .n:OJl)j{eH OTCTYlIaTb, eCJJH OH BCTpeT!1TC5I C 06paTHbiM
npeo6pa30BaH!1eM JIaIwaca, TaK KaK OT.n:eJlbHble pe3YJlbTaTbi I1Hor.n:a MO)j{HO
Hat!T11, HO y.n:OBJleTBOp!1TeJJbHble 06llJ,!1e MeTo.n:bl aBTOPY nOKa HeH3BeCTHbl.
26.5. 3KCnOHeHTbJ H MHOrO'IJJeHbl
f{or.n:a 06w.ee nOBe.n:eHHe 3a.n:al[!1 !1MeeT 9Kcno.HeHlJ,!1aJlbHblt! xapaKTep,
HCIlOJlb30BaH!1e MHOrOl[JleHa, B03Be.n:eHHoro B no.n:xo.n:51W.YIO CTeneHb, l[aCTO
DO,1lee y.n:06HO, l[eM cYMMa 9KcnoHeHT. MeTo.n: !1HTerpHpOBaHH5I raycca-
JIareppa !1 MeTO):(, !1CIlOJlb30BaHHbltl B nOCJJe):(Het! LJaCT!1 16.5, HJlJlIOCT-
p!1pYIOT 9TO BbICKa3bIBaH!1e. TaK KaK 06w.a51 TeXHHKa, He06xo):(I1Ma51 .n:Jl5I
peaJlH3alJ,!1!1 9Toro npe.n:JlO}KeH!151, Y)j{e H3JJO)j{eHa BO BTOpOtl l[aCT!1 KH!1rH,
HeT HJ}K.n:bl 06cY)j():(aTb ee .n:aJlbill e.
. 26.6. OCTa TO'IHble 'IJIeHbl
f{or.n:a cjJOPMYJJbl Hat!):(eHbl, eCTeCTBeHHO nO!1HTepeCOBaTbC5I OCTaTOqHblMH
l[JleHaMI1. B r.'1. 11 Mbl Hal[I1HaJl!1 C pa3J1o)j{eH!151 (paBeHcTBo (11.3-2)) npo-
113BOJlbHOt! cjJYHKlJ,I1!1 f (x) no cjJYHKlJ,I151M 1, x, ..., x m -l, ):(Jl5I KOTOpblX
<PopMY,1la 6blJla c):(eJJaHa TOl[HOt!. Eho pa3J10)j{eHl1e 0606llJ,eHO Ha np0l13-
BOJlbl-lbIe MHO)j{eCTB a cjJYHKlJ,!1t!; 113J10)j{eHl1e 9Toro Bonpoca MO)j(HO Hat!TH
B rrpeBocxo.n:HOt! (HO Tpy.n:HOt! ):(,1151 l[TeHH51) KHl1re XaycxoJl.n:epa «OCHOBbl
l[HCJleHHOrO aHaJll13a» [16].
Ha rrpaKTHKe nOJlb3a TaKI1X OCTaTOqHblX LJJJeHOB COMHI1TeJlbHa 11 OHI1
rrOKa pe):(KO ynoTpe6,7I51IOTC5I. OCTaTOl[Hble l[JleHbl, COOTBeTCTBYIOw.!1e cjJop-
MY JlaM ):(Jl5I cjJYHKlJ,Ht! C OrpaHHl[eHHbIM cneKTpoM, rrO-BI1):(!1MOMY, ew.e He
I1CC,7le):(OBaHbl.
r J1 A B A 27
OC06EHHOCTl1
27.1 BBe):(eHHe
B 7.4 6blJl OTMeqeH TOT cjJaKT, l[TO I1MeeTC5I, no cyw.eCTBY, TOJlbKO
TPH K.IJacca cjJYHKUHtI, KOTopble KaK K,7laCCbI HllBapl1aHTHbl OTHOCHTeJlbHO
C):(BHra He3aBHC!1Mot! nepeMeHHot!. 3TH TpH KJlaCCa 6blJJH paCCMOTpeHbl
BO BTOpOt! l[aCTI1, B rJlaBaX 21-25 !1 26 TpeTbet! qaCTH, nOCB5Iw.eHHbIX
{;OOTBeTCTBeHHO MHOrOl[JleHaM, cjJYHKUH5IM CPYPbe (sin nx, cos nx) 11 9KC-
nOHeHTaM.
3Hal[eHl-Ie I1HBapl1aHTHOCTH npH rrepeHoce He3aBHcHMot! nepeMeHHotl
Tep5leTC5I, eCJlH cjJYHKUH5I I1MeeT oc06eHHOCTb, TaK KaK paCIlOJJQ)KeHl1e OCo-
27.21
npYlMEP YlHTE fP AnA C OCOBEHHOCTbIO B BECKOHE4HOCT\1
345
6eHHOCTI-I .n:aeT eCTeCTBeHHOe HayaJJO Koop.n:l-lHaT. KpOMe Toro, oc06eHHOCTI-I,
xapaKTepH3Y!Olli,HeC5I CTpeMJIeHHeM 3Ha4eHI-lt! cjJYHKIJ,HH K 6eCKOIle4HOCTH
.n:JI5I KOHe4HblX 3Ha1.JeHHt! x, MOiKHO armpOKCHMHpOBaTb JJI-lLUb pau,I-IOHaJIb-
HbIMI-I cjJYHKIJ,I-I5IMH. Mbl HeeKOJIbKO pa3 6blJJI-I 6JII-I3KI-I K paCCMOTpeHl-l1O
oco6ellllOCTetJ 1-1 YKa3bIBaJJH B 16.5, 1.JTO I-IWOJIb30BaHl-le MHOjj{I-ITeJI5I .n:JJ5I
.n:etiCTBI-I5I e oc06ellllOCTb!O 06bl4110 JIYYLUe, 4eM HCfIOJIb30Balll-le a.n:.n:I-ITI-IB-
Horo 4JIella. B 1.9 .n:eJJaJIaeb 3aMella fIepeMeHHoro, YT06bl yerp alll-lTb
oc06e1l1lOCTI1.
CnOJIb30BaHl-le 1-13BeCTIIoro nOBe.n:eHI-I5I cjJYIIKU,HH B6JIH3H oc06eHHOCTH
CI-IJIbllo/nOMoraeT BbI4I-1CJIeHH5IM. 60JIbLUa51 yaCTb "Horo HCKYCCTBa nejj{HT
B 06JI em 4I1CTOt! MaTeMaTI-IKI-I H, cne.n:OBaTeJJbHO, 3a npe.n:ena!\m ,HOt!
KHl-lrH. OrpaHH1.JI-IMC5I n03TOMY .n:BYM5! HJIJI!OCTpaTHBHbIM!1 npHMepaMH, nOCJIe
1.Jero c.n:eJIaeM HeCKOJIbKO 06lli,HX 3aMe1.JaHl1ti. O.n:HaKO, KaK TOJIbKO KJJaCC
cjJYIIKUHti onpe.n:eJJeH, fIpOIJ,eCC Haxojj{.n:eHH5I OT.n:eJIbIlOrO npe.n:CTaBHTeJJ5"1
iHoro KJIaCCa CJIe.n:yeT fIO TOMY jj{e fIYTH, KOTOpblti HCnOJIb30BaJIC5! H paHbllle,
a 3I1aYHT, 06 9TOM MOjj{IIO 60JIbWe He rOBopHTb.
27.2. I1pllMep llHTerpaJIa C ocooeHHocTblO B oeCKOHeQHOCTR
f1pe.n:fIOJIOjj{I-IM, YTO paCCMaTpl-lBaeTC5I HHTerpaJI
x
f(x) = e x2 dx.
o
YI3BecTHo, qTO ero aCHMfITOTHqeCKOe pa3JIojj{elclHe eCTb
x 2
e
f(x)r-J 2x +...,
HO OHO HenpHro.n:Ho B6JIH3H x = O. Tor.n:a HafIHllleM
x
f(x) = e.,,2 dx = e x2 D (x)
o
H fIofIp06yeM onpe.n:eJIHTb D (x). f1po.n:HcjJcjJepeHIJ,HPyeM 9TO BbIpajj{eHue,
YT06bI fIOJIyql1Tb .n:HcjJcjJepeHUHaJIbHOe ypaBHeHHe:
D' (x) + 2xD (x) = I,
D(O)=O.
TefIepb HYjj{HO fIpoHHTerpHpOBaTb 4HCJIeHHO fIpocToe .n:HcjJcjJepeHllHa.7JbHOe
I
ypaBHeHl1e, H H3BeCTHO, YTO D (x) - 2x ' Kor.n:a x - co.
TaKHM 06pa30M, cjJYHKIJ,H5I D (x) XOpOLUO Be.n:eT ce65! Ha BeeM HHTep-
BaJIe 0 x co, H fI09TOMY JIYYIlIe H3yqaTh He 3a.n:aHHbIti HHTerpaJI
x
f(x) = e x2 dx = e x2 D (x),
o
346
OCOBEHHOCTI1
lIn. 27
a HHTerpan Il.OYCOHa
x
D (x) = e u - x 2 } e x2 dx.
a
Vl3 ::noro I1pHMepa BH.DHO, KaK npoCTo J1J-JOflla MO)f{HO BbJI.JHC,1HTb
I1HTerpaJJ C oc06eHHocTblO. Eho cnpaBe.n:JH:!BO 11 .n:JI5! CJIY'-IaeB, Kor.n:a
<jJYHKUH5! nJIOXO Be.QeT ce65!, OCTaBa5!Cb KOHeI.JHot!; TaK, MO)f{HO npl1MeH5!Tb
no.n:06HbJe XHTpOCTH, eCJIH Ha KOMIIJleKclloti I1JIOCKOCTH B6JIH3H .Qetlcnm-
TeJIbHot! OCI1 HMeeTC5! oco6eHHocTb. B npHBe.n:eHHoM npl1Mepe .n:eJIO CBe-
lIOCb K npoCToMY BBeD:eHHfO MHO)f{HTeJJ5!, YHH'-ITO)f{atOmero oco6eHHocTb,
11 HHTerpJ1pOBaHHIO D:H<p<pepeHIJ.HalIbHOro ypaBHeHH5!.
27.3. OcooeHHocTb B JIHHeMHOM JJ.HrprpepeHlJ.HaJIbHOM
ypaBHeHHH
VlHor.n,a 6bIBaeT, '-11'0 Tpe6yeTcH '-IHCJIeHHOe perneHHe JIHHetiHoro .n,HIjJ-
<pepeHlJ.HaJIbHOrO ypaBHeHH5J, KOTopoe I1MeeT oc06eHHocTb *) Ha OTpe3Ke
HHTerpl1pOBaHHH. O.n,liH 113 B03MO)f{HbIX cnoco60B npo.n,eJIaTb 31'0, I1pe.n,JlO-
)f{eHHbltI Bl1epBbIe npocpeccopOM TblOKH H CHOBa HJJJJIOCTPHPYIOlIIHtI YHH-
4TO)f{eHHe oc06eHHOCTH C nOMOliJ,blO MHO}KI1TeJJJ1, 3aKmO'laeTCJ1 B ClIe.n,ylO-
lueM. ITYCTb
y" + p (x)y = 0, y (x o ) = A. y' (x o ) = B,
(27.3-1)
fHe P (x) HMeeT oco6eHHOCTb B 06JIaCTH HHTerpHpOBaHHH, HanpHMep npH
..t = 0,
BbJ6epeM .n:1I5! cpatJHeHH5! ypamleHHe
u" + Q (x) 1l = 0,
(27.3-2)
HMelOmee H3BeCrHbIe perneHHH III (x) J1 ll (x) H oc06eHHOCTb Q (x) Toro
)f{e BH.n,a 11 B TOM )f{e MeCTe, liTO H P (x). HI1)f{e 6y.n:eT nOKa3aHO, KaK 31'0
MO}j\HO c.n,elIaTb. ITpe.n:nOJIO)f{H;Vl, .n:aJlee, 41'0
y (x) = a. (x) III (x) + p (x) 119 (x),
(27.3-3)
r.n:e a. (x) H p (x) - <PYHKIJ)1H, KOTopble ew.e H3.n:O onpe.n:eJII1Tb, HO npo
KOTopble npe.n:I1OJIaraeTC5I, 41'0 OHH 6Y.D:YT rlIa.n:KI1MI1. Ham MeTO.D: CJIe.n:yeT
Tenepb KlIaCCHI.JeCKoMY «MeTo.n,y BapHalJ.l11-I nOCT05lHHbIX». BBe.n:5! .n:Be HeH3-
neCTHbIe <PYHKUHI1 rl (x) H i3 (x), MbI MO)f{eM nOTpe60B3Tb BbJnOJIHeHH5J
(27.3-1) Heme OJJ,Horo YClIOBH5I, KOTopoe BbJ6epeM TaK:
a.'1l 1 + Wll = 0,
(27.3-4)
* j To eCTb KaKOH-JlH60 H3 K03Q4JH1J,HCHTOB ypaBHeHHH B6m13H HeKoropofi
T04KH He BeAeT CC6R KaK CTeneHHoR pH
7..31 OCOBEHHOCTb B Jll1HEI1HOM JJ.H<I><I>EPEHI.UIAJJbHOM YPABHEHHH 341
TaKI1M oopaaoM, HMeeM
y = C1./l 1 + U2'
y' = a.u; + u ,
y" = cm; + a.'u; + u + Wu .
I1011cTaBJl5I5I 9TI1 Bblpa)KeHI151 B (27.3-]), 110JlYlJaeM
(J. (u; + PUI) + (u + P(2) + a.'u; + Wu = O.
(27.3-5}
LlaJIee, I1cnOJlh3Y5I, lJTO UI 11 U 2 Y.lI.OBJleTBOp5IIOT (27.3-2), HaXO.ll.l1M
u; + PUj _ (P - Q) U1> u; + PU2 = (P - Q) U2'
TaKI1M o6pa30M, H3 (27.3-5) BblTeKaeT
a.'u; + Wu; = (Q - P) (a.UI + (2) = (Q - P)y.
Perua5I 9TO COBMeCTHO C (27.3-4), npHXO.ll.l1M K paBeHcTBaM
cl (lI I U; - U2ll;) = - ll2 (Q - P)y,
W (lljU; - U2 U ;) = U I (Q - P)y.
Ho lllU - u;u; - BpOHCKl1aH <PYHKIUlfi u l 11 U2' 11 J],Jl5I 9TorO cJlYlJa ,
(Ko9<P<Pl1IlHeH"f npH y' B ypaBHeHl-ll1 paBeH HY JlIO) OH eCTh KOHCTaHT8
W o *), r.lI.e
W o = UI (xo) ll (xo) - u 2 (xo) u; (xo).
W o JlerKO onpe.lI.eJJ5IeTcSJ no 113BeCTHblM peWeHI15IM U I (x) 11 U2 (x).
TaKI1M o6pa30M, Heo6xo.ll.I1MO peWI1Th ypaBHeHI1SJ
,_ I1dQ-P)y !Q'_ I1dQ-P)y
a.-- W ' ['- W '
o 0
y (x o ) = A, y' (xo) = B, y = a.UI + ll2'
(27.3-6)
Tpe6yeTCSJ Bbl6paTh Q (x) TaK, lJTo6hl MHO)KI-ITeJJh Q - P y6bIBaJl B6J11-131-1
oco6eHHocTI-I .lI.OCTaTO'mo 6bICTpO, lJTo6bI nepeKphlTh pOCT np0l13Be.ll.eHl1fi
ll2Y 11 UJ)' (I-IJlI-I, lJTO TO )Ke caMoe, npOH3Be.ll.eHl-lfi U , UIU 2 1-1 uD. ECJJI-I
9TO MO)KHO C.lI.eJlaTh, ypaBHeHI1SJ, onpe.ll.eJlSJIOLQl-le a (x) 11 (x) B6J11-1311
TOlJKI1, r.ll.e 6hlJla oco6eHHOCTh, 6Y.ll.YT BeCTli ce6SJ rOpa3.l1.0 JlY4rue.
I1OKa)l(eM Teneph, KaK MO)KHO Bbl6paTh U I (x) 11 U2 (x) .lI.JHI H8ruero
npHMepa. I1pe.ll.nOJIO)KI1M, lJTO
P(x)= .::.! +ao+alx+...
(a_I :f:. 0).
(27.3-7)
TaKHM 06pa30M, npe.n:nOJl8raeTC5I, lJTO eCJm y (0) :f:. 0, TO y" (0) 6eCI<O-
He'iHO, 11, 04eBI-I.lI.HO, HeB03MO)KHO armpOKCI1Ml1pOBaTh pewelme MHOr04J1eHOM
*) 3TO cJleilyer 1f3 reOpeMbJ JJHYBHJlJlR, COrJlaCHO KOTOpofi W (x) =
- JPI (x)dx
= Woe , rile PI - K03<jJ4JI1u,lIeHr npH y'. (Dpu.M. pea.)
:)48
OCOBEHHOCTI-'I
[f JJ. 27
C JIIo6ott TOlJHOCTblO B6JII13H TOIJI<H X = O. ITonpo6yeM 110JIO)KHTb
x X D - 1 ( )
111= l+ax+bx 2 = D(x) =X X.
(27.3-8J
Tor.lJ.a
u; = [2 (D-l)' + x CD-I)"] ( D t) U 1 )
( D ::) 111 == I)
I1JJH, I1Haqe,
" X [2 (D')2 - DD"J - 2DD'
11 1 = XD2 11 1 ,
Ho
D=l +ax+bx 2 , D'=a+2bx, D"=2b.
CJJe.lJ.OBaTeJlbHO, .lJ.JI5J MaJJblX X
u;
- 2a - 6bx + 2b 2 x 2
XD2
- 2a.- 6bx+2b 2 x 2
111 = xli +2ax+ (a 2 +2b1X 2 + ...J U1 =
= - 2 [ ; + (3b - 2a 2 ) +oo }ll
viTal{,
Q = _ 2; + (_ 6b + 4a 2 ) + (oo .) x + .. .
CpaBHlmaR Q 11 P, Bbl6HpaeM
- 2a = a_I'
a_I
a=-:z-,
I'IJIH
- 6b + 4a 2 = ao,
b - a l-aO
- 6
ITp!1 TaKOM Bbl60pe a 11 b B (27.3-8) HMeeM (HCrIOJIb3Yil (27.3-7))
Q-Pr--..> Cx+...
H Q - P y6hIBaeT .n;OCTaTOlJHO 6b1CTpO.
Tenepb onpe.lJ.eJlml 11 2 06blqHbiM o6pa30M:
x x
( ) _ ( ) \ _ ( ) \ 1+2a6+(a2+2b)62+2aW+b264 d o _
112 X - Ul X J ui (6) - Ul X ,) 62 v_
Xo Xo
[ 1 b 2 x 3 ]
=u 1 (x) -x+2a'lnlx\+(a2+2b)x+abx2+ .
OCTaBllleeC5J npoCTo. IToJlYlJI1BWa5JC5I CPYH)(IJ,H5J 111 (x) He HMeeT oco6eH-
HOCTH npH X = 0, a 112 (0) = - 1. Bropa51 npOH3BO.lJ.Ha5! CPYHKU,HH 112 (x)
6eCKOHelJHa npH x = O. IT pOH3Be.lJ.eH!1R (Q - P) lltY H (Q - P) 11 2 Y o6pa.
U(aIOTCH B HYJIb rrpH x = 0, qTO rr03BOJIfIer XOpOllJO muerpHpOBaTb ypaB-
27.41
OBillJ1E 3AME'1AHI151
349
HeHI1H (27.3-6) Mil orrpe.!J,eJIeHI1S1 a (x) 11 (x), TaK KaK oc06eHHOCTb co-
..lI.ep)!{I1TCSI TOJIbKO B U 2 I1 OT'1aC"fI1 MaCKI1PyeTcII HYJleM Q -- P. BTOT npl1-
Mep CHOBa nOKa3blBaeT, KaK nOJIe3HO Bbl.!J,eJII1Tb r JIaBHblrt '1JIeH pa3JIO)!{eHI1S1
B OKpeCTHOCTI1 oc06eHHOCTI1 H 3aTeM HC110JIb30BaTb 113BeCTHYIO Q)YHKlI,l1lO
.!J,JIH anl1pOKCHMall,HH B6JIH311 OC06eHHOCTI1.
27.4. OouUle 3aMeqaHHH
MbI l1pl1BeJIH .!J,Ba npl1Mepa Toro, KaK o6pamaTbcSI C oe06eHHocTIIMI1
B .!J,BYX '1aCTHbIX CJIYIJaiIx. B HaweM paCnOpiI)!{emIH Her pa3p360T3HHblX
npaBI1JI .!J,JISI o6w.ero CJIY'1aH, oc06eHHo .!J,JIiI Hem1HertHblX 33.!J,alJ. MHO roe
3aBI1CHT OT MaTeMaTI1IJeCKHX cl1oc06HocTert HC110JIHHTeJIII.
Ha COBpeMeHHb!X M3WHH3X C !JJ]aBalOw.ert 3aniITort 11 HO r.!J,a MQ)KHO,
l1HTerpl1pySI, nO.!J,ortTI1 K oc06eHHoCTI1 TaK 6JIH3KO, lJT06b! Ha OCTaBweMCII
.!J,O oc06011 TO'lKH oTpe3Ke JIerKO 6b!J10 C.!J,eJIaTb aHaJIHTI1IJecKYIO 3aMeHY.
J{or.!J,a oc06eHHOCTb npol1.!J,eHa, MO)!{HO CHOBa npl1CTynl1Tb K lJl1CJIeHHoMY
I1HTerpHpoBaHHIO. O.!J,HaKo 9TI1M CJIe.!J,yeT 110JIb30BaTbCSI JII111Ib Tor.!J,a, Kor.!J,a
JJ.pyrHe MeTO.!J,bl Tepl1iIT HeY.!J,aIJY, TaK KaK B 9TOM CJIYlJae lJaCTO TPY.!J,HO
Oll,eHI1BaTb 11 KOHTpOJIHpOBaTb TOlJHOCTb.
J{or.!J,a I1MeeTCH TaK Ha3blBaeMaSl «110.!J,BI1)!{HaiI» oc06eHHocTb, lJTO MO)!{eT
CJJYlJHTbCH B HeJII1HertHb!X 3a.!J,alJaX, oco6eHHo Ba)!{HO 6bITb BHHMaTeJIbHblM
K TOMY, KaK lJHCJIeHHble .!J,aHHb!e HC110JIb3YIOTCSI .!J,JJII nO.!J,60pa CPYHKlI,l1l1,
aonpOKCI1l\mpYIOw.el1 oc06eHHocTb. Bee )!{e TaKHe Bew.11 MorYT 6b!Tb H 6bI-
BaJIH C.!J,eJraHb! Y.!J,al!Ho; 9TO BOl1pOC CMeJIOCTI1 11 Tw.aTeJIbHOCT!1 Bb!lJ!1CJIeHl1tt.
4 ACT b TV
AJIrOPHTMbI 11 3BPI1CTM4ECKl1E METO.Ltbl
r J1 A R A 2R
HAXOlK.lI.EHHE HY JIEtt
28.1. AJIrOpHTMbl H aBpHCTHtfeCKne MeTO)l.bl
110 CI1X nop cj:JOPMYJJbl, KOTOpI.Ie MI.I paCCMa1'pHBaJJH, 3aJI,aBaJiH BCKO-
MYIO Bem14HHY B 5!BHOM B!1lle (I1CKJII04eHl'le COCTaBJ15!JIH TOJII.KO MeTOllbI
npOrH03a H KoppeKUHI1). OJll-IaKO B TaKHX cJIY'Ia5!x, KaK HaXO)!{,lLeHHe:
HYJIet1 CPYHKUHH I1JJH peweHHe CHCTeMI.I JIHHelhIbIX aJJre6pal14eCKI1X ypaB-
HeHHt1, BeJIH4DHI.I, KOTopI.Ie HallO 13 bI'IHCJIHTb, 3a.a:aHI.I HeHBHO. EC.TIll B 1'a-.
KO!! cHTyauHH 33,lLaH onpe,lLeJIeHHI.ItJ npouecc .QJI5! BbI411CJIeHH5! HeH3BeCT-
HbIX, TO rOBopn, 4TO 3aJI,aH a/l20pllln.AI.
Yl3BecTHO IHO)!{eCTBO 4aCTHI.tx MeTOJI,OB,npHMeH5!101HHXC5! B pa3JJt14--
HhlX KOHKpeTHbIX CJlY4a5!X, HO 4TO KaCaeTC5! 06lLLHX np!1HU!1nOB, KOTO-
pbIe MO)!{HO 6I.!J]0 6bl npHMeH5!Tb perYJl5!pHO, ,lJJ JlO, Ha,lLO CKa3aTb, Haxo-
JI,I1TC5! B npI1MI1THBHOM COCT05!HI1H. UeJib 9TO!! KI-HJm -- .QaTb 06Il1.He
MeTO,lLI.I H H.QeH, HCnOJIb3yeMbJe B BbI4HCJIHTeJIbHotJ npaKTHKe. T aK KaK
B 06JIaCTH aJIrOpHTMOB HX, nO-BH.QHMOMY, HeMHoro, TO HaM npH.Qe1'CH
.LI.OBOJIbCTBOBaTI.C5! .QJJ5! HeCKOJlI.KHX BI.I6paHHhIX aJJropHTMOB onHcaHHeM
Toro, 4TO Mo)IUIO, a 4ero HeJIb35! OT HI1X O)!{D.QaTb.
f{poMe TOro, .QJI5! 60JJblllDHCTBa 4aCTO BCTpe4alOIlI.HXC5! 3aJI,a4, Tpe-
6YlOIUHX npHMeHeHH5! I13BeCTHhIX aJIrOpHTMOB, Hanl1CaHbI CTaHJI,apT!-IbIe
nporpaMMI.I; 11 )leTaJlH MeTO)lOB MaJlO BOJIHYI01' nOTpe611TeJJ5!, eCJIH, KO-
He4HO, 01-1 nOHHM aeT, 4TO 3a pe3Y JIbTaTbI OH nOJIY4HT.
CywecTByeT MHOro CllTyaUIJ!!, KOr,lLa HeH3BeCTeH aJIrOpIJTM, KOTOphJIJ'
6YlleT .QaBaTb pe3YJIbTaT, HJII1 H3BeCTeH TOJIbKO MeTOJJ" COCT05!WHfi B nOJI-
HOM nepe6ope, Kor.a:a HeIJ3BeCTHO, lfTO 3aKOHlfHTC5! paHbwe: 3a.n:alfa, Ma-
IlJHHHOe BpeM5! IJJIIJ TepneIIHe 3aKa34IJKa.
B TaKotJ cIHyaUIJH npHXOLLIJTC5! npH6eraTb K J!1060MY 113 MeTo.n:OB,
KOTOPI.!!! BbIr JI5!,lLHT nOJJe3HI.IM. TIOJJO)!{eHIJe TlHll1lfHO JJ,JI5! 3aJI,a4 HaXO)!{-
JleHH51 MaKCHMYMOB HJIH MHHlIMYMOB <PYHKI.U!tI MI-IOfHX nepeMeHHbIX. CmUll-
KOM JJ,oporo HCCJIeJJ,OBaTb, CKalKeM, Bce JIOKaJJbHbJe MHHI1MYMbI lfeThIp-
Ha.n:uaTI1MepHOro npOCTpaHCTBa. MbI Heyl36eJI<H0 06palilaeMC51 J( npOBepKaM>
lJC5IKHX .lI.oraAOK, YAalfHbIX H.Qelt HJJH JII06bIX pa3YMHbIX cnOC060B [JOI1CKi:i
28.21
METO)]. )].EJJEHI151 nOnOJJAM )].JJ5I HAXmK)].EHI151 KOPH5I <PYHKl(Y!Y! 351
Tpe6yeMbIx BeJ1l14MH. TaKI-Ie npOIJ,eCCbJ Ha3bIBa!0TC5! 88pllCmll'leC1<llMll H
DY.lJ.Y"f\ paccMoTpeHbI B r JI. 31. OHl1 Jle)l{aT Ha nepe.IJ.HeM crpome I1HTe-
pecHblx cOBpe leHHbIx MCCJle.n.oBamlt! B 06J1aCTH .lIYMa!0WHX MaillHH H
pacn03HaBaHI15! o6pa30B H no Heo6xo.lIHMOCTH 6y.n.YT paccMOTpeHbI JlI1Illb
KpaTKo. TaKHM o6pa30M, ,na KHHra H.lIeT OT 5!BHbIX crOPMYJI K He5lBHbIM
q)QpMYJlaM, .lIJlSI KOTOpbIX H3BeCTeH MeTO.lI peilleHHSI (aJlrOpHTl'vl), M KOH-
lIaeTCSI npe.n.JlO)l{eHl1S1MH, KaCa!0WHMHCSI 1'01'0, 41'0 .lIeJlaTb C HeSlBHbIMI1
<popMYJlar.lH, .ll,J1SI KOTOpblX npaKTYJ4eCKHe aJ/ropHTMbl HeH3BeCTHb! (9BpH-
TH 4eCKI1e MeTo.n.bI).
28.2. MeTOJl, Jl,eJIeHHH nOnOJIaM Jl,JIH HaXOH<JI,eHHH KOpHH
<PYHKlI.IHI
ECJII1 .lIaHa HenpepbIBHaSl crYHKlI,HSI .lIet!CTBHTeJlbHOrO nepeMeHHoro x,
KOTopaSl npHHHMaeT OTpHlI,aTeJlbHOe 3Ha4eHHe npl1 x = a H nOJlO)J(HTeJlb-
Hoe 3Ha4eHHe npl1 x = b, TO H3BeCTHO, 41'0 Me)l{.n.y a H b cywecTByeT
TOlJ.Ka, B KOTOpOt! crYHKIJ,HSI 06paWaeTC5J B HYJlb. ECJlH pa3.lIeJll1Tb HHTep-
BaJl rrOnOJJaM H onpe.lIeJlHTb, nOJlO)l{HTeJibHa I1J1H OTpHlI,aTeJlbHa crYHKIJ,HSI
B TOlJ.Ke .lIeJleHfJ5I, TO TeM caMbIM Hat!.n.eTC5! nO.lIbIHTepBaJl, B KOTOpOM
1:PYHKIJ,I1S1 MeHSleT 3HaK (HJJH HYJlb crYHKIJ,HH). B npI1HIJ,Hne, nOB"ropHbIM
npI1 leHeHl1eM 91'01'0 MeTO.IJ.a (.n;eJIeHl1e nOIIOJl3M) MO)l{HO CKOJlb yro.n;HO
OJll13KO nO.lIot!TH K KOpH!0. TaK KaK Ka)l{.n.blt! illar .n.eJlI1T HHTepBaJl, B KO-
TOpOM JIe)l{HT KopeHb, nOnOJlaM, TO 10 illarOB YMeHbillaT HHTepBaJJ npH-
DJJH3HTeilbHo B 1000 pa3, 20 illarOB - B I 000000 pa3 11 T. .n.. ;::hOT
MeTO)]., KOTOphIt! npe.n.nOJlaraeT TOJlbKO HenpepbIBHoCTb 11 B03MO)l{HOCTb
BbI4I1CJIeHH5I crYHKIJ,I1H B JI!060t! TOlJ.Ke, BnOJIHe 9<jJcreKTI1BeH. CJle.n.yeT
CKa3aTb: «B npI1HlI.Hne», nOToMY lJ.TO cywecTBY!0T norpewHocTH BbI4I1C-
J1eHHt!.
D.aBalhe nOCMOTpI1M, KaK 9TH norpeWHOCTI1 OTpa)l{a!0TCH Ha npouecce
HaXO)l{)].eHI151 HYJlSI. [Jpe.n.nOJlO)l{I1M, lJ.TO npl1 BbIlJ.HCJleHHI1 3Ha4eHI15! crYHK-
11.1111 nOJJY4 aeTCSI He TOT BHaK 113-3a norpeWHOCTet! oKpyr JIeHH5!. PeaYJIb-
TaTOM 51BI1TCSI TO, 41'0 Ha CJle.n.Y!0w.eM ware 6y.n.eT paCCMOTpeHa He Ta
nOJlOBI1Ha I1HTepBaJla. O.n.HaKO, eCJII1 HeT .n.Pyrnx 0lllH60K B onpe.n.eJleHHI1
3HaKa crYHKlI,I1I1, Bcer.n.a 6y.n.eT nOJJY4aTbCSI, lJ.TO KopeHb aaKJI!0lJ.eH B no.n.bIH-
TepBaJle, O.n.I1H KOHell. KOToporo Haxo.n.I1TC5! B T04Ke, r.n.e 6bJJJ HeBepHO
onpe.n.eJleH 3HaK. Yl ':ITO BnOJIHe npI1eMJleMbIt! pe3YJlbTaT: MeCTOHaXO)l{.n.e-
.me KOpH51 onpe.n.eJlSleTC5J B npe.n.eJ/ax Hailll1X B03MO)l{HOCTet1 BbI4I1CJl5JTb
caMY crYHKIJ,I1!0. He60.lJbWa5! norpeWHOCTb He npl1BeJla K CJlI1WKOM 60Jlb-
molt oilll16Ke.
Ka)l{)J.bllt npe)J.JlaraeMblt! MeTO)J. )J.OJl)J(eH 06513aTeJlbHO HCCJle)J.oBaTbC5I
Ha yCTOt!4HBOCTb no OTHOWeHH!0 K 0llJl16KaM oKpyrJleHI151. MHOrfJe 113
HaH60Jlee 6bICTpbIX MeTo.n;OB npHBO)J.SlT K 60JlbillHM 01l1H6KaM, Kor)J.a OHH
pa60Ta!0T npH nOMexax. CJJe)J.ye1' nOM1H1Tb, 41'0 CJlYlJ.atiHO MO)l{HO OKa-
saTbCSI B6J1fJ3H HyJlSl H, CJle)J.OBa1'eJlbHO, OK333TbC51 nO)J.Bep)l{eHHbJMVI OillH6-
KaM 113-3a nOMex Ha JH060M 9TaJIe Bbl'!I1ClleHHI:!. OTMeTI1M ewe pa3, 4TO
352
HAXml<.1lEHI1E HYJJEt1'
I fJJ. 23
MeTO.ll: .lI:eJIeHlHI nOllOJJaM rpe6yeT TOJIbKO YMeHHSI Bbl4HCJJSlTb 31-Ja4eHl1S1
!jJYHKI.UH1 [1 He npe.ll:nOJJ3r3eT COCT3SJJeHTHI nporp3MMb! l1JJJl BbI4J.1CJJeHI1S1
npOH3BO.ll:Hot!.
YnpaiKHeHHe 28.2-1. CocTaBHTb 6J1oK-cxeMY rIpOrpaMMbl MerOll.a ll.e. eHHa
nOrIOJlaM.
BepOSlTHO, tlHTaTemO CP3SY npHJJ:eT B rOJIOBY, qTa MeTOJJ: JJ:e.'1emHl
nOnOJJaM MO)l{HO 3Ha'lHTeJJbHO YJJytJWHTb, eCJJl1 HCDOJIb30B3Tb JJ:JJ5I CJJe.l\ylO-
mero BbI'JI1CJJeHH5I He Cepe,DHHY OTpe3Ka, a TO 3HatJeHHe x, 13 KOTOpOM
.D:aeT HYJlb JIHHet!H351 HHTepnOJI5IUH5!
Me)l{.lI:Y ,QBYM5I fl3BeCTHblMH 3H34eH!1HMH
npOTHROnOJIO)l{HOro 3H3K3.
PHC. 28.3-1 nOK33bIBaeT CPYHK-
UHIO, .D:JlH KOTOpOt! MeTO.D: JJHHet!Hot!
I1InepnOJJ5IlJ.1111 6Y.lI:er HaBepH5lKa Me.n:-
JIeHHee, 4eM MeTO.D: .lJ,eJJeHH5I nOnOJl3M.
KOHe4Ho, MO)l{HO B03pa311Tb, 4TO .l\JJ5I
pa3YMHoro MHO)l{eCTBa !jJYHKUI1t! HHTep-
nOJl5lUHOHHWt! npouecc 6Y.ll:eT B cpe.lJ,-
HeM Tpe60BaTb HaxOW.D:eHll51 MeHhwero
PHC. 28.3-1. npHMep Mell.JIeHHOi1 4!1CJla 3Ha4eHI11i !jJYHKUI1I1. O.ll:H3KO OC-
CXOll.HMocrH. HORHoe tlOCTOHHCTBO MeTOtla tleJleHI1S1
nOnOJl3M COCTOHT B TOM, '-ITO 3.n:eCh H3-
BeCTHO 3apaHee, CKOJJhKO warOB nOTpe6yeTC5l, 4T06bl rrOJlY'JHTh 3a,QaI-IHYIO
To'mOCTb B rrOJJO)l{eHHI1 KOpH5I; .n:pyme MeTO.D:bI B Cpe.D:HeM MorYT Tpe60BaTb
MeHbllJerO '-IHCJla llIarOB, O,QHaKO HHKor.n:a HeJJb351 CKa3aTb, '-ITa 9TO 6y.n:er TaK
B KallitlOM KOHKpeTHOM CJlY4ae. Llet!CTBHTeJJhHO, KaK BI1,D,HO 113 pHC. 28.3-1,
MO)l{HO nocTpoHTh CPYHKUI1IO, .QJJ5I KOTOpoli He06xotlHMOe 'JHCJJO waros
6ytleT 60JJhllle JJlO60ro H3rrepe.D: 3atlaHHoro. LlJJ5I 9Toro tlOCTaTO'JHO yae-
JlH4HTb 3H3l/eHHe !jJYHl{IJJIH a TO'-/Ke x = b. Llpyrot! MeTO.n: JJl1Het!HOn
HHTepnOJl5lUHH, B KOTOpofi I1cnoJlb3YlOTC5I ABa nOCJle,D,HHX Bhl'JI1CJJeHHbIX
3H34eHf15I, eCJJH Kpl1Ba51 .D:OCTaTO'-lHO nJlOxa5l, MO)J(eT rrpUBeCTH npu rrOCJIe-
.lI:OBaTeJJbHblX 1lI3raX K ytlaJJeHHIO OT KOpH5I.
28.3. JJHHeHHaH UHTepnOJIHlJ.UH
I'(OJ
x=a
J:=b'
YnpaiKHeHHe 28.3-1. CocTaBHTb 6J1oK-cxeMY .lI.JlR MeTO.ll.a JlHHefiHOfi I1Hrep-
nOJlIIlJ,HH.
28.4. napaOOmiQeCKaH uHTepnOJIHlI.IHI
ECJ1I1 JlHHeliH3H HHTepnOJJ5IUHH JlY41l1e, qeM MeTO.Q .D.eJleHH5I nOnOJlaM,
TO, B03MOlliHO, elUe JJyqwe MeTOtl nOCTpOeHI15t napa60Jlbl no TpeM T04-
KaM. Bo BC5IKOM CJJyqae, ero cJle.ll:yeT paCCMOTpeTh.
kaK H B MeTO.Qe JJ:eJleHHH nOnOJJaM, npOH3BOJlbHO pa3JJ:eJJHM HHTepB3R
.D.Jl5I HaXO)l{tleHU5I TpeTbefi TOqKl!. (MO)l{eT 6b1Tb, npe.QIIOQTHTeJIbHee tlpyrHe
. 28.51
liEKOTOPbIE 0611U1E 3AME4AHK5J
353
CTpaTenm, HO Mh! He CTaHeM HX 06CY)I(lJ,aTb.) DOCTpm!M Terrepb napa-
60JIY, HCnOJIb3YH pa3AeJIeHHb!e pa3HOCTH H, HanpHMep, ljJOPMYJIY HblOToHa
(CM. 8.4). HYJJb 9TOti napa60JIbl, KOTOphltl nonaAaeT B I1HTepBaJI, AaeT
HaM lJeTBepTYIO T04KY.
3JJ.ecb ecn no KpattHett Mepe .!l.Be B03MO)J{HOCTI1: MO)J{HO B3HTb ABe
T04KH, HaH60JIee 6JH13KHe K TOJIbKO 4TO nOJIY'-IeHHOMY 3Ha'-leHI1IO KOpH5I,
HJlI1, 113 OCTOpO)J{HOCTI1, B3HTb .L\Be 6JII13KHe TO'-lKI1, B KOTOphlX ljJYHKUI151
HMeeT pa3Hhle 3HaKH. ABTop npeAn0411TaeT nOCJIeAHee Ha TOM OCHOBa-
EH!!, 'JTO OH JII06HT Hl\-IeTb HYJIb, KOTOpbltJ OH IHueT, 3aKJIIO'-IeHHbIM BHYTPH
H3BeCTHoro HHTepBaJIa 11 4T06hl 6b!JIa YBepeHHOCTb, '-11'0 napa60JIa HMeeT
AelJc-rBI1TeJIbHbIA HY JIb B 3TOM HHTepBaJIe. Ha nepBOM ware 06e B03MO)J{-
BOCTH AalOT OAHH H TOT )J{e pe3yJIbTaT, HO npH nOCJIe.LJ:YIOIllI1X warax
9TH Cf1oc06bl MorYT .LJ:aTh pa3JIH'-IHhle pe3YJIbTaTbJ.
DpeAOCTaBJI5IeM c1Y1Ta-remo npOBepHTb BJJH5IHl1e nOMex KSa.ll,paTH'-IHotl
HHTepnOJI5IUHH npH npH6JIH)J{eHHH K KOpHIO.
DOCJIeJJ.Hl1e .LJ:Ba MeTOAa HJ!.TIJOCTpI1PYIOT HeCKOJJbKO HeOnpeAeJIeHHhlt!
o6uJJ!J:! npl1Hlvm: 'teAl mOflbute Alemoo II 'teAl /lY'l.ute OH K-Q:JICemcfl,
meA{ XY:JICe OH A{o:JICem nOBecmu ce6fl B C/lY'lae OC/lOJICHeHldi C rjJYHK--
lI,lleiL OH MO)J{eT H B caMOM ANle OKa3aTbC5I xY)J{e npOCTb!X MeTO.LJ:OB
H, BepOHTHO, 6Y.LJ:eT 60JJee nOABep)J{eH nOMexaM.
HeKOTopoe 06111ee npeACraBJIeHl1e 0 TOM, '-ITO HaMH C.LJ:eJIaHO, MO)J{HO rro-
JIYlJHTb, BepHYBWHCb K lJeTblpeM OCHOBHblM BonpocaM, nOCTaBJJeHHblM B 7.1:
1. KaKoBhl Y3JIb!?
2. KaKoB KJIaCC ljJYHKUHtl?
3. KaKoB KprITepl1fi COfJIaCI151?
4. KaKoBa OWI16Ka?
Y3JIbi Bbl6HpaJIHCb 3.LJ:eCb rro XOAY AeJla, ,lI,JI5I aHaJII1TH'lecKoti 3aMeHbl
Hcno.lb30BaJIHCb HHTepnOJJ5Il.lHOHHbIe MHOrOlJJleHbl, a Mepotl ycrrexa CJIY-
)J{I1JIa oWH6Ka no x; AJI5I nepexo.L\a K oWH6Ke no Y lloTpe60BaJII-ICb 6hl
COBceM He60JIbWHe H3MeHeHH5I.
04eBH.QHo, .lI.JI5I HHTepIlOJI5Ill,HH MO)J{HO 6blJIO \-!cnOJIb30BaTb .L\pyrHe
KJIaCCbJ ljJYHKll,HJ:!, eCJIH OHH 60JIee OTBe'-laIOT IlpHpoAe H3Y'-IaeMbIX ljJYHK-
ll,HtJ. B HY)J{HbIX cJIYlJa5lx 9TO AeJ:!cTBHTeJIbHO CJIe.L\yeT AeJIaTb.
YnpamHeHHSI
28.4-1. HaqepTHre 6J1oK-cxeMY KBaJJ,panl'lHOH HHrepnOJliIlJ.HH, coxpaHRIOUJ,eH
nepeMeHY 3HaKa.
28.4-2. V1ccJleJJ,YHTe BJlHHHHe nOMex Ha MeroJJ, KBaJ],paTI1'IHOH HHTepnOJlHlJ.HU.
28.4-3. 4TO 6bI Bbl Ha3BaJll1 01ll116KOll, KOHTpOJlHpYIOUJ,et! IIH60 x, lIu60 .II?
28.5. HeKoTophle oOIll.lie 3aMeQaHIUI
Mw HCnOJJb30BaJIH JII1HeJ:!HYIO H KBa.L\paTHlJHYIO HHTepnOJJ5IUHIO AJI5I
arrnpoKCHMaUI1H HYJIeti ljJYHlUl.HH. O'-leSI1AHO, MO)J{HO HJJ:TH ,lI,aJIbWe H
CTpOHTb MHOrO'lJIeHbl 60JIee BblCOKI1X CTeneHet!. Bhl60p cTpaTerHI1 3aBfT-
CHT OT Toro, HaCKOJIbKO TPY.L\HO Bbl'-lHCJI5ITb ljJYHKlJ.HIO. ECJI\-! 3TO O'-leHb
12 P. B. XeMMHHI
354
HAXO}J{)lEHI1E HY JJEi%
IrJJ.2
TPY,lUW H Tpe6YIOTCSI, MO}J{eT 6bITb, 4aCbl MaWHHHOrO BpeMeHI1 Ha Ka}J{.QYIO
T04KY, TO, BepOSITHO, CJIe.QOBaJIO 6bl 06paT!1TbCSI K MHOr04JIeHaM 60JIee
BbICOKOt! CTeneHH, nOBblWaSl CTeneHb C Ka)/{.QOt! CJle.QYIOw.et! T04KOt!.
C .ilPyrot! CTOpOHbl, eCJIH <PYHKlI.HSI C4H"JaeTCSI JIerKO, TO JIYYWe npe.il-
n04eCTb KaKOt!-HI16Y.Qb npOCTOt! HCnbIT3HHblt! MerO.Q, K31{, HanpHMep, -'!e-
JIeHl1e nOnOJIaM.
06b14HO, KOrJl,a .QJIR H3XO}J{JI,eHHfJ HYJIet! <PYHKUHH y = Y (X) HcnOJIb-
3YlOr MHOr04JIeHbI, nCXO.ilHble Jl,aHHble 3nnpOKCI1MI1pYlOr MHOr04JIeHOM
OT X H 3aTeM npHMeHSlIOT 06paTHYIO HHTepnOJJSlIlHIO Jl,JISI HaXO}J{.QeHHSI
BYJISI. llaCTo CTOJIb }J{e pa3YMHo npe.QnOJIo}J{I1Tb, 4TO X ecrb MHorO'jJIeH
OT )I, 11 npHMeH5ITb npSlMYIO I1HTepnOJISIlJ,HIO. 6e3YCJIOBHO, 3TO HaMHoro
Jlerl\e.
Mbl p aCOIGTpeJlH TaK}J{e ( 7.1) MeTO.Q HblOToHa I1 B ynp a}J{HeHHI1
7.1-1 YKa3aJIH Ha 60JIee 06Il1.I1t! MeTO.Q, KOTOpbJt! HcnoJlb3yeT B'roPYlo
np0I13Bo.QHYIO. A. M. OCTpOBCKHt! B cBoet! BeJIHKOJIerIHot! KHHre [34)
npe.QnOJlaraeT, 4TO Bbl411CJleHHe <PYHKllI1H .QOJI}J{HO 6bITb npI1HSlTO 3a
eJl,HHI1UY pa60Tbl, a Bb14I1CJIeHne Ka}J{.ilot! np0I13BOJl,HOt! .ilOJI}J{HO paccMa-
TpHBaTbcSI KaK Jl,pyraSl e.ilI1Hnua pa60TbJ.
OTCIO.ila OH ouemmaeT mmet!HYIO l!HTepnOJIJJlI.I11O no Jl,BYM nOCJIe.QHI1M
BeJII14I1HaM B 1,618 eJl,l1HHlI.bl, HMeJJ IJ BH.Qy, 4TO Ka}J{.QbJt! war YBeJIl1'm-
BaeT 4HCJIO BepHblx 3HaKOB B 1,618 pa3a, B TO BpeMJJ K3K MeTOJl, HblOTOHa
oueHHBaeTCfJ B 1,414 eJlHHl1l1.bl Ha eJl, IHHlI.Y pa6oTbI. OTCIO.il3 M01KHO
CJI,e.r13Tb BbIBOJl" 4TO MeTOJl, JIHHet!Hot! HHTepnOJIJJUHI1 60JIee 3<p<peKTHBeH.
Ho, KaK HeOJl,HOKpaTHo nOJl,4epKHB3JIOCb, KOJIb CKOpO CPYHKUI15! BbI4I1C-
JleHa, nOCJle 3Toro 60JIbwet! 4aCTbIO eCTb n04TI1 Bce He06xoJl,I1MOe .QJI51
Bbl4HCJleHHSI np0I13BOJl,HOt!, H 4T06bl nOJJY411Tb ee, Tpe6yeTcSI ew.e 04eHb
HeMHoro pa60Tbl; TaKHM 06pa30M, eCJJH npHHHMaeTC5! BO BHHMaHI1e TOJlbKO
MaWI1HHOe BpeMSI, TO BnOJIHe BepOSlTHO, 4TO MeTOJl, HblOTOHa cpaBHHM
C JIHHet!Hot! I1HTepnOJI5!unet! I1JII1 Jl,a1Ke JIY4we ee, C .n:pyrot! CTOpOHbl,
MeT,O.n. HblOTOHa Tpe6yer aHaJlHTH4eCKoro HaXO)](.n:eHHSI npOH3Bo.n.Hofi
(a npH 3TOM 4eJIOBeK HepeJl,KO oWH6aeTC5!) I1 .n.onOJIHI1TeJIbHOrO nporpaM-
Ml1pOBaHHSI.
CJIe.QyeT OCBeT!1Tb ewe O.ilHH Bonpoc. 06bJ4HO OlleHHBalOT pa3J1H4Hble
MerO.QbJ no I1X KOHe4Hot! CKOpOCTH CXOJUJMocrH K HCrHHHOMY pemeHHIO
(B M3TeMaTI14eCKOM CMbICJIe) H npeHe6peraIOr BonpOCOM 0 TOM, CKOJ1bKO
warOB npH.QerC5! c.n:eJI3Tb, nR )](.Qe 4eM TaKaJJ CKOpOCTb 6y.n.eT .n:OCTHr-
HYTa. 11 CHOBa OnbIT nOKa3bIBaeT, 4TO HeKOTopble H3 HaH60JJee 6blCTpblX
MeTOlI.OB Tpe6YlOT 04eHb 60JlbWOrO BpeMeHI1 Ha no.n.rOTOBKY. f1pHMepbl,
npl1Bo.n:HMble B Y4e6HI1KaX, 06bl4HO I1MelOT OT 10 .n:o 20 .n:eCSlTH4HbIX
3HaKOB. O.QHaKo Ha npaKTHKe B 60JJbWl1HCTBe cJlY4aeB Tpe6yeTcSI OT
Tpex .QO llSlTH .n:eC51TI14HbIX 3HaKOB, TaK 4TO TaKHe llpHMepbl 4aCTO BBO.Q5!T
B 3a6J1Y)](.QeHHe.
HaKoHeu., MOJJlJaJIHBO npe.n.noJiaraJIOCb, 4TO HYJIb C caMoro HayaJ1a
H30JJHpOBaH H 4WraTeJIb XOTeJJ 6bl 3Harb, KaK c.n:eJJaTb 3TOT nepBblIJ cy-
w.eCTBeHHblt1 war. K C01KaJIeHI1IO, aBTOp He M01KeT CK33aTb Ha 3TOT C4eT
28.61
METOJI 6EPCTOY
355
HH4ero JleftCTBHTeJJbHO nORe3HOrO. 4aCTO 3aJl343 C3M3 no ce6e npH BRR-
M3TeRbHOM H3Y4eH1111 Jl3eT HeKOTOpylO I1Hrj.JOpM3UI1IO H cymeCTBYeT He-
CKORbKO M3TeMaTH4eCKHX TeopeM H3 9TOT npeLlMeT. Ho Bo06me OTLleRe-
HHe KopHeft eCTb flCKYCCTBO.
28.6. MeTO)); oepcToy ));JIH HaXOHi));eHIiH KOMnJIeKCHbIX KopHeif
MHorotJJIeHa
33Ll3'1a H3XO)l{LleHlUI KopHeft MHor04ReHOB B03HflK3eT LlOCT3T04HO
43CTO LlRH Toro, 4T06bI Onp3BJl3Tb ee TmaTeRbHoe H3Y4eHfle H p33pa-
60TKY cneUH3RbHbIX MeTOLlOB LlRR ee peweHHH. MbI 6YLleM paccM3TpH-
B3Tb TOJIbKO MHor04ReHbI C LleftCTBI-lTeRbHbIMfI K09rj.Jrj.JflufleHT3MfI, T3K KaK
06bI4HO C HflMfI fI CTaJJKHBaIOTCH Ha np3KTflKe. <t>YHLl3MeHTaJIbHbIM CBOft-
CTBOM LleftcTBI1TeRbHbIx MHOr04JJeHOB HBRHeTCH TO, 4TO OHI1 MorYT 6bITb
pa3JIO)l{eHbI H3 LleftCTBI-neJJbHbIe RHHeftHbIe H LletJCTBHTeJIbHbIe KBaLlp3TH4-
HbIe MHO)l{flTeRfI. TIpeLlnORO)l{flM, '1TO LleftcTBHTeJIbHbIe RHHeftHbIe MHO)l{H-
TeJIfI y)l{e YLlaJIeHbJ. KOMnReKCHbIMfI KOpHHMH CJIeLlyeT 3aHflM3TbC5I nOCRe
Toro, K?K Bce LleftcTBflTeJIbHbIe H3ftLleHbI.
Pa3JJI.J4HbIM 113BeCTHhIM MeTOLlaM HaXO)l{LleHIlH LlefiCTBHTeJIbHhIX JIIlHefi-
HbIX 11 KBaJlpanI4HbIX MHO)l{HTeJIeft MO)l{HO nOCBHTI1Tb ueJIYIO KHHry. TOT
rj.JaKT, 4TO cymecTByeT TaK MHoro MeToLloB, nOKa3bIB3eT, 4TO He CYULe-
CTByeT HI1 OLlHoro BnOJJHe Y LlOBReTBOpI-ITeJIbHOrO; Ka)l{LlbIfi MaTeM 3TllK
HMeeT CBOft c06CTBeHHbIfi H3JII06ReHHbIfJ MeTOLl flRI1 MeTOLlbl. TIO-BflLlIl-
MOMY, MeTOLl 6epCTOY ynoTpe6RHeTC5I Bo06me "lame JII060ro Llpyroro.
OH He HenorpewHM; BpeMeH3M11 OH MeLlJIeHHO CXOLlHTCH HJIH Ll3)1{e He
CXOLlflTCH BOBce, Ho B cpeLlHeM OH Ka)l{eTCH JJY4we JII060ro Llpyroro
MeTOLl3 *). BepoHTHo, 6116RHoTeK3 cTaHLlapTHbIx nporp3MM K3)1{LlOrO BbI-
4HCJIHTeJIbHOrO uelITp3 Bcer Lla I1MeeT HeKoTopylO y)l{e rOToBYIO npo-
rp3MMY, H BbI cKopee 6YLleTe HcnOJIb30BaTb ee, 4eM nHC3Tb CBOlOc06-
CTBeHHYIO. Ho Bce )l{e CTOHT 06CYLlHTb OLlHH MeTOLl, KOTOpblfi CHRbHO
noxo)l{ Ha MHome LlpyrHe.
TIYCTb LlaH MHOr04JIeH
P(X)= ao+ atx+ a x +...+ anx n .
(28.6- t)
TIpeLlnOJIO)l{HM, 4TO Y H3C eCTb K3KHe-RH60 Llor3LlKH 0 KB3Llp3TH4HOM
MHO)l{HTeJIe
x +px+q,
(28.6-2)
HBJIHlOmeMCH LleJII1TeReM 9Toro MHor04ReHa (BHa4aRe MO)l{HO nORar3Tb
p = q = 0, LITO ynpocTin nepBbIfi war). !-1cnORb3YH LleJIeHHe MHor04Jle-
*) Y!1orpe6IJreJleH raK)/{e MeTa)!. MIOJlJlepa, HeKoropble upe)!.UO!JHTalOf ,lte-
TO.il JIHHa.
12'
356
HAXO)!{llEHJ.1E HYJJEI1
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HOB (CM. 1.6), pa311eJlHM MHor04JIeH Ha MHO)!{HTeJlh, nOJIY4aH 4aCTHOe
H OCTaTOK, T, e,
n + n-I + +
anx an_Ix '" a o =
= (X2 + pX + q) (b n x n - 2 + bn_Ixn-a +. .. + b 2 ) + blx + boo
(28.6-3)
CMhICJI HHLleKCOB y b CTaHeT HCHhlM B .rraJlbHet1weM; 9TO 06JIer4aei CI1-
CTeMY 0603H84eHHt1. Ilo cxeMe .rreJIeHlH! MHOr04JJeHOB !1MeeM
J P qi an an_I a n _2 a n _3 ... a2 al ao
-I
qb n qb n _ 1 ". qb, qb a Q 0 2
pb n qb n _ 1 pb a pb 2
b n b n _ 1 b n2 b 2 ) b 1 b o
lle OCTaTOK paBeH
blx + boo
f{09cpqJl1ul1eHTbI CB513aHbl CJIe.l1YIOw.llMll aJIre6pall4ecKllMll ypaBHeHHSlMll:
b n = an'
b n _ 1 = an_I - pb n ,
b n _ 2 = a n _2 - pb n _ 1 - qb n ,
(28.6-4)
b n . k = an_k - pb n _ h + 1 -- qb n _ k + 2
(k=2,3,
... ,
n-l),
b o =a o -qb 2 .
J.1CKOMbltJ KB8.lIp8Tl14HbltJ MHO)!{flTeJlb nOJIY48eTCJI TOr.l18 11 rOJ/bKO
Tor.rra, Kor.rra OCTaTOK 6y.rreT TO)!{.rreCTBeHHO p8BeH HY JI 10, T. e.
b 1 =bo=O.
P8CC IOTpHM 9TH !W9cpcpI1Ul1eHTb! KaK cpYHKUHfI p 11 q,
b l = b l (p, q), b o = b o (p, q).
BOCflOJlb3yeMC5! Tel1epb .rrBYMepHbIM 8IJaJJOrOM MeTO.aa HblOTOHa
(CM 7.1) 11 pa3J10)!{!1l\! b l , b o B pH.rr T eJVlOpa B6J1I1311 BbI6paHHOi-! T04K!1
p, q. 0603Ha4HB I1CKOMoe perneHHe 4epe3 p*, q*, 110JlY4HM
b l (p*, q*)=O =b J (p, q)+ a:; I:1p+ ; I:1q+...,
bo!jJ*, q*)=O=bo(p, q)+ O!o I:1p+ O!o I:1q+...,
up uq
j (28.6-5)
r/1.e
Ap=p* -p,
Aq = q* - q
(21).6-6)
28.61
METOJ1 BEPCTOY
357
eyn oum6KH, KOTopble .a:OJDKHhl 6hITb .ll.06aBJIeHbl (rrpI16JIH>KeHHO) .a:JI5I
nOJIY4eHH5I CJle.a:Ylow.HX p 11 q. OT5paCbIBa51 B (28.6-5) Bce 4J1eHbl, KpoMe
JlI1Het1Hhlx, nOJIY4aeM napy JJHHet1HbIx ypaBHeHHt! lIJI5I npl1paw.eHI1t1, KOTO-
pble LlOJl)!{Hbl 5bITb CLleJIaHbl no p H q.
3a.a:a4a CBO.a:I1TC5I K HaXO)!{LleHI1IO 4aCTHbiX np0I13Bo.a:HbIX, KOTOp hIe
51BJl5IIOTC5I K09cpq)l1lI.l1eHTa !I1 npH Her13BeCTHbIX Ap 11 Aq. MO)!{HO 5bIJIO 51,1
LlaTb He50JIblllOe npHpaw.eHHe p H onpe.a:eJIHTb npHpaw.eHH51 b l H b o H
TO )!{e eaMoe npo.a:eJIaTb .a:Jl5I npHpaw.eHH5I q. f1pow.e, o.a:HaKO, Hat1TH !IX
aHaJII1TH4eCKH. LlHcpcpepeHlI.HpyeM ypaBHeHI1H (28.6-4) no p:
ab", _ °
dp- ,
ab",_\ '
-i)p -
ab n _, _ b
--(ffj- -- "'_I
b ab n
n-P iJp '
ab n _ 1
{J-
ap
ab n
-q dp '
iJb n _ k b dbn_k+l . ab"'_k+2
8P = - nHk+1 - P -a;;-- q iJp
db o _
7ip-
db,
-q ap'
ECJIH Terrepb o503Ha411Tb
ob k *
op =-cli,
(28.6-7)
TO IIOJIyqHM
Ch =0,
C I = b n
Ch_2 = bn-I
-PC ,
.,
- PCn_l
-qc ,
(28.6-8)
Ch.k = b n _ k + 1 - PC;_k+l - qC'h-k+2'
Cd -
-qc .
9TH ypaBHeHH5I I1MelOT TY )!{e CPOPMY, qTO 11 ypaBHeHI1H (28.6-4), KpoMe
c = 0, H MorYT 5bITb nOJIY4eHbI OTCIOLla, eCJII1 B ypaBHeHI15iX (28.6-4)
IlOLlCTaBI1Tb Ch_k B3aMeH b n _ k + 1 H b k B3aMeH ak_l' TOJJbKO nOCJIe.a:Hee
ypaBHeHl1e npHLleTcH npl1 9TOM CJJerKa nOLlnpaBI1Tb.
8TH 3aMe4aHH5i HaBo)].HT Ha MbICJIb f10BTOpHTb f1pOlI.eCC lJ.eJIeHH5i MHO-
rOqJIeHOB LlJI5i b (BMeCTO a), HCIlOJIb3Y51 TOT )Ke caMbIt! KBa.a:paTI14Hblt!
358
HA XO}!{llEHHE HY J1Eti
[TJJ. 28
MHOiKI1TeJJb, 4T06bJ !10JJY4I1Tb K03q:>qmUl1eHTbJ Ck. C11eJJaeM 3TO:
] p q l b n b n _ 1 b n _ 2 b n _ 3 b b J b o
qC n qC n _ 1 ... qC. qCa q C 2
PCn PCn_1 ' ., PC a P C 2
C n Cn_1 Cn_
. .. C 2 I C I Co
TaKI-IM 06pa30M, !10JlY4aeM
C n
=b n ,
Cl =b 1 -P C 2 -Q C 3'
CO =b o -Q C 2'
I
I
Cnk = b n _ k - PCn__k+1 - qC n __k+2' }
I
I
)
(28.6-9)
C n _ 1 = b n _ 1 - PCn'
C n __2 = b n _' J - PC n _ 2 - qc n ,
qacTHble npOH3BOJJ.Hble, KOTOpbJe HW.YTCH B (28.6-5), paBHbl (c Y4erOM
(28.6-7»
ob, *
(}p=-C\,
ob o *
iJp = - Co.
CpaBHHBaH (28.6-8) 11 (28.6-9), BI111I1M, 4TO
CLJ=Ck (k=n, n-l, ...,3,2),
c - CI - b l + PC2'
CJle.n.OBareJlbHO,
ob 1
7f;i = - C2,
; =-(CI-bl +PC2)'
(28.6-10)
PaCCMorpHM Te!1epb npouecc OTblCKaHlHl '1aCTHbIX npoH3BOJJ.HblX no q.
CHOBa JJ.l1cpcpepeHUl1pyeM (28.6-4) no p. 1'of11a
ob n _ 1 ob n 0
aq-= -P (j{j = ,
ob n _. _ _ b Obn_l dbn
oq - n -Pdq -q(}q'
iJb n _ k _ b db n _ k +. db n
---aq - - n. k+2 - P --;rq- - q -(f(]'
o = _ b 2 _ q db.
ilq ilq .
29.1]
BBEilEHYIE
359
IlOJJO)KI1M
ob k **
oq =-Ck'
(28.6-11)
Tor.n.a fJOJIY'IaeM
C *- 1 = 0,
':', bd+'-P':"+' -q':'-.+, I
- pC *- I
- qc *,
c "' 2=bn
(28.6-1 2)
**
Co
=b
-qd*.
TaI< KaK C * = C *-l = 0, TO .n.JI5I cpaHHeHI151 (28.6-9) H (28.6-12) HY)I(HO
110.n.CTaBHTb
Ct 2= ck
ct* = C - PCa.
(k=n, n-l, ..., 3),
LJacTHbie npoH3Bo.n.HbIe, KOTopbIe HY)I(HbI .n.JI5I (28.6-5), paBHbl
obi *,,. ob o ** ( )
=- =- =- =- - .
(-inK,
b] (p, q) = c /).p + cat::.q, }
b o (p, q) = (c] - b] - pc ) t::.p + (C2 - PCa) t::.q. (28.6-13)
PeweHHe STI1X .n.BYx ypaBHeHHtI .n.aeT nonp3BKY .n.JI5I KBa.n.paTH'IHOrO
MHO)l(HTeJI5I x 2 + px + q. CXO.n.HMOCTb,":"" eCJIH OHa HMeeT MeCTO, - KB3.n.-
paTH'IH351, T. e. oWH6KH npH .n.OCTaTO'lHO M3JIbIX 3Ha'leHl-J5IX B03BO.n.5ITC5I
B KBa.n.paT Ha Ka)l(.n.OM ware.
1TaK, nOJIY'lHJIC5I HTepaUHOHHbiti npouecc, KOTOpbItI, MbI Ha.n.eeMC5I,
6y.n.eT CXO.n.I-ITbC5I K KBa.n.p3T!-J'IHOMY MHO)l(HTemO Hcxo.n.Horo MHOrO'lJJeHa.
nOc. e Toro K3K OH 6y.n.eT Hatl.n.eH, MO)l(HO pa3.n.eJIHTb Ha Hero (nepBbIti war
.n.eJleHHH) H HCCJle.n.OBaTb '1aCTHOe KaK HOBbIti MHOrO'lJIeH TeM )l(e cnoc060M.
fJlABA29
CWCTEMW nWHEHHWX AnfE6PAWQECKWX YPABHEHHH
29.1. BBeJl,eHHe
3a.n.a'la peWeHI15I CHCTeMbI JJl1HetlHbIx aJIre6paH'IeCKI1X YP3BHeHI1tt B03-
HHKaeT O'leHb '1aCTO H npHBJJeKaeT BHHMaHHe MHOfHX I1CCJIe.n.oBaTeJIelt
B pe3YJJbTaTe HMeeTC5I MHO)l(eCTBO MeTo.n.OB, H TaK )l(e, K3K B cJIY'lae H3-
XO)l(.n.eHH5I KopHeIJ MHOrO'lJIeH3, Ha STY TeMY MO)l(HO HanHC3Tb ueJIYIO KHHry.
360
CJ1CTEl\1bT JJJ1HEI1HbTX An TESP AI14ECKI1X YP AiJHEHJ1 tr
rT JJ. 29
Mb! cocpe,llOT04HM CBoe BHHMaHHe Ha .nBYX WHpOKO HcnoJlb3yeMbIx Me-
To.nax, KOTopble THI1H4Hbl .nml .nBYX 60JlbWHX KJlaCCOB. Bb!60p MeTo.na, KO-
TOpbIIJ HaH60Jlee COOTBeTcTByeT .naHHot! MaUJHHe, eCTb CJlO)l{HbIt! Bonpoc.
rOBOpn, 4TO CI-ICTeMa JlHHetlHbIx ypaBHeHHfI 51BMleTCR nJloxo 06YCJlOB-
JleHHotJ, eCJlH, rpy60 rOBopR, ypaBHeHHR no 4TH JlHHetiHo 3aBI1CI1MbI. MHoro
YCHJlHt! ymJlo Ha !13ytfeHHHe Toro, KaK peman rrJloxo 06YCJlOBJJeHHble CH-
CTeMb!. O.nHaKo MO)l{HO nOCTaBHTb Bonpoc: Tpe6yeTcR JlI1 pewaTb TaKHe CH-
CTeM!>! B IIpaKTHtfeCKHX cHTyaU,l-lf/x? B KaKot! cpH3HtfeCKotJ C!-ITyaUHH OK a-
)l{YTCf/ none3HbIMH OTBeTbI, KOr./La OHH TaK OmYTlIMO 3aBIJCf/T OT K03cpcpIiUH-
eHTOB CHCTeMbI? 06bI4HO CnpaBe./LJII1BO cne.nYKJmee: BMeCTO OTBeTa HmYT MH-
HIJMaJJbHYIO CIJCTeMY nOtfTH JIHHet!HO He3aBIJCHMbIX ypaBHeHrIt!. B CBeTe ,not!
{-lHcpopMall.HH 3a.natfa MO)l{eT 6bITb JIytfme nOH5!Ta H 06bltfHO nepecpopMYJIH-
pOBaHa CHOBa 60JIee Y./LOBJIeTBOpHTeJIbHbIM 06pa30M. BnOJIHe Bep05!THO,
'HO nJIOXO 06YCJIOBJIeHHbIe CHCTeMbI ypaBHeHl1tJ, eCJm HCKJIlOlJHTb oWH6KH
oKpyr JleHHR H H3MepeHHR, HI3J1J1IOTCJI .net!cTBHTeJlbHO JlHHetJHO 3aBHCHMbJMH
H, CJle.nOBaTeJlbHO, He OTpa)l{alOT cpH3HtfeCKot! Cl-lTyall.HH.
29.2. MeTOJI. MCKJIIOIJemlH raycca
1-13 npfl.MbtX MeTOJIOB pemeHH5I CHCTeM JlHHetlHblx ypaBHeHIJtI, B opo-
TIJBOnOJIQ)J(HOCTb umepal1,UOH1ibt.M MeTO./LaM, HaH60JJee WHpOKO HCnOJIb-
3yeTcH MeTO.n [aycca. OH COCTOIJT B TOM, 4TO pemeHHe CHCTeMbJ JlHHet!-
HbIX ypaBHeHHtI ocymecTBJlHeTCR HCKJIKJlJeHHeM oepeMeHH!>IX no OtfepeJIIJ.
CY1ueCTByeT MHO)l{eCTBO pa3J1IJtfHbIX BapHaHTOB MeTo.na, HH)I{e 6y.neT pac-
CMOTpeH O.nI1H 113 HHX. TaK KaK MeTO.n RBJlReTCH OCHOBHbIM, paCCMOTpl1M
CHatfaJla OtfeHb npOCTot! npHMep. TIYCTb ./LaHa CHCTeMa:
x+ 2y+ 3z= 10,
X + 3y - 2z = 7,
2x- y+ z= 5.
BHa1.JaJle HCKJlf01.JaeM X BbllJfjTaHHeM nepBoro ypaBHeHHH, YMHO)l{eHHOrO
Ha nO,LlX0J15111iHe 4HCJJa H3 BToporo 11 TpeTbero ypaBHelllJtI aTO JIaeT
x+2y+3z=1O,
)1- 5z= - 3,
-5y-5z=-15.
3aTeM HCKJlI04aeM HeH3BeCTHoe y BbllfOTaHHeM BToporo ypaBHeHoH, YMHO-
)l{eHHOrO Ha - 5, H3 TpeTbero:
X+ 2y+ 3z= 10.
y - 5z = - 3,
- i30z = - 30.
29.21
METon I1CKnI04EHI151 fAYCCA
361
Tenepb 9TI1 ypaBHeHl151 peillalOTC Tal, Ha3bIBaeMbIM 06pamHbt.M XOaO.M>
Ha4HHa C KOHua. TIoCJIenHee ypaBHeHHe .!IaeT
z= I.
TIO.!ICTaBHB 130 BTopoe ypaBHeHHe, nOJIY4aeM
y=5- 3=2,
TIO.!ICTaBHB 9TH .!IBa 3HaqeHH5I 13 nepBoe, nOJIY'laeM
x = 10 - 4 - 3 = 3,
H Ha 9TOM peillelme KOH'-laeTC5I.
TIYCTb Tenepb HlVleeTC5I n ypaBHeHHti C n HeH3BeCTHblMH
all Xl + anX +.. _+ alnxn=bl,
a 21 xI + a 22 x 2 +. - - + a2nXn = b 2 ,
anixi + a n 2 x 2 +. . . + annx n = b n -
MbJ Ha'lHHaeM C «BblpaBHHBaHH5I» 3a.!Ia'lH: YMHO)l{aeM ypaBHeHH5I Ha HeKO-
TopbIe CTeneHH 10 TaK, 4T06bI Hal160JIbillee alj 13 KaJK.!IOM ypaBHeHHI1
HaXO.!II1J\OCb Me)K.!IY 1 H 10. ECJI\l 6Y.!I.er 06HapY)l{eHO, 4TO HaH60JIblimti
K09cpqmiJ,HeHT B Ka)l{.!IOM ypaBHeHI1H eCTb K09cpcpHUHeHT npH O.!IHOt! H
TOt! )!{e nepeMeHHoi1, HarrpHMep Xk' TO, eCTeCfBeHHO, CJIe.!IyeT raK)I{e 3a-
MeHHTb Xk' 4T06bI npHBeCTH HX B pa3YMHYlO 06JIacTb. Ha caMOM .!IeJIe
MO)!{eT rroHa.!Io6HTbC5I 3aMeHHTb HeCKOJIbKO Xk' 4T06bl coxpaHHTb pa3YM-
Hoe COOTHOilleHHe K09cpCPHUHeHTOB. BTO BblpaBHHBaHHe ycpe.!I.H5IeT 60JJb-
lime H MaJleHbKHe K09cpcpHUHeHTbI H He C03.!IaeT HHKaKI1X OillH60K oKpyr-
JJeHI151 npH C4eTe C rrJIaBalOllJ,et! 3aIHIToi1 (eCTeCTBeHHO, 4TO .!IJI5I MaillHHbl
C .!IBOH4HOti CHCTeMot! CJle.!IyeT l1crrOJlb30BaTb CTeneHH .!IBYX).
TIOCKOJIbKY au MO)!{eT OKa3aTbC5I paBHblM 0, HeT YBepeHHOCTH, 4TO
MO)!{HO HCnOJlb30BaTb nepBbIi1 K09cpcpHuHeHT. BMeCTO 9Toro HatineM Hal1-
60Jlb1lIHt! K09cpcpHuHeHT npH XI' CKa)!{eM akl' KOTOpblti Ha3bIBaeTC5I 2Jla8-
Hbt.M B/le.MeHtnO.M ypaBHeHl151. VIcno.%3yeM 9TO k-e ypaBHeHl1e, 4T06bl l1C-
KJII04HTb XI H3 .!Ipymx ypaBHel-IHti. ITOCKOJJbKY HYMepaUH5I ypaBHeHHti npo-
H3BOJIbHa, .!I,151 Y.!I06CTBa H3JIO)l{eHH5I f1peJUlOJIO)!{HM, 4TO 9TO ypaBHeHHe
H eCTb rrepBoe. (H.!ITfI fIO ypaBHeHI151M B fIaM5ITH MaillHHbl H MCKaTb 9Tor
HOMep - He CJlHillKOM .!Ioporo, XOT5I HeKOTopble, B03MO)l{HO, 'IT06bl .!IOCTH-
myrb Toro )!{e pe3YJIbTaTa, f1pe.!In04rYT f1pOCTO ynpaBJI5!Tb JIOrHKoti 3anaQH.)
Tenepb paCCMorpHM K09cpcpHuHeHTbi npH XO! 13 OCraBillI1XC5! ypaBHeHH5IX
H BocnoJlb3yeMc5I ypaBHeHHeM C HaH60JlbillHM TaKl1M K09cpcpl1Ul1eHTOM, 4TO-
6bI HCKJIlOlIl-1Tb ,x2 H3 npyrHx ypaBHeHl1ti. VI TaK l1,L1,eM .!IaJIbille OT O.!IHoro
Hel13BeCTHoro K .!IpyrOMY,
362
CI1CTEMbl nI1HEt%HbIX AJJ /ESP AI14ECKI1X YP A BHEHI1I1
[fJJ. 29
3.ZJ:eCb YMeCTHbJ HeKOTOpbJe 33Mel/3H!J]J. B npHHlI:Hne, eC.lJH CHCTeM8 /10-
CTaTOl/HO BeJJl-lKa, TO Hallle nepBOHa4aJIbHOe BbIpaBHHBaHHe MO)](eT nOJI-
HOCTbm HCl/e3HYTb H BpeMH OT BpeMeHH MO)](eT nOHaJl06HTbCH nOBTOpHOe
BbIpaBHHBaHHe. Ho MbI He 6Y/1eM paCCMaTpHBaTb TaKHe 60JJblllHe CHCTeMbJ.
ECJm' CHCTeMbI HMemT 50 HJJH 60JJee ypaBHeHHti, TO, npe)](lJ,e 4eM TpaTHTb
MalllHHHoe BpeMH, JJY4llle npOKOHCY JJbTHpOBanCH Y CneUHaJIHCTOB. 1J.e.'1o
B TOM, 4TO KOrlJ,a CHCTeMa npeBOCXOlJ,HT no pa3Mepy 10 HJJH 20 ypaBHe-
HHt!, 4aCTO Ha4HHamT n05IBJJHTbC5I HOBbIe 3rprpeKTbI H onaCHOCTH, B KOTO-
pble MbI 3.a:eCb BHI1KaTb He 6y.a:eM.
Kor.a:a Bb16I1paeTCH ypaBHeHHe, KOTopoe 6Y.ZJ:eT HCnOJJb30BaHO lJ,.7I5I HC-
!Wm4eHH5I CJJeJlymweti nepeMeHHot!, MO)]{HO CHa4aJJa pa3.a:eJJI1Tb Bce K03rp-
<pmmeHTbl ,noro ypaBHemlH Ha CTap1l1I1t1 K03cprp!'1QfleHT, TaK 4T06bl nep-
Bblti K03rpqnJJJ,!leHT CTaJI paBeH e.ZJ:flHflQe. Tor.a:a Dp!l 06paTHoM xO.IIe HeT Ha-
n:06HOCTH HH Ha 4TO .a:eJl!lTb. ECJJH .a:eJJeHl1e Tpe6yeT MHoro MaWI1HI!Oro
BpeMeHI1, TO MO)](HO n01J,CYHTaTb 06p8THym BeJII{lJljHY K Bbl6paHHOMY r JJaB-
HOMY 9JJeMeHTY 11 nOJlY4I1Tb MHO)](HTeJJI1 (Bce 1) .IIJJ5I HCKJJm4eHHH He.
!13BeCTHbIX H3 3Toro YP3BHeHH5I.
MO)]{HO 6bJJlO 6bJ nO.ZJ:yMaTb, 4TO CJIe.IIyeT, BMeCTO Toro 4T06bl 6paTb Xk
no.a:p5I.a:, I1CKaTb 04epe.a:Hoe Xk, BbI6!1p35I Ha!l60JIblllHt! K03rprpf1lJ,lIeHT 113 Bcex
QCTaBlllflXC5I; HO 3Ta .a:onOJlHflTeJIbHa5I TOHKOCTb KaK 6y.a:TO He YBeJIH4HBaer
TOl/HOCTH.
. 29.3. BapHaHTbl MeTO)(a faycca
CYIll,eCTByeT MHoro BapHaHToB MeTo.a:a raycca. O.ZJ:HaKo HaI160JIee 3Ha-
4HTeJIeH MeTO.a:, Ha3bIBaeMblt! MeTo.a:OM nOJIHOrO I1CKJIm4eHHH I1JJ1-1 MeTO.a:OM
faycca - }Kop.a:aHa. B 3TOM MeTO.IIe, Kor.a:a HCKJlm4amT Xk, TO HCKJIm4amT
ero fl3 Bcex ypaBHeHHtI, BK.'1m4aH H Te, KOTopbIe pa3pellleHbl .IIJIH npe.a:bl-
.a:YIll,HX rrepeMeHHblX. DpH 3TOM crroco6e HeT He06xo.a:I1MOCTI1 B 06paTHoM
XO.ZJ:e, VlHor.ZJ:a YTBep)](J(amT, 4TO 3TOT MeTO.II HMeeT npeHMywecTBa B T04-
HOCTH.
B CJIY4ae cHMMeTp!!T-lHbIX K03<fJrpHlI:HeHTOB, ec.'1H rJIaBHbJe 3JJeMeHTbI 6e-
pyTC5I H3 r JIaBHoti J(HarOHaJIH, pa60Ta MO)](eT 6bITb cOKpaUJ,eHa n04TH BlJ,Boe,
HO MO)](eT B03HHKHYTb HeKoTopaH nOTep5I T04HOCTII H3-3a nJJoxoro BbI60pa
r JlaBHbJX 9JIeMeHTOB. ECJlH M8Tp!lU3 IIOJJO)!(I1TeJIbHO orrpe.a:eJIeHH851, TO no-
TepH T04HOCTH MaJIb\.
MHorne BapHaHTbl MeTO.ZJ:a 06bl'-lHO HBJJHmTCH He3Ha4HTeJJbHbIMI1 H3Me-
HeHH MH, C.lIeJIaHHbJMH JJH y.a:OBJIeTBOpeH!I xapaKTepI1CTHKaM .IIaHHO M3-
UlI1Hbl (XOT5! OHH MorYT OKa3aTb 3aMeTHoe BJIHHH!le Ha nOJlYl/eHHbJe pe-
3YJJbTaTbl); n03TOMY OHM He ITO.a:XO.a:5IT .ZJ:JI5I 06CY)]{.IIeHH5I B Ha'-laJIbHOM Kypce.
XOpOlllI1M ITOC06!1eM HBJlHeTCH KHI1ra «COBpeMeHHbJe MeTo.ZJ:bJ Bbll/MCJJeHHH»
[29] *); B Hei1 B CRom 04epe.ZJ:b eCTb06uJ!lpHa51 6116mwrparp!lH ITO MHorHM
acneKTaM BbI4I1CJIeHI1t!, Blwm4aH CHCTeMbl ypaBHeHI1t1.
*) CM. raKJKe [161.
1\ 29.4]
METon r AYCCA-3At1nEJJ5!
363
29.4, MeTolI raycca -3aHiJ.eJIH
B npOTI1BOnOJI01KHOCTb l1p5!MblM MeTo.naM peweHlHI, TaKHM KaK MeTO.n
faycca, cyw.ecTBYlOT HTepaUHoHHbIe MeTo.nbl, TaKHe KaK MeTO.n faycca-
3atLa.eJI5!. TIepecTaBHM ypaBHeHH5I B npHMepe 29.2 TaK, 'JI06bl 60JIbWl1fi
K03cpcpHuHeHT nepBoro ypaBHeHH5I 6blJI 6hl npl1 x, 60JIbWI1tJ K03cpcpHllHeHT
BToporo ypaBHeHH5! l1PH Y H TpeTbero - npH z;
2x- Y+ z= 5,
x+3y-2z= 7,
x + 2 Y + 3z = 10.
HaYHHaeM c l1pH6JIH)!(eHHOrO peWeHl-I5I, CKa)!(eM, x = y = z = 0, V!cnOJIb-
3yeM Tel1epb l1epBoe ypaBHeIII1e .nJI5! HaXO)!(.neHH5I HOBoro 3HayeHH5I
_ 5+y-z _ 5
x- 2 -:-2'
5
6ep51 X = 2' z = 0, pewaeM BTopoe ypaBHeHl1e OTHOCHTeJIbHO y
7 -x+2z 3
y= 3 =2'
HaKOHell, HC110JIb3YH 3Tn Bbl4nCJIeHHble 3HaqeHn5!, pewaeM TpeTbe ypaB-
HeHHe OTHOCHTeJIbHO z
10-x-2y 3
z= 3 -"2'
9TH Tpl1 BeJII1411Hbl .nalOT HOBoe l1pl16.rIl1JKeHl1e 11 MOJKHO 110BTOpHTb ll.I1KJI:
555
x=2' y =2' Z =6'
CJIe.nYlOw.ee nOBTOpeHl1e J1aeT
10 16
x=3=3,33-+ 3, Y=g= 1,780-+2,
28
z = 27 - 1,04 -+ 1.
3aMeTI1M, qTO CXOJ1HMOCTb Me.nJIeHHa5l. ECJII1 B03HHKaeT oWH6Ka, TO
OHa MO)!(eT nOBJIH5ITb Ha YHCJIO warOB, HO He BJII1HeT (B npHHllHl1e) Ha
KOHeqHblt1 OTBeT. 06bI4HO l1po.nOJI)!(alOT 11Tepa1l11l1 .no Tex nop, nOKa 113 Ie-
HeHH5! He CTaHYT J10CTaTOYHO MaJIbl; 4TO 9TO MO)!(eT 03HayaTb no OTHO-
WeHi-IlO K OTBeTY - YJKe .npyrofi BOl1pOC.
06W.Hfi CJIyqafi BO MHorOM TaKofi )!(e. YpaBHeHH5I paC110JIaralOTC5I TaK,
4T06bl 60JIbllme K09cpcpl1UHeHTbl 6bIJIn Ha r JIaBHOfi J1narOHam-I, ECJJn 9TH
4JIeHW rJIaBHOfi J1HarOHaJII1 .nOCTaT04HO cnJIbHO l1peBocxo.n5lT .npyrne
K09cpcpHll,HeHTbl ypaBHeHI1H, TO CXO.nHMOCTb rapaHTnpOBaHa, B npOTnBHOM
CJIyqae - Her. BOT O.nHO J10CTaTOYHoe YCJIOBHe:
I au I> I ail 1+ i ai2 ! +, ., + I aU_1 I + i aii+! I + .. . + I ain I
J1JI5! Bcex i (l1pe.nnOJIaraeTC5I, 4TO cnCTeMa He pacna.naeTCH Ha .nBe He3a-
BncnMble cnCTeMbI ypaBHeHHtJ),
364
CI1CTEMbl JJI1HEHHblX AJJ rE6P AWIECKI1X YP ABHEHI1H
!fJJ.29
ECJJl1 B CHCTeMe HMeeTC5! MHOrO HY JIeB blX K03cpq m U H eHTOB aii' TO
3TOT MeTO.a rrpe.arr04THTeJIbHee MeTo.aa HCKJII04eHH!I. MeTO.a faycca-
3at!.aeJJ5! HCIlOJIb3yeT cyw.eCTBOBaHHe HYJletl, B TO Bpe:ll5! KaK MeTO.a
BCKJII04eHH5! - HeT.
CYlueCTByeT MHO)J(eCTBO pa3JIW{Hb1X BapHaHTOB mepaIJ,110HHOro MeTOJJ:a.
rIpH MeJJ:JIeHHot! CXOJJ:HMOCTI1 Mbl CTapaeMC5! yraJJ:aTb, HCrrOJIb3Y!l pa3JIH4-
Hble TeXHWleCKHe npHeMbI, r.ae OHa Ha4HHaeTC5!, H nonaCTb Ty.aa 3a OJJ:HH
mar. TaK Ha3bIBaeMbie pe/taKcal1,UOHHbte MeTO.1I.bl 5!B0HIOTCH MeTo.aa:VIH
TaKoro THrra. 3a .1I.eTaJlHMH CHOBa OTCblJIaeM 4HTaTeJIH K [29].
29.5. nOBblIIIeHHal! TOqHOCTb
ECJm HMeeTCH HeKoTopblti MeTO.a peIlJeHHH CHCTeMbJ ypaBHeHHft H
nY}K!IO rrOBblCHTb T04HOCTb, TO MO)J(HO BOCI10.1b30BaTbCH CJIeJJ:YIOw.HM npH-
eMOM. fIo.n:CTaBHM B ypaBHeHHe Hat!.n:eHHble BeJIII4HHbl x H Bbl4HCJIHM pa3-
HOCTH, I1CITOJIb3Y5J .n:BOftHYIO T04HOCTh. 3aTeM CHOBa peIlJHM CHCTeMY ypaB-
Helmt!, rro.aCTaBJJH5! rrOJIY4eHHble pa3HOCTI1 B rrpaBble 4aCTlJ ypaBHeHHt!.
CYMMa HOB oro peweHHH co CTapblM .aaeT 60JIee T04Hoe peIlJeHHe. JIerKO
BH.n:eTb, nOlJeMY 3TO TaK. fIepBoe MHO)J(eCTBO 3Ha t jeHHIJ x(\) y.n:OBJJeTBO-
pHeT ypaBHeHI1HM C I1paBblMH 4aCT5JMH, paBHblMH b i MHHYC pa3HOCTb ri
B i-M ypaBHeHHH, TorJJ:a KaK BTopoe peWeII!1e X(2) y.n:OB.1eTBOpHeT I1PH-
6JJH3HTe.1JbHO ypaBHeHH5JM C pa3HOCTHMH B rrp2BblX 4aCT5!X. C.1e.n:OB2TeJlbHO,
.\;(\) + X(2) y.aOBJleTBOpHeT TeM )J(e caMblM ypaBHeHH5JM, HO C CYMMOti .n:Byx
I1paBblX 4acTeIJ:
(b i - ri)+rj =b j .
29.6. 06rn:Me 3aMeqaHMI!
O'leBHJJ:HO, 3.aeCb He HY)J(HO 6bIJI0 6paTb HHKaKoro KJIaCCa CPYHKIJ,Hft
D:JHI aHaJIHTH4eCKot! 3aMeHbl. E.aHHCTBeHHblt! YMecTHblft Borrpoc H3 cxeMbl
I'JI. 7 - 4eTBepTb1IJ: KaK08a mO'iHOcmb? XOTl1M JIH MbJ, 4T06bl Xi 6blJIH
TO'lHbIMH? XOTHM JIH Mbl, 4T06bl pa3HOCTI1 6blJII1 MaJIbl? XOTHM JIH Mbl,
4T06bl KaKaH-TO CI1CTeMa ypaBHeHHii, JJ:JHI KOTOpOtt 3TO no.n:C4l1TaHHoe
peWeHl1e 5!BJUleTC5I T04HblM peIlJeHHeM, 6blJla 6J1113Ka K Ha'l3J1bHOii cnCTeMe?
IIoc,1e.aHee yCJloBHe 3BY'lI1T HeCKOJIbKO crpaHHO, TaK KaK K09cpcpHUHeHTOB
n (n + I), a OTBeTOB TOJlbKO n; n HX OTKJIOHeHl1ti OT I1CTHHHoro pewe-
HIm MorYT 6bITb pacnpe.1I.eJIeHbl cpe.n:n n (n + I) K03cpcpnUHeHTOB MHorHMH
11 MHorl1MI1 cnoc06aMH, H OTKJlOHeHH5I npaBblX 4acTeti ecn Bcero JIHWb
4aCTHblt! CJIY4at!. TaKHM 06pa30M, Bonpoc B 9TOM CJIY4ae cyw.eCTBeHHO
OTJ1H'laeTC5I OT Toro )J(e caMOro Borrpoca .n:J151 HYJleft MHOr04J1eHa, r.n:e MI,l
LlOJI)J(Hbl 6blJIH pacrrpe.n:eJlI1Tb n OWH60K B HYJI!lX cpe.aH n K09cp<pHUHeH-
TOB. fIpe.D:CTaBJI!leTC5!, 4TO Ha Bonpoc 0 T04HOCTI1 B .n:aIlHOM CJIY'lae OT-
BeTHTh TPYJJ:Hee, 4eM B 60JlbWHHCTBe lJ.pyrnx CJlY4aeB. YaCTO JIYlJille
Bcero CBH.aeTeJIbCTBYIOT 0 T04HOCTH MaJIeHbKHe pa3HOCTH no OTHOilleHHIO
K .aaHHhIM rrpaBblM 4aCTHM.
,30.2J
05PAUlEHI1E MATPI1UbI METOJJ.OM I1CKJJI04EHI151 no fAYCCY
365
f'JJABA 30
OEPAI.I.lEHHE MATPHU 11 COI>CTBEHHbIE 3HA4EHH5I
30.1. BBell.eHlle
TUlaTeJJbHOe 113Y4eHl-Je MaTpHlI. Be)J,eTCH Y}I{e )J,aBHO, H TIpO HHX H3Be-
CTHO 01.JeHb MHoro. OHH BCTpe1.JaIOTCH H BO MHomx pa3JJI1'JHbIX <p1-I3H1.Je-
CKHX 3a)J,a1.Jax. B pe3YJJbTaTe Ha Eny TeMY I-IMeeTCH orpoMHaH JIHTepaTypa
H MHOr01.JHCJJeHHbJe MeTO)J,bI )J,JJH pa3JJI-I1.JHbIX 3a)J,a1.J. PaccMoTpHM JJI-Illlb
HeCKOJJbKO TIpl1MepOB I-I OTOlllJleM 1.JI1TaTeJJH 3a .lI.aJJbHetllllI-IMI1 CBe.ll.eHY1HMY1
K cJJe.ll.YIOUlHM KHHfaM: PaJJbCTOH H YI-IJJ<p [37], «CoBpeMeHHbIe BbI1.JI-ICJJH-
TeJJbHbIe MeTO)J,bI» [29] H XaycxoJJ)J,ep [16], f JJaBHbIM 06pa30M K 6I-I6JJl-Io-
rpa<pHH B )J,BYX TIOCJJe.n:HHX.
MbJ TIpe)J,TIOJJaraeM, 1.JTO 1.JI-ITaTeJJb 3HaKOM C 9JJeMeHTapHoIJ TeopHetl
MaTpHlI.,
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YaCTo 6bIBaeT Heo6xo.ll.HMO 06paTHTb KBa.n:paTHYIO MaTpI-IlI.Y
A = (aij)'
npe.u.c1aBHM ce6e npHMoyro.1JbHYIO MaTpHlI.Y
( au al '" al n I 0
a l .a .:. .a
anI an '" ann 0 0
)
KOTopaH nOJJYtlaeTCH, eCJJH K A npl1mICaTb e.n:HHH1.JHYIO MaTpHlI.Y I.
ITpI-IMeHI-IM Tenepb MeTO.n: HCKJJI01.JeHI-IH faycca, He 3a60T51Cb 0 npaBblX
cToJJ61I.ax. Kor.u.a raYCCOBO I-ICKJJI01.JeHHe 3aKOH1.JHTCH, nOJJY4HM
c 0 b ll bl b'")
:-:. 0 b'Jl b'J n .
1 b nl bn'J '" bnn
YTBep}l{.u.aeTC5t, 4TO MaTpI-IlI.a B
( bll b12 .., b In \
B = b 21 .b'J .... . b'Jn )
\U nJ Un'J . .. bnnl
366
OSP AillEHf1E MATPYlU 11 COSCTBEHHbIE 3HA4EHf151
[rJJ. 30
eCTb MaTpl1lJ.a, 06paTHa51 K A. 4T06bl y6e.lU1TbC5I B 9TOM, 3aMeTHM, lfTO
KaM,IJ,blfi war npouecca HCKJlIOlfeHH5I 9KBHBaJleHTeH YMHOMeHHIO CJleBa Ha
HeKOTOPYIO MaTpHUY. Dp0I13Be,IJ,eHHe Bcex 9TI1X JleBblX MaTpHU eCTb, Olfe-
BH,IJ,HO, A-1, nOTOMY lfTO OHO npHBO,IJ,HT A K e,IJ,HHl1lfHOfi MaTpHue!. Ho,
6Y.l.LYlfH npl1MeHeHO K n npaBblM CToJl6uaM, 9TO np0l13Be.l.LeHHe .l.LeJlaeT 113
e.aI1HI1YHoti MaTpl1Ubl MaTpl1uy B:
A-I. 1= B.
CJle.l.LOBaTeJJbHO, B 11 eCTb 06paTHa51 K A MaTpl1ua.
30.3. 3aJJ:aqa HaXQ}KJJ:eHHH coocTBeHHblx 3HaqeHHH
MHome 3a,IJ,aYH npHBO,IJ,51T K CI1CTeMe ypaBHeHI1t1:
a1Jxl + a12X2 + + alnXn = AX!>
a21xl + a22x2 + + a2nXn = AX2'
a n l X l+ a n2 X 2+ .., +annXn=AXn,
HJlU
Ax = AX,
r.l.Le A Hell3BeCTHO. 3TI1 ypaBHeHI151 COBMeCTHbI TOr.l.La '11 TOJJbKO TOr.ll.a.
KOf.ll.a
a l1 - A a 12 °In
= a2] 022 -- A °2n =0.
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anI a n 2 ann - A
Onpe,IJ,eJJI1TeJJb eCTb MHOrOYJleH OT A CTeneHI1 n, 11, B npl1HUl1ne, eCJlH
BbIlfHCJlI1Tb 9TOT onpe.l.LeJlI1TeJJb ,IJ,Jl5I n + 1 3HalfeHI151 A, MOMHO HCnOJlb30-
BaTb 9TH 3HalfeHI151 .l.LJl5I HaXOM,IJ,eHI151 MHOrOlfJleHa
(A) = O.
TaKHM nYTeM HaXO.l.L5ITC5I, B KOHI-Le KOHUOB, H KOpHI1 MHOrOYJleHa CA).
BnpolfeM, 01l1l16KH oKpyr JleHH5I II 60JlbWOe lfl1CJlO He06xo.aHMb1X BbIYHC-
JleHl1fi 06bllfHO 3aCTaBJl5IIOT OTKa3aTbC5I OT 3Toro MeTo.aa B nOJlb3Y ,IJ,pyrHx.
I-1HOr,IJ,a HYMHO rOJlbKO Hal160JlbWee no MO,IJ,YJlIO 3Haqe/1I1e A. Dpe.l.L-
nOJlOMHM, YTO A n.efiCTBI1TeJlbHoe. B TaKOM cJlYlfae MOMHO .lI.eIlCTBOBaTb
cJle.aYlOilll1M 06pa30M. B03bMeM npOH3BOJlbHbitl seKTop
v=CJ
30.31
3AlJ:A'-IA HAXO)J{!lEHI151 COECTBEHHbIX 3HA4EHI1it
367
113BecTHo, 4TO eCJlH C06CTBeHHble 3HaqeHH5I pa3J1H4Hbl, TO COOTBeTCTSYKJ-
llJ.He C06CTBeHHble BeKTOpbl (peWeHlHI CHCTeMbl ypasHeHI111 .aJl51 COOTBeT-
CTBYIOllJ.HX C06CTSeHHblX 3Ha4eHI1I1) 06pa3YIOT nOJlHYIO CHCTeMY, 1'. e. JlIO-
6011 BeKTOp Y MO)J{HO npe.aCTaBHTb B SI1.ae
Y=CJ,)II + C2Y2+"'+ cnYn>
r.ae Yi - C06CTBeHHhle seKTophl, a Ci - KOHCTaHTbI.
YMHO)J{I1M 06e 4aCf!1 paseHCTsa Ha A
n n
Ay = 2: Ci AYi= 2: Ci AiYi,
i=I i=1
H Bo06me
n
A"y = '\:1 C. A k y ,.
L.J I l
i=1
Tenepb, ecmr I AI I> I Ai I .aJl51 i = 2, 3, . . ., n, TO npH k 00
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Ak+l ) , C A k+l y
11 1 A
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6Y!leT YMHO)J{aTbC5I Ha O.aHO H TO )J{e 4HCJlO AI'
Ha I1P aKTHKe peKoMeH.ayeTc5I HOpMHposaTb nOJJY4alOllJ.HI1C5I BeKTop Ha
Ka)l{.aOM ware TaK, 4T06bl HaH60JlbWa51 KOMnOHeHTa ero 6hlJla pasHa 1.
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wero 110 MO.ay JlIO C06CTBemIOrO 3Ha4eHH5I. TIpH 3TOM 3ao.aHO nOJlY4aeTC5I
c06CTBeHHbll1 seK'rop.
(=hoT npouecc, 04eBI1.aHO, MO)J{HO YCKOPI1Tb, eCJlI1 npe.aBapHTeJlbHO.
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A 2 , A4, AS, ..., A 2 ".
Tor.aa, npHMeH5I5I
n
A 2" - '\:1 ) 2 k
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i=1
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Borrpoc 0 TOM, 4TO 6y.aeT, eCJlu CJlY4al1HO CI = 0, Hecepbe3eH. ECJlH
.aa)J{e 9TO H 6blJlO BepHO BHa4aJle, TO CJlY4aI1Ha51 oW116Ka c.aeJiaeT 9TO
CJJaraeMOe HeHYJleBblM n03)1{e 11, s KOHu.e KOHUOS, OHO CTaHeT .lI:OMI1HI1-
368
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B03MO)KHOCTb np05JBI1ThC5I BTOPOMY no MO.ll,YJ/lO co6cneHHOMY 3Ha4eHHID,
rrpv.xon:!1T B rOJJOBY nOIJTH K a)j(.lI,o MY. 3'10 .lI,et-lCTBHTeJlbHO MO)KHO C.lI,eJlaTb,
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rrpeBpaTHTC5I B WYM, TaK lJTO Ka)K.Qoe c, e.ll,YIOIHee c06CTBeHl-lOe 3Ha4eHHe
6Y.lI,eT onpe.ll,eJJeHO Bce C MeHbwetl TOlJHOCTblO.
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MepHO Talwe:
If 2, ..., 10.
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c HHMI1 Ha1l.0 6hITh OCTOpO)KHbIM !-I3-3a BHyrpeHHetl HeycTot!lJl1BOCTH 3a-
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30BaTbC\I rrpHMepaMH.
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HHMYM nYTeM nOJJHOrO rrepe60pa Bcex KOM6I1HaU!1H; HO Mbl He cor J/aCHbl
IlJJaT!1Tb TaKOH ueHOH 3a MHHHMYM, TaK KaK Bbl!1rpbllll Toro He CTOI1T.
Pellll1Tb, KaK .n:aJ1eKO MbI XOTI1M l1.n:m, MomHO TOJ1bKO B Kam.aOM KOHKpeT-
HOM CJJYlJae. Ho Ha HblHelliHeM )Tarre (I1CTOpH!1) HCn0J1b30BaHI151 BbllJHC-
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Teopl1eii Haxom.n:eH!1S1 3J1rOpHTMOB YMeHbllleHI1H 06w.eii .n:JJI1Hbl, '-IeM HrHO-
pl1pOBaTb ee; C 3a.aalJaMI1 TaJ\Oro me nma IIpH.QeTC5I BCTpeTHTbC5I ew.e
MHoro pa3, 11 HaBepHoe lJTO-HI16YAb, '-IeMY MomHO HaYlJI1TbC5I B 9TOM
C.JIYlJae, 1I0MQ)KeT HaM B 6y.ayw.eM.
Ue.JIb 9TOtl KHl1f11 He r J1y60Koe llpOHI1KHOBeHHe B :ny 06.JIaCTb, a
CKopee 0630p HeCKOJ1bKHX B03MomHblX MeTo.n:OB, KOTopble IIpI1XO.a5lT
B rOJJOBY IIPH pellleH!1I1 pa3Hblx 3a.aalJ Mo.n:e J mpoBaHI1S1. 3a.n:alJa Ha-
xom.n:eHH5I MllHI1MYMa - 9TO peaJlbHa5l, BamHa51 II lJaCTO OlJeHb TPYJ1-
Hag 3a.n:alJ3 11, HeCOl'iIHeHHO, O.n:H3 H3 Tex, BHI1M3H1le K KOTOpblM 6y.n:eT
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KaK J1llHetJHoe 11 .lU1HaMlllJeCKCe npOrpaMMllpOB3Hlle, KOTopb1e Bbl-
pOCJ1H 113 IIOI1CKa H311JiYlJllII1X H.JIll no KpatlHetl Mepe XOpOllll1X pe-
WeH!1t1.
3.n:eCb, o.aHaKO, Ha.n:o c.n:e,'laTb OJ1HO 3aMelJaHl1e. B Tex C!1Tyaull51x,
KOTopble CJ1l1ll1KOM rpOM03.n:KI1 .LJ,J151 0lITI1MH3aUl1H B TOlJHOM CMbICJ1e,
e03Hl1KaeT 60ilblllOtl C06JJa3H 1l0lIp060BaTb OnTHM1l311pOBaTb He60Jlbllll1e
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BbIllie npl1Mepe MO)j(HO IIblT3TbCSI 0lITI1MI131lpOBaTb He60iIblllI1e 6nOKl1
cxeM. OnblT, KameTCSI, nOKa3blBaeT, lJTO 9TO B 06w.eM 1I.'lOxa51 TaKTI1Ka;
01lTI1M1l3aUI151 MaJ1blX 6JIOKOB np!1Bo.QI1T K 60J1bWllM IIOTep5JM rrpl1 OKOH-
qaTeJIbHOM HX coe.n:HHeHHI1. VI3 :noro npaBllJ1a, HeCOMHeHHo, eCTb MHoro
lICKJ1l0lJeHI1t1, HO B UeJJOM, KaJKeTCSI, OHO cOOTBeTcTByeT H3BeCTHOMY
npHHUllny CI1CTeMOTeXHHK!1: «He CJIl1ll1KOM CTapaTb SI OllTHM1l3HpOBaTb
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374
HEKOTOPbIE npl1MEPbI MO.1lEJlI1POBAHI15J
[fJJ. 31
31.3. TIpHMep MO.1leJIHpOBaHIHI CKJIa.1lCKIfX Onepall.IfH
TIpHMep, KOTOpbIti 6y.n:eT onHcaH B 3TOM naparpacpe, OTHOCHTC5I
cKopee K TOMY, 4TO 06bl4HO Ha3bIBalOT npaKTH4ecKHMH npHJlO)f{eHH5IMH,
4eM K 06JIacTI-J HaY4Hblx HCCJIe.n:oBaHHtI, HO OH 3aTp arHBaeT MHOrHe
np06JJeMbI, BCTalOlIJ,He B 06bI4HOti HaY4HOti pa6OTe. y13 3Toro peaJIbHOrO
npHMepa MO)f{HO H3BJIe4b p5l.n: onpe.n:eJIeHHbIX ypOKOB, H OT4aCTl1 pa.n:H
HHX 6bIJI BKJII04eH HMeHHO 3TOT npHMep.
Ha paHHeti CTa.n:HH npaKTHKH Be.n:eHH5I CKJIa.n:CKOro X0351t1CTBa O.QHOti
1-13 KOMnaHI-lti 6blJIO npOBe.QeHO MO.n:eJIHpOBaHl-le Ha BbI4HCJIHTeJlbHOti
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1. Kor.n:a 3aKa3bIBaTb HOBOe 060py.n:oBaHHe?
2. CKOJIbKO o60py.n:oBaHH5I 3aKa3bIBaTb?
3. UeHa OTCp04KH B BbII10JJHeHHH 3aKa3a.
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3aKa30B nO TeJIecpOHY, a He nHCbMeHHO, CKopetillieti 06pa60TKOti nOJIY4eH-
HbIX TOBapOB, rJJaBHbIM 06pa30M Ha CTa.n:I-IH npOBepKH Ka4eCTBa H .QOCTaB-
KH B .QetlCTBYIOllI.He XpaHI1JIHllI.a.
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pacxo.n:bI (H HaJJOrH). C .n:pyrotl CropOHbI, )f{eJIaTeJIbHO H36e)f{aTb «npOCToeB»,
Kor.n:a HY)f{HOrO npe.n:MeTa HeT Ha CKJIa.n:e, CJIe.QOBaTeJIbHO, HY)f{HO CTpeMHTh-
C5I HMeTb 60JIbllIHe 3anaCb!. 04eBH.n:HO, HeKOTOpa51 06llI.a51 Mepa CTOHMOCTH,
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cpaBHHBaTb 06a 06CT05lTeJIbCTBa H CY.QHTb 0 paBHOBeCHH .n:BYX 3THX
npOTHBOnOJIO)f{HbIX CHJI.
B npHHIJ,Hne Mo.n:eJIl1pOBaHHe 6bIJIO 04eHb npOCTbIM. MbI 3anHCbIBaJJIJ
B naM5ITI-I MallIHHbI KOJIl1'leCTBO HaJJH4HOrO TOBapa B Ha4aJIe HCCJIe.n:yeMoro
nepHo.n:a. 3aTeM npOH3Bo.n:HJJHCb .n:etlcTBHTeJIbHO Ha6JIIO.QaBllIHeC5I BbI.n:a4H
H nOnOJJHeHH5I CKJIa.Qa, KOTopbIe npOHcxo.n:I-IJII-I Ha npOT5I)f{eHHH BoceMHa.Q-
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npOBep5lJIOCb, He nopa JIH .QeJIaTb HOBbIti 3aKa3. ECJIH .Qa, TO pe1lI3JIOCb,
CKOJIbKO 3aKa3aTb, p33MellI.aJIHCb 3aKa3bI H BbI4HCJI5IJIOCb, Kor.n:a OHII 6y-
AYT nOJIY4eHbI. TIoCJIeAH5I5I OnepaIJ,H5I BKJII04aeT CJIY'-IatiHYIO Beml4HHY
C pacnpe.n:eJIeHHeM DyaccoHa (3.n:eCb Ha 3TOM MO)f{HO He OCTaHaBJIIIBaTbC5I,
nOCKOJIbKY CJIY4atiHbie BeJIH4HHbI 6y.n:YT paCCMOTpeHbI B CJIe.n:ylOllI.etl rJl3Be).
KOfi:J:a HacTynaJJO BpeM5I BbII10JIHeHH5I 3aKa3a, ero npH6aBJl5IJIH K HMelOllI.eMYC5I
KOJJH4eCTBY, npH6aBJJ5I5I cooTBeTcTBYIOllI.ee 4HCJIO B 514etlKY, co.n:ep)f{allI.Ylo
41-1CJlO HaJIH4HbIX npe.QMeToB. Depe.n: Ka)f{.Qoti BbI.Qa4etl .n:eJIaJIaCb npoBepKa,
OCTaJJOCb JIH 4TO-HH6Y.Qb Ha CKJIa.n:e, II eCJlH HeT, TO perHCTpHpOBaJlC5I
npOCTotl, 11 HaAO 6blJlO CJIe.n:HTb, 4epe3 KaKoe BpeM5I CKJIa.n: nOnOJIH5IJIC5I.
3\.41
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375
HaKOHeu, '-Iepe3 paBHbIe IIpOMe)J{YTJ{H Bbl1Ie'-l3TbIBaJIOCb HMeIOIlJ,eeC5I
KOJIH'-IeCTBO BMeCTe co cnHCKOM IIpOCTOeB H HX JIHTeJIbHOCTblO.
TIporoH5I5I 3TY MO eJJb JJ5I pa3HMX npaBHJI COCTaBJIeHI151 3aKa30B KaK
no BpeMeHH, TaK H no 06'beMY HX, a TaK)J{e H3MeH5I5I cpe Hee BpeM5I 0)J{11-
aHH5I 3aKa3aHHbIX TOBapOB, MO)J{HO cpaBHHTb pe3YJIbTaTbI IIPH pa3JIH'-IHott
T3KTHKe, H B npHHUI1I1e MO)J{HO 6bIJIO CKa3aTb, '-ITO npOH30WJIO 6bI no
I1CTe'-leHHI1 18 MeC5IUeB, eCJJH CJIe OBaTb Ka)J{ OMY 113 npOBepeHHbIX npa-
BI1.lJ. O).J,HaKO B 3TOM paCCY)J{ eHI1H eCTb O HO c.lJa60e MeCTO. 3a pe3.lJb-
HbIti nepl1o B 18 MeC5IueB 6bIJIH OTMe'-leHbI CJIY'-latiHbIe npOCTOI1, 11 1I0TOM
He OTMe'-l3.lJOCb HOBbIX 3351BOK O Tex nop, nOKa TOB3p He IIOCTYII3JI.
TeM C3MbIM MO eJIl1pOBaHl1e 6bIJIO He BIIO.lJHe TO'-lHbIM.
D.pyrott He OCTaTOK BbI5ICH5IeTCH, eCJJI1 IIbITaTbCSJ Y3HaTb He TO, '-ITO
CJIY411JIOCb 6bl, 3 '-ITO IIpOl13oti eT B 6y yw.eM. Yl3-3a Toro, 4TO MO.11e.lJb
Bbl.11a4 H nOnOJIHeHl1ti He nOCTpoeHa, 3 I1cnO.lJb30BaJJOCb Ha6.lJIO.11aBweecSJ
B .11eticfBHTeJJbHOCTH, He y aeTc5I 1I0CTaBHTb ce651 B TI1nH4HYIO 6Y.11ym.ylO
CHTY3IlHIO. 5e3 COMHeHI151, no 4acTHoMY nOJIO)J{eHHIO eJl B 33.llaHHbIe 18 Me.
CHueB MO)J{HO 6bI.lJ0 CJIY'-IatiHO OT.11aTb IIpe.11nO'.JTeHHe O.11HOti <popMYJIe, B TO
BpeMH K3K Ka4eCTBO .11pyroti BbIwe. D.aJIbHetiwee TeOpeTl1'-1eCKOe HCCJIe-
.llOBaHHe MOrJIO 6bI aTb JJY'-IWl1ti pe3Y.lJbTaT, HO 3Toro C.11eJIaHO He 6bIJIO.
KaK )J{e MO)J{HO OaeHI1Tb 3TY pa60TY? B 60JIbWOti CTeIIeHI1 OHa oae.
HI1BaeTC5I nOTpe6I1TeJJ5IMH pe3YJIbTaTa. OHH C'-Il1TaJIH ce651 (H .11etiCTBH-
TeJIbHO 6bI.lJI1) HH)J{eHepaMH-npaKTI1KaMH 11 He CJIHWKOM Bepl1JII1 B TeOpl1lO.
BbI'II1C.lJeHl1e Toro, '-ITO npOH30WJIO 6bI HMeHHO 3a Te 18 MeC5IaeB, eCJJH
6bI OHf-I npl1.l1:ep)J{I1BaJ/Hcb pyrOti TaKTHKH (.lla)J{e KOr.lla 6bIJIO YKa3aHo
Ha He.11OCTaTO'-lHOCTb 3Toro), 3Ha'-ll1JIO .11JJ5I HHX rop a3.110 60JJbUIe, '-IeM
TeOpeTH'-IeCKHe HCCJJe.110BaHI151.. MbI He 6Y.11eM 06cY)J{.11aTb, KaKoti no.11xo
JIY'-lwe, TaK KaK y Ka)J{.11oro 113 HI1X .eCTb CBOH He.11OCTaTKH, a OCTaHO-
BI1MC5I Ha TOM O'-leBH.11HOM YTBep)J{.11eHl1l1, '-ITO 6ecnO/le3HO 8bt'UlC/lflmb
pe3Y/lbmam 8 mmcoii ifjop.Me, 8 Komopoii nompe611me/lb He noozom08-
/leH e20 UCnO/lb308amb. To, '-ITO BbI c06HpaeTecb BbI'-IHCJ/5ITb, .1l0JI)J{HO
.11OCTaTO'-lHO xopowo nO.11XO.11HTb BaweMY 3aKa3'-1HKY,
CJIe yeT ynoM5IHYTb eII1.e O.llI1H MOMeHT B 3TOti HCTOpHI1. XOTSJ H3C
YBep5lJIH, '-ITO 3anHCH Be YTcSJ nOJIHO 11 TO'-lHO, TeM He MeHee B HI1X
6bIJII1 Hatl eHbI npOTI1BOpe'-lH5I. TaKoBo 06w.ee nOJJO)J{eHHe: O'-leHb TPY.11HO
HatiTH TO'-lHYIO perl1CTpaal11O npowJJoro, oc06eHHo eCJ/l1 OHa eJJaeTC5I
BPY4HYIO, 11 He CJIe yeT npl1HHMaTb Ha Bepy '-IY)J{l1e CJIOBa; '-IT06b1 y6e-
.llHTbC5I, '-ITO aHHbIe cooTBeTcTBYIOT TOMY, '-ITO 0 HHX rOBOp5lT, H3.l1.0
np0I13BO I1Tb COOTBeTCTBYIOII1.YIO l1X npOBepKY. 50JIbWOe KOJIl1'-1eCTBO
T04HblX .11aHHbIx - Bew.b '-Ipe3BbI'-IattHo pe.11Ka5l.
31.4. TpexMepHble KpeCTlIKH-HOJUlKH
3atlMeMc5I Tenepb MO eJIHpOBaHl1eM HeCKOJIbKO pyroro TIm a, n XOTSI
OHO C<POPMYJIl1pOBaHO B <POp Me HrpbI, npHHIJ.HnbI ero BeCbMa npaKTI1-
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376
HEKOTOPbTE nPMMEPbl MOilEJJMPOBAHI151
[UI. 31
TpeXMepHOM KyGe C peGpoM B 'IeTblpe KJJeTKI1. UeJJb - nOCTaBHTb 'JeTbI-
pe KpeCTHKa Ha OJl.Hoti np5JMoti npe)l{.rre, qeM npOHIBHlJKY y.rraCTCfI no-
CTaBHTb Ha O.rrHOti np5JMoti 'IeTblpe HOJlHKa (npe.rrnOJJaraeTCH, 4TO
IJI1TaTeJlb 3HaKOM C I1rpoti B KpeCTI1KH-HOJJHKI1 Ha KBa.rrpaTe 3 X 3
KJleTKI1 ).
B .rrBYMepHoM cJJYlJae .rrJl5I ,not! Hrpbl 113BeC'fHa onpeJl,eJJeHHafI CTpa-,
TerH5J, HO 9TO, nO-BHJI,I1MOMY, He TaK Jl,Jl5I Tpex 113MepeHHtL TaKHM
06pa30M, BCTaeT 3a.rra4a Bwpa60TaTb npaKTH'IeCKHH cnoco6 BN60pa
cJlenYIO[JJ,ero XOJl,a. B npHHlI.Rne MO)/{HO nepenpoGoBaTb Bce B03MO)KHble
npOJIOmKeHHfI I1rpbl, HO 9TOT nYTb, M5JrKO rOBOpS!, 113Hypl1TeJlbHblti. B 3TOM
CMbJCJIe Hrpa nOXO)Ka Ha MHOrl1e BCTpe'laIOUJ,l1eCfI Ha npaKTI1Ke cl1TyalI.lm.
B npHHQHne eCTb B03MO)l{HOCTb nOJIHOrO nepeGopa Bcex KOM611Halwti, HO
npaKTI1'1eCKI1 3Ta B03MO)J{HOCTb He.rrOCTI1)1{I1Ma 11 HaM npI1XO.lJ.HTCSJ pa3pa-
6aTbIBaTb .llPyrHe MeTo.n:bI pe[JJeHH5J 3a.n:a'lH. Dpel1MymeCTBO H3Y4eHHS!
HrpbI, a He npaKTl1IJeCKI1X 3anal! COCTOI1T B TOM, 4TO Hrpa Wl1pOKO
H3BeCTHa, npOCTa 11 'JeTKO C<POPMYJJl1pOBaHa, TOrJl,a KaK MHorHe npaKTH-
qeCKHe 33J1,a'JH H30611J1YIOT MeJJKI1MH .n:eTaJl5JMH, KOTopbIe TPYJI,HO 113JIaraTb
11 KOTOpbJe Hw,ero He J106aBJI5JIOT K nOHHMaHl11O Toro, KaK pewaTb
3a.n:a4Y.
HaqHeM C Toro, 'JTO npHfIllli1eM HOMepa KJleTKaM Ky6a (pI1C. 31.4-1).
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21 \ 22 \ 23 \ 24
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i 29 1 30 I 31 I 32
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41 \ 42 \ 43 \ 44
45 \ 46 \ 471 48
PHC. 31.4-1.
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61 I 62\ 63 I 64
I 1 I
TO HOMepa HX KJIeTOK 06pa3YJOT apH<pMeT114eeKYIO nporpeccl1Io. 06paTHoe
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KOTOpbIX Jl,Be lmarOHaJlbHble, - Bcero 40 np5IMbIx.
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OJlflOtl 113 Ka)J{lI.ot1 60KOBOti).
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u l1e K Jll1aMerpaJlbHO npoTHBOfIOJIOjj{HOtJ BepWI1He Ky6a.
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)l{eHHfi Ha npHMblX. ITocJle KaJKJJ.OrO xOJJ.a CJleJJ.yeT H3MeHHTb 3aTpOHYTble
npHMbJe. OlJ,eHKa, KOTOpaH nplH1HCbJBaeTCH Ka)l{JJ.ofi JlHHHH, paBHa CYMMe
HOMepOB KJleTOK, 3aHHTbJX Ha 9T0l1 npHMofi nepBblM HrpOKOM, eCJlH Ha
Hen HeT KJleTOK, 3aHHTblX BTOpblM HrpOKOM, TaKaH )l{e CYMMa C MHHyCOM,
ecm-l Ha npHMofi KJJeTKH 3aHHTbl TOJlbKO BTOpblM HrpOKOM, H HYJlb,
eCJI1-l Ha npHMOn CTOHT H KpeCTHKH H HOJlI1KH. Elll.e HY)KHO KaK-TO OTMe-
'-IaTb COBceM nYCTble npHMbJe.
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H Bbmrpal1Te.
2. ECJm Y npoTHBHHKa eCTb TPH HOJlHKa Ha npHMotl, 3al1MHTe 4eT-
BepTYJO KJIeTKY, 4T06bl He JJ:aTb eMY BblHrpaTb.
3. ECJlH eCTb KJIeTKa, B KOTOpOtl nepeceKaJOTcH XOTH 6bl JJ.Be npHMble,
Ha Ka)l{JJ:otl H3 KOTOpblX CTOHT no JJ.Ba KpeCTI-lKa H HeT HOJlHKOB, cbIrpal1Te
B «BI1J1KY» H BbmrpaeTe Ha cJleJJ.YJOlll.eM xo.n.y.
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KHpoBaTb, a TO OH BbmrpaeT.
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4J1eHbI 6Y.LI.YT MOHOTOHHO B03paCTan, TO ee He XOTeJlOCb 6bl C4UTaTb
cJIY4allHot!, XOTH B HeKOTopOM CMbICJIe OHa CTOJJb }Ke CJlY'Iat!Ha, KaK H
JlI06a5I .LI.pyraR KOHKpeTHa5I nOCJIe.LI.OBaTeJIbHOCTb. 04eBH.LI.HO, Tpe6YlOTC5I
eu e KaKHe-TO CBot!CTBa.
fIepBbIM .n:eJJOM, rOBopHTCH, 4TO 110CJJe.n:OBaTeJIbHOCTb lI:OJJ}KHa 6bITb
110CTpoeHa CJlY4at!HbIM C110C060M. Ho 4TO 3TO TaKoe? Be.LI.b BCRKHt!
nOp5I.LI.OK CTOJlb )\(e CJIY4aeH, KaK 11 Jl106ot! .LI.pyrot!. fIpI1XO.LI.HTCH cKa3aTb,
4TO He .LI.OJJ}KHa 6bITb BH.LI.Ha KaKa5I-HH6y.n:b 3aKOHOMepHOCTb B cnoco6e,
KOTOpbIM BbJ6paHa nOCJIe.n:OBaTeJIbHOCTb. 01J:!-IH 113 3aKOHOB, lI:JIR KOTOpbIX
MQ)l(HO YCTpOI1Tb I1pOBepKY, - 3aKOH KOppeJl5II1.HI1 4!1CJIa 11 CJIe.LI.YIOw.ero
3a HHM; MO}KHO JlH I1pe.LI.CKa3aTb CJIe.LI.ylOmee 4HCJlO 110 npe.n:bI.LI.yme-
MY? TaKHM 06pa30M, eCJIH Xi - CJlY4at!Hble 4HCJla, TO .LI.JlH Haillett
382
CJJYlJAHHbIE YI1CJJA 11 METO)lbl MOHTE-KAPJlO
IfJJ. 32
110CJIeJlOB3TeJIbHOCTI1 JlOJI)KHO BbI110JIHHTbC5I paBeHCTBO
!(Xi - ; ) (Xi+!- )=O,
i
r.rre B311TO cpe.rrHee X = !/ . VlMeH .rrOCTaT04HO MHOrO BpeMeHH, MO)KHO
nplWYM3Tb OyeHb !HOrO B03MO)KHbIX npOBepOK, TaK MHorO, 4TO He Bce
HX y.rraeTCH npHMeHHTb.
B 9TOM Bonpoce He06xo.rrHMO CTaTb Ha npaKTHyecKylO T04KY 3pemI5I:
eCJIl1 no OTHOWeHllIQ K .rraHHOMY 4aCTHOMY npHMeHeH1110 3aKOH He3aMeTeH,
TO .rrJJH .rr3HHoro npHMeHeHI1H 411CJIa CJIYY3tJHbl, 11 Mbl ,LI,OJI)KHbl 9tHM Y,Ll,O-
lJJ1eTBopHTbCH. B caMOM .rreJIe, C npaKTI14eCKotJ T04KI1 3peHH5I, 6blTb
MO)KeT, 11 He Tpe6yeTcH I1MeTb JletJCTBHTeJlbHO CJIY4atJHYlO nOCJJe.rrOBa-
TeJIbHOCTb, a npe,Ll,n04TI1TeJlbHee 60JIee r JIa.rrKoe pacnpe.rreJJeHHe, HCYep-
nhlBalOlllee 4aCTb cOBoKynHocTH 6e3 tlOBTOpeHl1tJ.
Cnoc06, KOTOpblM 06bl4HO HCIIOJIb3YlOtCH CJIY4at!Hble nOCJle.rrOBaTeJlb-
HOCTI1 BO MHorHX npHJIO)KeHHHX, COCTOHT B TOM, 4T06bI f1blTaTbCH oue-
HI1Tb CTaTHcTHKy 60JlbWOt! COBoKynHOCTH C06b1THt! no MaJlot! Bbl60pKe.
IT09TOMY X04eTC5I f10JlYlJI1Tb «CBepXTl1nI14HYIO» BbI60PKY TaK, lJT06bl ,LI,JIH
MaJIot! Bbl60pKH I1MeTb B peaJIbHOM CJIY4ae CTa6uJIbHOCTb, KaK ,LI,JlH 60Jlb-
WOt!. XOqeTC5I HMeTb 60JIee rJla,Ll,KOe pacnpe.rreJleHl1e, qeM 06b14HO, lJTo6bl
6blJIO He C. HWKOM MHoro 60JlbW11X Bbl6poCOB - pOBHO CTOJlbKO, CKOJJbKO
Ha,Ll,O, lJT06bl He BnaCTb B npOTHBonOJIOiKHYIO OWH6KY CJIHWKOM OJlHO-
P0,Ll,HOtJ cOBoKynHoCTH. Mbl XOTHM, eCJlI1 9TO B03MO)KHO, yCTpOI1Tb BblpOB-
HeHHYIO, OlJl1IUeHHYIO H npOBepeHHYIO nOCJle,Ll,OBaTeJJbHOCTb CJJyqatJHbIX
lJHCeJl.
BblCKa3aHHble 3aMe4aHI1H He YHI1BepCaJlbHbl; OHI1 HeBepHbl, HanpI1Mep,
eCJII1 Tpe6yeTcH oueHHTb rpaHl1Ubi 113MeHeHl1H KaKot!-HH6Y,LI,b BeJJl1lJI1Hbl.
32.2. feHepHpoBaHHe CJIYlJaUHblX IJHCeJI B MaUHIHe,
paooTalOmeu B Jl,BOI1IJHOii CI1CTeMe
Hal160Jlee WHpOKO ynOTpe6J1HeTc5I CJle,Ll,YIOIllHt! MeTO.rr nOJl)'lJeHHH
CJIYlJatJHblx lJHCeJI;
x n +! = pX n (no MO,Ll,YJIIO .rrBa B CTeneHH .lI.JlHHbl IaWHHHOro CJlOBa.)
TaKHM o6pa30M, HcnoJlb3yeTc5I O.1l.HO YMHO)KeHHe Ha Ka)K,LI,Oe lJHCJJO H
MJIa.rrWHe pa3p5l,Ll,bl npOH3Be,Ll,eHHH 6epYTc5I B Ka4eCTBe cJle,Ll,ylOwero cJlY-
yatJHoro 4I1CJla. OCHOBHOI1 Bonpoc COCTOHT B TOM, 4TO B3HTb 3a p H
KaK Bbl6paTb Xo.
OTBeT Ha 9TH BonpOCbl H .1I.OKa3aTeJlbCTBO Toro, 4TO ,LI,aHHblt! MeTO.1l.
11.0 nOBTOpeHI151 ,LI,aeT ,LI,J1I1HHble nOCJIe,Ll,OBaTeJIbHOCTI1 CJIYlJal1Hblx 'lHCeJI,
OCHOBbIBalOTCH Ha HeKOTopblX pe3YJJhTaTaX reoprm qHCeJJ. D.JlH 06JJer4e-
HI151 BbIBOJla *) (ueJIb KOToporo - npO,Ll,eMOHC"rpl1pOBaTb MetOJI 11 pe3yJlb-
'") 3aHMcrBOBa/JHoro y M, !leSH.
32.21
fEHEPI1POBAHI1E CJJY4A! Hh!X 411CEJJ B llB0I14H0I1 MAI!JI1HE
383
TaTbI) paCCMOTpHM 331l34Y 1I0JIY4eHIUI CJIY43tJHhIX l\l-IceJI H3 MamHHe,
pa60TaIOm.etJ C llBOH4HblMH 4HCJIaMH. Pe3Y JIbTaTbl llJI5I lleCHTI1'IHOtJ MamlJHbl
npHBelleHbl B CJIe1lYIOm.eM naparpacpe.
06bIl\HaH B TeOpHI-I 41-1CeJI 3a111-1Cb
x = a(mod nz)
03Ha4aeT, 41'0 X - a 1leJIl1TCH Ha nz.
ECJII-I H3ma Maml1Ha k-pa3pHllHaH H 6epYTcH 110CJIe.ll.HHe k pa3pH1l0B
np0113Be1leHI1H (Y.lI.BOeHHotJ 1lJII1Hhl) px n , TO
xo=a,
X n + 1 = PX n (med 2k)
(k 3). }
(32.2- I)
ECJII1 a lJ.eJIHTCH Ha 2, TO nOCJle1l0BaTeJlbHOCTb 6Y.ll.eT 9KBHBa" eHTHa
TaKotJ:
a
YO=2'
Yn+I - PYn (mod 2 kl ),
KOTopaH Ha caMOM 1leJJe cooTBeTCTByeT 60JJee KOPOTKOMY Mal11l1HHoMY
CJIOBY, II 3Ha'JIIT, 31leCb He HcnOJlb3yIOTCH Bce B03MO)J(HOCTI-I Hal11etJ MamHHbL
f109TOMY B03bMeM B Kal\eCTBe a He4eTHoe 4I1CJJO.
AHaJlOrl14HO P 1l0JJ)J(HO 6bITb He4eTHbIM, TaK KaK eCJlI1 6bl OHO 6hlJJO
4eTHbIM, TO
xk+J = pk+la = 0
11 C ,noro MeCTa Mbl 6hl I1MeJ!11 Tpl1Bl1aJIbHYIO nOCJIe1l0BaTeJIbHOCTb
HYJIetJ, '
llaJIee, Bce He4eTHble 411CJIa p MQ}J(HO 3aflHcaTb 01lHHM H3 cfloco60B:
8t- 3,
8t-l,
8t+ 1,
8t+3.
T e 0 p eM a 1. EC/l11 P = 8t + 1, mo
p2k-3 = I (mod 2"),
(32.2-2)
m. e. nopflooTC p He 60JlbUJe, 'lei" 2"-3, 11 fl8JlflemCfl iJeJlllmeJle.M
'lllCJla 2 k - 3 .
II. 0 K a 3 a l' e JI beT B O. YTBep)J(D:eHl1e TeopeMbI 9KBI1BaJIeHTHO TOMY,
41'0 pa3HOCTb p2k-3 - I lleJJl1TCH Ha 2 k . TaI\ I\aK
TO
a 2 -1 =(a+ I)(a-I),
p2k-3 _ I = (p2k-4 + I) (p2"-5 + I)
(p + l)(p - 1). (32.2-3)
llJIH Ka)J(.lI.otJ CKo6KI1 (i I)
2'
p2i + 1 =(8t + 1)2' + 1 = 1 +(1 + 8t)2 i =(1 + 1)+ 2)-I)"Q;t"8",
"=1
384
CJJYlJAHHbIE lJl1CJIA 11 METO)J.bl MOHTE-KAPJIO
[Tn. 32
OTC10.lla 2 .lleJll1T p ' + J H H3 (32.2-3) cJJe.a:yer, 4TO 2'H .a:eJllfT
(p k-4 + 1) (p H + I) ... (p + I). (32.2-4)
11a,1ee,
(p + I )(p - 1) = (p - 1) = (8t + I) - 1 = 16 (4t + t),
noaToMY (p + I) (p - I) .a:eJIHTC5I H3 2 4 11 H3 (32.2-3), Y'-lI1TbIB3S! (32.2-4),
"k k-a
nOJlY'1aeM, '-ITO 2 4 .2 -4 = 2 .lleJlI1T (p - 1).
T e 0 p e M a 2, EC/lll P = 8t =!: 3, mo
p,!IH =1= I (mod 2"), p2k-2 = 1 (mod 2 k ),
m. e. nopfiaOK p paeeH 2k- .
11 ° K a 3 a 1 e J! beT B 0. I-1cnOJ!b3YR (32.2-3) B aTOM cJ!y t l8e, nony-
43eM
2'
p ' + I = I + (3 + 8t) ' = 1 + 3 ' + 2: ( + IlC:itR8k3( '-k).
k='
Ho
2'
(1 + 3 i) = 1 + (4 _1)2' = 1 + (1 - 4)2' = 1 + 1 + 2:: CZ t (_ 4)k.
k=l
OTC10.ll3
2 i
p ' + I = 2 + C i [3 I-ktk. 2 k + (- l)k]4"
k=1
H p2' + 1 .lleJJHTCll H3 2, HO He .lleJJH1C5I Ha 4.
CJJe.a:osaTeJlbHO, 113 paBeHcTB3 (32.2-3) nOJlylJ3eM, '110 2 k - 4 lleJIHT
(p2k-1 + 1) ( p2 k- 5 + I). . .(p'! + I), HO 2 k - 3 He .l1.eJlI1T aTo np0I13 B e.lleHl1e.
33MeTHJ\I, '-ITa
(p + I)(p - I) = 8 (8t + 6t + 1)= 8 X (He'-leTHOe '-II1CJJo).
TaKBM o6pa30M, B JIl060M cJ/Y43e o -t I)(p -- 1) lleJlHTCSI Ha 2 3 11 He
lleJWTCR Ha 2 4 . CJ/e.a:OBiJTeJ/bHO, (V - J) peJ/BTCR Ha 2 k - 1 , HO He .a:e-
JI 111 C\I H3 2 k . VlHhIMH CJIOB3MH,
p IH of= 1 (mod 2k).
3am-lweM Tenepb
p k-1 _ 1 = (p k-2 + 1) (p /H _ I),
k-3 k k-3
np11'1eM 2 .a:eJ111T p + J 11 2 -I HeJnIT p2 - 1, TaK 4TO 2 k .lI.eJIliT
11:& np0l13Be.l1eHHe 11
p k-2 = I (mod 2").
32.2]
fEHEPVlPOBAHVlE CJJY4AHHblX 41-1CEJJ B ltB0l14HOH MA1JIVlHE
385
T e 0 p e M a 3, EC/zu p = St - 3, mo nocAeoo8ameAbftOCmb xo,
Xl' ..., X,2k-2_1)' nopO:J/CoeHHafl ifJoPMYAOii (32.2-1), eemb He1Comopafl
nepeemaH081Ca nOC/lefJo8ame/lbHOemU
1, 5, 9,
... ,
2 k - 3, eellu a = l (mod 4),
Ullll
3,7, 11, ..., 2 k -l, eellll a = 3 (mod 4).
D. 0 K a 3 aTe JJ b C T B O. PaCCMOTpl1M 3Ha4eHl1e (apn), n = 0, 1,...,
k-2
2 2 - 1. Pa3HOCTb Me)l{lty 110CJleltOBaTeJlbHblMH 4J1eHaMI1
apn+l _ apn = apn (p _ 1)= apn (8t - 4)
.lleJlHTCH Ha 4. Ho Y)I{e 113BeCTHO (TeopeMa 2), 4TO nOp5l.ll0K 4HCJIa p
paBeH 2 k - 2 . CJle.llOBaTeJIbHO, nOJIY4aIOTC5I 2 k - 2 pa3JI\14Hb1X 4J1eHOB, pa3-
HOCTH KOTOpblX lteJlflTCfI Ha 4, H3 4ero CJleltyeT YTBep)l{lteHHe TeOpeMbI.
vITaK, B 3aBHCHMOCTH OT Bbl60pa a H npl1 yCJlOBHH, '1TO P = 8t - 3
(.llJlfl mo60ro t), MO)\{HO nOJIY'1HTb nepeCTaHoBKY OltHOti 113 ltBYX nOCJJe-
.llOBaTeJIbHOCTet!, YKa3aHHblX B TeopeMe 3. DpH BblBOlte :)Toro cpaKTa X n
ClJl1TaJIHCb IJ,eJlbIMH 4nCJIaMH; npn nCnOJlb30BaHHH HX CJIeBa CTaBHTC5I
ltB0l14Hafl 3an5lTa51 H HCnOJIb3YIOTC5I 4HCJIa
x n .2- k .
Ho 3Ha4eHHe t HeXOpOllJo Bbl6HpaTb COBceM rrpOH3BOJIbHO. HanpHMep
3Ha4eHHe t = I ltaeT p = 5, H eCJlH r.Qe-.HH6yltb BCTpeTHTC5I MaJleHbKOe X n ,
CKa)j{eM IO-k, TO 3a HHM rrOJIY4HTCfI ltJlHHHaSl nOCJleltOBaTeJIbHOCTb rro-
creneHHo B03paCTalOllJ,HX 4HCeJl. Bo H36e)\{aHHe :nHX HenpHSlTHOCTeti
MO)l{HO 6blJI0 6bl Bbl6parb raKoe t, 4T06bl CTapllJl1e pa3p5lltbl p' T k
OblJlH HJlH 00 I, HJIH 0 10. DOltp06HOro H3YlJeHH5I rrpeHMYllJ,eCfB Bbl60pa
pa3JII14HblX 3HalJeHHtI t .QJI5I PJ)JIY4eHH5I pa3JJYIlJHblX nepeCTaHOBOK Tex )I{e
caMbIX 4HCeJl, nO-BH,IUI!\!OMy,AIeT.
D.JI5I .QaHHoro p = (8t -f 3) Mbl nOJlY4HJJH .QBa IJ,HKJla, Ka)\(.Qbltl 113
KOTOpblX HC4epnblBaeT lJeTBepTb Bcex HMelOllJ,HXC5I B MallJI1He lJHCeJI.
OnaBllJaSlC5I nOJlOBHHa 4HCeJI BXOltHT B MeHbllJHe IJ,HKJIbl, 3aBI1CflllJ,He OT
Bbl60pa a.
ECJlI1 B nOCJleltOBaTeJlbHOCT5IX TeopeMbl 3 onyCHITb .QBa rrOCJIeLI.HHX
.llBOH4HbIX pa3p5lLI.a, TO nOJlYlJeHHa51 nOCJleltOB areJlbHOCTb 6y LI.er nepe-
craHoBKoti rrOJIHoro IJ,HKJla .QJlHHbl 2 k - 2 : 0, 1, 2, ..., 2 k - 1 - I. ECJlI1 He
onyCTHTb LI.Ba nOCJIeltHHX pa3pfl.Qa, TO nOCJleLI.HHtI pa3pfl.Q Bcer.Qa 6Y.QeT 1
H, KOHetIHO, Re 6Y.QeT cJlY4atlHb1M. Depexolt51 OT npaBblx pa3p5l.QoB
K JJeBblM H npOBep5l51 pa3Hble Bcrpe4aIOllJ,HeCfI pa3p5lltbl, HaXOJJ:HM Bce
MeHbwe H MeHbllJe 3aKoHoMepHocTeti. D03TOMY rrpHH5ITO He paCClJl1Tbl-
6aTb Ha CJlYlJa HOCTb HeCKOJlbKHX rrOCJIe.QHHX pa3p5lJJ:OB.
386
CJ1YYAt1"HbIE YI1CJJA 11 METOJlbI MOHTE-KAPJJO
[f JJ. 32
32.3. reHepHpoBaHHe CJ1YtIaHHbIX tIuceJl
Ha )l,eCSITOtIHOM MamOHe
OCHOBHbIe pe3Y JIbTaTbI B 9TOt! 06JJaCTl1 113J10)J\eHbl B CTaTbe Mow-
I'vtaHa *), r,lle np!1BOJl5!TC5! cjJopMYJIa lI.JI5! S-3HaYHotllI.eC5!T!14HOt! MallHIHbI (8 4)
P = 7 4 k+\
KOTOpa5! !1MeeT nep!10lI. JlJIHHbI 5. lOs-a, a TaK)J\e pe3Y JIbTaTbI npOBepKI-!
ee Ha 0lI.HHHaJlUaTI13Ha4Hot1 M aUI!1 He. llpyrotl BbI60p **) P =
=76768779754638671 877 (npI1BeJleHHOe no MOJlYJIIO 10S) .naeT MaK-
C!1MaJIbHNe nepHOJlN.
ECTb H lI.aJIbHet!WHe !1CCJIeJlOBaHH5!, HCIlOJIb3YlOuIHe 60JIee CJJO)J\Hble
cjJOPMYJlbI. 4>OPMYJlY .aJJ5! JlBOHIJHOt! MaIllf1HbJ.
X n +l = (2 a + 1) x n + c (mod 2a )
MO)J\HO HatlT!1 B CTaTbe POTeH6epra ***).
M3yYaJI!1Cb TaK)J\e cjJOPMYJIN Bf1Jla
x n + 1 = aX n + Xn_l'
5ICHO, lfTO np06J1eMa nOJlYlfeHH5! CJlyYat!HotJ BeJIHIJHHbl Ha MaIllUHe 6blCTpO
pa3BHBaeTC5! H He I'vIO)!{eT 3.aeCb 6bITb I1CIJepnaHa. MbI OCTaBJI5!eM 9TY
TeMY J])Hl .aaJIbHetlIU!1X I1CCJIe.aOBaHHt1.
32.4. .l1.pyrHe pacnpe)J,eJleHHSI
3Ha5!, KaK nOJ1yyaTb paBHoMepHo pacnpe.aeJIeHHbIe CJ1Y'Jat1HbIe IJHCJIa,
MO)l{HO reHepnpOBaTb JlpyrHe pacnpepeJleH!1H. M3BeCTHO MHoro pa3JIHIJHblX
cnoc060B. PaCCMOTpI1M HeKOTopbIe 113 HI1X.
TIpeJlnOJIO)l{HM, 4TO Tpe6yeTC5! pacnpe.lIeJIeHl1e fey); KaK nOJ1Y4I1Th
ero H3 paBHoMepHoro? 0.11[1/1 [13 CfToc060B - rrpr-rpaBHHTb .lIBe cjJYHKllH[j
pacnpepeJJeHH5!, paBHOMepHYIO WIll nepeMeHHot1 x H )I{eJIaeMYIO .lIJIf! y:
x y
l.dx=V(y)dy=P(y)=x.
o 0
TaK!1M 06pa30M, eCJIH MO)!{HO HatiTH !j:JYHKUI1IO, o6paTHYlO K F (y), TO nMeeM
y = p- 1 (P (y» = p-I (x).
B npI1Huvme, 9TO Bce, 4TO Tpe6yeTC5!.
*) Jack M 0 s h man, The Generation of Pseudo-random Numbers on a
Decimal Machine, J. Assoc. Computing Machinery, vol. 1, pp. 88-91, 1954,
**) npHHa)],Jle)l(aIl\f1ti E. H. I'HJl6epTY.
***) A. Rot e n b erg A. New Pseudo-random Number Generator,
J. Assoc. Computer Machinery, vol. 7, pp. 75-77, 1960. CM. raK)/{e J. Certain,
On Sequences of Pseudo-random Numbers of Maximal Lenght, J. Assoc. Com-
puter Machinery, vol. 5, pp. 353-356, 1958.
32.41
J],PYfI1E PACOPEJ],EJIEHf151
381
flpOCTblM npHMepOM ::noro cnoc06a HBJlHeTCH 9KCIlOHeHllHaJlbHOe pac-
npeJl,eJleHl1e
f(Y) = e- Y ,
"
x=' e- YdY = 1 - e-->',
o
(32.4-1 )
fIJlI1
e- Y = I-x.
Ha npaKTI1Ke MO}\{HO 3aMeHI1Tb CJlY4atiHYIO BeJll14HHY 1 - Xj BeJl11411-
HOti Xj, TaK 4TO e -Yj = Xj I1J1H
yj=-lnxj.
3Ta CJIY4atiHaH BeJIH411Ha ynOTpe6J15!JlaCb C 60Jlbllll1M ycnexOM.
ECJlH npHMeHHTb 9TOT MeTOJJ. K HopMaJlbHOMY pacnpeJJ.eJleHHIO
}__ e- Y ", TO npl1LleTCH npI16J1I1}\{aTb CPYHKllHIO, 06paTHYlO eff (HanpI1Mep,
.' 21t
PO racTI1Hrcy (13], CTp. 191). Ho OI1blT nOKa3blBaeT, 4TO AJIH 06paTHblx
<PYHKUl1ti pacnpeAeJleHI1H npH6J1H}\{eHI15! HHorAa BeAp K cepbe3HblM
OWI16KaM.
BMeCTo Toro 4T06b1 6paTb 06paTHYlO CPYHKUHlO, MO}\{HO BOCnOJlb30-
fJaTbC5! TeM XOpOIllO 113BeCTHblM cpaKToM, 4TO cYMMa CpaBHI1TeJlbHO He-
60JlbWOrO 411CJla cJlY4atiHblx BeJIl1lJl1H C JlJ06blM pacnpeLleJleHl1eM 06b14HO
naeT OlJeHb xopoillee npI16J1I1}\{eHl1e K HopMaJlbHoMY paCnpeLleJJeHI1IO. D.JI5!
paBHoMepHoro pacnpeLleJleHH5! Tpe6yeTcH cYMMa OKOJJO AeCHTI1 4I1CeJI,
lJT06bl nOJIYlJl1Tb n04TI1 HOpMaJlbHOe pacnpeJl,eJ1eHl1e. B Jl,eC5!THlJHOt! Ma-
IllI1He npHH5!TO 6paTb LlBeHaLluaTb lJHCeJl, lJT06bl 1136e)KaTb B03MO}\{HOrO
B3aI1MOLlet!CTBI15! Me}\{LlY MeXaHI13MOM reHepl1pOBaHI15! H npOll,eCCOM CJIo}\{e-
HI15!. CYMMa He LlaeT 13 cpeLlHeM HYJI5!, 11 n09TOMY HY}\{HO BbllJHTaTb He06xo-
LlI1MYIO BeJll1lJI1HY: 5, eCJlH CKJlaJl,blBaJlI1Cb Jl,eC5!Tb 4HCeJl, 11 6, eCJlI1 cJlarae-
MbIX 6bJJJO LlBeH3LlLl.aTb. Yl.cnOJIb3YH pe3YJJbTaTbl 2. 9, nOJlYlJaeM LlHC-
nepcHIO :>
1= (n=10, 12).
Bbl6HpaH n paBHblM 12, nOJlYlJaeM LlHcnepcHIO 1, lJTO 5!BJI5!eTC5! BTO-
pblM LlOBOLlOM 13 nOJ!b3Y n = 12.
B cJlYlJae nyaccoHoBCKoro paCnpeLleJleHH5I ycnelllHo ynOTpe6JJ5IeTC5I
cJleJl,YIOu:(Ht! cooc06. HaxoLlHM TaKoe k, lJT06bl LlJl51 Xj U3 paBHoMepHofo
pacnpeJl,eJleHH5I BbII10JIH5!JJHCb HepaBeHcTBa
XIX . .. Xk;:=: e- m , 'XIX , .. Xk+l < e- m ,
cae m - cpeLlHee uyaccoHoBcKoro pacopeLleJIeHH5I. Eho k H eCTh HY}\{HOe
4HCJlO.
388
CJJY'-IAJ1HblE '-IHCJJA H METonbl MOHTE-KAPJIO
[rJJ. 32'
ECJJl1 m MaJlO, 3Ta npolJ,e.llypa 3KOHOMH4Ha, HO npH .llOCTaT04HO 60Jlb-
IllI1X m OHa 3aHHMaeT MHoro BpeMeHH H He CJlHlllKOM xopOllla. Bnp04eM,
eCJlH m .llOCTaT04HO BeJJl1KO, TO pacnpe.lleJleHHe fIyaccoHa .llOBOJlbHO T04HO
1/pv.6JJH)KaeTC5J HOpMaJlbHbIM.
32.5. MeTO,ll. MOHTe-KapJIo
Mbl npHBeJlH HeKoTopble HB MHOfilX H3BeCTHblX CTIoe060B nO.lIY4eHHR
CJJY4atlHblx BeJJl14HH C pa3HblMH pacnpe.lleJleHH5JMI1 H Tenepb npHCTynaeM
({ onHeaHHIQ B03MO)KHbIX CTIoco6oB HX HCnOJJb30BaHH5J. Ha3BaHHe «:VleTO.llbJ
MOHTe-KapJlo» .llJl5I MeTO.llOB, el1CTeMaTH4eeKH HCnOJJb3YIQlllHX c.'Jyu.atlHYIO
SeJJH4HHY, BOCXO.llHT K nOeJle.llHHM ronaM BTOpOtl MHpOBOtl BOtlHW, KOr.ll3
cpOH HetlMaH H Y JlaM HCTIOJlb30BaJlH CJJY4atlHble 4HClla .llJl5J MOll.eJll1pOB3HIHI
1/0Be.lleHlHI HetlTpOHoB. KOM6HHHpoBaHHoe HenOJlb30BaHlie M3111HH e JJIQ.llbMH
.llJJ5J MO.lleJJHpOBaHH5J eJlY4atlHoro npoueeca Bbl30BOB Ha TeJlecpoHHot!
eTaHlJ,HH 6b/JI0 OCYllleCTBJleHO He n03)Ke 1926 rO.lla, XOT5J B Te nHH 3TO
Ha3b1B3J10Cb (a Koe-rne H Tenepb H33bIBaeTe5J) <6pocaHHeM» no aHaJlOrHH
e 6poeaHHeM KoeTH .llJl5I nOJJY4eHH5J eJJyqatlHblx 4HCeJl. Ho eaMa Hne51
BoexonHT no MeHbllletl Mepe K cppaHuY3eKoMY eCTeCTBOHenblTaTeJlIQ 6IQcp-
<POHY, KOTOpbIIl 3aMeTHJI, 4TO eeJlH HrOJlKY 6poeaTb cJlY4atlHblM 06pa30M
Ha pa3J1I1HoBaHHYIQ nJlOeKOeTb, TO Bep05JTHOeTb Toro, 4TO HrOJlKa nepe-
ee4eTC5J C KaKotl-HH6Y.llb JIlIHl1etl, CB5J3aHa C 4HCJlOM 'It.
Y CTaHOBHM 9TOT cpaKT. fIyeTb .llaHO ceMetiCTBo paBHOOTCT05JlllHX na-
paJlJleJlbHblX np5JMbIX JJl1Hl1t1 e paCeT05JHHeM I Me)K.llY HliMH. fIyeTb .llJJl-lH3
+-
(C 7/2
a)
Tt/2 9'
PHC. a2.5-1.
HrOJlKH 1< 1. Kor.lla HrOJlKY 6pocalOT, ueHTp ee MO)KeT ynaeTb OT np5JMofi
H3 JlI060M paCeT05JHHH OT 0 no 1/ . fIYCTb 3TO 6Y.lleT nepeMeHHa5J x.
PacowTpHM, n3JIee, yroJl 'f HaKJlOHa HrOJlKI1 K np5JMblM (pHC. 32.5-1, a).
fIepeMeHHble x U 'f MO)KHO C4UTaTb eJlY4atlHblMH H He3aBHCI1MbIMH.
YeJlOBHe nepeee4eHHH HrOJlKH e OAHOt! H3 JlHHHt! eeTb
x < cos 'f (- ; < cp < ; ).
pHe. 32.5-1,6 3alllTpHXOBaHa 06J1aCTb, orpaHH4eHHa5J KpHBOfi
B KOTOPYIQ .llOJJ)KHa nona.1l.aTb T04Ka C KOOp.1I.HHaTaMH (cp, x).
Ha
I
X=2 cos cp,
\\ 32.61
ElliE onHA VlJJJlroCTPAl\1-HI METonA MOHTE-KAPJI0
389
OTHorneHMe
B03MO)J{HbiX
nJIOtllanH ::nol1 06J1aCTM K nJIOtllanH Bcero np5lMoyrOJIbHHKa
TOlJeK eCTb Bep05lTHOCTb nepeCelJeHH5I. Eho OTHorneHHe paBHO
1t/2
COS 'fd'f
p= -1t/2
I
2 1t
1t 1t 2
-1r!'2
21
cos r.pdcp=-.
1t
TaKHM 06pa30M, MO)J{HO paccMaTpHBaTb paccY)J{neHH5I 6rorprpOHa KaK
onpeneJIeHHe 3HalJeHI-I5I 'It MeTOnOM MOHTe-KapJIo. ECJII-I 6bI npo 11; 6blJ!0
H3BeCTHO TOJlbKO, lJTO OHO JIe)J{HT Me)J{ny I 11 10, 3TO 6bIJI 6bl npeKpacHbII1
cnoc06 06HapY)J{HTb, lJTO 'It npHMepHo paBHo 3. 3aTpanIB HeKOTopbII1 TPYJl.
H CneJI3B MHoro 6pocaHHJ1, Mbl MOl' JJH 6bl nOJ/YlJHTb 'It npHMepHo 3,1.
Ho, 4T06bl noJIYlJl1Tb 66JJbrnylO TOlJHOCTb, HY)J{HO 6blJIO 6bl npoBonl1Tb
np5lMbIe, 113Mep5lTb nJII1HY I1rOJlKH, pa3611paTb cJ/YlJao cnopHoro nepece-
lJeHH5I H T. n. C TOlJHOCTblO 60JJbrnel1, 4eM B Harnl1X CI1JIax, JII160 npo-.
H3BonHTb HeBep05lTHO 60JlbillOe 4HCJlO 6pocaHHI1. 11 3TO eCTb 061lJ,HI1
<j.JaKT. MeTon MOHTe-KapJIo, 6bITb MO)J{eT, JIY4111e npYrHx Ha npeJl,BapH-
TeJlbHol1 CTanI1H, KOma OH nOMoraeT rrOJlYlJl1Tb 06tllee rrpenCTaBJleHHe
o CI1TyaU.HI1, HO eCJIH Tpe6yeTc5I rrOJIylJHTb TOlJHble pe3YJIl,TaTl,l, u.eHHOCTl,
ero 3HalJHTeJIbHO MeHbllle.
l1JlJJIOCTpI1PY51 9TO 3aMe4aHHe, npl1BeneM npl1Mep H3 rrpaKTHKH aBTopa.
Korna TOJIl,KO HalJaJII1 n05lBJl5ITl,C5I BbllJl1CJIl1TeJll,Hble MalllHHl,l, 6l,lJla rrpen-
JIO)J{eHa 3ana4a, KOTOPYIO B rrepBOHa4aJll,Hot1 aHaJIHTH4eCKol1 rpopMe
6blJlO 61,1 TpynHo pellll1Tl, H Ha C3MblX 61,lCTpblX COBpeMeHH1,IX MalllHHax,
HO OKa3aJIOCb, lJTO H3 caMOM neJIe ::no 3analJa 0 nBH)J{eJUH-I HOHa B 3JIeK-
Tpl1lJeCKOM nOJIe B ra3e. MOneJlHpOBaHl1e no MeTony MOHTe-KapJIo
C 10000 'IaCTI1U naJIO rparpHI( pacnpene.'1eHI151 CKOp0CTH BnOJIl, nOJI5I 11
nepneHnl1KYJI5IpHO K HeMY. nOCJIe TOro KaK rpH311K nepeCTaJI )J{aJIOBaT1,C5I
Ha HH3KYIO TOlJHOCTb, OH CKa3aJI He4TO Bp0Jl,e: «XM.... Eho noxo)J{e Ha
3JIJIHrrTHlJeCKOe pacnpeJl,eJIeHHe MaKCBeJIJIa, TOJIl,KO CJIerKa CJI,BHHYToe.
XM. . .». 11 9TO naJIO eMY KJIlOlJ K aHaJII1Tl1lJeCKOMY pellleHHIO 3analJl1,-
TOJIl,KO TaK H 61,lJlH I1CnOJIl,30BaH1,l lJl1CJleHH1,le pe3Y JIl,TaTbl MOneJIl1pOBaHI151.
K CO)l(aJIeHHIO, Ha caMOM neJIe MeTon MOHTe-KapJIo B 60JIl,lllHHCTBe CJIY-
lJaeB HCnOJIb3yeTc5I TOJIl,KO Torna, Korna npOlJHe MeTOJl.1,1 Y)Ke Hc'!epnaH1,I,
a B 9TH X 06CT05lTeJIl,CTBaX OH Tep5leT HeKOTopl,le CBOI1 npeHMYtlleCTBa.
YnpamHeHHe 32.5-1. OnHc3rb nporpaMMY. MOJ\eJlHpYIOIl\YlO 3a,IJ,a'lY DlO4J-
4JoHana U,H4JPOBOH MalllI1He.
32.6. EIII.e onHa IiJIJIIOCTpaIJ.HH MeTona MOHTe-KapJIo
06bI'IHO peaJIl,Ha51 3ana'!a BKJIIO'!aeT 3JIeMeHT CJIY'Iat1HocTH, 'ITO Ha-
T3J1KHBaeT H3 I1CnOJIl,30BaHHe CJIy'!aI1H1,IX lJHCeJI B MOJl,emlPYlOtlleM npo-
uecce. Ho HHor Jl,a, KaK B rrpHMepe C HI' JIoit DlOrprpOHa, He06xoJl.HMO Hat1TH.
390
CllYlJAI1HbIE lJl-1CJJA 1-1 METOLlbI MOHTE-KAPllO
[fll.32
.apyrylO <POPMYJlI1pOBKY 3an,alJH, lJT06b1 OHa y)f{e co.n:ep)f{aJIa CJIYlJat!HbIt!
9JIeMeHT. (HeKOTopbIe .n:alOT HM5I «MOHTe-KapJIo» TOJIbKO TaKHM CJIYlJaHM.)
DpHBe.n:eM eIll.e CJIe.n:ylOIlI.Ht! npHMep HCnOJIb30BaHH5I CTaTHCTHlJeCKOrO
no.n:xo.n:a.
B nCHXOJIorHlJeCKOM 3KcnepHMeHTe rpynne H3 nHTH lJeJJOBeK, CH.n:HU\HX
3a CTOJIOM, 6 blJI a .n:aHa 3a.n:alJa .n:JI5I COBMeCTHoro perneHH5I. vl3YlJaJIOCb
H OCHOBHOM nOBe.n:eHHe HX B 3aBHCI1MOCTH OT H3MeHeHHH KaHaJIOB CBH311
Me)f{.n:y HHMI1. OKa3aJIOCb, lJTO HeKOTOpa51 opraHH3alJ.H5I 3TOt! CBH3H LlaJIa
.JIYlJIllHt! pe3Y JIbTaT, lJeM .n:pyrl1e.
O.n:HaKO npH 3TOM He 6bIJIO OTBeTa Ha Bonpoc: «HaCKOJIbKO XOPOWH
'5b1JII1 CHCTeMbI CB5I3H?» To, lJTO H3I11ep5lJIOCb, 6bIJIO HX B3aI1MHOe COOT-
HOIlleHI1e. ECTeCTBeHHO 3a.n:aTb Bonpoc: «A eCJIH 6bI JIIO.n:I1 .n:et!cTBOBaJII1
CJIYlJat!HbIM 06pa30M, KaKOB 6bIJI 6bI pe3YJIbTaT?» B HeKoTopolll CMbICJIe
HCCJIe.n:OBaHHe CJIY1.Jat1Horo nOBe.n:eHI151 .n:aeT a6coJIIOTHYlO Mepy ycnexa
OpraI-IH3alJ.1i1i I1X pa60TbI. TaKot1 3KcnepliMeHT TaK)f{e 5ICHee OTpa)f{aeT
<CTPYKTypHble 3<p<peKTbI, nOJIYlJaIOIlI.HeC5I 6JJaro.n:apH HaJIHlJl110 Tex HJIIi
HHbIX KaHaJJOB CBH31i.
TaKI1M 06pa30M, I1Hor.n:a CTaTHCTHlJeCKIit1 no.n:xo.n:, He HBJlH5ICb HIiKOHM
06pa30M lJaCTblO Mo.n:eJJIi IicCJle.n:yeMot1 CIiCTeMbI, MO)f{eT TeM He MeHee BHeCTIi
BKJIa.n: B nOHlilllaHl1e pa60TbI Mo.n:eJIIi. He 06H3aTeJIbHO npe.n:noJIaraTb, lJTO
. IO.n:1i .n:et1CTBYIOT CJJYlJat1HbIM 06pa30M, HO OTBeT Ha Bonpoc: «L[TO
(SbIJlO 6bI, eCJII1 6bI OHIi .n:et1cTBOBaJIIi CJIYlJat1HO?» - np05lCH5IeT Ha6mo-
JJ.aeMYIO KapTIiHY.
32.7. MeTo)!. }l{YJIIiKa
BBIi.n:y nony JIHpHOCTH KpaCOlJHbIX CJJOB «MoHTe-KapJJo» MbI npHHHJIIi
:n:pyroe )f{I1BOnHCHoe BbIpa)f{eHl1e «MeTO.n: )f{YJIIiKa» *) .n:JI5I onlicaHHH HCnOJIb-
30BaHIi51 B BbIlJI1CJIeHH5IX no MOHTe-KapJIo KOppeJIl1pOBaHHblX BeJIHlJI1H.
Y.n:alJHOe IiCnOJlb30BaHHe KOppenl1pOBaHHblx (oc06eHHo OTpl1lJ.aTeJIbHO
KOppeJlI1pOBaHHbIx) BeJIl1lJl1H MO)f{eT CI1JIbHO YMeHbllll1Tb 061>eM BbIlJHCJIe-
HIit1 npl1 MO.ll.eJIlipOBaHl11i no MOHTe-KapJIo.
KaK y)f{e lJaCTO cJIYlJaJIOCb, 06cY)f{.n:a51 pa3JJI1lJHbIe BbllJl1CJIHTeJIbHbIe
'BOnpOCbI, Mbl 6bICTpO yrJIy6HJII1Cb B cTaTHcTI1KY Ii BbIHY)f{.n:eHbI npl13HaTb,
lJTO 3Ta TeMa BbIXO.n:I1T 3a paMKH BBo.n:Horo Kypca no MeTO.ll.aM BbllJHCJIe-
lmt1. Ho BbIro.n:a TaKI1X MeTo.n:OB B cJIYlJa5lx, Kor.n:a IiX 1II0)f{HO 3aCTaBI1Tb
pa60TaTb, CJII1IllKOM BeJIHKa, lJT06bI He CKa3aTb 0 HUX XOTb HeCKOJIbKO
CJlOB. I1pHBe.ll.eM JII1Wb O.n:HH npl1Mep. XOMMepCJII-I Ii MOpTOH **) om1Cbl-
BalOT npl1MeHeHl1e OTpHlJ.aTeJIbHO KOppeJJI1pOBaHHbIX BeJIHlJI1H B 3Kcnepli-
*) flpIlHa,IJ.JleJKHT, nO-BII)I;HMOMY, npo4J. TblOKI1 H IIcnoJlb3yercR )l;IIR 0603Ha-
<JeHf1R JlI060ro MerO)l;a YCKopeHHR BbIlJHCJIeHl1ti nepeXO,IJ.OM K 9KBHBaJleHTHOH
3a)l;a'le.
**) J. M. H 0 m mer s 1 e y and K. W. M 0 r ton, A New Monte Carlo
Techique: Antithetic Variates, Proc. Cambridge Phil. Soc., vol. 52, pI. 3, pp.
449-457, 1956.
N+ 1.11
BA}!{HOCTh BOnpOCA
391
MeHTe 5fOcpcpOHa, OHH CHa4aJla 6ePYT JlHe }!{eCTKO CKpeIlJJeHHbIe KpeCTOM
HrOJlKH H nOKa3bJBafOT, 4TO fIOJlY4aeTC51 8bJIlrpbfW 8 4HCJle He06xo.rrHMb1X
6pOCaHI111 8 12,2 pa3a (O,UHO 6pocaHI1e 11 perl1CTpalm51 nepeCelJeHI1IJ
KpeCTa 3aC411TbJBafOTCIJ 3a ,UBa 6pocaHI1IJ C perHCTpaUl1el1 npOCTol1l1rJlbI),
T. e. MO}!{HO O}!{H,Uan, 4TO ,UJlIJ 1I0JIY4eHI1IJ TOI1 JKe TOlJHOCTH C KpecToM
npH,UeTCIJ c,UeJIaTb B 12,2 pa3a MeHbwe 6pocaHI1t1. Kor,Ua OHI1 nepeWJlH
K TpeM I1rOJlKaM, nepeCeKafOUJ,I-IMCIJ no,U pa8HbIMH yr JlaMI1 (H C4HTaJII1 O,UHO
6pocaHI1e 3a Tpl1), 6bIJI ,UOCTl1fHYT 8bmrpbIW B 44,3 pa3a; ,UJlIJ 4eTbIpex
8hJIlrpbIW 6bIJI 8 107,2 pa3a. TOT JKe pe3YJIbTaT MOJKHO nOJIY4HTb,
CKpenl1B I1rOJIKI1 B BI1,Ue npa811,TIbHOrO TpeyrOJIbHHKa. LI.aJIee, OHI1 nOKa-
3hIBafOT, 4TO C nOMomblO HeKoTopbIX MeTO,UOB CTaTHCTHKH MOJKHO nOJIY4I1Tb.
eme 60JIbWI111 8bmrpblw. yITaK, npeJK,Ue 4eM Ha411HaTb BbIlJl1CJIeHHIJ no
MOHTe-KapJJo, cOBeTyeM nOJIY4HTb KOHCYJIbTaUl1fO y xopowero CTaTI1C-
TI1Ka, 4T06bl 1I0cMoTpeTb, lJeM OH MOJKeT 1I0M04b.
I'J1ABA N+l
I1CKYCCTBO BbIIJI1CJISITb J],JISI I1H)I{EHEPOB 11 YIJEHbIX
N + 1.1. Ba}f{HOCTb Bonpoca
B KHHry no MeTo,UaM BbI4I1C.7JeHHtI He npl1HIJTO 8KJIIOlJaTb rJIaBY Ha'
Heonpe,UeJIeHHYfO, o6mylO TeMY: KaK no.rrxo.rrI1Tb K 3a,UalJe 11 pewan ee,
eCJII1 peweHHe Tpe6yeT HCIIOJIb30BaHHIJ BbI4I1CJII-ITeJlbHbIX MaWHH. CaM
3arOJI080I{ :not! r JIaBbI ,UBycMbICJJeH - ero MO}!{HO nOHI1MaTb KaK a'upe-
COBaHHbItI lJeJIOBeKY, 8hIlJHCJIIJIOmeMY ,UJI5I UHJKeHepOB H Y4eHbIX, TaK H
K HI1M CaMI1M; Ha caMOM ,UeJIe HMeeTCIJ B 811.rry 11 TO 11 .rrpyroe.
He CJIe.rryeT OTHOCHTbC5I Hecepbe3Ho K :not! TeMe TOJIbKO H3-3a TOro,
4TO 8 HeKOTopbIX MeCTax H3JIaraeTCR CKopee JII14HOe MHeHl1e, 4eM YCTa-
H08JIeHHbIe cpaKTbl. ABTOPy OHa npe.rrCTaBJIIJeTC5I 60JIee 8aJKHotl, lJeM
MIWflJe KOHKpeTHble pe3YJIbTaTbf, H3JIOJKeHHbIe B .rrpyrlIx 4aCTIJX KHHrlI.
B HaCTOIJw.ee BpeMIJ BC5I TeMa npe.rrcTa8JI5IeT c060t! cKopee HCKyCCT80,
HeJKeJII1 HaYKY, HO 3TO flOJIOJKeHl1e 6hICTpO MeHIJeTCIJ, TaK KaK caMO Ha-
JII1411e 8bIlJHCJIHTe,TIbHbIX MaWI1H c.rreJIaJIO B03MOJKHOt! MeXaHI13aUHIO HeKO-
TOphiX npoueCCOB, npo KOTopbIe Kor .n.a-TO nOJIaraJIH, 4TO OHH Tpe6YIOT
4eJlOBelJeCKot! MhICJII1, H Tenepb B 3TOt! 8aJKHot! 06J1aCTH lIpo.n.OJIJKaIOTCIJ
aKTI1BHble nOHCKH.
4eM 60JIbW MhI Y3HaeM 0 TOM, KaK MbI pewaeM 3a.n.a411, TeM 60JIb-
wylO lJaCTb pa60TbI MOJKHO 6y.n.eT nepeJlO)l(l1Tb Ha MaWHHbl. YlCKYCCTBO
peWeHH5I 3a,Ua4 Ha BbIlJHCJIHTeJIbHbIX MaWHHax HHTepeCHO H caMO 110 ce6e;
OHO MOJKeT nOMOlJb 80 MHorHX cl1TyaUI1IJx H CHJIbHO YBeJlWmTb ueHHOCTb,
MaWHHHhlX BbIlII1CJIeHI1t!,
50JlbUlHHCTBO Y4eHhlX H36ermOT 3aHHMaTbC5I HCCJle.n.OnaHHeM Toro, liTO,
Bo06w.e r080p5l, MOJKHO Ha3BaTb npOl eCCOM T80plleCKoro MbIWJJeHH5I, HO,.
392
I1CKYCCTBO BbI4I1CJJQTb J\JJq VlH)!{EHEPOB VI Y4EHbiX
[W. N+I
'eCTb HeCKOJlbKO 3aMe4aTeJlbHbiX HCKJlI04eHHt!. «MeTO.n:» ApxHMe.n:a, 110-
./KaJ/yN, O.n:I1H H3 C8MbiX paHHHX IIpHMepOB T8KI1X HCKlIlO'IeHHt! CpeJII1
MaTeMaTHKOB, a Tenepb I1MeeTCH em.e COBpeMeHHaH KlIaCCI14eCKaH KHl1ra
Dot!a «l{aK pemaTb 3a.n:a4Y» [36]. O.n:HaKo 06a 3THX aBTopa 3aHHMalI!1Cb
pemeHHeM T04HO cq)opMYiIBpoBaHHblx 3a.n:a4, Tor.n:a KaK HaM mnepeCHO,
'iTO .n:eJlaTb, Kor.n:a 3a.n:a4a nOCTaBileHa He4eTKO 11 CTOJIb )!{e HeonpeLle-
J1eHHbl YCJJOBH5I, KOTOpbJM JJ:OJl)!{Hbl YJJ:OBJleTBOp5lTb pe3YJJbT8TbI. 11,eBH3
3TOt! KHHro
ll,e.llb pUCi{em08 - He i{UC.Ila, U nOHU.MUHUe
nOKa3bIBaeT, HaCKO.lIbKO LUl1pOKa Hama 06J1acTb HCC.lJe.n:OBaHI1t!.
. N + 1.2. 4TO Mbl cOOl:lpaeMCSI tl;eJlaTb C oTBeToM'?
ITotia B CBoet! KHl1re «l{aK pemaTb 3a.n:a4Y» [36] nOKa3blBaeT, KaK
B a}f(HO rrOH5ITb YC.lJOBI151. ABTop 3TOt! KHl1f11 B pe3Y.lJbTaTe MHoro.lIeTHefi
npaKTI1KH BbI4HC.lJeHI1t! Ha 3aKa3411Ka npHWe.lI K y6e)!{.n:eHHIO, 4TO 06bI4HO
'TTepBblM .n:eJJOM C.lIeLlyeT no.n:YMaTb: «A 4TO Mbl c06l1paeMc5I .n:eJJaTb C OT-
BeToM?» l3y.n:YT JlI1 BbI4I1C.lJeHHbIe Be.lI04HHbl .n:etlcTBI1TeJJbHO OTBe4aTb Ha
Bonpoc, KOTOpbIH HaM 3a.n:aH? Bce JJO OHI1 HaM HY}f(HbI? MO}f(eT 6blTb,
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