О приближенном решении систем нелинейных уравнений
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| Datum: | 1953 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | Russisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
1953
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/7731 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512716804849664 |
|---|---|
| author | Давиденко, Д. Ф. Давиденко, Д. Ф. |
| author_facet | Давиденко, Д. Ф. Давиденко, Д. Ф. |
| author_sort | Давиденко, Д. Ф. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-08-15T09:55:16Z |
| description | - |
| first_indexed | 2026-03-24T03:33:13Z |
| format | Article |
| fulltext |
1953 YKPAI1HCKI111 .MATEMATI14ECKI111 )I(YPHAJI
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HbiX np11 l!I1CJieHHOM 11HTerpHpOB3HH11 CHCTeMbl ( 1,4), 113 ,O:OCT3T0l!HO M3JIOH
OKpeCTHOCTI1 T0l[KI1 (x1il, x~l, .• . , x~2, i,;), T. e. 3HaqeHI15I Bl1,0:3
xl =x\i) + oxl, x2=x~il + ox~, ... , x,.=x~l + oxn> l.=i.;+ oA..
HaqaJibi-IOe 3HaqeHI1e ,o:JIH t BbJ611paeM npo113BOJibHO, B3HB, Hanp11Mep, t0 = 0.
Dpoil,o:H HeKoTopyi-O oKpecTHOCTb TOlJKI1 (xiil, x~l, . .. , x~l, A.;), CHOBa
B03Bpaw,aeMC5I K CI1CTeMe ( 1 ,4) 11 Dp0,0:0JI)K3eM l[I1CJieHHO 11HTerp11pOB3Tb ee
( eCJI I1 3TO He06XO,O:HMO), np11l!eM 33 H3l[3JibHbie 3HaqeHI15I X, (r = 1, 2, ... , n)
H J. 6epeM DOCJie,O:Hl-0!-0 113 CI1CTeM 11X 3H3LJeHI1H, DOJiyqeHHblX Dp11 l!I1CJieHHOM
m-nerp11pOB3HliH CI1CTeMbl ( 1 ,8) .
3Hal[eHI15I X, (r = 1, 2, ... , n) 11 J., DOJiyqeHHbJe l[I1CJieHHbJM 11HTerp11-
pOB3I-IHeM CI1CTeMbi (1 ,8), 11 6y,o:yT 11CKOJ\IblMI1 Dp116JII1.tKeHHb!MI1 peweHI15IMH
CI1CTeMbl ( 1, 1) B OKpeCTI-IOCTI1 TOLJKI1 (x~il, X~11, .•. , x~l , A;).
3,o:ecb Heo6xo,o:11:-.10 3aMeTI1Tb, qyo np11 npoxo.tK,o:eHHI1 qepe3 TOl[KY
( {i) (i) .(;) } ) dl ( x 1 , x 2 , ••• , Xn, ·i Dp0113BO,O:H35I df Mel-meT 3l-13K T3K KaK B CI1JIY ycJIOBI15I
( 1 ,7) 0!-13 B 3TOH TOlJKe o6paw,aeTC5I B H.)' Jib) 11, C.lJe,O:OBaTeJibHO, J. l-13lJI1H3eT
y6biBaTb. Do3TOMY npa nepexo,o:e oT CI1CTeMbi n + 1 ypaBHeHI1H (1,8) K CI1-
CTeMe n ypaBHel-11111 ( 1,4) He06XO,O:HMO ruar 11HTerp11pOB3!-1115I h Bbi6paTb
OTp11II,3TeJibHbiM, T. e. 113MeHI1Tb HanpaBJie!-IHe mnerp11pOB3HI15I. 3TO o6ecne-
4HT H3MeHeHHe napaMeTpa }, B TOM HanpaBJie!-11111, B K3KOM OR Me!-15!JIC5I ,0:0
nepexo,o:a K cncTeMe n ypasHei-IHH ( 1,4).
Boo6w,e HanpasJieHI1e 11HTerp11posaHI1H 6y,o:eT MeHHTbCH Ka.tK,O:biH pa3,
K3K TOJ!bKO Mbl npOXO,O:I1M qepe3 TOlJKI1 Kp11BOH (1 ,5), B KOTOpblX Bb!DOJI
HHeTCH YCJIOBI1e (1,7).
3a~leTI1M T3K.tKe, l[TO B CI1JIY yCJIOBI1H
Llr (xiil, x~l, ... , x~l, ), i) =f= 0, r= 1, 2, ... , n,
ypaBHel-1115! ( 1 ,4) B OKpeCTHOCTI1 T0l[KI1 (x\il, x~l, .•• , x~;l, i.;) MO.IKHO pa3pe-
dxr dA.
lll11Tb OTHOCI1TeJibHO -d (r=1, 2, ... , k-1, k+1, ... , n) 11 -d , T. e. B35!Tb
xk xk
33 He33BI1CI1MY!-O nepeMeHHY!-0 JI!-06oe XJ?, 11 B 3TOH OKpeCTHOCTI1 BMeCTO
CI1CTeMbl ( 1 ,8) l[I1CJieHHO np011HTerp11pOB3 Tb CI1CTeMy
dxr=Ll,(xl> x2,···· Xn, 'A))' r=l,2, ... , k-1, k+l, . . . , n,
dxk Llk(xi. x2,· .. , Xm J.
d'A L1 (xll x 2, .. . , x,, l)
dxic = Lli;(xh X2,· • • , Xm A)·
2. DycTb Tenepb B TO_lJKe (x~il , x~;>, ... , x~>, J.i) o6JiaCTI1 G, HBJIH!-0-
w,eilcH peweHI1eM CHCTeMbi ( 1, I), BbiDOJIHH!-OTCH cJie,o:yi-Ow,l1e ycJIOBI1H:
L1 (xy>, x~il, . .. , x~l, A;)= 0,
L1r(xiil, x~>, ... , x~>, ).;)=0, r=1, 2, ... , n. (I ,9)
TaKI1e TOl[KI1, KOTOpbre o,o:HospeMeHHO y,o:osJieTBOpHI-OT ycJIOB11HM ( 1 ,9)
11 (I, I), 6y,o:eM H33biBaTb oco6biMI1 ToqKaMI1 11HTerpaJibHOH Kp11BOH (I ,5) 1•
1 3n! OC06hie TO'IKH O T,ll,eJibHOii UHTe,rpaJibHOH KpUBOH, T. e. OCOObie T O'IKH, BC1Cpe
'laiOWHCCH B ,ll,HcpcpepeHl(HaJ!bHOH reoMeTJinil, He CJie,n:yeT CMeillHBaTb C OC06biMH TOtJKaMH
,n:ncpcpepenu.naJibHhiX ypasuennii.
6. YKp3HHCKHH MaTeMaT. )l.;:ypuaJI, T. V, N2 2. 201
Ol!eBHJIHO, liTO B 3TOM CJiyqae TaKMe l!HCJieHHhiM HHTerpHpoBaHHeM
CHCTeMbl (1,4) HeJib35I 6y}leT HaihH perueHHH CHCTeMbl (1,1) B OKpeCTHOCTH
oco6oR TOl!KH. ITo3ToMy, l!T06br HaRTH yKa3aHHbie perneHHH, Heo6XOJIHMO
B OKpeCTHOCTH OC060R TOl!KH BMeCTO CHCTeMbl ( 1 ,4) l!HCJieHHO rrpOHHTerpH
pOBaTb CHCTeMy ( 1,8), T. e. ITOCTYIH!Tb, B006Ill,e rOBOpH, aHaJIOrHl!HO rrpe,llbi
JIYill,eMy CJIYllaiO.
ITpn 9TOM HeKoTopoe OCJIOMHeHne 6yJieT TOJihKO B TOM c.nyqae, ecmf
OC06aH TOl!Ka HBJIHeTCH TOl!KOR CaMorrepecel!eHHH KpHBOR ( 1 ,5) .
B caMoM ,LI.eJie, B MOMeHT rrpoxoMJieHim oco6oR TOl!KH rrpOH3BOJI-
dx, dA
Hb!e df (r = 1, 2, ... , n) H df B CHCTeMe (1,8) MeHHIOT CBOH 3HaKH (Tar-:
KaK B CHJIY ycJIOBHR ( 1 ,9) OHH B i3TOR TO liKe o6palll,a!OTC5I B HYJib) H, CJie
JIOBaTeJibHO, x, (r = 1, 2, .. . , n) H A. MeHHJOT HarrpaBJieHHe cBoero H3Me
HeHHH, B TO BpeMH KaK B ,lleRCTBHTeJibHOCTH OHH JIOJIMHbl MeHHTbCH B rrpe)K
HeM HarrpaBJieHHH.
iB ,llaHHOM CJiyl!ae rrpH l!HCJieHHOM HHTerpHpOBaHHH CHCTeMbl ( 1 ,8) ITOCJie
rrpOXO)K,ll,eHHH OC060R TOl!KH CJie,ll,yeT H3MeHHTb 3HaKH BbiWeyKa3aHHbiX rrpo
H3BOJlHbiX Ha IIpOTHBOIIOJIO)KHbie . .3n:IM CaMbiM Mbl o6ecrrel!HM H3MeHeHHe
X, (r = 1, 2, ... , n) H A ITOCJie OC060}I TOl!KH B TOM Me HarrpaBJieHHH,
ll KaKOM OHH H3MeHHJIHCb JlO OC060R TOl!KH.
§ 2.
Ope,li,JiaraeMhiR MeTOJl BeChMa rrpocTo rrpHMeHHM TaKMe H K HaxoM,ll,e
HHJO IIJ2H6JIHMeHHbiX perneHHR CHCTeM aJire6paHl!eCKHX H TpaHCUeH,ll,eHTHbiX
ypaBHeHHR
(2, 1)
KOTOpbie He CO,ll,epMaT rrapaMeTpa A.
B 3TOM cJiyqae Heo6xo,li,HMO TOJibKO rrpe,li,BapHTeJihHO rrpeo6pa3oB aTh
CHCTeMy (2, 1) K BHJlY ( 1,1), paCCMOTpeHHOMY B § l.
C 3TOR ueJihJO B CHCTeMy (2, 1) BBOJlHM rrapaMeTp A. TaK, l!T06br :
1) rrpn A. = 1 rrpeo6pa3oBaHHaH cHcTeMa
<J?.k(x !, X2, ... , Xm A.) = 0, k = 1, 2, ... , n
o6pall}.aJiaCb B HCXOJlHYIO (2, 1);
2) rrpH A = 0 MOMHO 6hiJIO 6e3 3aTpy,li,HeHHR HaRTH ee perueHHe
x 1 = xi0', X2 = x&0l, ..• , Xn = x~0'.
EcJIH npH 3TOM !IJYHKUHOHaJibHbiR onpe,ll,eJIHTeJih
J- D(q;l, f{J2,• • ., fPn)
- D(x1, x~, ... , Xn)
(2,2)
OKaMeTCH paBHbiM HYJIIO B TOl!Ke (x~0l, X~0), • •• , x~l, 0), TO CJie,ll,yeT BBeCTH
rrapaMeTp A KaKHM-JIH60 JlPYrHM CIIOC060M, O,li,HaKO TaK, l!T06bi YCJIOBHH
1) H 2) BbiiTOJIHHJIHCb.
C rrpeo6pa30BaHHOR CHCTeMOR ( 2,2) IIOCTyrraeM aHaJIOrHl!HO BbiUieH3JIO
)KeHHOMY (§ 1), rrpHl!eM IIOJIYlleHHYIO CHCTeMy JlH¢¢epeHUHaJihHbiX ypaBHe
HHR l!IfCJieHHO HHTerpHpyeM Ha HHTepBaJie 0 < }, < l. B Kal!eCTBe Hal!aJibHbiX
21J2
YCJIOBHH 6epeM 3Hai.JeHHfl X1, X 2, • .• , Xn, COOTB€TCTBYIOW.He A= 0. 3Hal!eHH51
X1, X 2, ••• , Xn npH }, = 1 6y.D.yT HCKOMbiM perneHHeM CHCT€Mbl (2,1).
3 a Me lJ a H He. 3cpcpeKTHBHOCTb perneHHfl CHCT€Mbl (2, 1) B 3Hai.JHT€Jlb
HOH Mepe 3aBHCHT OT cnoco6a BBe.n.eHHfl napaMeTpa A.
§ 3.
LJ.JIH HJ!JllOCTpaUHH 113J!OllieHHOfO MeTO.LJ.a paCCMOTpHM ,D,Ba np!1Mepa.
1. OycTb .n.aHo ypaBHeHHe
A. 11--z
arcsin - -2 /t- ~%+0,047x-0,094A.-1,844=0.
X X
(3, 1)
Ope.n.noJiolliHM, liTo Tpe6yeTcH HaHTH npH6JIHMeHHbre peweHHfl 3Toro
ypaBH€HHfl .Ll.Jlfl TaKHX 3Hal!eHHH napaMeTpa A:
0, 1, 2, 3, ... , 10.
Opo.n.Hcj:>cj:>epeHUHpoBaB JieByiO l!aCTb ypaBHeHHH ( 3,1) no }, 11 pa3pern!1B
" dx rt-.rt-.
OTHOCHT€JlbHO Dp0!13BO.LJ.!-10!1 d)., , DOJIYliHM .Ll.H't''t'epeHUHaJib!-10€ ypaBH€HHe
dx o,094x2VT--.l:a-x(x+2A.)
d).= 0,047x!Vxt-).%-},(x+2l).
(3,2)
YpaBHeHHe (3,2) l!HCJieHHo HHTerpHpyeM MeTO.Ll.OM A.n.aMca-lllTepMepa
Ha HHTepBaJie 0 <A< 10 DpH Hal!aJibHOM YCJIOBHH
A= 0, X= 81,78723,
KOTOpoe HaXO.LJ.HM Henocpe.LJ.CTBeHHO H3 ypaBHeHHfl ( 3,1), IlOJiarafl B HeM
A= 0.
OrpaHHl!HBaHCb npu Bbil!HCJieHHHX IlflTbiO .n.ecHTHl!HbiMH 3HaKaMH, war
HHTerpupoBaHHH h Bbr6upaeM paBHbiM 0,5. Opu TaKC!M Bbi6ope h pa3HOCTH
TpeTbefO DOpH.D.Ka B cpopMyJie ( 1 ,6) He OKa3biBa!OT BJIHHHHH Ha pe3yJibTaT,
H Mbl HX He YliHTbiBaeM npH Bbll!HCJieHHHX.
Pe3yJibTaTbi Bbil!HCJieHHH npnse.n.eHbi B Ta6JI. 1.
Hal!aJio Ta6JIHUbi cocTaBJIHeM cnoco6oM nocJie.n.osaTeJibHbiX npH6-
JIHllieHHH 1.
OoJiyl!eHHbie 3Hal!eHHH x,
2, 3, . . . , 10, H 6y.LJ.yT HCKOMb!MH
COOTB€TCTBY!OW.He 3Ha li€HHHM
peWeHHHMH ypaBH€HHfl (3, 1).
A= 0, 1,
OorpeWHOCTb, .LJ.ODylll,eHH3fl npH Bbli.JHCJieHHHX, He npeBbiWaeT 10-4 _
3aMeTHM, liTO npHMeHeHHe JIJ06oro H3 cymecTBY!Oil.I.HX MeTO,li,OB K pewe
HHIO ypaBH€HHfl (3, 1) Il0Tpe6oBaJIO 6b! perneHHfl .LJ.€C51TH ypaBHeHHH COOT
BeTCTBeHHO .Ll.Jlfl Kalli.LJ.OfO 3a.D.aHHOfO 3!-Iat.IeHHfl napaMeTpa }, H, CJie.LJ.OBa
TeJibHO, 6oJibillOH 3aTpaTbi speMeHH. B t.IaCTHOCTH, npu peweHHH 3Toro ypas
HeHHH MeTO.LJ.OM HTepaUHH He06XO.Ll.HMO, npelli.LJ.e BCero, .Ll.Jlfl Kalli.LJ.OfO A rpa
cpHli€CKH onpe.LJ.eJIHTb .LJ.OCTaTOl!HO TOI.JHbie HCXO.LJ.Hbl€ DpH6JIHllieHHfl K pewe
HHflM, KOTOpb!e o6ecnel!HJIH 6bl npeo6pa30BaHHe ypaBH€HHfl (3, 1) K BH.Ll.Y
x = .w(x, A;) (i = 1, 2, ... , 10)
c co6JIIO.D.eHneM yCJioBHH
I A. H. K p bl Jl 0 B, [6], CTp. 301.
203
( Cl\1. BBe,ll,emJe), 'ITO B ,ll,aHHOI\1 C.lJy'Iae rrpe,li,CTaBJHieT 3Hal!I1TeJibHbie Tpy.n
HOCTH.
n }. X
0 0
I
81,78723
1 0,5 82,65716
0 0 81,78823
1 0,5 8'2,65775
2 1,0 83,52945
3 1,5 84,40237
4 2,0 85,27647
5 2,5 86,15 I83
6 3,0 87,02836
7 3,5 87,90618
8 4,0 88,78517
9 4,5 89,66542
10 5,0 90,54686
I I 5,5 91,42956 .
12 6,0 92,31346
13 6,5 93,19858
14 7,0 94,08492
15 7,5 94,97246
16 8,0 95,86120
17 8,5 96,75 I I4
18 9,0 97,64226
19 9,5 98,53!58
20 10,0 99,42805
I
dx
0,~6993
0,87052
0,87170
0,87292
0,87410
0,87536
0,87653
0,87782
0,87R99
0,88025
0,88144
0,88270
0,88390
0,88512
0,88634
0,88754
0,88874
0,88994
0,891 12
0,89232
0,89347
· dx
'Y)=h
d?.
0,86993
0,87111
0,86993
0,87111
0,8723I
0,87351
0,87473
0,87594
0,87717
0,87839
0,87962
0,88084
0,88207
0,88329
0,88451
0,8857.3
0,88694
0,88814
0,88934
0,89053
0,89172
0,89289
Ta6Ji nua 1
0,00118
0,00118 2
0,00 120 0
O,OOI20 2
0,00122 -- 1
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0,00123 -I
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0,00123 -I
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O,OOI22 0
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0,00121 - 1
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0,00120 -1
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0,00117
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He06XO.[LHMO, B006me fOBOpH, MHOfOKpaTHOe ITOBTOpeHHe rrpou.ecca HTe
paiJ.HH.
f1p e.n:JJaraeMb!M MeTO,li,OM, KaK Mbl yMe y6e,li,HJJHCb, ypaBHeHHe (3,1)
p ewaeTCH ,li,OBOJJbHO rrpOCTO H He Tpe6yeT 60JiblllOI':f 3aTpaTbl BpeMeHH, TaK
KaK rroJiyqaeM cpasy, o.nHo sa .npyrH~!. peweHHH ,li,JJH scex sHa 'leHHH },.
2. B Kal!eCTBe BTOporo rrpHMepa B03bMeM CHCTeMy .nsyx ypaBHeHH.il:
y - sin x + I ,32 = 0, 1
COSy- X+ 0,85 = 0. (3,3)
f1 ycTb Tpe6yeTCH HaHTH ee perneHHe C TO'IHOCTb!O ,li,O 10- 4 •
I1peo6pa3yeM rrpeM.n:e scero CHcTeMy ( 3,3) K BH.LLY ( 1,1), .LLJJH qero
BBO,li,HM rrapaMeTp }, CJJe,ll,yiOIUHM o6pa30M :
y + },(1,32 - sin x) = 0,
COSy -X + 0,85 = 0. (3 ,4)
f1oJJyl!eHHa51 CHCTeMa ypaBHeHHH rrpH A= 1 COBITa,ll,aeT C HCXO,li,HOH
(3,3), a rrpH }, = 0 oHa HMeeT pemeHHe
X = 1 ,85; y = 0.
I 9TOT npHMep Mbl B351JIH H3 KHHrH ,U)!{. CKap6opo [1].
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1. LI. M . C K a p 6 o p o, LlHcJieHHbre MeTOJlbi MaTeMaTHqecrwro aHam-r3a, foCT'ex-
H3llaT, 1934.
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CHCTCM HCm!HeHHb!X ypaBHeHHH, Jl. YqeH. 3an. riCJl. HH-Ta, 28 (1939).
3. H. Curry, The method o.f stee-pest descent fo-n non-linear minimization
problems, Quarterly o f Appl. Mathern., 2, N9 3 (1944).
4. Jl. B. K a !-1 Top o B H q, <PyHKuHoHaJibHbTli aHa.rrH3 H npHKJianHa>I MaTeMaTHKa,
YcnexH MaTeM. HayK, T . III, Bbrn. 6 (28) (1948).
5. E. Lahaye, Sur !a resolution des systemes d'equations transcendantes, A cad.
Roy. Belgique Bull. Cl. Sci (5), 34 (1948).
6. A. H. K p bl JI o B, JleKuHH o npH6JIHMeH'HhTX BbrqHcJieHH>IX, M.-Jl., 1950.
I1oJiytreHa 8 neKa6pH I 952 r.
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|
| id | umjimathkievua-article-7731 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus |
| last_indexed | 2026-03-24T03:33:13Z |
| publishDate | 1953 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/9a/13f8cfd77a2ee87d15e7487c0bd7229a.pdf |
| spelling | umjimathkievua-article-77312023-08-15T09:55:16Z О приближенном решении систем нелинейных уравнений О приближенном решении систем нелинейных уравнений Давиденко, Д. Ф. Давиденко, Д. Ф. - Institute of Mathematics, NAS of Ukraine 1953-05-28 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7731 Ukrains’kyi Matematychnyi Zhurnal; Vol. 5 No. 2 (1953); 196-206 Український математичний журнал; Том 5 № 2 (1953); 196-206 1027-3190 rus https://umj.imath.kiev.ua/index.php/umj/article/view/7731/9427 Copyright (c) 1953 Д. Ф. Давиденко |
| spellingShingle | Давиденко, Д. Ф. Давиденко, Д. Ф. О приближенном решении систем нелинейных уравнений |
| title | О приближенном решении систем нелинейных уравнений |
| title_alt | О приближенном решении систем нелинейных уравнений |
| title_full | О приближенном решении систем нелинейных уравнений |
| title_fullStr | О приближенном решении систем нелинейных уравнений |
| title_full_unstemmed | О приближенном решении систем нелинейных уравнений |
| title_short | О приближенном решении систем нелинейных уравнений |
| title_sort | о приближенном решении систем нелинейных уравнений |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7731 |
| work_keys_str_mv | AT davidenkodf opribližennomrešeniisistemnelinejnyhuravnenij AT davidenkodf opribližennomrešeniisistemnelinejnyhuravnenij |