Pll-2002-87

E TI tKHnKoB O 11 IOnnarneB M E IOJInarneBa

AnATITI1BHhIH AJIrOPI1TM BhIQI1CJIEHKH

ltDYHKUI1I1 HA JII1TIlIIl1UEBOH rPAHI1UE

TPEXMEPHOrO TEJIA

TIO 3AnAHHOMY rPAnl1EHTY HEro nPHMEHEHHE B MArHHTOCTATHKE

~l~ (11 v~J ~Nte rfO 1~elt-- Ill t HanpaBJ1eHO B laquo)KypHaJI Bbl4HCJlHTeJlbHOM MaTeMaTHKH

H MaTeMaTH4eCKOM qJH3HKHraquo

2002

BueneHHe

ITPH lJHCneHHOM peIlleHHH npaKTHlJeCKHX 33IlalJ HaH60nee BruKHbIM BOnpOshy

COM SlBnSleTCSI TOlJHOCTb nonyqaeMblx peIlleHHH n03TOMY B nOCnenHee BpeMS BCe

60nbille BHHMaHlliI yneJ1SleTcSI pa3BHTHIO lJHCneHHblX MeTOnOB n03BOnSiIOIlJHX KOHshy

TpOnHpOBaTb TOlJHOCTb BbllJHCnemfH 3neCb cnenyeT OTMeTHTb nporpecc Il11S1 nllshy

HeHHblX 3aualJ 3J1J1HnTHlJeCKOrO THna H nHHeHHblX 33IlalJ TeOpnH ynpyroCTH CBSIshy

3aHHblH C pa3BHTHeM auanTHBHblX hp-anropHTMoB B MeTone KOHelJHblX 3neMeHTOB

B CBSl3H C 3THM YKalKeM nHOHepCKHe pa60Thl [123] B OTnHlJlle OT nByMepHblx

3aualJ rne eCTb XOPOiliO pa3pa6oTaHHble nonXonbl HanpHMep [45] B TpexMepHblx

33IlalJaX cHTyauHSI 60nee cnOlKHaSI B KalJeCTBe npHMepa npuBeneM cnell)lOIlJ)lO

HenHHeHH)lO KpaeBYJO 33IlalJy nYCTb o6nacTb n c R3 orpaHHlJeHa nnnUlHueBoil

rpaHHueH [6] H n = n u nl rne 06naCTH n H n1 pa3lleJ1SllOTCSI BHYTpeHHeH rpashy

HHuell r en KOTOPasr TaKlKe SlBJ1SIeTCSI nHnUlHueBoH Tpe6yeTcSI HaHTI poundpYHKumf

u Ul YllOBneTBOpSiIOIlJHe ypaBHeHIDIM

3Eax a

(ai(X u lU)) + a(x u lu) 0 x E n (1)

3 a aUl 3 (2)i~l ax (aij(x) ax + i~ x E n l

3

U Ul = 0 L(ai(x u lu) (3) i=1

3 au u = u(X) x E anjr L aij(X)- Qi Ul(X) x E anIr (4)

ij=l aXj flle lu - rpanHeHT CPyHKUHH u a (Xi Qi - KOMnOHeHTbI BeKTopOB BHernHHX HOpshy

Mane K COOTBeTCTBYIOLQHM 06JIaCTSIM TIPH 3TOM paBeHcTBa (3) Ii (4) nOHIIMashy

IOTCSI B CMblCJIe Cne)lOB [6] COOTBeTCTBYlOlllllX cpYHKunii EY)leM npe)lnOnaraTb lJTO

Bce K03cpqmmfeHTbI B ypaBHeHHSlX (1 )(2) BellleCTBeHHbIe It BMeCTe C 3anaHHblMIi

CPYHKUHSlMH il Ul YllOBJ1eTBOpSiIOT ycnoBIiSlM Heo6xo)lHMbIM )lJ1S1 OllH03HalJHOii

pa3pewliMocTU 3aualJli (1)-(4) Tenepb npe)lnOnOJKIfM lJTO B 06JlaCTII n1 113BeCTHa

nOCTaTOlJHO rmlllKaSI BeKTop-cpYHKUHSI G KOTOPaSI HMeeT BaJKHOe caMOCTOSlTeJlbHOe

3HalJeHHe a ee KOMnOHeHTbI Gi (i 123) Y)lOBJ1eTBOpSlIOT ypaBHeHluo

a (aij(x) Gj ) +aXi

To eCTb eCJlH BBeCTIf CPYHKlUflO cent TaKYIO lJTO Icent GB nb TO cent 6Y)leT SlBJlSlTbCSI

lJaCTHblM perneHHeM ypaBHeHHSI (2) Torua 3aualJY (1)-(4) MOJKHO ccpopMymtpOBaTb

2

1

OTHOCl1TellhHO CPYHKUl1H U B 0611aCTl1 0 l1 Ul = Ul - cent B 0611aCTl1 0 1 IIpl1lJeM B

0611aCTJ1 n1 nOllyqaeTcH OJlHOpOJlHOe ypaBHeHl1e )JJlH CPYHKUl1l1 Ul IImnoMY eCnl1

BeKTOp a BhIlJl1CllHeTCH 60nee TOlJHO 6hICTpee lJeM CPYHKURH f B 06naCTl1 fh l1

npl1 )TOM ynpomaeTCH npouecc nOCTpOeHJ1H TpeXMepHOH CeTKl1 TO nepecpopMYshy

l1l1pOBKa 3alalJl1 (1)-(4) l11H Hel13BeCTHhIX U Ul SlBJ1SIeTCH OnpaBlaHHoH OlHaKO

TOlJHOCTh pellIeHl1H HOBOH 3alalJl1 6YleT 3aBl1CeTh OT TOlJHOCTl1 BhIlJl1CneHl1H CPYHKshy

Ul1l1 cent KOTopaSl 6YleT BXOJll1Th B ycnOBl1H Ha rpaHl1ue r HacToSlmaH pa60Ta nOCBHmeHa pa3pa60TKe 3cpcpeKTl1BHOrO MeTOla onpeJleneshy

Hl1S1 CPYHKUI1l1 cent Ha l1l1nllIl1ueBOH rpaHl1Ue r TpexMepHoro Tena ecnl1 3anaHa JlOCTashy

TOlJHO rnalKaH BeKTop-cpYHKUl1H a= vcent l1 B HeKoTopoH TOlJKe Xo E r l13BeCTHO

lJTO cent(xo) = cento - 3alaHHOe 3HalJeHl1e 3analJa BhIlJl1CneHl1H CPYHKUl1l1 cent B 06shy

meM cnyqae CBOll1TCH K pellIeHl1IO Henl1HeHHOro ypaBHeHl1H Ha OCHOBe TeOpl1l1

MOHOTOHHhIX onepaTopOB npl1 eCTeCTBeHHhIX ycnOBl1HX Ha cpyHKUl1IO cent JlOKa3hIBashy

eTCH CYI1leCTBOBaHl1e ell1HCTBeHHoro 0606meHHoro pellIeHl1H 3Toro ypaBHeHHSI H

CXOll1MOCTh KOHelJHo-meMeHTHhIX npH6nH)KeHHH K 3TOMY pellIeHl1IO IIpe)JJlarashy

eMhIH alanTHBHhIH anropHTM Ha OCHOBe MeTOla KOHelJHhIX meMeHTOB n03BonHeT

nonyqaTh pellIeHl1e C HeKOTopOH 3anaHHOH TOlJHOCThIO Pa3pa6oTaHHhIH MeTOl

l1MeeT Ba)KHOe npMeHeHe B MarHHTOCTaTHKe nOCKonhKY laeT B03MO)KHOCTh C

Heo6xoJll1MOH TOlJHOCThIO BhIlJl1CJ1SITh nOTeHUHan OT 06MOTKH Ha rpaHHue pa3lena

cpeJl C pa311HlJHhlMH MarHl1THhIMl1 xapaKTepHCTHKaMl1 AJIroPHTM anpo6HpOBancSi

npH paClJeTaX TpeXMepHhlX MarHl1THhIX noneH HeCKonhKHX MarHl1THhIX CHCTeM

B TOM lJl1cne l11S1 cneKTpOMeTpHlJeCKoro MarHHTa lHnOnhHoro THna H 60nhllIoro

COJIeHOHllanhHoro MarHHTa cpH3HlJeCKOrO 3KCnepMeHTa L3 (UEPH )KeHeBa)

YpaBHeHHe H eIO pa3peWHMOCTL

IIOllYQl1M ypaBHeHHe llJISi onpeleneHHSI CPYHKUl1H cent H lOK3)KeM ero pa3pellIHshy

MOCTh I3YleM npelnOnaraTh lJTO r - OlHOCBH3HaSI rpaHHua OTMeTHM lJTO Ha

npaKTHKe lJaCTO 6hlBaeT nOHHTHO KaK MHorOCBSl3HYIO rpaHHUY 061gtelHHHTh B 0Jlshy

HOC BSI3Hyro

ILrIH nHnllIl1ueBOH rpaHHUhI nOlJTl1 BCIOJly TO eCTh Ha ee manKHX lJaCTHX

MO)KHO onpeJlenHTh eJlHHHlJHhle opToroHanhHhIe BeKTophI KacaTenhHOH 6HHOPshy

ManH l1 BHellIHeH HopManH KOTophIe o6pa3yIOT nOKanhHyro npaByro leKaPTOBY

CHCTeMY KOOpll1HaT 3TH BeKTophI 0603HalJHM lJepe3 S pound ii COOTBeTCTBeHHO B TaKOH CHCTeMe KOOp)ll1HaT )lOCTaTOlJHO fnanKHH BeKTOp GH3 npOCTpaHCTBa rp)HKshy

UHH K = a Vcent cent E L2(r) Vcent E (L2(r))3 l1MeeT KOMnOHeHThI Gs Gt Gn

Qepe3 aT 0603HalJHM BeKTOp

3

C KOMrrOHeHTaMM GSl Gt bull A )J)15I LlOCTaTOYHO maLIKoH ~yHKUMM J yepe3 7rJ 0603HaYMM BeKTop

7rJ s oflos + f oflOt KOMrrOHeHTaMM KOToporo 5IBJUIIOTC5I rrpOM3BOUHhle rro HarrpaBJIeHMSlM r M f

J TIpeUrrOJIO)KMM YTO rpaHMua f pa36MTa Ha HeKOTOphle KOHeYHhle )JIeMeHThl

onpeLleJIeHHhlM CTaHuapTHhlM 06pa30M [6J npM 3TOM TOYKa 70 E f B KOTOPOH

3aUaHO 3HayeHMe 1gt0 5IBJI5IeTC5I Y3JIOM 3JIeMeHTa B 3aBMCMMOCTM OT rrOBeUeHM5I

3aUaHHOH BeKTOp-~YHKUMM G pa3L1eJIMM 3TM 3JIeMeHThl Ha LlBe rpyrrrrhi M f 2

K f1 OTHeceM TaKMe 3JIeMeHThI Ha KOTOPhlX BeJIMYMHa IGrl 60JIhWe BeJIMYMHhl

IGnl rLle Gn -HOpMaJIhHa5I KOMrrOHeHTa BeKTopa G H Ha060pOT K f2 OTHeceM

TaKMe 3JIeMeHThI Ha KOTOPhIX BeJIMYMHa IGnl HaMHoro 60JIhWe BeJIW-IMHhI IGrl K f 1 OTHeceM TaK)Ke 3JIeMeHThI K KOTOPhlM OTHOCMTC5I Y3eJI C TOYKOH Xo HMeHHO Ha

3TMX 3JIeMeHTax orrpeLleJIMM maLlKYID ~MHMTHYIO ~YHKUMIO X TaKYIO YTO X(TO) 1gt0 M X 0 BO Bcex OCTaJIhHhlX y3JIax TIyCTh 1gt1 1gt - X X E fl Torua)]JI5I

~YHKUMM 1gt1 rrOJIyYaeM ypaBHeHMe

(5)

M YCJIOBMe

1gt1(pound0) = 0 Xo E fl (6)

)LrI5I 3JIeMeHTOB rpyrrrrhI f2 KpoMe ypaBHeHM5I 7r1gt = Gn Ba)KHO MCrrOJIh30BaTh

ypaBHeHMe)]JI5I rrpOM3BOLlHOH rro HOPMaJIM OT ~YHKUMM 1gt rrOCKOJIhKY Ha f2 ~YHKshy

UM5I Gn rrpMHMMaeT 60JIhllme 3HaIIeHM5I yeM ~YHKUM5I Gr TI03TOMY Tpe6y5I

BhlIlOJIHeHMSI paBeHCTBa 71gt G MCrrOJIh3yeM OTHOllIeHMe ~YHKUMM Gn K 81gt1 on - rrpoM3BoUHOH rro HOPMaJIM OT ~yHKUMlI 1gt HMeeM

3TO OTHOllleHMe UOJI)KHO 6hlTh paBHO eUMHMue BBeueM ~YHKUMIO

Torua )]JISI x E f2 rrOJIyYMM ypaBHeHMe

(7)

TaKMM 06pa30M MCXOUHa5I 3aUaYa CBOUMTC5I K CMCTeMe (5)-(7) Jln5I perneHM5I

CMCTeMhI (5)-(7) MeTOUOM KOHeYHhlX 3JIeMeHTOB CHaYaJla BBeueM BCnOMOrdTenhshy

Hhle 0603HayeHM5I onpeUeJIMM cOOTBeTcTBYIOwee o606weHHoe peweHMe a 3aTeM

MCCJIeuyeM pa3pewMMocTh ypaBHeHM5I )]JI5I o606weHHoro peWeHMSI M CXOUHMOCTh

peweHHSI UMcKpeTM30BaHHoro ypaBHeHml K o606weHHoMY

4

BueneHHe

ITPH lJHCneHHOM peIlleHHH npaKTHlJeCKHX 33IlalJ HaH60nee BruKHbIM BOnpOshy

COM SlBnSleTCSI TOlJHOCTb nonyqaeMblx peIlleHHH n03TOMY B nOCnenHee BpeMS BCe

60nbille BHHMaHlliI yneJ1SleTcSI pa3BHTHIO lJHCneHHblX MeTOnOB n03BOnSiIOIlJHX KOHshy

TpOnHpOBaTb TOlJHOCTb BbllJHCnemfH 3neCb cnenyeT OTMeTHTb nporpecc Il11S1 nllshy

HeHHblX 3aualJ 3J1J1HnTHlJeCKOrO THna H nHHeHHblX 33IlalJ TeOpnH ynpyroCTH CBSIshy

3aHHblH C pa3BHTHeM auanTHBHblX hp-anropHTMoB B MeTone KOHelJHblX 3neMeHTOB

B CBSl3H C 3THM YKalKeM nHOHepCKHe pa60Thl [123] B OTnHlJlle OT nByMepHblx

3aualJ rne eCTb XOPOiliO pa3pa6oTaHHble nonXonbl HanpHMep [45] B TpexMepHblx

33IlalJaX cHTyauHSI 60nee cnOlKHaSI B KalJeCTBe npHMepa npuBeneM cnell)lOIlJ)lO

HenHHeHH)lO KpaeBYJO 33IlalJy nYCTb o6nacTb n c R3 orpaHHlJeHa nnnUlHueBoil

rpaHHueH [6] H n = n u nl rne 06naCTH n H n1 pa3lleJ1SllOTCSI BHYTpeHHeH rpashy

HHuell r en KOTOPasr TaKlKe SlBJ1SIeTCSI nHnUlHueBoH Tpe6yeTcSI HaHTI poundpYHKumf

u Ul YllOBneTBOpSiIOIlJHe ypaBHeHIDIM

3Eax a

(ai(X u lU)) + a(x u lu) 0 x E n (1)

3 a aUl 3 (2)i~l ax (aij(x) ax + i~ x E n l

3

U Ul = 0 L(ai(x u lu) (3) i=1

3 au u = u(X) x E anjr L aij(X)- Qi Ul(X) x E anIr (4)

ij=l aXj flle lu - rpanHeHT CPyHKUHH u a (Xi Qi - KOMnOHeHTbI BeKTopOB BHernHHX HOpshy

Mane K COOTBeTCTBYIOLQHM 06JIaCTSIM TIPH 3TOM paBeHcTBa (3) Ii (4) nOHIIMashy

IOTCSI B CMblCJIe Cne)lOB [6] COOTBeTCTBYlOlllllX cpYHKunii EY)leM npe)lnOnaraTb lJTO

Bce K03cpqmmfeHTbI B ypaBHeHHSlX (1 )(2) BellleCTBeHHbIe It BMeCTe C 3anaHHblMIi

CPYHKUHSlMH il Ul YllOBJ1eTBOpSiIOT ycnoBIiSlM Heo6xo)lHMbIM )lJ1S1 OllH03HalJHOii

pa3pewliMocTU 3aualJli (1)-(4) Tenepb npe)lnOnOJKIfM lJTO B 06JlaCTII n1 113BeCTHa

nOCTaTOlJHO rmlllKaSI BeKTop-cpYHKUHSI G KOTOPaSI HMeeT BaJKHOe caMOCTOSlTeJlbHOe

3HalJeHHe a ee KOMnOHeHTbI Gi (i 123) Y)lOBJ1eTBOpSlIOT ypaBHeHluo

a (aij(x) Gj ) +aXi

To eCTb eCJlH BBeCTIf CPYHKlUflO cent TaKYIO lJTO Icent GB nb TO cent 6Y)leT SlBJlSlTbCSI

lJaCTHblM perneHHeM ypaBHeHHSI (2) Torua 3aualJY (1)-(4) MOJKHO ccpopMymtpOBaTb

2

1

OTHOCl1TellhHO CPYHKUl1H U B 0611aCTl1 0 l1 Ul = Ul - cent B 0611aCTl1 0 1 IIpl1lJeM B

0611aCTJ1 n1 nOllyqaeTcH OJlHOpOJlHOe ypaBHeHl1e )JJlH CPYHKUl1l1 Ul IImnoMY eCnl1

BeKTOp a BhIlJl1CllHeTCH 60nee TOlJHO 6hICTpee lJeM CPYHKURH f B 06naCTl1 fh l1

npl1 )TOM ynpomaeTCH npouecc nOCTpOeHJ1H TpeXMepHOH CeTKl1 TO nepecpopMYshy

l1l1pOBKa 3alalJl1 (1)-(4) l11H Hel13BeCTHhIX U Ul SlBJ1SIeTCH OnpaBlaHHoH OlHaKO

TOlJHOCTh pellIeHl1H HOBOH 3alalJl1 6YleT 3aBl1CeTh OT TOlJHOCTl1 BhIlJl1CneHl1H CPYHKshy

Ul1l1 cent KOTopaSl 6YleT BXOJll1Th B ycnOBl1H Ha rpaHl1ue r HacToSlmaH pa60Ta nOCBHmeHa pa3pa60TKe 3cpcpeKTl1BHOrO MeTOla onpeJleneshy

Hl1S1 CPYHKUI1l1 cent Ha l1l1nllIl1ueBOH rpaHl1Ue r TpexMepHoro Tena ecnl1 3anaHa JlOCTashy

TOlJHO rnalKaH BeKTop-cpYHKUl1H a= vcent l1 B HeKoTopoH TOlJKe Xo E r l13BeCTHO

lJTO cent(xo) = cento - 3alaHHOe 3HalJeHl1e 3analJa BhIlJl1CneHl1H CPYHKUl1l1 cent B 06shy

meM cnyqae CBOll1TCH K pellIeHl1IO Henl1HeHHOro ypaBHeHl1H Ha OCHOBe TeOpl1l1

MOHOTOHHhIX onepaTopOB npl1 eCTeCTBeHHhIX ycnOBl1HX Ha cpyHKUl1IO cent JlOKa3hIBashy

eTCH CYI1leCTBOBaHl1e ell1HCTBeHHoro 0606meHHoro pellIeHl1H 3Toro ypaBHeHHSI H

CXOll1MOCTh KOHelJHo-meMeHTHhIX npH6nH)KeHHH K 3TOMY pellIeHl1IO IIpe)JJlarashy

eMhIH alanTHBHhIH anropHTM Ha OCHOBe MeTOla KOHelJHhIX meMeHTOB n03BonHeT

nonyqaTh pellIeHl1e C HeKOTopOH 3anaHHOH TOlJHOCThIO Pa3pa6oTaHHhIH MeTOl

l1MeeT Ba)KHOe npMeHeHe B MarHHTOCTaTHKe nOCKonhKY laeT B03MO)KHOCTh C

Heo6xoJll1MOH TOlJHOCThIO BhIlJl1CJ1SITh nOTeHUHan OT 06MOTKH Ha rpaHHue pa3lena

cpeJl C pa311HlJHhlMH MarHl1THhIMl1 xapaKTepHCTHKaMl1 AJIroPHTM anpo6HpOBancSi

npH paClJeTaX TpeXMepHhlX MarHl1THhIX noneH HeCKonhKHX MarHl1THhIX CHCTeM

B TOM lJl1cne l11S1 cneKTpOMeTpHlJeCKoro MarHHTa lHnOnhHoro THna H 60nhllIoro

COJIeHOHllanhHoro MarHHTa cpH3HlJeCKOrO 3KCnepMeHTa L3 (UEPH )KeHeBa)

YpaBHeHHe H eIO pa3peWHMOCTL

IIOllYQl1M ypaBHeHHe llJISi onpeleneHHSI CPYHKUl1H cent H lOK3)KeM ero pa3pellIHshy

MOCTh I3YleM npelnOnaraTh lJTO r - OlHOCBH3HaSI rpaHHua OTMeTHM lJTO Ha

npaKTHKe lJaCTO 6hlBaeT nOHHTHO KaK MHorOCBSl3HYIO rpaHHUY 061gtelHHHTh B 0Jlshy

HOC BSI3Hyro

ILrIH nHnllIl1ueBOH rpaHHUhI nOlJTl1 BCIOJly TO eCTh Ha ee manKHX lJaCTHX

MO)KHO onpeJlenHTh eJlHHHlJHhle opToroHanhHhIe BeKTophI KacaTenhHOH 6HHOPshy

ManH l1 BHellIHeH HopManH KOTophIe o6pa3yIOT nOKanhHyro npaByro leKaPTOBY

CHCTeMY KOOpll1HaT 3TH BeKTophI 0603HalJHM lJepe3 S pound ii COOTBeTCTBeHHO B TaKOH CHCTeMe KOOp)ll1HaT )lOCTaTOlJHO fnanKHH BeKTOp GH3 npOCTpaHCTBa rp)HKshy

UHH K = a Vcent cent E L2(r) Vcent E (L2(r))3 l1MeeT KOMnOHeHThI Gs Gt Gn

Qepe3 aT 0603HalJHM BeKTOp

3

C KOMrrOHeHTaMM GSl Gt bull A )J)15I LlOCTaTOYHO maLIKoH ~yHKUMM J yepe3 7rJ 0603HaYMM BeKTop

7rJ s oflos + f oflOt KOMrrOHeHTaMM KOToporo 5IBJUIIOTC5I rrpOM3BOUHhle rro HarrpaBJIeHMSlM r M f

J TIpeUrrOJIO)KMM YTO rpaHMua f pa36MTa Ha HeKOTOphle KOHeYHhle )JIeMeHThl

onpeLleJIeHHhlM CTaHuapTHhlM 06pa30M [6J npM 3TOM TOYKa 70 E f B KOTOPOH

3aUaHO 3HayeHMe 1gt0 5IBJI5IeTC5I Y3JIOM 3JIeMeHTa B 3aBMCMMOCTM OT rrOBeUeHM5I

3aUaHHOH BeKTOp-~YHKUMM G pa3L1eJIMM 3TM 3JIeMeHThl Ha LlBe rpyrrrrhi M f 2

K f1 OTHeceM TaKMe 3JIeMeHThI Ha KOTOPhlX BeJIMYMHa IGrl 60JIhWe BeJIMYMHhl

IGnl rLle Gn -HOpMaJIhHa5I KOMrrOHeHTa BeKTopa G H Ha060pOT K f2 OTHeceM

TaKMe 3JIeMeHThI Ha KOTOPhIX BeJIMYMHa IGnl HaMHoro 60JIhWe BeJIW-IMHhI IGrl K f 1 OTHeceM TaK)Ke 3JIeMeHThI K KOTOPhlM OTHOCMTC5I Y3eJI C TOYKOH Xo HMeHHO Ha

3TMX 3JIeMeHTax orrpeLleJIMM maLlKYID ~MHMTHYIO ~YHKUMIO X TaKYIO YTO X(TO) 1gt0 M X 0 BO Bcex OCTaJIhHhlX y3JIax TIyCTh 1gt1 1gt - X X E fl Torua)]JI5I

~YHKUMM 1gt1 rrOJIyYaeM ypaBHeHMe

(5)

M YCJIOBMe

1gt1(pound0) = 0 Xo E fl (6)

)LrI5I 3JIeMeHTOB rpyrrrrhI f2 KpoMe ypaBHeHM5I 7r1gt = Gn Ba)KHO MCrrOJIh30BaTh

ypaBHeHMe)]JI5I rrpOM3BOLlHOH rro HOPMaJIM OT ~YHKUMM 1gt rrOCKOJIhKY Ha f2 ~YHKshy

UM5I Gn rrpMHMMaeT 60JIhllme 3HaIIeHM5I yeM ~YHKUM5I Gr TI03TOMY Tpe6y5I

BhlIlOJIHeHMSI paBeHCTBa 71gt G MCrrOJIh3yeM OTHOllIeHMe ~YHKUMM Gn K 81gt1 on - rrpoM3BoUHOH rro HOPMaJIM OT ~yHKUMlI 1gt HMeeM

3TO OTHOllleHMe UOJI)KHO 6hlTh paBHO eUMHMue BBeueM ~YHKUMIO

Torua )]JISI x E f2 rrOJIyYMM ypaBHeHMe

(7)

TaKMM 06pa30M MCXOUHa5I 3aUaYa CBOUMTC5I K CMCTeMe (5)-(7) Jln5I perneHM5I

CMCTeMhI (5)-(7) MeTOUOM KOHeYHhlX 3JIeMeHTOB CHaYaJla BBeueM BCnOMOrdTenhshy

Hhle 0603HayeHM5I onpeUeJIMM cOOTBeTcTBYIOwee o606weHHoe peweHMe a 3aTeM

MCCJIeuyeM pa3pewMMocTh ypaBHeHM5I )]JI5I o606weHHoro peWeHMSI M CXOUHMOCTh

peweHHSI UMcKpeTM30BaHHoro ypaBHeHml K o606weHHoMY

4

1

OTHOCl1TellhHO CPYHKUl1H U B 0611aCTl1 0 l1 Ul = Ul - cent B 0611aCTl1 0 1 IIpl1lJeM B

0611aCTJ1 n1 nOllyqaeTcH OJlHOpOJlHOe ypaBHeHl1e )JJlH CPYHKUl1l1 Ul IImnoMY eCnl1

BeKTOp a BhIlJl1CllHeTCH 60nee TOlJHO 6hICTpee lJeM CPYHKURH f B 06naCTl1 fh l1

npl1 )TOM ynpomaeTCH npouecc nOCTpOeHJ1H TpeXMepHOH CeTKl1 TO nepecpopMYshy

l1l1pOBKa 3alalJl1 (1)-(4) l11H Hel13BeCTHhIX U Ul SlBJ1SIeTCH OnpaBlaHHoH OlHaKO

TOlJHOCTh pellIeHl1H HOBOH 3alalJl1 6YleT 3aBl1CeTh OT TOlJHOCTl1 BhIlJl1CneHl1H CPYHKshy

Ul1l1 cent KOTopaSl 6YleT BXOJll1Th B ycnOBl1H Ha rpaHl1ue r HacToSlmaH pa60Ta nOCBHmeHa pa3pa60TKe 3cpcpeKTl1BHOrO MeTOla onpeJleneshy

Hl1S1 CPYHKUI1l1 cent Ha l1l1nllIl1ueBOH rpaHl1Ue r TpexMepHoro Tena ecnl1 3anaHa JlOCTashy

TOlJHO rnalKaH BeKTop-cpYHKUl1H a= vcent l1 B HeKoTopoH TOlJKe Xo E r l13BeCTHO

lJTO cent(xo) = cento - 3alaHHOe 3HalJeHl1e 3analJa BhIlJl1CneHl1H CPYHKUl1l1 cent B 06shy

meM cnyqae CBOll1TCH K pellIeHl1IO Henl1HeHHOro ypaBHeHl1H Ha OCHOBe TeOpl1l1

MOHOTOHHhIX onepaTopOB npl1 eCTeCTBeHHhIX ycnOBl1HX Ha cpyHKUl1IO cent JlOKa3hIBashy

eTCH CYI1leCTBOBaHl1e ell1HCTBeHHoro 0606meHHoro pellIeHl1H 3Toro ypaBHeHHSI H

CXOll1MOCTh KOHelJHo-meMeHTHhIX npH6nH)KeHHH K 3TOMY pellIeHl1IO IIpe)JJlarashy

eMhIH alanTHBHhIH anropHTM Ha OCHOBe MeTOla KOHelJHhIX meMeHTOB n03BonHeT

nonyqaTh pellIeHl1e C HeKOTopOH 3anaHHOH TOlJHOCThIO Pa3pa6oTaHHhIH MeTOl

l1MeeT Ba)KHOe npMeHeHe B MarHHTOCTaTHKe nOCKonhKY laeT B03MO)KHOCTh C

Heo6xoJll1MOH TOlJHOCThIO BhIlJl1CJ1SITh nOTeHUHan OT 06MOTKH Ha rpaHHue pa3lena

cpeJl C pa311HlJHhlMH MarHl1THhIMl1 xapaKTepHCTHKaMl1 AJIroPHTM anpo6HpOBancSi

npH paClJeTaX TpeXMepHhlX MarHl1THhIX noneH HeCKonhKHX MarHl1THhIX CHCTeM

B TOM lJl1cne l11S1 cneKTpOMeTpHlJeCKoro MarHHTa lHnOnhHoro THna H 60nhllIoro

COJIeHOHllanhHoro MarHHTa cpH3HlJeCKOrO 3KCnepMeHTa L3 (UEPH )KeHeBa)

YpaBHeHHe H eIO pa3peWHMOCTL

IIOllYQl1M ypaBHeHHe llJISi onpeleneHHSI CPYHKUl1H cent H lOK3)KeM ero pa3pellIHshy

MOCTh I3YleM npelnOnaraTh lJTO r - OlHOCBH3HaSI rpaHHua OTMeTHM lJTO Ha

npaKTHKe lJaCTO 6hlBaeT nOHHTHO KaK MHorOCBSl3HYIO rpaHHUY 061gtelHHHTh B 0Jlshy

HOC BSI3Hyro

ILrIH nHnllIl1ueBOH rpaHHUhI nOlJTl1 BCIOJly TO eCTh Ha ee manKHX lJaCTHX

MO)KHO onpeJlenHTh eJlHHHlJHhle opToroHanhHhIe BeKTophI KacaTenhHOH 6HHOPshy

ManH l1 BHellIHeH HopManH KOTophIe o6pa3yIOT nOKanhHyro npaByro leKaPTOBY

CHCTeMY KOOpll1HaT 3TH BeKTophI 0603HalJHM lJepe3 S pound ii COOTBeTCTBeHHO B TaKOH CHCTeMe KOOp)ll1HaT )lOCTaTOlJHO fnanKHH BeKTOp GH3 npOCTpaHCTBa rp)HKshy

UHH K = a Vcent cent E L2(r) Vcent E (L2(r))3 l1MeeT KOMnOHeHThI Gs Gt Gn

Qepe3 aT 0603HalJHM BeKTOp

3

C KOMrrOHeHTaMM GSl Gt bull A )J)15I LlOCTaTOYHO maLIKoH ~yHKUMM J yepe3 7rJ 0603HaYMM BeKTop

7rJ s oflos + f oflOt KOMrrOHeHTaMM KOToporo 5IBJUIIOTC5I rrpOM3BOUHhle rro HarrpaBJIeHMSlM r M f

J TIpeUrrOJIO)KMM YTO rpaHMua f pa36MTa Ha HeKOTOphle KOHeYHhle )JIeMeHThl

onpeLleJIeHHhlM CTaHuapTHhlM 06pa30M [6J npM 3TOM TOYKa 70 E f B KOTOPOH

3aUaHO 3HayeHMe 1gt0 5IBJI5IeTC5I Y3JIOM 3JIeMeHTa B 3aBMCMMOCTM OT rrOBeUeHM5I

3aUaHHOH BeKTOp-~YHKUMM G pa3L1eJIMM 3TM 3JIeMeHThl Ha LlBe rpyrrrrhi M f 2

K f1 OTHeceM TaKMe 3JIeMeHThI Ha KOTOPhlX BeJIMYMHa IGrl 60JIhWe BeJIMYMHhl

IGnl rLle Gn -HOpMaJIhHa5I KOMrrOHeHTa BeKTopa G H Ha060pOT K f2 OTHeceM

TaKMe 3JIeMeHThI Ha KOTOPhIX BeJIMYMHa IGnl HaMHoro 60JIhWe BeJIW-IMHhI IGrl K f 1 OTHeceM TaK)Ke 3JIeMeHThI K KOTOPhlM OTHOCMTC5I Y3eJI C TOYKOH Xo HMeHHO Ha

3TMX 3JIeMeHTax orrpeLleJIMM maLlKYID ~MHMTHYIO ~YHKUMIO X TaKYIO YTO X(TO) 1gt0 M X 0 BO Bcex OCTaJIhHhlX y3JIax TIyCTh 1gt1 1gt - X X E fl Torua)]JI5I

~YHKUMM 1gt1 rrOJIyYaeM ypaBHeHMe

(5)

M YCJIOBMe

1gt1(pound0) = 0 Xo E fl (6)

)LrI5I 3JIeMeHTOB rpyrrrrhI f2 KpoMe ypaBHeHM5I 7r1gt = Gn Ba)KHO MCrrOJIh30BaTh

ypaBHeHMe)]JI5I rrpOM3BOLlHOH rro HOPMaJIM OT ~YHKUMM 1gt rrOCKOJIhKY Ha f2 ~YHKshy

UM5I Gn rrpMHMMaeT 60JIhllme 3HaIIeHM5I yeM ~YHKUM5I Gr TI03TOMY Tpe6y5I

BhlIlOJIHeHMSI paBeHCTBa 71gt G MCrrOJIh3yeM OTHOllIeHMe ~YHKUMM Gn K 81gt1 on - rrpoM3BoUHOH rro HOPMaJIM OT ~yHKUMlI 1gt HMeeM

3TO OTHOllleHMe UOJI)KHO 6hlTh paBHO eUMHMue BBeueM ~YHKUMIO

Torua )]JISI x E f2 rrOJIyYMM ypaBHeHMe

(7)

TaKMM 06pa30M MCXOUHa5I 3aUaYa CBOUMTC5I K CMCTeMe (5)-(7) Jln5I perneHM5I

CMCTeMhI (5)-(7) MeTOUOM KOHeYHhlX 3JIeMeHTOB CHaYaJla BBeueM BCnOMOrdTenhshy

Hhle 0603HayeHM5I onpeUeJIMM cOOTBeTcTBYIOwee o606weHHoe peweHMe a 3aTeM

MCCJIeuyeM pa3pewMMocTh ypaBHeHM5I )]JI5I o606weHHoro peWeHMSI M CXOUHMOCTh

peweHHSI UMcKpeTM30BaHHoro ypaBHeHml K o606weHHoMY

4

C KOMrrOHeHTaMM GSl Gt bull A )J)15I LlOCTaTOYHO maLIKoH ~yHKUMM J yepe3 7rJ 0603HaYMM BeKTop

7rJ s oflos + f oflOt KOMrrOHeHTaMM KOToporo 5IBJUIIOTC5I rrpOM3BOUHhle rro HarrpaBJIeHMSlM r M f

J TIpeUrrOJIO)KMM YTO rpaHMua f pa36MTa Ha HeKOTOphle KOHeYHhle )JIeMeHThl

onpeLleJIeHHhlM CTaHuapTHhlM 06pa30M [6J npM 3TOM TOYKa 70 E f B KOTOPOH

3aUaHO 3HayeHMe 1gt0 5IBJI5IeTC5I Y3JIOM 3JIeMeHTa B 3aBMCMMOCTM OT rrOBeUeHM5I

3aUaHHOH BeKTOp-~YHKUMM G pa3L1eJIMM 3TM 3JIeMeHThl Ha LlBe rpyrrrrhi M f 2

K f1 OTHeceM TaKMe 3JIeMeHThI Ha KOTOPhlX BeJIMYMHa IGrl 60JIhWe BeJIMYMHhl

IGnl rLle Gn -HOpMaJIhHa5I KOMrrOHeHTa BeKTopa G H Ha060pOT K f2 OTHeceM

TaKMe 3JIeMeHThI Ha KOTOPhIX BeJIMYMHa IGnl HaMHoro 60JIhWe BeJIW-IMHhI IGrl K f 1 OTHeceM TaK)Ke 3JIeMeHThI K KOTOPhlM OTHOCMTC5I Y3eJI C TOYKOH Xo HMeHHO Ha

3TMX 3JIeMeHTax orrpeLleJIMM maLlKYID ~MHMTHYIO ~YHKUMIO X TaKYIO YTO X(TO) 1gt0 M X 0 BO Bcex OCTaJIhHhlX y3JIax TIyCTh 1gt1 1gt - X X E fl Torua)]JI5I

~YHKUMM 1gt1 rrOJIyYaeM ypaBHeHMe

(5)

M YCJIOBMe

1gt1(pound0) = 0 Xo E fl (6)

)LrI5I 3JIeMeHTOB rpyrrrrhI f2 KpoMe ypaBHeHM5I 7r1gt = Gn Ba)KHO MCrrOJIh30BaTh

ypaBHeHMe)]JI5I rrpOM3BOLlHOH rro HOPMaJIM OT ~YHKUMM 1gt rrOCKOJIhKY Ha f2 ~YHKshy

UM5I Gn rrpMHMMaeT 60JIhllme 3HaIIeHM5I yeM ~YHKUM5I Gr TI03TOMY Tpe6y5I

BhlIlOJIHeHMSI paBeHCTBa 71gt G MCrrOJIh3yeM OTHOllIeHMe ~YHKUMM Gn K 81gt1 on - rrpoM3BoUHOH rro HOPMaJIM OT ~yHKUMlI 1gt HMeeM

3TO OTHOllleHMe UOJI)KHO 6hlTh paBHO eUMHMue BBeueM ~YHKUMIO

Torua )]JISI x E f2 rrOJIyYMM ypaBHeHMe

(7)

TaKMM 06pa30M MCXOUHa5I 3aUaYa CBOUMTC5I K CMCTeMe (5)-(7) Jln5I perneHM5I

CMCTeMhI (5)-(7) MeTOUOM KOHeYHhlX 3JIeMeHTOB CHaYaJla BBeueM BCnOMOrdTenhshy

Hhle 0603HayeHM5I onpeUeJIMM cOOTBeTcTBYIOwee o606weHHoe peweHMe a 3aTeM

MCCJIeuyeM pa3pewMMocTh ypaBHeHM5I )]JI5I o606weHHoro peWeHMSI M CXOUHMOCTh

peweHHSI UMcKpeTM30BaHHoro ypaBHeHml K o606weHHoMY

4

OnpeueJlHM rHJlb6ep1OBO npOCTpaHCTBO CPYHKUHH

wi (f) u U E L2(f) VTu E (L2(f))2

CO CKaJllIPHblM npOH3BeueHHeM

H HOPMOH

Ilullwl(l) ((u uhtJ(l)) 12 U1) E Wi(f)

nYCTb fo - IIaCTb rpaHHUbl f a Lo - HeKOTopIDI He3aMKHYTIDI KpHBasI Ha rpaHHue

f H BbmOJlH5IeTC5I OllHO H3 YCJlOBHH

J udS = 0 HJlH JudI = 0 (8) 10 Lo

Torlla TaKOH Kllacc CPYHKUHH H3 Wi(f) 6yueM 0603HaIJaTb IJepe3 Wio(f) MHO)Keshy

CTBO CPYHKUHI1 H3 Wio(f) 51BJ151eTC5I nOllnpocTpaHcTBoM B Wi(f) JJn51 CPYHKUHH

U v E Wio(f) MO)KHO BBeCTH cKamlpHoe npmnBelleHHe

(u v hVio(r) = (VTU VTV)(L2(f))2

H cooTBeTcTBYIOIJJYIO HOPMY KOTOpa51 6yueT gtKBHBaJleHTHa [71 HopMe npOCTpaHshy

CTBa Wi(f) 0606IJJeHHb1M peweHHeM CHCTeMbl (5)-(7) 6YlleM Ha3blBaTb CPYHKUHIO u E

Wio(f) YllOBJleTBOp5l1OIJJyJO HHTerpaJlbHOMY TO)KJleCTBY

jVTUmiddotVTvdS+ j p(G IVTul)(VTu-GT)middotVTvdS j (GT VTXJmiddot VTvdS (9) fl 12 11

npH G I( H Jl106011 CPYHKUHH v E Wio(f) JJn51 HCCJleuOBaHH5I pa3peUlHMocTH ypaBHeHH5I (9) paccMoTpHM BcnOMordTeJlbshy

Hble onepaTopbl Bl H B2 KOTopble 33U3UHM C nOMOIJJblO paBeHcTB

(B1UlVI)wiWd == jVTUImiddotVTU1dS UtVI E Wi(f) 11

(B2U2V2)W(I2) == j p(GIVTll21)(VTU2 GT)middot VTV2dS [2

1 I I npH U2V2 W2(f2) H C E K OIJeBHllHO IJTO BI W2o(f t ) ----+ W2o(f) KpOMe

Tom Bl - nOJlO)KHTeJlbHO-OnpelleJleHHbIH onepaTOp B Wdo(f) nOKa)KeM IJTO Bz Wdo(f2) ----+ Wio(f 2 ) JJn51 U E Wdo(f2 ) HMeeM

p(G IVTUn ~ 2 nOCKOJlbKY IG VTul ~ ICnl nOgtTOMY

(Bzu v) w (I2) lt 2middot Ilullwio([d IIvllwlo(Ld 2middot IIGTII(lJ2(lt)2 Il vlln1o(ld

5

uPYfHe CBOHCTBa OnepaTOpa B2 CCPOPMYJlHpOBaHbl B CJleJlYIOlueH JleMMe

JIeMMa 1 OnepaTop B2 Wio(r2) -lgt Wio(r2 ) 51BJ151eTC5I

1) CHJlbHO MOHOTOHHbIM TO eCTb BblTIOJlH5IeTC5I HepaBeHCTBO

cu V E lvio(r2 )

2) JlHnllIHU-HenpepbIBHbIM TO eCTh BblTIOJlH5IeTC5I HepaBeHCTBO

fJle 01 = 2 + max(IIGn l) G E K XE r 2

)loKa3aTeJlhCTBO

UOKaJKeM CHJlhH)IO MOHOTOHHOCTb YlMeeM )lJl51 CU V E Wio(r2)

J[p(G IVTcul)(vu - GT )

r 2

(10)

)ln51 ynpolUeHH5I 3anHCH BBeJleM 0603Ha4eHH5I

I1CnOJlb3Y5I 3TH 0603Ha4eHHSI npeo6pa3yeM BhlpaJKeHHe B (10) CJleJlYlOlUHM o6pashy

30M

(B2U B2v u v)WiO(r2) J[(p(O) - 1)0 - (p(V) - I)V] (0 - V)dS+ [2

+ J (0 - V)(U - V)dS J[(p(U) -1)(IUI2-0V)+(p(V)-I)(IVI2-UV)]dS+ 12 I2

+llu vll~Vio(fd

TaK KaK OmiddotV ~ ~(IUI2 + IVI2) TO

(B2U B2v u V)WlO(r2) ~ J[(p(U) 1)(IU12 - ~IUI2 ~IVI2)+ r 2

+(p(V) 1)(IV12 - ~IOI2 - ~IVI2)ldS + lIu vll~io(12)

(12) J (101 2 IVI2)(p(U) - p(V)dS + Ilu vll~l20(r)2 12

6

ilOCKonbKY

p(U) - p(V) - ~(IUI IVI) (11) - IUlivl

TO nOnblHTerpaJIbHoe BbIpa)[(eHlfe BCerna HeOTplflJaTenbHO ilOnOMY nonyqaeM

To eCTb B2- ClfnbHO MOHOTOHHblH onepaTop

ilOKa)[(eM QTO B2 - nlfnWlflJ-HenpepbIBHbIH onepaTop HMeeM

B2vll wio(r2 ) = sup I(B2u IIwll wiO(r2) =1

= SUp I j[p(U)U - p(V)V] 7rWdSI ~ IIwll wiO(r2)=1 r 2

~ SUp I j[p(U)IU - VI + Ip(U) - p(V)IIVIJI7rw ldSI IIwll wJO(r2)=1 r2

ilOCKOnbKY lUI = 17r U 61 ~ IGnl1f TaK)[(e IVI ~ IGnl 1f3 (11) cnenyeT

Ip(O) p(V)1 lt 110nV11 il03TOMY 1f3 nocnenHero HepaBeHcTBa If 1f3 HepaBeHcTBa KOWIf-BYHSlKOBCKOro noshy

nyqaeM

OTKyna CJIenyeT nlfnWlflJ-HenpepbIBHocTb onepaTopa B2 C nOCTOSlHHOii 01 2 + max(IIGnl) J1eMMa nOKa3aHa xEr2

ilYCTb H = Wio(rl ) x Wio(r2) Jth CBOHCTB onepaTopOB Bl If B2 cnenyroT

CBoiIcTBa onepaTopa B H ~ H 33J(aBaeMOro paBeHCTBOM

(Bu V)H

rne U (UlU2) v = (VlV2) E Wio(r l ) x Wio(r2 ) OH SlBJISleTCSI ClfJIbHO

MOHOTOHHblM If JIlfnWlfU-HenpepblBHbIM il03TOMY 1f3 o6meii TeopeMbl [7] 0 pa3pe-middot

WlfMOCTIf ypaBHeHlfH C TaKlfMIf onepaTopaMIf nonyqaeM cymecTBoBaHlfe o6paTHoro

OnepaTopa B-1 H ~ H ilepeiineM K nlfcKpeTlf3alJ1f1f ypaBHeHlfSl (9) ilYCTb Ni(x) i 12 mshy

KOHeQHO-3neMeHTHbIe 6a3lfCHbie 4lHKlJlflf onpeneneHHble Ha 3neMeHTax rpynn

7

r 1 H r 2 npHQeM 3TH CPYHKUHH 06pa3)lOT nOl1H)lO CHCTeMY l1HHeHHO He3aBHCHMbIX

311eMeHTOB B H TIpH611lf)KeHHOe pellIeHHe ypaBHeHHx (9) centm 6yneM HCKaTb B BHne

m A

centm L centiNi(X) X E r i=l

rne centi- HeH3BeCTHbie 3HaQeHHX CPYHKUHH centm B Y311aX 311eMeHTOB TIonCTaBJlXX B

ypaBHeHHe (9) BMeCTO U Bblpa)KeHHe)])lX centm H HCII0l1b3YX B KaQeCTBe v CPYHKUHH

Ni B COOTBeTCTBHH C MeTOnOM 6y6HoBa-ranepKHHa HMeeM CHCTeMY ypaBHeHHH

1p(xG IVTcentml)(i centjVTNj shy GT)middot VTNidS 1vTX V TNidS (12) f j=l fl

rne i 12 m 3necb )])lX yno6cTBa 3aIIHCH

p(x

3aMelaHUe 1 CHcTeMY (12) TaK)Ke MO)KHO IIOl1YQHTb IIYTeM CPOPMaJlbHOro nHcpcpepeHUHpoBashy

HHX CPYHKUHOHaJla

- VT(centm + X)1 2dS + 11IGT V TcentmI2 + (Gn - alG V Tcentml)2]dS f2

no K03CPCPHUHeHTaM centi i 12 m 3necb a - 3HaK KOMIIOHeHTbl Gn bull

Cl1enCTBHeM CBOHCTB onepaTopa B COrJIaCHO TeopHH MOHOTOHHblX oIIepaTopoB

[7] 5IBJl5leTC5I cyruecTBoBaHHe enHHCTBeHHoro pellIeHH5I centm CHCTeMbI (12) Hero

CXOnHMOCTb B IIpOCTpaHCTBe H K CPYHKUHH U KOTOP851 51BJI5leTC5I enHHCTBeHHbIM

pellIeHHeM ypaBHeHHSI (9) KpOMe Torn pellIeHHe ypaBHeHH5I (12) MO)KeT 6blTb

IIOJIyqeHo C IIOMOIqblO HTepaUHOHHoro rrpouecca

(k) -w vTx VTNidS + w p(x IVTcentm I)GT middot VTNidS (13)1 1

r 1 r

JIJI5I i 12 m 3necb rrapaMeTp w E (02at) k - HTepaUHoHHbIM HHneKC

k 01 I(mI pellIeHHSI CHCTeMbl (12) - centm H pellIeHH5I ypaBHeHH5I (9) - U

8

--------------------------------------- -------

2

3llCCb Hm JUmCHHOC nOllnpOCTpaHCTBO B H COCT05lmCC U3 )nCMCHTOB BHJla m L oiNi(X) mc oi HCKOTOPbIC K03QJqmuucHTbI )InsI OUCHKU nOrpCIlIHOCTU 1=1 06bI l iHO UCnOnb3YlOT HCpaBCHCTBO

inf U - v II JJ S II u I u II H (14)lIEHm

rllC Iu QJYHKUU5I-UHTcpnOflSIHT U3 H m 3auaBaCMa5I paBCHCTBOM I u( x) =

UiNi (x) 3llCCb Ui - 3Ha-lCHU5I QJYHKUUU U B Y3l1ax 3nCMCHTOB

3aMCTUM -ITO ccnu f 2 -nycToc MHO)KCCTBO TO nonyqacM nUHCHHbIH cnyqaH B

KOTOPOM ypaBHcHUC lIn51 0606mcHHoro pCrnCHU5I UMCCT BUll

jVrUVrVdS j(Gr - VrX) VrvdS UV E Wio(f) (15) [ r

Torlla U3 06mcH TCOpUU MCTOlla KOHC4HblX 3nCMCHTOB lIn51 3nnunTU4CCKUX 3ashy

lla4 [61 lln5I ypaBHcHu5I (15) CnCllYCT CllUHCTBCHHOCTb pCrnCHU5I U B npocTpaHcTBc

Wio(r) cymecTBoBaHHe ellHHCTBeHHoro perneHH5I COOTBcTcTByJOmeH CHCTeMbI

nonyqaeMoH MeTOllOM Ey6HoBa-ranepKHHa a TaK)Ke CXOJlHMOCTb npH6nH)KeHHOrO

perneHH5I cPm K 0606meHHoMY perneHHIO u TIPH 3TOM cnpaBelInHBa oueHKa aHashy

norH4HaSi (14)

Iullwl20 (1) (16)

iln51 HTepaUHoHHoro npouecca (13) peKOMeHJlyeTc5I Hcnonb30BaTb B Ka-leCTBe

Ha-lanbHbIX 3Ha4eHHH A(O) 0 w 1 H p-(x G IV A(O) I) = 1 Bo MHorHXPm rVm bull

cnyqa5lx 3TOro 6bIBaeT JlOCTaTO-lHO 4T06bI perneHHe CHCTeMbI OnpellenHnOCb 3a

OllHY HTepaUHIO

AlJanTHBHLIH aJIfOPHTM

TO-lHOCTb nOJlyqaeMoro npH6nH)KeHHoro perneHHSI ltPm KaK nOKa3bIBalOT HCshy

paBeHCTBa (14)(16) 3aBHCHT OT BbI60pa 6a3HCHbIX QJYHKUHH Nji 12 m TIOJl06paTb TaKUe 6a3HCHble QJYHKUHH -IT06bI cpa3y nonyqHTb He06xOllHMYlO T04shy

HOCTb KaK npaBHno He YllaeTCSI TImnOMY npHXOllHTCSI HaXOllHTb HeCKonbKO

npH6JU1)KeHHbIX perneHHH TaKHM 06pa30M 4T06bl Ka)KJlOe HOBoe np1l6J1H)KCHHOe

perneHHe 6bInO 6nH)Ke K T04HOMY perneHHIO ypaBHeHlISI 4eM npellbIllymee TIpH

3TOM nOCTpoeHHe 6a311CHbIX QJYHKUHH llJ151 Ka)KJlOrO HOBOro pCrneHH5I OCHOBbIBashy

eTCSI Ha aHanH3e nOBelleHH5I npeJlbIllymllx npH6nH)KeHHbIX peweHlIH C HCnOJlb30shy

BaHHeM HeKoToporo auanTlIBHoro anropHTMa B HaH60nee YHHBepcanbHblX auashy

I1THBHbIX anropHTMax 6a3HCHble QJYHKUHH llJ1S1 HOBO[O npH6m1)KeHHOrO perneHHSI

MOryT CTPOHTbCSI cpa3y JlByMSI cnoc06aMH OllHOBpeMeHHO UepBblii cnoco6 3ashy

KnlO4aeTCSI B TOM 4TO pa3MepbI KOHe4HbiX 3neMeHTOB YMeHbwalOTCSI B 3aBHcHMOshy

CTH OT anpHopHblX oueHOK H nOBelleHH5I npCllblJlymero npH6Ju-f)KeHHoro pellIeHH5I

9

uPYfHe CBOHCTBa OnepaTOpa B2 CCPOPMYJlHpOBaHbl B CJleJlYIOlueH JleMMe

JIeMMa 1 OnepaTop B2 Wio(r2) -lgt Wio(r2 ) 51BJ151eTC5I

1) CHJlbHO MOHOTOHHbIM TO eCTb BblTIOJlH5IeTC5I HepaBeHCTBO

cu V E lvio(r2 )

2) JlHnllIHU-HenpepbIBHbIM TO eCTh BblTIOJlH5IeTC5I HepaBeHCTBO

fJle 01 = 2 + max(IIGn l) G E K XE r 2

)loKa3aTeJlhCTBO

UOKaJKeM CHJlhH)IO MOHOTOHHOCTb YlMeeM )lJl51 CU V E Wio(r2)

J[p(G IVTcul)(vu - GT )

r 2

(10)

)ln51 ynpolUeHH5I 3anHCH BBeJleM 0603Ha4eHH5I

I1CnOJlb3Y5I 3TH 0603Ha4eHHSI npeo6pa3yeM BhlpaJKeHHe B (10) CJleJlYlOlUHM o6pashy

30M

(B2U B2v u v)WiO(r2) J[(p(O) - 1)0 - (p(V) - I)V] (0 - V)dS+ [2

+ J (0 - V)(U - V)dS J[(p(U) -1)(IUI2-0V)+(p(V)-I)(IVI2-UV)]dS+ 12 I2

+llu vll~Vio(fd

TaK KaK OmiddotV ~ ~(IUI2 + IVI2) TO

(B2U B2v u V)WlO(r2) ~ J[(p(U) 1)(IU12 - ~IUI2 ~IVI2)+ r 2

+(p(V) 1)(IV12 - ~IOI2 - ~IVI2)ldS + lIu vll~io(12)

(12) J (101 2 IVI2)(p(U) - p(V)dS + Ilu vll~l20(r)2 12

6

ilOCKonbKY

p(U) - p(V) - ~(IUI IVI) (11) - IUlivl

TO nOnblHTerpaJIbHoe BbIpa)[(eHlfe BCerna HeOTplflJaTenbHO ilOnOMY nonyqaeM

To eCTb B2- ClfnbHO MOHOTOHHblH onepaTop

ilOKa)[(eM QTO B2 - nlfnWlflJ-HenpepbIBHbIH onepaTop HMeeM

B2vll wio(r2 ) = sup I(B2u IIwll wiO(r2) =1

= SUp I j[p(U)U - p(V)V] 7rWdSI ~ IIwll wiO(r2)=1 r 2

~ SUp I j[p(U)IU - VI + Ip(U) - p(V)IIVIJI7rw ldSI IIwll wJO(r2)=1 r2

ilOCKOnbKY lUI = 17r U 61 ~ IGnl1f TaK)[(e IVI ~ IGnl 1f3 (11) cnenyeT

Ip(O) p(V)1 lt 110nV11 il03TOMY 1f3 nocnenHero HepaBeHcTBa If 1f3 HepaBeHcTBa KOWIf-BYHSlKOBCKOro noshy

nyqaeM

OTKyna CJIenyeT nlfnWlflJ-HenpepbIBHocTb onepaTopa B2 C nOCTOSlHHOii 01 2 + max(IIGnl) J1eMMa nOKa3aHa xEr2

ilYCTb H = Wio(rl ) x Wio(r2) Jth CBOHCTB onepaTopOB Bl If B2 cnenyroT

CBoiIcTBa onepaTopa B H ~ H 33J(aBaeMOro paBeHCTBOM

(Bu V)H

rne U (UlU2) v = (VlV2) E Wio(r l ) x Wio(r2 ) OH SlBJISleTCSI ClfJIbHO

MOHOTOHHblM If JIlfnWlfU-HenpepblBHbIM il03TOMY 1f3 o6meii TeopeMbl [7] 0 pa3pe-middot

WlfMOCTIf ypaBHeHlfH C TaKlfMIf onepaTopaMIf nonyqaeM cymecTBoBaHlfe o6paTHoro

OnepaTopa B-1 H ~ H ilepeiineM K nlfcKpeTlf3alJ1f1f ypaBHeHlfSl (9) ilYCTb Ni(x) i 12 mshy

KOHeQHO-3neMeHTHbIe 6a3lfCHbie 4lHKlJlflf onpeneneHHble Ha 3neMeHTax rpynn

7

r 1 H r 2 npHQeM 3TH CPYHKUHH 06pa3)lOT nOl1H)lO CHCTeMY l1HHeHHO He3aBHCHMbIX

311eMeHTOB B H TIpH611lf)KeHHOe pellIeHHe ypaBHeHHx (9) centm 6yneM HCKaTb B BHne

m A

centm L centiNi(X) X E r i=l

rne centi- HeH3BeCTHbie 3HaQeHHX CPYHKUHH centm B Y311aX 311eMeHTOB TIonCTaBJlXX B

ypaBHeHHe (9) BMeCTO U Bblpa)KeHHe)])lX centm H HCII0l1b3YX B KaQeCTBe v CPYHKUHH

Ni B COOTBeTCTBHH C MeTOnOM 6y6HoBa-ranepKHHa HMeeM CHCTeMY ypaBHeHHH

1p(xG IVTcentml)(i centjVTNj shy GT)middot VTNidS 1vTX V TNidS (12) f j=l fl

rne i 12 m 3necb )])lX yno6cTBa 3aIIHCH

p(x

3aMelaHUe 1 CHcTeMY (12) TaK)Ke MO)KHO IIOl1YQHTb IIYTeM CPOPMaJlbHOro nHcpcpepeHUHpoBashy

HHX CPYHKUHOHaJla

- VT(centm + X)1 2dS + 11IGT V TcentmI2 + (Gn - alG V Tcentml)2]dS f2

no K03CPCPHUHeHTaM centi i 12 m 3necb a - 3HaK KOMIIOHeHTbl Gn bull

Cl1enCTBHeM CBOHCTB onepaTopa B COrJIaCHO TeopHH MOHOTOHHblX oIIepaTopoB

[7] 5IBJl5leTC5I cyruecTBoBaHHe enHHCTBeHHoro pellIeHH5I centm CHCTeMbI (12) Hero

CXOnHMOCTb B IIpOCTpaHCTBe H K CPYHKUHH U KOTOP851 51BJI5leTC5I enHHCTBeHHbIM

pellIeHHeM ypaBHeHHSI (9) KpOMe Torn pellIeHHe ypaBHeHH5I (12) MO)KeT 6blTb

IIOJIyqeHo C IIOMOIqblO HTepaUHOHHoro rrpouecca

(k) -w vTx VTNidS + w p(x IVTcentm I)GT middot VTNidS (13)1 1

r 1 r

JIJI5I i 12 m 3necb rrapaMeTp w E (02at) k - HTepaUHoHHbIM HHneKC

k 01 I(mI pellIeHHSI CHCTeMbl (12) - centm H pellIeHH5I ypaBHeHH5I (9) - U

8

--------------------------------------- -------

2

3llCCb Hm JUmCHHOC nOllnpOCTpaHCTBO B H COCT05lmCC U3 )nCMCHTOB BHJla m L oiNi(X) mc oi HCKOTOPbIC K03QJqmuucHTbI )InsI OUCHKU nOrpCIlIHOCTU 1=1 06bI l iHO UCnOnb3YlOT HCpaBCHCTBO

inf U - v II JJ S II u I u II H (14)lIEHm

rllC Iu QJYHKUU5I-UHTcpnOflSIHT U3 H m 3auaBaCMa5I paBCHCTBOM I u( x) =

UiNi (x) 3llCCb Ui - 3Ha-lCHU5I QJYHKUUU U B Y3l1ax 3nCMCHTOB

3aMCTUM -ITO ccnu f 2 -nycToc MHO)KCCTBO TO nonyqacM nUHCHHbIH cnyqaH B

KOTOPOM ypaBHcHUC lIn51 0606mcHHoro pCrnCHU5I UMCCT BUll

jVrUVrVdS j(Gr - VrX) VrvdS UV E Wio(f) (15) [ r

Torlla U3 06mcH TCOpUU MCTOlla KOHC4HblX 3nCMCHTOB lIn51 3nnunTU4CCKUX 3ashy

lla4 [61 lln5I ypaBHcHu5I (15) CnCllYCT CllUHCTBCHHOCTb pCrnCHU5I U B npocTpaHcTBc

Wio(r) cymecTBoBaHHe ellHHCTBeHHoro perneHH5I COOTBcTcTByJOmeH CHCTeMbI

nonyqaeMoH MeTOllOM Ey6HoBa-ranepKHHa a TaK)Ke CXOJlHMOCTb npH6nH)KeHHOrO

perneHH5I cPm K 0606meHHoMY perneHHIO u TIPH 3TOM cnpaBelInHBa oueHKa aHashy

norH4HaSi (14)

Iullwl20 (1) (16)

iln51 HTepaUHoHHoro npouecca (13) peKOMeHJlyeTc5I Hcnonb30BaTb B Ka-leCTBe

Ha-lanbHbIX 3Ha4eHHH A(O) 0 w 1 H p-(x G IV A(O) I) = 1 Bo MHorHXPm rVm bull

cnyqa5lx 3TOro 6bIBaeT JlOCTaTO-lHO 4T06bI perneHHe CHCTeMbI OnpellenHnOCb 3a

OllHY HTepaUHIO

AlJanTHBHLIH aJIfOPHTM

TO-lHOCTb nOJlyqaeMoro npH6nH)KeHHoro perneHHSI ltPm KaK nOKa3bIBalOT HCshy

paBeHCTBa (14)(16) 3aBHCHT OT BbI60pa 6a3HCHbIX QJYHKUHH Nji 12 m TIOJl06paTb TaKUe 6a3HCHble QJYHKUHH -IT06bI cpa3y nonyqHTb He06xOllHMYlO T04shy

HOCTb KaK npaBHno He YllaeTCSI TImnOMY npHXOllHTCSI HaXOllHTb HeCKonbKO

npH6JU1)KeHHbIX perneHHH TaKHM 06pa30M 4T06bl Ka)KJlOe HOBoe np1l6J1H)KCHHOe

perneHHe 6bInO 6nH)Ke K T04HOMY perneHHIO ypaBHeHlISI 4eM npellbIllymee TIpH

3TOM nOCTpoeHHe 6a311CHbIX QJYHKUHH llJ151 Ka)KJlOrO HOBOro pCrneHH5I OCHOBbIBashy

eTCSI Ha aHanH3e nOBelleHH5I npeJlbIllymllx npH6nH)KeHHbIX peweHlIH C HCnOJlb30shy

BaHHeM HeKoToporo auanTlIBHoro anropHTMa B HaH60nee YHHBepcanbHblX auashy

I1THBHbIX anropHTMax 6a3HCHble QJYHKUHH llJ1S1 HOBO[O npH6m1)KeHHOrO perneHHSI

MOryT CTPOHTbCSI cpa3y JlByMSI cnoc06aMH OllHOBpeMeHHO UepBblii cnoco6 3ashy

KnlO4aeTCSI B TOM 4TO pa3MepbI KOHe4HbiX 3neMeHTOB YMeHbwalOTCSI B 3aBHcHMOshy

CTH OT anpHopHblX oueHOK H nOBelleHH5I npCllblJlymero npH6Ju-f)KeHHoro pellIeHH5I

9

ilOCKonbKY

p(U) - p(V) - ~(IUI IVI) (11) - IUlivl

TO nOnblHTerpaJIbHoe BbIpa)[(eHlfe BCerna HeOTplflJaTenbHO ilOnOMY nonyqaeM

To eCTb B2- ClfnbHO MOHOTOHHblH onepaTop

ilOKa)[(eM QTO B2 - nlfnWlflJ-HenpepbIBHbIH onepaTop HMeeM

B2vll wio(r2 ) = sup I(B2u IIwll wiO(r2) =1

= SUp I j[p(U)U - p(V)V] 7rWdSI ~ IIwll wiO(r2)=1 r 2

~ SUp I j[p(U)IU - VI + Ip(U) - p(V)IIVIJI7rw ldSI IIwll wJO(r2)=1 r2

ilOCKOnbKY lUI = 17r U 61 ~ IGnl1f TaK)[(e IVI ~ IGnl 1f3 (11) cnenyeT

Ip(O) p(V)1 lt 110nV11 il03TOMY 1f3 nocnenHero HepaBeHcTBa If 1f3 HepaBeHcTBa KOWIf-BYHSlKOBCKOro noshy

nyqaeM

OTKyna CJIenyeT nlfnWlflJ-HenpepbIBHocTb onepaTopa B2 C nOCTOSlHHOii 01 2 + max(IIGnl) J1eMMa nOKa3aHa xEr2

ilYCTb H = Wio(rl ) x Wio(r2) Jth CBOHCTB onepaTopOB Bl If B2 cnenyroT

CBoiIcTBa onepaTopa B H ~ H 33J(aBaeMOro paBeHCTBOM

(Bu V)H

rne U (UlU2) v = (VlV2) E Wio(r l ) x Wio(r2 ) OH SlBJISleTCSI ClfJIbHO

MOHOTOHHblM If JIlfnWlfU-HenpepblBHbIM il03TOMY 1f3 o6meii TeopeMbl [7] 0 pa3pe-middot

WlfMOCTIf ypaBHeHlfH C TaKlfMIf onepaTopaMIf nonyqaeM cymecTBoBaHlfe o6paTHoro

OnepaTopa B-1 H ~ H ilepeiineM K nlfcKpeTlf3alJ1f1f ypaBHeHlfSl (9) ilYCTb Ni(x) i 12 mshy

KOHeQHO-3neMeHTHbIe 6a3lfCHbie 4lHKlJlflf onpeneneHHble Ha 3neMeHTax rpynn

7

r 1 H r 2 npHQeM 3TH CPYHKUHH 06pa3)lOT nOl1H)lO CHCTeMY l1HHeHHO He3aBHCHMbIX

311eMeHTOB B H TIpH611lf)KeHHOe pellIeHHe ypaBHeHHx (9) centm 6yneM HCKaTb B BHne

m A

centm L centiNi(X) X E r i=l

rne centi- HeH3BeCTHbie 3HaQeHHX CPYHKUHH centm B Y311aX 311eMeHTOB TIonCTaBJlXX B

ypaBHeHHe (9) BMeCTO U Bblpa)KeHHe)])lX centm H HCII0l1b3YX B KaQeCTBe v CPYHKUHH

Ni B COOTBeTCTBHH C MeTOnOM 6y6HoBa-ranepKHHa HMeeM CHCTeMY ypaBHeHHH

1p(xG IVTcentml)(i centjVTNj shy GT)middot VTNidS 1vTX V TNidS (12) f j=l fl

rne i 12 m 3necb )])lX yno6cTBa 3aIIHCH

p(x

3aMelaHUe 1 CHcTeMY (12) TaK)Ke MO)KHO IIOl1YQHTb IIYTeM CPOPMaJlbHOro nHcpcpepeHUHpoBashy

HHX CPYHKUHOHaJla

- VT(centm + X)1 2dS + 11IGT V TcentmI2 + (Gn - alG V Tcentml)2]dS f2

no K03CPCPHUHeHTaM centi i 12 m 3necb a - 3HaK KOMIIOHeHTbl Gn bull

Cl1enCTBHeM CBOHCTB onepaTopa B COrJIaCHO TeopHH MOHOTOHHblX oIIepaTopoB

[7] 5IBJl5leTC5I cyruecTBoBaHHe enHHCTBeHHoro pellIeHH5I centm CHCTeMbI (12) Hero

CXOnHMOCTb B IIpOCTpaHCTBe H K CPYHKUHH U KOTOP851 51BJI5leTC5I enHHCTBeHHbIM

pellIeHHeM ypaBHeHHSI (9) KpOMe Torn pellIeHHe ypaBHeHH5I (12) MO)KeT 6blTb

IIOJIyqeHo C IIOMOIqblO HTepaUHOHHoro rrpouecca

(k) -w vTx VTNidS + w p(x IVTcentm I)GT middot VTNidS (13)1 1

r 1 r

JIJI5I i 12 m 3necb rrapaMeTp w E (02at) k - HTepaUHoHHbIM HHneKC

k 01 I(mI pellIeHHSI CHCTeMbl (12) - centm H pellIeHH5I ypaBHeHH5I (9) - U

8

--------------------------------------- -------

2

3llCCb Hm JUmCHHOC nOllnpOCTpaHCTBO B H COCT05lmCC U3 )nCMCHTOB BHJla m L oiNi(X) mc oi HCKOTOPbIC K03QJqmuucHTbI )InsI OUCHKU nOrpCIlIHOCTU 1=1 06bI l iHO UCnOnb3YlOT HCpaBCHCTBO

inf U - v II JJ S II u I u II H (14)lIEHm

rllC Iu QJYHKUU5I-UHTcpnOflSIHT U3 H m 3auaBaCMa5I paBCHCTBOM I u( x) =

UiNi (x) 3llCCb Ui - 3Ha-lCHU5I QJYHKUUU U B Y3l1ax 3nCMCHTOB

3aMCTUM -ITO ccnu f 2 -nycToc MHO)KCCTBO TO nonyqacM nUHCHHbIH cnyqaH B

KOTOPOM ypaBHcHUC lIn51 0606mcHHoro pCrnCHU5I UMCCT BUll

jVrUVrVdS j(Gr - VrX) VrvdS UV E Wio(f) (15) [ r

Torlla U3 06mcH TCOpUU MCTOlla KOHC4HblX 3nCMCHTOB lIn51 3nnunTU4CCKUX 3ashy

lla4 [61 lln5I ypaBHcHu5I (15) CnCllYCT CllUHCTBCHHOCTb pCrnCHU5I U B npocTpaHcTBc

Wio(r) cymecTBoBaHHe ellHHCTBeHHoro perneHH5I COOTBcTcTByJOmeH CHCTeMbI

nonyqaeMoH MeTOllOM Ey6HoBa-ranepKHHa a TaK)Ke CXOJlHMOCTb npH6nH)KeHHOrO

perneHH5I cPm K 0606meHHoMY perneHHIO u TIPH 3TOM cnpaBelInHBa oueHKa aHashy

norH4HaSi (14)

Iullwl20 (1) (16)

iln51 HTepaUHoHHoro npouecca (13) peKOMeHJlyeTc5I Hcnonb30BaTb B Ka-leCTBe

Ha-lanbHbIX 3Ha4eHHH A(O) 0 w 1 H p-(x G IV A(O) I) = 1 Bo MHorHXPm rVm bull

cnyqa5lx 3TOro 6bIBaeT JlOCTaTO-lHO 4T06bI perneHHe CHCTeMbI OnpellenHnOCb 3a

OllHY HTepaUHIO

AlJanTHBHLIH aJIfOPHTM

TO-lHOCTb nOJlyqaeMoro npH6nH)KeHHoro perneHHSI ltPm KaK nOKa3bIBalOT HCshy

paBeHCTBa (14)(16) 3aBHCHT OT BbI60pa 6a3HCHbIX QJYHKUHH Nji 12 m TIOJl06paTb TaKUe 6a3HCHble QJYHKUHH -IT06bI cpa3y nonyqHTb He06xOllHMYlO T04shy

HOCTb KaK npaBHno He YllaeTCSI TImnOMY npHXOllHTCSI HaXOllHTb HeCKonbKO

npH6JU1)KeHHbIX perneHHH TaKHM 06pa30M 4T06bl Ka)KJlOe HOBoe np1l6J1H)KCHHOe

perneHHe 6bInO 6nH)Ke K T04HOMY perneHHIO ypaBHeHlISI 4eM npellbIllymee TIpH

3TOM nOCTpoeHHe 6a311CHbIX QJYHKUHH llJ151 Ka)KJlOrO HOBOro pCrneHH5I OCHOBbIBashy

eTCSI Ha aHanH3e nOBelleHH5I npeJlbIllymllx npH6nH)KeHHbIX peweHlIH C HCnOJlb30shy

BaHHeM HeKoToporo auanTlIBHoro anropHTMa B HaH60nee YHHBepcanbHblX auashy

I1THBHbIX anropHTMax 6a3HCHble QJYHKUHH llJ1S1 HOBO[O npH6m1)KeHHOrO perneHHSI

MOryT CTPOHTbCSI cpa3y JlByMSI cnoc06aMH OllHOBpeMeHHO UepBblii cnoco6 3ashy

KnlO4aeTCSI B TOM 4TO pa3MepbI KOHe4HbiX 3neMeHTOB YMeHbwalOTCSI B 3aBHcHMOshy

CTH OT anpHopHblX oueHOK H nOBelleHH5I npCllblJlymero npH6Ju-f)KeHHoro pellIeHH5I

9

r 1 H r 2 npHQeM 3TH CPYHKUHH 06pa3)lOT nOl1H)lO CHCTeMY l1HHeHHO He3aBHCHMbIX

311eMeHTOB B H TIpH611lf)KeHHOe pellIeHHe ypaBHeHHx (9) centm 6yneM HCKaTb B BHne

m A

centm L centiNi(X) X E r i=l

rne centi- HeH3BeCTHbie 3HaQeHHX CPYHKUHH centm B Y311aX 311eMeHTOB TIonCTaBJlXX B

ypaBHeHHe (9) BMeCTO U Bblpa)KeHHe)])lX centm H HCII0l1b3YX B KaQeCTBe v CPYHKUHH

Ni B COOTBeTCTBHH C MeTOnOM 6y6HoBa-ranepKHHa HMeeM CHCTeMY ypaBHeHHH

1p(xG IVTcentml)(i centjVTNj shy GT)middot VTNidS 1vTX V TNidS (12) f j=l fl

rne i 12 m 3necb )])lX yno6cTBa 3aIIHCH

p(x

3aMelaHUe 1 CHcTeMY (12) TaK)Ke MO)KHO IIOl1YQHTb IIYTeM CPOPMaJlbHOro nHcpcpepeHUHpoBashy

HHX CPYHKUHOHaJla

- VT(centm + X)1 2dS + 11IGT V TcentmI2 + (Gn - alG V Tcentml)2]dS f2

no K03CPCPHUHeHTaM centi i 12 m 3necb a - 3HaK KOMIIOHeHTbl Gn bull

Cl1enCTBHeM CBOHCTB onepaTopa B COrJIaCHO TeopHH MOHOTOHHblX oIIepaTopoB

[7] 5IBJl5leTC5I cyruecTBoBaHHe enHHCTBeHHoro pellIeHH5I centm CHCTeMbI (12) Hero

CXOnHMOCTb B IIpOCTpaHCTBe H K CPYHKUHH U KOTOP851 51BJI5leTC5I enHHCTBeHHbIM

pellIeHHeM ypaBHeHHSI (9) KpOMe Torn pellIeHHe ypaBHeHH5I (12) MO)KeT 6blTb

IIOJIyqeHo C IIOMOIqblO HTepaUHOHHoro rrpouecca

(k) -w vTx VTNidS + w p(x IVTcentm I)GT middot VTNidS (13)1 1

r 1 r

JIJI5I i 12 m 3necb rrapaMeTp w E (02at) k - HTepaUHoHHbIM HHneKC

k 01 I(mI pellIeHHSI CHCTeMbl (12) - centm H pellIeHH5I ypaBHeHH5I (9) - U

8

--------------------------------------- -------

2

3llCCb Hm JUmCHHOC nOllnpOCTpaHCTBO B H COCT05lmCC U3 )nCMCHTOB BHJla m L oiNi(X) mc oi HCKOTOPbIC K03QJqmuucHTbI )InsI OUCHKU nOrpCIlIHOCTU 1=1 06bI l iHO UCnOnb3YlOT HCpaBCHCTBO

inf U - v II JJ S II u I u II H (14)lIEHm

rllC Iu QJYHKUU5I-UHTcpnOflSIHT U3 H m 3auaBaCMa5I paBCHCTBOM I u( x) =

UiNi (x) 3llCCb Ui - 3Ha-lCHU5I QJYHKUUU U B Y3l1ax 3nCMCHTOB

3aMCTUM -ITO ccnu f 2 -nycToc MHO)KCCTBO TO nonyqacM nUHCHHbIH cnyqaH B

KOTOPOM ypaBHcHUC lIn51 0606mcHHoro pCrnCHU5I UMCCT BUll

jVrUVrVdS j(Gr - VrX) VrvdS UV E Wio(f) (15) [ r

Torlla U3 06mcH TCOpUU MCTOlla KOHC4HblX 3nCMCHTOB lIn51 3nnunTU4CCKUX 3ashy

lla4 [61 lln5I ypaBHcHu5I (15) CnCllYCT CllUHCTBCHHOCTb pCrnCHU5I U B npocTpaHcTBc

Wio(r) cymecTBoBaHHe ellHHCTBeHHoro perneHH5I COOTBcTcTByJOmeH CHCTeMbI

nonyqaeMoH MeTOllOM Ey6HoBa-ranepKHHa a TaK)Ke CXOJlHMOCTb npH6nH)KeHHOrO

perneHH5I cPm K 0606meHHoMY perneHHIO u TIPH 3TOM cnpaBelInHBa oueHKa aHashy

norH4HaSi (14)

Iullwl20 (1) (16)

iln51 HTepaUHoHHoro npouecca (13) peKOMeHJlyeTc5I Hcnonb30BaTb B Ka-leCTBe

Ha-lanbHbIX 3Ha4eHHH A(O) 0 w 1 H p-(x G IV A(O) I) = 1 Bo MHorHXPm rVm bull

cnyqa5lx 3TOro 6bIBaeT JlOCTaTO-lHO 4T06bI perneHHe CHCTeMbI OnpellenHnOCb 3a

OllHY HTepaUHIO

AlJanTHBHLIH aJIfOPHTM

TO-lHOCTb nOJlyqaeMoro npH6nH)KeHHoro perneHHSI ltPm KaK nOKa3bIBalOT HCshy

paBeHCTBa (14)(16) 3aBHCHT OT BbI60pa 6a3HCHbIX QJYHKUHH Nji 12 m TIOJl06paTb TaKUe 6a3HCHble QJYHKUHH -IT06bI cpa3y nonyqHTb He06xOllHMYlO T04shy

HOCTb KaK npaBHno He YllaeTCSI TImnOMY npHXOllHTCSI HaXOllHTb HeCKonbKO

npH6JU1)KeHHbIX perneHHH TaKHM 06pa30M 4T06bl Ka)KJlOe HOBoe np1l6J1H)KCHHOe

perneHHe 6bInO 6nH)Ke K T04HOMY perneHHIO ypaBHeHlISI 4eM npellbIllymee TIpH

3TOM nOCTpoeHHe 6a311CHbIX QJYHKUHH llJ151 Ka)KJlOrO HOBOro pCrneHH5I OCHOBbIBashy

eTCSI Ha aHanH3e nOBelleHH5I npeJlbIllymllx npH6nH)KeHHbIX peweHlIH C HCnOJlb30shy

BaHHeM HeKoToporo auanTlIBHoro anropHTMa B HaH60nee YHHBepcanbHblX auashy

I1THBHbIX anropHTMax 6a3HCHble QJYHKUHH llJ1S1 HOBO[O npH6m1)KeHHOrO perneHHSI

MOryT CTPOHTbCSI cpa3y JlByMSI cnoc06aMH OllHOBpeMeHHO UepBblii cnoco6 3ashy

KnlO4aeTCSI B TOM 4TO pa3MepbI KOHe4HbiX 3neMeHTOB YMeHbwalOTCSI B 3aBHcHMOshy

CTH OT anpHopHblX oueHOK H nOBelleHH5I npCllblJlymero npH6Ju-f)KeHHoro pellIeHH5I

9

--------------------------------------- -------

2

3llCCb Hm JUmCHHOC nOllnpOCTpaHCTBO B H COCT05lmCC U3 )nCMCHTOB BHJla m L oiNi(X) mc oi HCKOTOPbIC K03QJqmuucHTbI )InsI OUCHKU nOrpCIlIHOCTU 1=1 06bI l iHO UCnOnb3YlOT HCpaBCHCTBO

inf U - v II JJ S II u I u II H (14)lIEHm

rllC Iu QJYHKUU5I-UHTcpnOflSIHT U3 H m 3auaBaCMa5I paBCHCTBOM I u( x) =

UiNi (x) 3llCCb Ui - 3Ha-lCHU5I QJYHKUUU U B Y3l1ax 3nCMCHTOB

3aMCTUM -ITO ccnu f 2 -nycToc MHO)KCCTBO TO nonyqacM nUHCHHbIH cnyqaH B

KOTOPOM ypaBHcHUC lIn51 0606mcHHoro pCrnCHU5I UMCCT BUll

jVrUVrVdS j(Gr - VrX) VrvdS UV E Wio(f) (15) [ r

Torlla U3 06mcH TCOpUU MCTOlla KOHC4HblX 3nCMCHTOB lIn51 3nnunTU4CCKUX 3ashy

lla4 [61 lln5I ypaBHcHu5I (15) CnCllYCT CllUHCTBCHHOCTb pCrnCHU5I U B npocTpaHcTBc

Wio(r) cymecTBoBaHHe ellHHCTBeHHoro perneHH5I COOTBcTcTByJOmeH CHCTeMbI

nonyqaeMoH MeTOllOM Ey6HoBa-ranepKHHa a TaK)Ke CXOJlHMOCTb npH6nH)KeHHOrO

perneHH5I cPm K 0606meHHoMY perneHHIO u TIPH 3TOM cnpaBelInHBa oueHKa aHashy

norH4HaSi (14)

Iullwl20 (1) (16)

iln51 HTepaUHoHHoro npouecca (13) peKOMeHJlyeTc5I Hcnonb30BaTb B Ka-leCTBe

Ha-lanbHbIX 3Ha4eHHH A(O) 0 w 1 H p-(x G IV A(O) I) = 1 Bo MHorHXPm rVm bull

cnyqa5lx 3TOro 6bIBaeT JlOCTaTO-lHO 4T06bI perneHHe CHCTeMbI OnpellenHnOCb 3a

OllHY HTepaUHIO

AlJanTHBHLIH aJIfOPHTM

TO-lHOCTb nOJlyqaeMoro npH6nH)KeHHoro perneHHSI ltPm KaK nOKa3bIBalOT HCshy

paBeHCTBa (14)(16) 3aBHCHT OT BbI60pa 6a3HCHbIX QJYHKUHH Nji 12 m TIOJl06paTb TaKUe 6a3HCHble QJYHKUHH -IT06bI cpa3y nonyqHTb He06xOllHMYlO T04shy

HOCTb KaK npaBHno He YllaeTCSI TImnOMY npHXOllHTCSI HaXOllHTb HeCKonbKO

npH6JU1)KeHHbIX perneHHH TaKHM 06pa30M 4T06bl Ka)KJlOe HOBoe np1l6J1H)KCHHOe

perneHHe 6bInO 6nH)Ke K T04HOMY perneHHIO ypaBHeHlISI 4eM npellbIllymee TIpH

3TOM nOCTpoeHHe 6a311CHbIX QJYHKUHH llJ151 Ka)KJlOrO HOBOro pCrneHH5I OCHOBbIBashy

eTCSI Ha aHanH3e nOBelleHH5I npeJlbIllymllx npH6nH)KeHHbIX peweHlIH C HCnOJlb30shy

BaHHeM HeKoToporo auanTlIBHoro anropHTMa B HaH60nee YHHBepcanbHblX auashy

I1THBHbIX anropHTMax 6a3HCHble QJYHKUHH llJ1S1 HOBO[O npH6m1)KeHHOrO perneHHSI

MOryT CTPOHTbCSI cpa3y JlByMSI cnoc06aMH OllHOBpeMeHHO UepBblii cnoco6 3ashy

KnlO4aeTCSI B TOM 4TO pa3MepbI KOHe4HbiX 3neMeHTOB YMeHbwalOTCSI B 3aBHcHMOshy

CTH OT anpHopHblX oueHOK H nOBelleHH5I npCllblJlymero npH6Ju-f)KeHHoro pellIeHH5I

9

a CTeneHb MHoroqneHOB C nOMombIO KOTOPbIX npenCTaBJIeHbI COOTBeTCTByIOmHe

6a3HCHbIe ltPYHKUHH OCTaeTC51 HeH3MeHHOH Bo BTOPOM cnoc06e YBeJmqHBaeTC51

CTeneHb MHoroqneHOB B 3aBHCHMOCTH 01 anpHopHbIX oueHOK H nOBeneHH51 pashy

Hee Hai1ueHHoro npH6JIH)KeHHOro pellleHH51 a pa3MepbI KOHeqHblX meMeHTOB Ha

KOTOPblX OHH onpeneJIeHbI He MeH51IOTC51 HCnOJIb30BaHHe 3THX nBYX cnoc060B

B ananTHBHOM anropHTMe nOMoraeT CTPOHTb 6bICTpOCXOll5lmHeCSI npH6nH)I(eHHbIe

pellleHH51 npH OTHOCHTeJIbHO He60JIblllHX BblqHCJIHTenbHblX 3aTpaTax

KpOMe TaKOrO 06mero nonXona OTMeTHM KOHKpeTHble oc06eHHOCTH 3anaqH

onpeneJleHH51 npH6JIH)I(eHHoro pellleHHSI ypaBHeHHSI (9) C HeKOTOpOH 3anaHHOH

TOqHOCTbIO

Bo-nepBblx 3anaqa onpeneJIeHH51 ltPYHKUHH U 5lBJI51eTCSI qaCTbIO 06meH 3auaqH

BHUa (1)-(4) H n03ToMY anropHTM onpeneJIeHH51 npH6JIH)I(eHHOrO pellleHH51 centm nOn)l(eH cornaCOBblBaTbCSI co cTpaTerHeH pellleHHSI 06meH 3auaQH BO-BTOPbIX H3shy

BecTHO TOQHOe 3HaQeHHe rpanHeHTa HCKOMOH ltPYHKUHH - BeKTop G H n03ToMY ero

MO)l(HO HCnOJIb30BaTbllJ15l oueHKH pa3HocTH TOQHOro H npH6nH)I(eHHoro pellleHHSI

Ilpe)KUe QeM CltP0PMYJIHpOBaTb pa3pa60TaHHbIH aBTopaMH ananTHBHbIH anroshy

pHTM BBeneM He06xonHMbIe 0603HaQeHHSI H npHBeneM HeKoTopbIe yrBep)I(UeHH51

B COOTBeTCTBHH C 06meH 3anaQeH (1 )-(4) 6yneM npennonaraTb QTO 06JIaCTb

0 = 0 U 0 1 pa36HTa CTaHUapTHbIM 06pa30M [6] Ha BbITIYKJIbie lllecTHrpaHHHKH H

B COOTBeTCTBHH C 3THM pa36HeHHeM rpaHHua r COCTaBJIeHa H3 BbITIYKJIbIX QeTbIshy

pexyrOJIbHHKOB

Ka)l()~oH TOQKe x = (Xl x2 X3) QeTblpexyroJIbHHKa rl C r C HOMepOM l 6yneM

CTaBHTb B COOTBeTCTBHe TOQKY ~ = (66) KBanpaTa E2 [-11) x [-11] C

nOMOmlIO npe06pa30BaHH51

4

Xi (14) L xik bull (1 + 6k~d(1 + 6k6) i = 123 (17) k=l

rne KoopnHHaTa x(k) (XIk X2k X3k) yrJIOBOH BepIllHHbI C HOMepOM k QeTbIshy

pexyrOJIbHHKa rl cooTBeTcTByeT BeplllHHe KBanpaTa E2 C KoopnHHaTOH ~(k) =

(6k 6J TaKHe npe06pa30BaHH51 KaK (17) Ha3bIBaIOT H30napaMeTpHQeCKJfMH

HX CBOHCTBa paccMoTpeHbI B [69] BbITIYKJIOCTb QeTbIpexyroJIbHHKa II 5lBJI5leTCH

nOCTaTOQHbIM YCJIOBHeM llJ15l onpeneJIeHH51 06paTHoro npe06pa30BaHH51 KpOMe

Toro 6yneT HeBblpO)I(UeHHOH MaTpHua

axshyJ- _2 - 123 (18) - O~i ~J=

3 shyrne oXjo6 (j = 123) - KOMnOHeHThI BeKTopa fi x T2 fj = E ikOXkO~j

k=l j 12 Korna x E rl Il03TOMYllJ15l nOCTaTOQHO rnallKoH $YHKUHH U MO)l(HO

onpeneJIHTb Vu(x) J-IV~U(~) 3lleCb Ve - rpanHeHT no nepeMeHHblM 666

10

)JJ151 auarrTHBHOrO anropHTMa 6Y)JeM HCrrOJlh30BaTh qeThlpe THrra qeThlpeXshy

yronhHblX )J1eMeHTOB 3TH meMeHThl 6Y)JeM 3a)JaBaTh CPYHKllH5IMH CPOPMhI (6a3HCshy

HhlMH CPYHKllH5IMH orrpeueneHHhlMH Ha OuHOM meMeHTe) If Y3J1aMH Ha KBa)JpaTe

E2 KOTOphle npe06pa30BaHHeM 06paTHhIM K (17) CTaB5ITC5I B COOTBeTCTBHe TOqshy

KaM qeThlpexyronhHHKa rpaHHllhI r TIepBhIM THrrOM meMeHTOB 6yueM Ha3hlBaTb narpamKeBbl meMeHThl C 6 Y3J1aMH

HMelOwHe CPYHKllHH CPOPMbl B rrepeMeHHblX (6 6) E E2 cneuYIOwero BMa

N(12)(() = Q(1k) (31 bull cfh k I ~ (3If32 I 1 12 k = 1 6

131~2

me 161 31 + 32 6162 - HeOTpHllaTenbHhle llenble qHCna a K03cpcpHllHeHThl

Q~~~ 161 2 Jlj15I KalKJlOrO Y3J1a C HOMepoM k onpeueJ15IIOTC5I KaK pemeHHe

CHCTeMhl N~12) (e(j)) 6(e(k) e(j)) j = 1 6 3uech 6(e(kl e(j)) = 1 ecnH

6k = 6j H 6k 6j B rrpoTHBHOM cnyqae 6(e(k)e(j)) = O PacnOnO)KeHHe

Y3J10B rrOKa3aHO Ha pHC 1 TIPH TaKOM 3auaHHH Y3J10B CHCTeMa Jlj15I onpeueshy

neHH5I 6a3HcHbiX CPYHKllHH ouH03HaqHO pa3pemHMa 3TOT THfl 3neMeHTOB 6yueT

Hcnonh30BaTbC5I JlJ15I BcnOMOraTeJ1hHbIX BblqHCneHHH a He Jlj15I nOCTpOeHH5I npHshy

6J11DKeHHhlX pellleHHH

BTOPbIM THnOM 3neMeHTOB 6yueM Ha3hIBaTh 3neMeHTbI Y3J1bl KOTOPhlX rronyqashy

IOTC5I rrpH upo6neHHH qeTblpexyroJ1bHI1Ka Ha qeTblpe KalKJlblH H3 KOTOPbIX 6yueT

narpaH)KeBhlM 3neMeHTOM C 9 Y3J1aMH cooTBeTcTsYIOWHMH y3J1aM KBa)JpaTa E2

KOTophIe H306p3)KeHhl Ha pHC 2 TaKHe 3neMeHThi 6yuYT HMeTh B cyMMe 25 pa3shy

nHqHbIX Y3nOB H nOKanhHO orrpeueneHHhle CPYHKllHH cpOPMhl TIpHBeueM CPYHKllHH

CPOpMbI HarrpHMep Jlj15I KBaupaTa 0 6 6 1

(2k)(2(~ Q3132 11

1J1~2

(2k) (2( 1)2(2 - 1)2+Q21 11 I lt2

(2 k) rue k HOMep y3na rrpHHaJl)1e)Kawero 3TOMY KBaupaTy K03cpCPHllHeHThl Q31P2

onpeueJ15IIOTC5I KaK pemeHIfe CHCTeMbl N~22)(e(j)) = 6(e(k)e(j)) JlJ15I Y3JIOB C HOshy

MepaMH k j H3 3Toro )Ke KBMpaTa AHanOrHqHO rronyqaeM 6a311CHhie CPYHKllHH

JlJ15I upyrnx Tpex KBaupaTOB H3 E2bull CHCTeMhl JlJ15I orrpe)JeneHH5I 6a3HCHhlX CPYHKshy

IlI1H OJJH03HaqHO pa3pemHMhI 3TOT THrr 3neMeHTOB 6YJJeM Hcnonb30BaTh Jlj15I

HaXOlKJleHH5I rrpH6nH)KeHHOrO pemeHH5I CHCTeMhl (12) 0 TOM KaK HCKnlOql1Th

H3 CHCTeMhI (12) HeH3BecTHhle 3HaqeHH5I COOTBeTCTBYIOlllHe BHYTpeHHHM Y3J1aM

3THX 3neMeHTOB 6YJJeT nOJJpo6HO CKa3aHO B n05ICHeHHH K 5-MY mary a)JarrTHBHoro

anropHTMa

II

)JJ151 auarrTHBHOrO anropHTMa 6Y)JeM HCrrOJlh30BaTh qeThlpe THrra qeThlpeXshy

yronhHblX )J1eMeHTOB 3TH meMeHThl 6Y)JeM 3a)JaBaTh CPYHKllH5IMH CPOPMhI (6a3HCshy

HhlMH CPYHKllH5IMH orrpeueneHHhlMH Ha OuHOM meMeHTe) If Y3J1aMH Ha KBa)JpaTe

E2 KOTOphle npe06pa30BaHHeM 06paTHhIM K (17) CTaB5ITC5I B COOTBeTCTBHe TOqshy

KaM qeThlpexyronhHHKa rpaHHllhI r TIepBhIM THrrOM meMeHTOB 6yueM Ha3hlBaTb narpamKeBbl meMeHThl C 6 Y3J1aMH

HMelOwHe CPYHKllHH CPOPMbl B rrepeMeHHblX (6 6) E E2 cneuYIOwero BMa

N(12)(() = Q(1k) (31 bull cfh k I ~ (3If32 I 1 12 k = 1 6

131~2

me 161 31 + 32 6162 - HeOTpHllaTenbHhle llenble qHCna a K03cpcpHllHeHThl

Q~~~ 161 2 Jlj15I KalKJlOrO Y3J1a C HOMepoM k onpeueJ15IIOTC5I KaK pemeHHe

CHCTeMhl N~12) (e(j)) 6(e(k) e(j)) j = 1 6 3uech 6(e(kl e(j)) = 1 ecnH

6k = 6j H 6k 6j B rrpoTHBHOM cnyqae 6(e(k)e(j)) = O PacnOnO)KeHHe

Y3J10B rrOKa3aHO Ha pHC 1 TIPH TaKOM 3auaHHH Y3J10B CHCTeMa Jlj15I onpeueshy

neHH5I 6a3HcHbiX CPYHKllHH ouH03HaqHO pa3pemHMa 3TOT THfl 3neMeHTOB 6yueT

Hcnonh30BaTbC5I JlJ15I BcnOMOraTeJ1hHbIX BblqHCneHHH a He Jlj15I nOCTpOeHH5I npHshy

6J11DKeHHhlX pellleHHH

BTOPbIM THnOM 3neMeHTOB 6yueM Ha3hIBaTh 3neMeHTbI Y3J1bl KOTOPhlX rronyqashy

IOTC5I rrpH upo6neHHH qeTblpexyroJ1bHI1Ka Ha qeTblpe KalKJlblH H3 KOTOPbIX 6yueT

narpaH)KeBhlM 3neMeHTOM C 9 Y3J1aMH cooTBeTcTsYIOWHMH y3J1aM KBa)JpaTa E2

KOTophIe H306p3)KeHhl Ha pHC 2 TaKHe 3neMeHThi 6yuYT HMeTh B cyMMe 25 pa3shy

nHqHbIX Y3nOB H nOKanhHO orrpeueneHHhle CPYHKllHH cpOPMhl TIpHBeueM CPYHKllHH

CPOpMbI HarrpHMep Jlj15I KBaupaTa 0 6 6 1

(2k)(2(~ Q3132 11

1J1~2

(2k) (2( 1)2(2 - 1)2+Q21 11 I lt2

(2 k) rue k HOMep y3na rrpHHaJl)1e)Kawero 3TOMY KBaupaTy K03cpCPHllHeHThl Q31P2

onpeueJ15IIOTC5I KaK pemeHIfe CHCTeMbl N~22)(e(j)) = 6(e(k)e(j)) JlJ15I Y3JIOB C HOshy

MepaMH k j H3 3Toro )Ke KBMpaTa AHanOrHqHO rronyqaeM 6a311CHhie CPYHKllHH

JlJ15I upyrnx Tpex KBaupaTOB H3 E2bull CHCTeMhl JlJ15I orrpe)JeneHH5I 6a3HCHhlX CPYHKshy

IlI1H OJJH03HaqHO pa3pemHMhI 3TOT THrr 3neMeHTOB 6YJJeM Hcnonb30BaTh Jlj15I

HaXOlKJleHH5I rrpH6nH)KeHHOrO pemeHH5I CHCTeMhl (12) 0 TOM KaK HCKnlOql1Th

H3 CHCTeMhI (12) HeH3BecTHhle 3HaqeHH5I COOTBeTCTBYIOlllHe BHYTpeHHHM Y3J1aM

3THX 3neMeHTOB 6YJJeT nOJJpo6HO CKa3aHO B n05ICHeHHH K 5-MY mary a)JarrTHBHoro

anropHTMa

II

TpeThliM TlinOM 3JIeMeHTOB 6Y)1eM Ha3hIBaTh CepeH)1linOBhI 3JIeMeHThI C 17 Y3shy

JIaMli lt1JYHKlllili CPOPMhI TaKIiX 3JIeMeHTOB B nepeMeHHhlx (6 6) E E2 IiMelOT

BIi)]

N (34)(() _ (3k) (f3t (32 + (3k) (4 ( + (3k) ( (4k I - LJ af3tf32 11 12 a41 11 12 a14 1112 k 1 17

1f314

r)1e K03cpcplilllieHThI a~ )lJISI Ka)KllOrO Y3JIa C HOMepOM k Onpe)1eJISlIOTCSI nYTeM

perneHIiSl COOTBeTCTByromeH CIiCTeMhl aHaJIOrnqHO TOMY KaK 6h1JIO OnliCaHO B

CJIyqae 3JIeMeHTOB nepBOrO Tlina PaCnOJIOJKeHlie Y3JIOB nOKa3aHO Ha pliC 2 3TOT Tlin imeMeHTOB TaKJKe 6Y)1eM IiCnOJIh30BaTh )1JISI nocTpoeHIiSl npIi6JIIiJKeHHOrO

perneHIiSl CIiCTeMhI (12) HaKoHell qeTBepThlM TlinOM 3JIeMeHTOB 6Y)1eM Ha3hIBaTh JIarpaHJKeBhl 3JIeMeHThI

C 15 Y3JIaMli KOTophle nOJIyqeHhI )106aBJIeHlieM )1ByX )10nOJIHIiTeJIhHhIX Y3JIOB K Y3shy

JIaM Ky6aTypHoH cpOpMyJIhl TOqHOH )1JISI MHoroqneHOB )10 nSlTOH CTeneHIi BKJIlOqlishy

TeJIhHO [8] PacnOJIOJKeHlie Y3JIOB nOKa3aHO Ha pliC 1 lt1JYHKlllili CPOPMhl IiMelOT

BIi)]

N~44)(e) = L a~~ efl e~2 k = 1 15 1f314

r)1e K03cpcplilllieHThi a~~ 131S 4 )1JISI KaJK)1Oro Y3JIa C HOMepoM k onpe)1eshy

JISIIOTCSI KaK perneHlie COOTBeTCTBYlOmeH CIiCTeMhl N~44)(e(j)) = 8(e(k) e(j)) j 1 15 )JjISI 3a)1aHHhIX Y3JIOB 3Ta clicreMa 0)1H03HaqHO pa3pernliMa 3TOT

Tlin 3JIeMeHTOB 6Y)1eM IiCnOJIh30BaTh )1JISI BcnOMOraTeJlhHhlX BhlqIiCJIeHIiH a He )1JISI

nOCTpoeHIiSl npli6JIIiJKeHHoro perneHIiSl CIiCTeMhl (12) C lleJIhlO onlicaHIiSl CBOHCTB Ka)KllOro Tlina 3JIeMeHTOB BBe)1eM 0603HaqeHliSl

)1JISI CPYHKllIiH-IiHTepnOJISlHTOB onpe)1eJIeHHhIX Ha qeThlpexyroJIhHIiKe 1 C r mk (k)()I kn U L UiNi n (x) i=l

3)1eCh HH)1eKC k 3a)1aeT Tlin 3JIeMeHTa ffik - qliCJIO Y3JIOB 3JIeMeHTa THna k Ui

Ni(kn) - 3HaqeHlie CPYHKllliH U B i-M Y3JIe t Hi-51 6a3liCHaSI CPYHKllliSl Ha 3JIeMeHTe

COOTBeTCTBeHHO

MeTO)1hI IiCCJIe)10BaHHSI annpOKcHMaUIiOHHhIX CBOHCTB KOHeqHhIX 3JIeMeHTOB

1i3BeCTHhI H npIiBe)1eHhl HanpliMep B [96] B pe3YJIhTaTe IiX npHMeHeHIiSl npli

HeKOTophlX OrpaHliqeHHSlX Ha qeThIpexyrOJIhHIiK 1 c r B npe)1nOJIOJKeHIiIi qTO

CPYHKUIiSl U E W+l (1) nOJIyqalOTcSI oueHKIi ornli6KH HHTepnOJISlllli1i Bli)]a

Iu I(kn)ulwnn) Cmiddot hn L JIDiuI 2dS12

11I=n+l Tl

12

r)le I IHri hd - nOJIYHopMa B npOCTpaHCTe Wl Crl) TO eCTb )lJ1S1 QJYHKllHH

V E W1Crl) Ivl~vnTd = (VTv V TV)(L2(Yl)P h - )lHaMeTp meMeHTa C - noshy

JIQ)KHTeJIbHaSI KOHCTaHTa He3aBHCHMaSl OT h H u DiU = 8 ITJI11j8xi18x8xjJ TJ = (TJl TJ2 TJ3) ITJI = TJl + 2 + 3 3)leCb TJi (i = 123) - HeoTpHllaTeJIbHble

ueJIble lIHCJIa TIOCKOJIbKY TaKHe oueHKH 01lm6KH HHTepnOJ1SluHH 3aBHCSlT H OT

hnBeJIHlIHHbl H OT BeJIHlIHHbl CYMMbl lIaCTHblX npOH3BO)lHbIX nOpSlllKa n + 1 TO

eCTeCTBeHHO npH nOCTpoeHHH 6a3HCHbIX QJYHKUH yqHTbIBaTb 06e 3aBHCHMOCTH

TeM 60JIee 3TO He06xo)lHMO eCJIH paCCMaTpHBalOTCSI aHaJIHTHlIeCKHe QJYHKllHH C

oc06eHHOCTSlMH Y TaKHX QJYHKUH lIaCTHble npOH3BO)lHbIe 60JIblllero nOpSlllKa MOshy

ryT 6blTb HaMHOrO 60JIbllle no BeJIHlIHHe lIeM lIaCTHble npOH3BO)lHbIe MeHblllero

nOpSl)lKa

33Me1J3HHe 2 CJIe)lyeT OTMeTHTb liTO nOJIe3HbIMH MOryT 6blTb OueHKH 01llH6KH HHTepnOJISluHH

CHH3Y TIYCTb)lJ1S1 paCCMaTpHBaeMblX THnOB 3JIeMeHTOB (k = 1234)

mk

Hmbj(rl) = v E Wi(rl) 11 = L Vi Ni(kl1X) x E rl (i=l iii)

rue Vi (1ltS i ltS rnk) - HeKOTopble K03QJQJHuHeHTbI Torna eCJIH U E Wi(rl) TO

cnpaBeuJIHBO HepaBeHCTBO

(19)

UOKa)l(eM HepaBeHCTBO (19) I1MeeM

Ill - I(kn)lllwihd = IIVTll - V T(I(kll)ll) II (L2hl))2 =

k= IIVT(ll + Cj) - VT(~(lli + Cj)N 11r)II(L2 hd)2

i=l

rue Cj - npOH3BOJIbHasi nOCTOSlHHaSl Bbi6epeM Cj TaK lIT06bl 11) + Cj = 0 rue Ij

- 3HalleHHe QJYHKUHH U B Y3JIe x(j) E Ti 0603HallHM U = U - Uj Torua Uj = degIi

nOJIyqaeM

Ill - ](kn)lllwihd = IIVTu - V T(I(k1l)U) II (L 2C-1l))2 =

= Iu(x) - iij Nyn) (x) - I UiNi(kn)(J)IHlhl) ~ (i=l iolj)

~ inf Iii - vlWdhd = inf Iu - VIHihl)IIEilmk j hd VEIImk (Yl)

Bbl6HPaSI TaKO Y3eJI 3JIeMeHTa rl npH KOTOPOM nOCJIeuHee HepaBeHcTBo 6yueT

MaKCHMaJIbHbIM nOJIYlIaeM OueHKY (19)

13

OrcJOna CJIenyeT BbIBOn qTO nOrpelliHOCTb HHTepnOJISIUIUI Ha igtJ1eMeHTe Ii MOJKHO OueHHBaTb CHH3Y HCnOJIb3Y5I npH6JIHJKeHHble pellleHH5I nBYMepHblX Kpashy

eBblX 3auaq )lHpHXJ1e nOCKOJIbKY npH pellleHHH TaKHX 3auaq Ha 3JIeMeHTe Ii MeTOshy

nOM By6HoBa-raJIepKHHa nOJIyqaJOTC5I oueHKH norpelliHOCTH KaK B npaBOH qaCTH

HepaBeHCTBa (19) TaKOH nonxon 51BJ151eTC5I YHHBepCaJIbHbIM HO lJ151 paccMaTpHshy

BaeMOH 3auaqH 60JIee 3Q(peKTHBHbIH cnoc06 JIOKaJIbHOH oueHKH norpelliHocTH

OCHOBaH Ha BbIpaJKeHHH 3TOH norpelliHocTH Qepe3 QaCTHble npoH3BonHble 3allaHshy

HOro BeKTopa

KaK 06bNHO Qepe3 c(n)(E2) n 2 0 6yneM 0603HaQaTb npocTpaHcTBo QJYHKshy

UHH onpeneJIeHHbIX Ha E2 KOTopble HMeJOT HenpepbIBHble QaCTHble npoH3BonHbie

no nop5lJlKa n BKJ1l0QHTeJIbHO TIPH 3TOM C(O)(E2) == C(E2) - npocTpaHcTBo Heshy

npepbIBHbIX QJYHKUHH C HOPMOH Ilvllc(E2) max Iv(~)I rne v E C(E2)~EEz

)Jn51 nOCTpOeHH5I auanTHBHoro aJIrOpHTMa nOKaJKeM oueHKY norpelliHocTH HHshy

TepnOJ15lUHH KOTOPyro yn06Ho BblQHCJ15ITb TIPH 3TOM 6yneM HCnOJIb30BaTb rnaushy

KOCTb 3auaHHoro BeKTopa G JIeMMa 2

TIyCTb u E c(n+l)(E2) rne n = 2 Korna k npHHHMaeT 3HaQeHH5I 1 HIm 2 H

n = 4 Korna k npHHHMaeT 3HaQeHH5I 3 HJIH 4 Torna npH ~ E E2 CnpaBelJ1HBO

HepaBeHcTBo

lu [(kn)ul dk L (IIDffG~1IlC(E2) + IIDffG61IC(E2)) (20) IPI=n

3neCb I~I = ~l + ~2 ~l~l HeoTpHuaTeJIbHble ueJIble QHCJIa G~j = aua~j j = 12 a K03QJQJHUHeHTbI dk (1 k 4) npHHHMaJOT 3HaQeHH5I d1 = 3316 d2 12 d3 d4 = 83

LloKalaTeJIbCTBO

OrMcTHM CBOHCTBO QJYHKUHH-HHTepnOJI5IHTa KOTopoe 6yneT HCnOJIb30B3TbC5I

npH nOKa3aTCJIbCTBC [(kn)u 51BJ151eTC5I HHTepnOJ15lUHOHHbIM MHOroqnCHOM elJHHshy

CTBeHHblM [8] )JJ15I K3JKJlOro 3HaQeHH5I k H COOTBeTCTBYJOlllerO 3HaQeHHSI n TIoshy

3TOMY eCJIH p(kl) (~) MHOfOqneH CTeneHH l n )JJ15I 3auaHHbIX 3HaQeHHH k TO

[(kn)p(kl) p(kl)(~) TIYCTb u(~) E c(n+l) (E2) fne n = 2 KOflla k = 1 11JIH

k = 2 H n 4 KOfna k = 3 HJ1H k 4 PaccMoTpHM norpelliHocTb v(kn)

v(kn)(~) = u(~) - I uiNikn)(~) ~ E E2 i=l

ltlgtYHKUHIO U B Y3J1ax ~(i) npencTaBHM B BHlJe p5lna TeHJIOpa B Y3J1e ~(j)

Ui = L Dffu(~(j))(~(i) - ~(j))3~ + R(n+l)(~(i) ~(j)) IPIn

14

OrcJOna CJIenyeT BbIBOn qTO nOrpelliHOCTb HHTepnOJISIUIUI Ha igtJ1eMeHTe Ii MOJKHO OueHHBaTb CHH3Y HCnOJIb3Y5I npH6JIHJKeHHble pellleHH5I nBYMepHblX Kpashy

eBblX 3auaq )lHpHXJ1e nOCKOJIbKY npH pellleHHH TaKHX 3auaq Ha 3JIeMeHTe Ii MeTOshy

nOM By6HoBa-raJIepKHHa nOJIyqaJOTC5I oueHKH norpelliHOCTH KaK B npaBOH qaCTH

HepaBeHCTBa (19) TaKOH nonxon 51BJ151eTC5I YHHBepCaJIbHbIM HO lJ151 paccMaTpHshy

BaeMOH 3auaqH 60JIee 3Q(peKTHBHbIH cnoc06 JIOKaJIbHOH oueHKH norpelliHocTH

OCHOBaH Ha BbIpaJKeHHH 3TOH norpelliHocTH Qepe3 QaCTHble npoH3BonHble 3allaHshy

HOro BeKTopa

KaK 06bNHO Qepe3 c(n)(E2) n 2 0 6yneM 0603HaQaTb npocTpaHcTBo QJYHKshy

UHH onpeneJIeHHbIX Ha E2 KOTopble HMeJOT HenpepbIBHble QaCTHble npoH3BonHbie

no nop5lJlKa n BKJ1l0QHTeJIbHO TIPH 3TOM C(O)(E2) == C(E2) - npocTpaHcTBo Heshy

npepbIBHbIX QJYHKUHH C HOPMOH Ilvllc(E2) max Iv(~)I rne v E C(E2)~EEz

)Jn51 nOCTpOeHH5I auanTHBHoro aJIrOpHTMa nOKaJKeM oueHKY norpelliHocTH HHshy

TepnOJ15lUHH KOTOPyro yn06Ho BblQHCJ15ITb TIPH 3TOM 6yneM HCnOJIb30BaTb rnaushy

KOCTb 3auaHHoro BeKTopa G JIeMMa 2

TIyCTb u E c(n+l)(E2) rne n = 2 Korna k npHHHMaeT 3HaQeHH5I 1 HIm 2 H

n = 4 Korna k npHHHMaeT 3HaQeHH5I 3 HJIH 4 Torna npH ~ E E2 CnpaBelJ1HBO

HepaBeHcTBo

lu [(kn)ul dk L (IIDffG~1IlC(E2) + IIDffG61IC(E2)) (20) IPI=n

3neCb I~I = ~l + ~2 ~l~l HeoTpHuaTeJIbHble ueJIble QHCJIa G~j = aua~j j = 12 a K03QJQJHUHeHTbI dk (1 k 4) npHHHMaJOT 3HaQeHH5I d1 = 3316 d2 12 d3 d4 = 83

LloKalaTeJIbCTBO

OrMcTHM CBOHCTBO QJYHKUHH-HHTepnOJI5IHTa KOTopoe 6yneT HCnOJIb30B3TbC5I

npH nOKa3aTCJIbCTBC [(kn)u 51BJ151eTC5I HHTepnOJ15lUHOHHbIM MHOroqnCHOM elJHHshy

CTBeHHblM [8] )JJ15I K3JKJlOro 3HaQeHH5I k H COOTBeTCTBYJOlllerO 3HaQeHHSI n TIoshy

3TOMY eCJIH p(kl) (~) MHOfOqneH CTeneHH l n )JJ15I 3auaHHbIX 3HaQeHHH k TO

[(kn)p(kl) p(kl)(~) TIYCTb u(~) E c(n+l) (E2) fne n = 2 KOflla k = 1 11JIH

k = 2 H n 4 KOfna k = 3 HJ1H k 4 PaccMoTpHM norpelliHocTb v(kn)

v(kn)(~) = u(~) - I uiNikn)(~) ~ E E2 i=l

ltlgtYHKUHIO U B Y3J1ax ~(i) npencTaBHM B BHlJe p5lna TeHJIOpa B Y3J1e ~(j)

Ui = L Dffu(~(j))(~(i) - ~(j))3~ + R(n+l)(~(i) ~(j)) IPIn

14

R(II+1)(((i) (W) = I Dtulaquo((j) + (Jilaquo((i) - ~(jraquo))(((i) ~(j)lp 0 lt (Ji lt 1 18I=n+ I

3)leCb 3 (81 P2) 181 = 81 + P2 PI P2 - HeOTpHuaTeJIbHbIe ueJIbIe qHCJIa

8 = (3132 Du = al8lua~fla~2 KpOMe Toro

(((i) (j)l (6i-6j)81(6i 6j)fi2

~(j) + (Ji(~(i) - ~(jraquo) (6j + (Ji(6i - 6j)6j + (Ji(6i - 6j))

I10)lCTaBJIJIH TaKoe rrpe)lCTaBJIeHHe ltPYHKUHH U B BbIpameHHe JlJIJI rrorpeIIIHOCTH

rrOJIyqHM

V(kn)(~) = U(~) - I[ L Dfu(~(jraquo)(~(i) - ~(j)lpl]Nn)(~)_ i=l 18I$n

- I R(n+1)(~(i) ~(jraquo)Ni(kn)(~) i=1

3JleCb B KBaJlpaTHbIX cKo6Kax CTOHT MHOrOqJIeH CTerreHH n n03TOMY yqHTbIshy

BaH CBoitcTBO ltPYHKUHH-HHTeprrOJIJIHTa 6YJleM HMeTb

lIaJIee HCrrOJIb3YH pa3JIOmeHHe ltPYHKUHH U B pH)l TeiiJIopa B TOqKe ~(j) rronyshy

V(kn)(o = R(n+1) (~ ~(jraquo) I R(n+1) (~(i) ~(jraquo)Nlkn) (0 i=l

OueHHM ltPYHKUHIO v (k n) yqHTbIBaH qTO INlkn) (~) I 1 JlJIJI Beex 1 i rnk bull

BBeJleM TaKme 0603HaqeHHe

I1MecM JlJIJI JII060H TOqKII ( E

IV(kn)(~)1 IR(n+l)(~~(j))1 + l~tt~k IR(1I+1)(~(i)~(jraquo) Ni(kn)(OI

L p IIDfullC(E2 ) 1I88(~ ~(jraquo)IIC(E2)+18I=n+1

+ L pIIDfuIIC(E2) ~ax (88laquo((i)~(jraquo)) = 18I=n+1 l$r$mk

= L c~~~21IDfUIlC(E2) (21) 18I=n+l

15

c1~~ 11~(3(~~(j))lIc(EJ) + 1~~k (~(3(~(i)~(j)))

npH KOHKpeTHblX n H k OTMeTHM qTO C1~~2 ((32)(312(32 + (32)f32) (3 338

c1~~ 2(J 1 C1~~2 = C1~~2 = (2(J)(2(312f32 ) 163 npHqeM MaKCHMlliIbHble

K03rpqmlJHeHTbI nOJIyqaIOTc5I npH CMernaHHblX npOH3BOllHblX IT03TOMY eCJIH lUISI

CMernaHHblX npOH3BOlHblX npH (31 (31 2 1 HCnOJIb30BaTb OlJeHKY

al(3l u 1 al(3I-1G~ al(3I-1G~

Ia~fla~g21 2 (I a~f1-1a~~21 + Ia~fla~g2~11) me G~j =aua~j j 12 TO H3 (21) nOJIyqaeM HepaBeHcTBo (20)

uJUI OlJeHKH norpelliHocTH HHTepnOJISIlJHH (20) Tpe6yeTc5I BblqHCJIeHHe qaCTHblX

npOH3BOlHbiX BToporo H qeTBepToro nop5lllKa OT BeKTopa G(~) = J G(x) rle J - MaTpHlJa onpeleJIeHHaSI B (18) ITPH 3TOM eCTeCTBeHHO npHMemITb rpOpMyJIbI

C TeMH JKe 3HaqeHH5IMH BeKTopa G KOTopble He06xolHMbI lJI5I rpOpMHpOBaHH5I

CHCTeMbI ypaBHeHHH (12) C nOMOIllbIO Ky6aTypHblx rpOpMyJI IT03TOMY lJI5I npHshy

6JIHJKeHHOrO BblqHCJIeHHSI qaCTHblX npOH3BOlHbIX G(~) 6YleM HCnOJIb30BaTb 3JIeshy

MeHTbl nepBoro H qeTBepToro THnOB TaKHM 06pa30M lJI5I qaCTHblX npOH3BOlHbiX

D~G(~) DtG(~) 6YlleM npHMeH5ITb annpOKCHMalJHH

G(~(j))D~NP2)(~) (22)

(23)

Uanee npHBeleM 3llanTHBHbIH anropHTM lJI5I pellieHH5I ypaBHeHHSI (9) H l3llHM

He06xolHMble nOSlCHeHHSI AJIropHTM COCTOHT H3 CJIel)lOllJHX marOB

1) Un51 33llaHHOrO pa36HeHHSI rpaHHlJbI r Ha qeTblpexyrOJIbHHKH reHepHpyeTcSI

HHrpopMalJHSI lJ)1S1 npelCTaBJIeHH5I pa3peJKeHHOH MaTpHlJbI CHCTeMbl (12) ITPH 3TOM

npelnOJIaraeTCSI qTO Ka)KlblH qeTblpexyrOJIbHHK HMeeT no 17 Y3J10B cooTBeTcTBYshy

IOllJHX 3JIeMeHTaM TpeTbero THna

2) IJjrn BCex qeTblpexyrOJIbHHKOB rpaHHI~bI r B Y3J1ax COOTBeTCTBYIOllJHX Y3JIaM

3JIeMeHTOB qeTBepToro THna B KOTOPblX ellJe He onpeleJIeH BeKTOp G npOBO)lSlTCSI

ero BblqHCJIeHHSI B 3aBHCHMOCTH OT nOBeleHHSI KOMnOHeHT BeKTopa G rpaHHlJa

r pa3leJISleTCSI Ha rpaHHlJy r 1 H Ha rpaHHlJy KaK 3TO onHcaHO B pa3leJIe 1 3) UnSl qeTbIpexyroJIbHHKOB rpaHHlJbI r C nOMOIllbIO rpOpMyJI (20)(22) H (20)(23)

BblqHCJISIIOTCSI OlJeHKH norpemHOCTH Bce qeTbIpexyroJIbHHKH pa3leJISiIOTC5I Ha l(Ba

16

KJIacca K nCpBOMY KJIaccy OTHOCSlTCSI lJCTblpcxyrOJlbHHKH JIJ1SI KOTOPblX OuCHKa

nOrpCWHOCTH V(34) MCHbWC OUCHKH nOrpCWHOCTH V(22) Ko BTOPOMY KJIaccy

OTHOCSlTCSI BCC OCTaJIbHhlC lJCTbIPCXyrOJlbHHKH rpaHHubl r 4) ILrrn BCCX lJCTblpcxyrOJlbHHKOB rpaHH11bl r BbIlJHCJlSlIOTCSI 3JlCMCHTHbIC MaTpH11bl

H gtJlCMCHTHbIC BCKTOPbl H3 KOTOPblX qOpMHpyIOTCSI MaTpHua H npaBaSI lJaCTb CHshy

CTCMbI (12) npH gtTOM Ha lJCTbIpcxyrOJlbHHKaX ncpBoro KJIacca HCnOJlb3yIOTCSI

qYHKUHH qOPMbl gtJlCMCHTOB TpcTbcro THna a )lJlSl lJCTblpCXyroJlbHHKOB BToporo

KJIacca gtJlCMCHTbI BTOpOro THna KpoMc Toro JIJ1SI BbIlJHCJlCHH HHTCrpaJIOB no

lJCTbIpcxyroJlbHHKaM ncpBoro KJIacca HCnOJlb3YlOTCSI Ky6aTypHblc q0PMYJlbl C 13 Y3JlaMH KOTOPblC CCTb Y gtJlCMCHTOB lJCTBCpToro THna a )lJlSl BbllJHCJlCHHSI HHTcrpashy

JlOB no lJCTblpcxyrOJlbHHKaM BToporo KJIacca HCnOJlb3YCTCSI COCTaBHaSI q0pMYJla H3

lJCTbIPCX Ky6aTypHbIX qOpMyJl C 13 Y3JlaMH

5) PcwacTcSI CHCTCMa (12) C nOMOlllblO HTCpaunoHHoro npo11ccca (13) 6) lJjISI Ka)K)Joro lJCTbIpcxyrOJlbHHKa Ii C r BbllJHCJlSlCTCSI BCJlHlJHHa Ci - OTHOCHshy

TCJlbHaSl pa3HH11a MC)I()lY TOlJHblM PCWCHHCM U H npH6JlH)(CHHbIM centm

VcentmlhdSjJIIGI11dS il

rUc centm rcentm+ ri (7IG rcentml 3UCCb (7 - 3HaK KOMnOHCHTbI Gn IIGIIt IGsl +IGtl +IGnl aHaJIorHlJHO BbllJHCJlSlCTCSI IIG centmlli npH gtTOM HCnOJlbshy

3YlOTCSI TC iKC Ky6aTYPHblC q0PMYJlbl lJTO H Ha ware 4) H BbllJHCJlCHHblC Ha warc

2) 3HalJCHHSI BCKTopa G ECJlH 3analJa C 3aUaHHoH TOlJHOCTblO CIllC HC pCWCHa TO

npOBO)lHTCSI YTOlJHCHHOC pa36HCHHC lJCTbIpCXyroJlbHHKOB COCTaBJISllOlllHX nOBCpxshy

HOCTb r H np011cUypa CHOBa nOBTopSlCTCSI C wara 1)

UauHM nOSlCHCHHSI )lJlSl Ka)l()loro wara anropHTMa

OTHOCHTCJlbHO ncpBoro wara CJlC)lYCT 3aMCTHTb lJTO HCnOJlb30BaHHC gtJlCMCHTOB

C pa3HOH CTcnCHblO HHTCpnOJISluHH HO C OUHHaKOBbIMH Y3JlaMH 06JlCflJaCT 3aUaHHC

pa3pC)(CHHOM CTpyKTYPbl MaTpHubI CHCTCMbI (12) Ha BTOPOM warc pa3UCJISlIOTCSI gtJleMCHTbl C JlHHCHHblMH ( rpaHJma r 1) H HCshy

JlHHCHHbIMH ( rpaHHua r 2) ypaBHcHHSlMH BbIlJHCJlCHHblC 3Ha4CHliSI BCKTopa G Ha

BTOPOM warc )laJICC HCnOJlb3yIOTCSI Ha TPCTbCM lJCTBCPTOM H WCCTOM warax aJIshy

ropHTMa BOJlCC Toro CCJl nOCJle ncpBoro PCWCHHSI CHCTCMbl (12) Hco6xoUHMaSl

TOlJHOCTb HC 6Y)lCT UOCTHrHYTa TO npH YTOlJHCHHOM pa36HCHHH OUHoro lJCTbIPCXshy

yrOJlbHHKa ncpBoro KJ1aCca Ha lJCTblPC Hco6xoUHMO 6YUCT BbllJHCJlSlTb BCKTOP G B HOBbIX 32 Y3Jlax H3 KOTOPbIX KaK nOKa3aHO Ha pHC 3 TOJlbKO 22 SlBJ1SllOTCSI

BHYTPCHHHMH AHaJIOrHlJHO -)lJlSl lJCTblpcxyroJlbHHKOB BTOpOro KJ1acca

Ha TPCTbCM ware aJIropHTMa Bbl6HpacTcSI HaH60Jlcc noUxOUSllllaSl annpoKcHMashy

11HSI )lJlSl HCH3BCCTHOH qYHKUHH B 3aBHCHMOCTH OT 3HalJeHH KOMnOHCHT BeKTopa

G H BbllJHCJlSlCMbIX OUCHOK norpcwHoCTH

17

Ha lIeTBepTOM ware aJIropHTMa Ky6aTYPHhie CPOPMYJIhl KaK H annpOKCHMaUH5I

HeH3BeCTHOH CPYHKUHH Ha TpeTheM ware BhI6HpaIOTC5I B 3aBHCHMOCTH OT 3HalleHHH

KOMnOHeHT BeKTopa GH BhlllHCJI5IeMhIX oueHOK norpellIHOCTH

OnHweM nOJlp06HO npouecc HCKJIlOqeHH5I HeH3BeCTHhIX COOTBeTCTBYIOIUHX

BHYTpeHHHM Y3JIaM )l1151 meMeHTOB BToporo KJIaCca npH peweHHH CHCTeMhl (13) Pa3JleJIHM Bce HeH3BeCTHhie CHCTeMhI Ha JlBe rpynnhl npHqeM KO BTOPOH rpynne

OTHeceM HeH3BeCTHhle cooTBeTcTByIOIUHe BHyTpeHHHM Y3JIaM KpoMe UeHTpaJIhshy

HhIX B COOTBeTCTBHH C TaKHM pa3JleJIeHHeM cHCTeMY (13) MOiKHO 3anHcaTh B

BHJle (k+l) + A (k+l) b(k)A llPl I2P2 WI

(k+l) + A (k+l) - b(k)A21Pl 22P2 - W 2 (24)

B 3TOH CHCTeMe MaTpHua A22 6YJleT 6JIOqHO JlHarOHaJIhHOH MaTpHueH C 6JIOKaMH

pa3MepHocThlO 8 x 8 06paIUa51 3Ty MaTpHUY H HCKJIlOlIa5I BTOPyIO rpynny HeH3shy

BeCTHhIX nOJIyqaeM HOBYIO cHcTeMY

(25)

ITpHBeJleHHe CHCTeMhl (24) K CHCTeMe (25) BhInOJIH5IeTC5I Ha ypoBHe 3JIeMeHTHhIX

MaTpHU H BeKTopOB TO eCTh CHCTeMa (25) COCTaBJI5IeTC5I H3 ypaBHeHHH onpeJleshy

JIeHHhIX Ha 3JIeMeHTax BToporo THna H HMelOIUHX BHJl

rJle all a12 a22 a21 - 3JIeMeHTHhie MaTpHUhl PI - BeKTOp HeH3BeCTHhIX COOTshy

BeTCTBYIOIUHH rpaHHqHhlM H ueHTpaJIhHOMY Y3JIaM 3JIeMeHTa bi b2 3JIeMeHTHhle

BeKTophI npaBOH qaCTH

OTHOCHTeJIhHO nOCJIeJlHerO mara aJIrOpHTMa CJIeJlyeT 3aMeTHTh qTO nOCTposhy

eHHe yrOqHeHHOro pa36HeHH5I qeThlpexyroJIhHHKOB rpaHHUhI r 3aBHCHT OT Tpyshy

JlOeMKOCTR BhlqHCJIeHH5I BeKTopa C H eCJIH BhlqHCJIeHHe BeKTopa C Tpe6yeT

OTHOCHTeJIhHO 60JIhlllHX 3aTpaT npoueccopHoro BpeMeHH 3BM TO )KeJIaTenhHO

npOBOJlHTh pa36HeHH5I B COOTBeTCTBHH co cxeMOH pHC 3 ITpHBeJleM JlBa MOJleJIhHhIX npHMepa JlJI5I JleMOHCTpaurIH CBOHCTB npeJlJIaraeshy

MOro aJIropHTMa ITpHMep nepBhlH nycTh eJlHHHqHhIH Ky6 E3 [-1 1] xl] x 1] pa36HT Ha 6 nHpaMHJl Ka)K)Ja5I H3 KOTOPhIX HMeeT BepllIHHy B ueHshy

Tpe Ky6a a OCHOBaHHeM - rpaHh Ky6a B Ka)KJlOH nHPaMHJle 33JlaHa CPYHKUH5I

U = xrk JIH60 U x~k JIH60 U = x5k (k HaTYPaJIhHoe) TaK qTO Ha rpaHHue

Ky6a U = 1 H VrU = O ECJIH B OJlHOH yrJIOBOH TOqKe E3 33JlaTh TOqHOe 3Haqeshy

HHe CPYHKUHH U a Ha rpaH5IX - VrU TO TOqHOe peWeHHe Ha nOBepXHOCTH Ky6a

OnpeJleJUleTC5I 3a OJlHy HTepaUHIO npH HCnOJIh30BaHHH caMOro rpy6oro pa36HeHH5I

middot18

c1~~ 11~(3(~~(j))lIc(EJ) + 1~~k (~(3(~(i)~(j)))

npH KOHKpeTHblX n H k OTMeTHM qTO C1~~2 ((32)(312(32 + (32)f32) (3 338

c1~~ 2(J 1 C1~~2 = C1~~2 = (2(J)(2(312f32 ) 163 npHqeM MaKCHMlliIbHble

K03rpqmlJHeHTbI nOJIyqaIOTc5I npH CMernaHHblX npOH3BOllHblX IT03TOMY eCJIH lUISI

CMernaHHblX npOH3BOlHblX npH (31 (31 2 1 HCnOJIb30BaTb OlJeHKY

al(3l u 1 al(3I-1G~ al(3I-1G~

Ia~fla~g21 2 (I a~f1-1a~~21 + Ia~fla~g2~11) me G~j =aua~j j 12 TO H3 (21) nOJIyqaeM HepaBeHcTBo (20)

uJUI OlJeHKH norpelliHocTH HHTepnOJISIlJHH (20) Tpe6yeTc5I BblqHCJIeHHe qaCTHblX

npOH3BOlHbiX BToporo H qeTBepToro nop5lllKa OT BeKTopa G(~) = J G(x) rle J - MaTpHlJa onpeleJIeHHaSI B (18) ITPH 3TOM eCTeCTBeHHO npHMemITb rpOpMyJIbI

C TeMH JKe 3HaqeHH5IMH BeKTopa G KOTopble He06xolHMbI lJI5I rpOpMHpOBaHH5I

CHCTeMbI ypaBHeHHH (12) C nOMOIllbIO Ky6aTypHblx rpOpMyJI IT03TOMY lJI5I npHshy

6JIHJKeHHOrO BblqHCJIeHHSI qaCTHblX npOH3BOlHbIX G(~) 6YleM HCnOJIb30BaTb 3JIeshy

MeHTbl nepBoro H qeTBepToro THnOB TaKHM 06pa30M lJI5I qaCTHblX npOH3BOlHbiX

D~G(~) DtG(~) 6YlleM npHMeH5ITb annpOKCHMalJHH

G(~(j))D~NP2)(~) (22)

(23)

Uanee npHBeleM 3llanTHBHbIH anropHTM lJI5I pellieHH5I ypaBHeHHSI (9) H l3llHM

He06xolHMble nOSlCHeHHSI AJIropHTM COCTOHT H3 CJIel)lOllJHX marOB

1) Un51 33llaHHOrO pa36HeHHSI rpaHHlJbI r Ha qeTblpexyrOJIbHHKH reHepHpyeTcSI

HHrpopMalJHSI lJ)1S1 npelCTaBJIeHH5I pa3peJKeHHOH MaTpHlJbI CHCTeMbl (12) ITPH 3TOM

npelnOJIaraeTCSI qTO Ka)KlblH qeTblpexyrOJIbHHK HMeeT no 17 Y3J10B cooTBeTcTBYshy

IOllJHX 3JIeMeHTaM TpeTbero THna

2) IJjrn BCex qeTblpexyrOJIbHHKOB rpaHHI~bI r B Y3J1ax COOTBeTCTBYIOllJHX Y3JIaM

3JIeMeHTOB qeTBepToro THna B KOTOPblX ellJe He onpeleJIeH BeKTOp G npOBO)lSlTCSI

ero BblqHCJIeHHSI B 3aBHCHMOCTH OT nOBeleHHSI KOMnOHeHT BeKTopa G rpaHHlJa

r pa3leJISleTCSI Ha rpaHHlJy r 1 H Ha rpaHHlJy KaK 3TO onHcaHO B pa3leJIe 1 3) UnSl qeTbIpexyroJIbHHKOB rpaHHlJbI r C nOMOIllbIO rpOpMyJI (20)(22) H (20)(23)

BblqHCJISIIOTCSI OlJeHKH norpemHOCTH Bce qeTbIpexyroJIbHHKH pa3leJISiIOTC5I Ha l(Ba

16

KJIacca K nCpBOMY KJIaccy OTHOCSlTCSI lJCTblpcxyrOJlbHHKH JIJ1SI KOTOPblX OuCHKa

nOrpCWHOCTH V(34) MCHbWC OUCHKH nOrpCWHOCTH V(22) Ko BTOPOMY KJIaccy

OTHOCSlTCSI BCC OCTaJIbHhlC lJCTbIPCXyrOJlbHHKH rpaHHubl r 4) ILrrn BCCX lJCTblpcxyrOJlbHHKOB rpaHH11bl r BbIlJHCJlSlIOTCSI 3JlCMCHTHbIC MaTpH11bl

H gtJlCMCHTHbIC BCKTOPbl H3 KOTOPblX qOpMHpyIOTCSI MaTpHua H npaBaSI lJaCTb CHshy

CTCMbI (12) npH gtTOM Ha lJCTbIpcxyrOJlbHHKaX ncpBoro KJIacca HCnOJlb3yIOTCSI

qYHKUHH qOPMbl gtJlCMCHTOB TpcTbcro THna a )lJlSl lJCTblpCXyroJlbHHKOB BToporo

KJIacca gtJlCMCHTbI BTOpOro THna KpoMc Toro JIJ1SI BbIlJHCJlCHH HHTCrpaJIOB no

lJCTbIpcxyroJlbHHKaM ncpBoro KJIacca HCnOJlb3YlOTCSI Ky6aTypHblc q0PMYJlbl C 13 Y3JlaMH KOTOPblC CCTb Y gtJlCMCHTOB lJCTBCpToro THna a )lJlSl BbllJHCJlCHHSI HHTcrpashy

JlOB no lJCTblpcxyrOJlbHHKaM BToporo KJIacca HCnOJlb3YCTCSI COCTaBHaSI q0pMYJla H3

lJCTbIPCX Ky6aTypHbIX qOpMyJl C 13 Y3JlaMH

5) PcwacTcSI CHCTCMa (12) C nOMOlllblO HTCpaunoHHoro npo11ccca (13) 6) lJjISI Ka)K)Joro lJCTbIpcxyrOJlbHHKa Ii C r BbllJHCJlSlCTCSI BCJlHlJHHa Ci - OTHOCHshy

TCJlbHaSl pa3HH11a MC)I()lY TOlJHblM PCWCHHCM U H npH6JlH)(CHHbIM centm

VcentmlhdSjJIIGI11dS il

rUc centm rcentm+ ri (7IG rcentml 3UCCb (7 - 3HaK KOMnOHCHTbI Gn IIGIIt IGsl +IGtl +IGnl aHaJIorHlJHO BbllJHCJlSlCTCSI IIG centmlli npH gtTOM HCnOJlbshy

3YlOTCSI TC iKC Ky6aTYPHblC q0PMYJlbl lJTO H Ha ware 4) H BbllJHCJlCHHblC Ha warc

2) 3HalJCHHSI BCKTopa G ECJlH 3analJa C 3aUaHHoH TOlJHOCTblO CIllC HC pCWCHa TO

npOBO)lHTCSI YTOlJHCHHOC pa36HCHHC lJCTbIpCXyroJlbHHKOB COCTaBJISllOlllHX nOBCpxshy

HOCTb r H np011cUypa CHOBa nOBTopSlCTCSI C wara 1)

UauHM nOSlCHCHHSI )lJlSl Ka)l()loro wara anropHTMa

OTHOCHTCJlbHO ncpBoro wara CJlC)lYCT 3aMCTHTb lJTO HCnOJlb30BaHHC gtJlCMCHTOB

C pa3HOH CTcnCHblO HHTCpnOJISluHH HO C OUHHaKOBbIMH Y3JlaMH 06JlCflJaCT 3aUaHHC

pa3pC)(CHHOM CTpyKTYPbl MaTpHubI CHCTCMbI (12) Ha BTOPOM warc pa3UCJISlIOTCSI gtJleMCHTbl C JlHHCHHblMH ( rpaHJma r 1) H HCshy

JlHHCHHbIMH ( rpaHHua r 2) ypaBHcHHSlMH BbIlJHCJlCHHblC 3Ha4CHliSI BCKTopa G Ha

BTOPOM warc )laJICC HCnOJlb3yIOTCSI Ha TPCTbCM lJCTBCPTOM H WCCTOM warax aJIshy

ropHTMa BOJlCC Toro CCJl nOCJle ncpBoro PCWCHHSI CHCTCMbl (12) Hco6xoUHMaSl

TOlJHOCTb HC 6Y)lCT UOCTHrHYTa TO npH YTOlJHCHHOM pa36HCHHH OUHoro lJCTbIPCXshy

yrOJlbHHKa ncpBoro KJ1aCca Ha lJCTblPC Hco6xoUHMO 6YUCT BbllJHCJlSlTb BCKTOP G B HOBbIX 32 Y3Jlax H3 KOTOPbIX KaK nOKa3aHO Ha pHC 3 TOJlbKO 22 SlBJ1SllOTCSI

BHYTPCHHHMH AHaJIOrHlJHO -)lJlSl lJCTblpcxyroJlbHHKOB BTOpOro KJ1acca

Ha TPCTbCM ware aJIropHTMa Bbl6HpacTcSI HaH60Jlcc noUxOUSllllaSl annpoKcHMashy

11HSI )lJlSl HCH3BCCTHOH qYHKUHH B 3aBHCHMOCTH OT 3HalJeHH KOMnOHCHT BeKTopa

G H BbllJHCJlSlCMbIX OUCHOK norpcwHoCTH

17

Ha lIeTBepTOM ware aJIropHTMa Ky6aTYPHhie CPOPMYJIhl KaK H annpOKCHMaUH5I

HeH3BeCTHOH CPYHKUHH Ha TpeTheM ware BhI6HpaIOTC5I B 3aBHCHMOCTH OT 3HalleHHH

KOMnOHeHT BeKTopa GH BhlllHCJI5IeMhIX oueHOK norpellIHOCTH

OnHweM nOJlp06HO npouecc HCKJIlOqeHH5I HeH3BeCTHhIX COOTBeTCTBYIOIUHX

BHYTpeHHHM Y3JIaM )l1151 meMeHTOB BToporo KJIaCca npH peweHHH CHCTeMhl (13) Pa3JleJIHM Bce HeH3BeCTHhie CHCTeMhI Ha JlBe rpynnhl npHqeM KO BTOPOH rpynne

OTHeceM HeH3BeCTHhle cooTBeTcTByIOIUHe BHyTpeHHHM Y3JIaM KpoMe UeHTpaJIhshy

HhIX B COOTBeTCTBHH C TaKHM pa3JleJIeHHeM cHCTeMY (13) MOiKHO 3anHcaTh B

BHJle (k+l) + A (k+l) b(k)A llPl I2P2 WI

(k+l) + A (k+l) - b(k)A21Pl 22P2 - W 2 (24)

B 3TOH CHCTeMe MaTpHua A22 6YJleT 6JIOqHO JlHarOHaJIhHOH MaTpHueH C 6JIOKaMH

pa3MepHocThlO 8 x 8 06paIUa51 3Ty MaTpHUY H HCKJIlOlIa5I BTOPyIO rpynny HeH3shy

BeCTHhIX nOJIyqaeM HOBYIO cHcTeMY

(25)

ITpHBeJleHHe CHCTeMhl (24) K CHCTeMe (25) BhInOJIH5IeTC5I Ha ypoBHe 3JIeMeHTHhIX

MaTpHU H BeKTopOB TO eCTh CHCTeMa (25) COCTaBJI5IeTC5I H3 ypaBHeHHH onpeJleshy

JIeHHhIX Ha 3JIeMeHTax BToporo THna H HMelOIUHX BHJl

rJle all a12 a22 a21 - 3JIeMeHTHhie MaTpHUhl PI - BeKTOp HeH3BeCTHhIX COOTshy

BeTCTBYIOIUHH rpaHHqHhlM H ueHTpaJIhHOMY Y3JIaM 3JIeMeHTa bi b2 3JIeMeHTHhle

BeKTophI npaBOH qaCTH

OTHOCHTeJIhHO nOCJIeJlHerO mara aJIrOpHTMa CJIeJlyeT 3aMeTHTh qTO nOCTposhy

eHHe yrOqHeHHOro pa36HeHH5I qeThlpexyroJIhHHKOB rpaHHUhI r 3aBHCHT OT Tpyshy

JlOeMKOCTR BhlqHCJIeHH5I BeKTopa C H eCJIH BhlqHCJIeHHe BeKTopa C Tpe6yeT

OTHOCHTeJIhHO 60JIhlllHX 3aTpaT npoueccopHoro BpeMeHH 3BM TO )KeJIaTenhHO

npOBOJlHTh pa36HeHH5I B COOTBeTCTBHH co cxeMOH pHC 3 ITpHBeJleM JlBa MOJleJIhHhIX npHMepa JlJI5I JleMOHCTpaurIH CBOHCTB npeJlJIaraeshy

MOro aJIropHTMa ITpHMep nepBhlH nycTh eJlHHHqHhIH Ky6 E3 [-1 1] xl] x 1] pa36HT Ha 6 nHpaMHJl Ka)K)Ja5I H3 KOTOPhIX HMeeT BepllIHHy B ueHshy

Tpe Ky6a a OCHOBaHHeM - rpaHh Ky6a B Ka)KJlOH nHPaMHJle 33JlaHa CPYHKUH5I

U = xrk JIH60 U x~k JIH60 U = x5k (k HaTYPaJIhHoe) TaK qTO Ha rpaHHue

Ky6a U = 1 H VrU = O ECJIH B OJlHOH yrJIOBOH TOqKe E3 33JlaTh TOqHOe 3Haqeshy

HHe CPYHKUHH U a Ha rpaH5IX - VrU TO TOqHOe peWeHHe Ha nOBepXHOCTH Ky6a

OnpeJleJUleTC5I 3a OJlHy HTepaUHIO npH HCnOJIh30BaHHH caMOro rpy6oro pa36HeHH5I

middot18

KJIacca K nCpBOMY KJIaccy OTHOCSlTCSI lJCTblpcxyrOJlbHHKH JIJ1SI KOTOPblX OuCHKa

nOrpCWHOCTH V(34) MCHbWC OUCHKH nOrpCWHOCTH V(22) Ko BTOPOMY KJIaccy

OTHOCSlTCSI BCC OCTaJIbHhlC lJCTbIPCXyrOJlbHHKH rpaHHubl r 4) ILrrn BCCX lJCTblpcxyrOJlbHHKOB rpaHH11bl r BbIlJHCJlSlIOTCSI 3JlCMCHTHbIC MaTpH11bl

H gtJlCMCHTHbIC BCKTOPbl H3 KOTOPblX qOpMHpyIOTCSI MaTpHua H npaBaSI lJaCTb CHshy

CTCMbI (12) npH gtTOM Ha lJCTbIpcxyrOJlbHHKaX ncpBoro KJIacca HCnOJlb3yIOTCSI

qYHKUHH qOPMbl gtJlCMCHTOB TpcTbcro THna a )lJlSl lJCTblpCXyroJlbHHKOB BToporo

KJIacca gtJlCMCHTbI BTOpOro THna KpoMc Toro JIJ1SI BbIlJHCJlCHH HHTCrpaJIOB no

lJCTbIpcxyroJlbHHKaM ncpBoro KJIacca HCnOJlb3YlOTCSI Ky6aTypHblc q0PMYJlbl C 13 Y3JlaMH KOTOPblC CCTb Y gtJlCMCHTOB lJCTBCpToro THna a )lJlSl BbllJHCJlCHHSI HHTcrpashy

JlOB no lJCTblpcxyrOJlbHHKaM BToporo KJIacca HCnOJlb3YCTCSI COCTaBHaSI q0pMYJla H3

lJCTbIPCX Ky6aTypHbIX qOpMyJl C 13 Y3JlaMH

5) PcwacTcSI CHCTCMa (12) C nOMOlllblO HTCpaunoHHoro npo11ccca (13) 6) lJjISI Ka)K)Joro lJCTbIpcxyrOJlbHHKa Ii C r BbllJHCJlSlCTCSI BCJlHlJHHa Ci - OTHOCHshy

TCJlbHaSl pa3HH11a MC)I()lY TOlJHblM PCWCHHCM U H npH6JlH)(CHHbIM centm

VcentmlhdSjJIIGI11dS il

rUc centm rcentm+ ri (7IG rcentml 3UCCb (7 - 3HaK KOMnOHCHTbI Gn IIGIIt IGsl +IGtl +IGnl aHaJIorHlJHO BbllJHCJlSlCTCSI IIG centmlli npH gtTOM HCnOJlbshy

3YlOTCSI TC iKC Ky6aTYPHblC q0PMYJlbl lJTO H Ha ware 4) H BbllJHCJlCHHblC Ha warc

2) 3HalJCHHSI BCKTopa G ECJlH 3analJa C 3aUaHHoH TOlJHOCTblO CIllC HC pCWCHa TO

npOBO)lHTCSI YTOlJHCHHOC pa36HCHHC lJCTbIpCXyroJlbHHKOB COCTaBJISllOlllHX nOBCpxshy

HOCTb r H np011cUypa CHOBa nOBTopSlCTCSI C wara 1)

UauHM nOSlCHCHHSI )lJlSl Ka)l()loro wara anropHTMa

OTHOCHTCJlbHO ncpBoro wara CJlC)lYCT 3aMCTHTb lJTO HCnOJlb30BaHHC gtJlCMCHTOB

C pa3HOH CTcnCHblO HHTCpnOJISluHH HO C OUHHaKOBbIMH Y3JlaMH 06JlCflJaCT 3aUaHHC

pa3pC)(CHHOM CTpyKTYPbl MaTpHubI CHCTCMbI (12) Ha BTOPOM warc pa3UCJISlIOTCSI gtJleMCHTbl C JlHHCHHblMH ( rpaHJma r 1) H HCshy

JlHHCHHbIMH ( rpaHHua r 2) ypaBHcHHSlMH BbIlJHCJlCHHblC 3Ha4CHliSI BCKTopa G Ha

BTOPOM warc )laJICC HCnOJlb3yIOTCSI Ha TPCTbCM lJCTBCPTOM H WCCTOM warax aJIshy

ropHTMa BOJlCC Toro CCJl nOCJle ncpBoro PCWCHHSI CHCTCMbl (12) Hco6xoUHMaSl

TOlJHOCTb HC 6Y)lCT UOCTHrHYTa TO npH YTOlJHCHHOM pa36HCHHH OUHoro lJCTbIPCXshy

yrOJlbHHKa ncpBoro KJ1aCca Ha lJCTblPC Hco6xoUHMO 6YUCT BbllJHCJlSlTb BCKTOP G B HOBbIX 32 Y3Jlax H3 KOTOPbIX KaK nOKa3aHO Ha pHC 3 TOJlbKO 22 SlBJ1SllOTCSI

BHYTPCHHHMH AHaJIOrHlJHO -)lJlSl lJCTblpcxyroJlbHHKOB BTOpOro KJ1acca

Ha TPCTbCM ware aJIropHTMa Bbl6HpacTcSI HaH60Jlcc noUxOUSllllaSl annpoKcHMashy

11HSI )lJlSl HCH3BCCTHOH qYHKUHH B 3aBHCHMOCTH OT 3HalJeHH KOMnOHCHT BeKTopa

G H BbllJHCJlSlCMbIX OUCHOK norpcwHoCTH

17

Ha lIeTBepTOM ware aJIropHTMa Ky6aTYPHhie CPOPMYJIhl KaK H annpOKCHMaUH5I

HeH3BeCTHOH CPYHKUHH Ha TpeTheM ware BhI6HpaIOTC5I B 3aBHCHMOCTH OT 3HalleHHH

KOMnOHeHT BeKTopa GH BhlllHCJI5IeMhIX oueHOK norpellIHOCTH

OnHweM nOJlp06HO npouecc HCKJIlOqeHH5I HeH3BeCTHhIX COOTBeTCTBYIOIUHX

BHYTpeHHHM Y3JIaM )l1151 meMeHTOB BToporo KJIaCca npH peweHHH CHCTeMhl (13) Pa3JleJIHM Bce HeH3BeCTHhie CHCTeMhI Ha JlBe rpynnhl npHqeM KO BTOPOH rpynne

OTHeceM HeH3BeCTHhle cooTBeTcTByIOIUHe BHyTpeHHHM Y3JIaM KpoMe UeHTpaJIhshy

HhIX B COOTBeTCTBHH C TaKHM pa3JleJIeHHeM cHCTeMY (13) MOiKHO 3anHcaTh B

BHJle (k+l) + A (k+l) b(k)A llPl I2P2 WI

(k+l) + A (k+l) - b(k)A21Pl 22P2 - W 2 (24)

B 3TOH CHCTeMe MaTpHua A22 6YJleT 6JIOqHO JlHarOHaJIhHOH MaTpHueH C 6JIOKaMH

pa3MepHocThlO 8 x 8 06paIUa51 3Ty MaTpHUY H HCKJIlOlIa5I BTOPyIO rpynny HeH3shy

BeCTHhIX nOJIyqaeM HOBYIO cHcTeMY

(25)

ITpHBeJleHHe CHCTeMhl (24) K CHCTeMe (25) BhInOJIH5IeTC5I Ha ypoBHe 3JIeMeHTHhIX

MaTpHU H BeKTopOB TO eCTh CHCTeMa (25) COCTaBJI5IeTC5I H3 ypaBHeHHH onpeJleshy

JIeHHhIX Ha 3JIeMeHTax BToporo THna H HMelOIUHX BHJl

rJle all a12 a22 a21 - 3JIeMeHTHhie MaTpHUhl PI - BeKTOp HeH3BeCTHhIX COOTshy

BeTCTBYIOIUHH rpaHHqHhlM H ueHTpaJIhHOMY Y3JIaM 3JIeMeHTa bi b2 3JIeMeHTHhle

BeKTophI npaBOH qaCTH

OTHOCHTeJIhHO nOCJIeJlHerO mara aJIrOpHTMa CJIeJlyeT 3aMeTHTh qTO nOCTposhy

eHHe yrOqHeHHOro pa36HeHH5I qeThlpexyroJIhHHKOB rpaHHUhI r 3aBHCHT OT Tpyshy

JlOeMKOCTR BhlqHCJIeHH5I BeKTopa C H eCJIH BhlqHCJIeHHe BeKTopa C Tpe6yeT

OTHOCHTeJIhHO 60JIhlllHX 3aTpaT npoueccopHoro BpeMeHH 3BM TO )KeJIaTenhHO

npOBOJlHTh pa36HeHH5I B COOTBeTCTBHH co cxeMOH pHC 3 ITpHBeJleM JlBa MOJleJIhHhIX npHMepa JlJI5I JleMOHCTpaurIH CBOHCTB npeJlJIaraeshy

MOro aJIropHTMa ITpHMep nepBhlH nycTh eJlHHHqHhIH Ky6 E3 [-1 1] xl] x 1] pa36HT Ha 6 nHpaMHJl Ka)K)Ja5I H3 KOTOPhIX HMeeT BepllIHHy B ueHshy

Tpe Ky6a a OCHOBaHHeM - rpaHh Ky6a B Ka)KJlOH nHPaMHJle 33JlaHa CPYHKUH5I

U = xrk JIH60 U x~k JIH60 U = x5k (k HaTYPaJIhHoe) TaK qTO Ha rpaHHue

Ky6a U = 1 H VrU = O ECJIH B OJlHOH yrJIOBOH TOqKe E3 33JlaTh TOqHOe 3Haqeshy

HHe CPYHKUHH U a Ha rpaH5IX - VrU TO TOqHOe peWeHHe Ha nOBepXHOCTH Ky6a

OnpeJleJUleTC5I 3a OJlHy HTepaUHIO npH HCnOJIh30BaHHH caMOro rpy6oro pa36HeHH5I

middot18

Ha lIeTBepTOM ware aJIropHTMa Ky6aTYPHhie CPOPMYJIhl KaK H annpOKCHMaUH5I

HeH3BeCTHOH CPYHKUHH Ha TpeTheM ware BhI6HpaIOTC5I B 3aBHCHMOCTH OT 3HalleHHH

KOMnOHeHT BeKTopa GH BhlllHCJI5IeMhIX oueHOK norpellIHOCTH

OnHweM nOJlp06HO npouecc HCKJIlOqeHH5I HeH3BeCTHhIX COOTBeTCTBYIOIUHX

BHYTpeHHHM Y3JIaM )l1151 meMeHTOB BToporo KJIaCca npH peweHHH CHCTeMhl (13) Pa3JleJIHM Bce HeH3BeCTHhie CHCTeMhI Ha JlBe rpynnhl npHqeM KO BTOPOH rpynne

OTHeceM HeH3BeCTHhle cooTBeTcTByIOIUHe BHyTpeHHHM Y3JIaM KpoMe UeHTpaJIhshy

HhIX B COOTBeTCTBHH C TaKHM pa3JleJIeHHeM cHCTeMY (13) MOiKHO 3anHcaTh B

BHJle (k+l) + A (k+l) b(k)A llPl I2P2 WI

(k+l) + A (k+l) - b(k)A21Pl 22P2 - W 2 (24)

B 3TOH CHCTeMe MaTpHua A22 6YJleT 6JIOqHO JlHarOHaJIhHOH MaTpHueH C 6JIOKaMH

pa3MepHocThlO 8 x 8 06paIUa51 3Ty MaTpHUY H HCKJIlOlIa5I BTOPyIO rpynny HeH3shy

BeCTHhIX nOJIyqaeM HOBYIO cHcTeMY

(25)

ITpHBeJleHHe CHCTeMhl (24) K CHCTeMe (25) BhInOJIH5IeTC5I Ha ypoBHe 3JIeMeHTHhIX

MaTpHU H BeKTopOB TO eCTh CHCTeMa (25) COCTaBJI5IeTC5I H3 ypaBHeHHH onpeJleshy

JIeHHhIX Ha 3JIeMeHTax BToporo THna H HMelOIUHX BHJl

rJle all a12 a22 a21 - 3JIeMeHTHhie MaTpHUhl PI - BeKTOp HeH3BeCTHhIX COOTshy

BeTCTBYIOIUHH rpaHHqHhlM H ueHTpaJIhHOMY Y3JIaM 3JIeMeHTa bi b2 3JIeMeHTHhle

BeKTophI npaBOH qaCTH

OTHOCHTeJIhHO nOCJIeJlHerO mara aJIrOpHTMa CJIeJlyeT 3aMeTHTh qTO nOCTposhy

eHHe yrOqHeHHOro pa36HeHH5I qeThlpexyroJIhHHKOB rpaHHUhI r 3aBHCHT OT Tpyshy

JlOeMKOCTR BhlqHCJIeHH5I BeKTopa C H eCJIH BhlqHCJIeHHe BeKTopa C Tpe6yeT

OTHOCHTeJIhHO 60JIhlllHX 3aTpaT npoueccopHoro BpeMeHH 3BM TO )KeJIaTenhHO

npOBOJlHTh pa36HeHH5I B COOTBeTCTBHH co cxeMOH pHC 3 ITpHBeJleM JlBa MOJleJIhHhIX npHMepa JlJI5I JleMOHCTpaurIH CBOHCTB npeJlJIaraeshy

MOro aJIropHTMa ITpHMep nepBhlH nycTh eJlHHHqHhIH Ky6 E3 [-1 1] xl] x 1] pa36HT Ha 6 nHpaMHJl Ka)K)Ja5I H3 KOTOPhIX HMeeT BepllIHHy B ueHshy

Tpe Ky6a a OCHOBaHHeM - rpaHh Ky6a B Ka)KJlOH nHPaMHJle 33JlaHa CPYHKUH5I

U = xrk JIH60 U x~k JIH60 U = x5k (k HaTYPaJIhHoe) TaK qTO Ha rpaHHue

Ky6a U = 1 H VrU = O ECJIH B OJlHOH yrJIOBOH TOqKe E3 33JlaTh TOqHOe 3Haqeshy

HHe CPYHKUHH U a Ha rpaH5IX - VrU TO TOqHOe peWeHHe Ha nOBepXHOCTH Ky6a

OnpeJleJUleTC5I 3a OJlHy HTepaUHIO npH HCnOJIh30BaHHH caMOro rpy6oro pa36HeHH5I

middot18

npHMep BTOPOH nYCTb B yrJIOBOH TOtIKe E3 3anaHO TOtIHOe 3HatIeHHe CP)HKUHH

u= In((l l-Xl+X~+x5) aHa rpamIX 33llaH Vru TOf)]aAJUI onpe)]eJIeHH5I npHshy

6JIHJKeHHOfO perneHH5I )]OCTaTOtIHO O)]HOH HTepaUHH CXO)]HMOCTb npH6JIHXeHHbIX

pellIeHHH K TOtIHOMY npe)]CTaBneHa B Ta6JIHue 1 CJIe)]yeT TaKJKe OTMeTHTb tITO

ecnH 3anaTb B O)]HOM YlJle E3 3HatIeHHe centYHKUHH U H Ha rpamlx Ky6a V rU TO

TOtIHOe perneHHe 3a O)]HY HTepaUHIO npH caMOM rpy60M pa36HeHHH HaXO)]HTC5I AJUI

nI06bIX centYHKUHH U HMeIOlUHX BH)] MHOfOtIJIeHOB BXO)J5llUHX B 6a3HCHbIe centYHKUHH

npH6JIHJKeHHoro perneHH5I TaK npH Hcnonb30BaHHH TOJIbKO 3JIeMeHTOB BTOpOro

THna (KJIacc 2) TOtIHOe perneHHe 6y)]eM HMeTb )])151 MHOfOtIJIeHOB BH)]a xiI xrxr f)]e 1171 2 a TaKJKe xtxt2 H xtx~2 f)]e 1 kl k2 3 kl =I k2 CooTBeTshy

CTBeHHO npH HCnOJIb30BaHHH TOJIbKO 3JIeMeHTOB TpeTbefo THna (KJIacc I) TOtIHOe

pellIeHHe 6y)]eM nOJIyqaTb eCJIH U 6y)]eT MHOfOtIJIeHOM BH)]a xilxrxr npH ITI 4 HJIH xtxk2 f)]e 1 k 1k2 3 kl =I k2

Ta6J1Hua 1

HOPMbI

lIu shy cPmIIC(r) max Cl

Ilu cPmllC(f) max Cl

Ilu - centmIIC(r) max Cl

32x32

2690E-08 955IE-07

1181E-08 5456E-05

2690E-08 955IE-07

3 llpHMeHeHHe B MarHHTocraTHKe

IIycTh 0- 06beM 3aHHMaeMhIH H30TponHOH MafHHTHOH cpe)]OH a 0 1- o6heM C

HeMafHHTHOH cpe)]oH TOf)]a ypaBHeHH5I MafHHTOCTaTHKH HMeIOT Clle)WIOIIJHH BH)]

[1011] 113 0 1 x fi = 0 13 JLJLofi x EO

113=0 Vxfi=J 13 -LO fi x E OJ

Inmiddot 0 [11 x fi]r 0 lim 1131 =0 lim lill =0Ixl-+oo IxJ-+oo

r)]e jj ii - BeKTopbI HH)]YKUHH 11 Hanp5l)KeHHOCTH MafHHTHOfO nOJl5l 11- MarHHTHasI

npOHHuaeMOCTb cpe)]bI - MO)KeT 33llaBaTbC5I KaK centYHKUH5I OT Ifi I HJIH KaK centYHK UH5I

19

OT IHI f-Lo- MaI1lHTHasI nOCTOHHHaH J - 3a11aHHaH BeKTop-cpy-HKllHH nJIOTHOCTH

TOKa OTJIH4HaH OT HyJIH B HeKOTopOH 06JIaCTH Os COl bull B CJIyqae OTCYTCTBHH

MaI1lHTHOH Cpe)]bl pemeHHe 3a11a4H HMeeT BH)]

IloSTOMY npH BBe)]eHHH CKaJJHPHblX nOTeHIlHaJJOB IJ cp no cpopMyJIaM 7 IJ B MaI1lHTHOH cpe)]e H 7 cp - jj BHe MarnHTHOH cpe)]bl nOJIyqalOT [12] ]] cJIe)]yshy

lOUJyID KpaeByID 3a11a4y

7 f-L(I7IJI)7IJ 0 x E 0 7 7cp 0 x E 0 1 (26)

iimiddot (f-L(I7IJI)7IJ 7cp jjS) = 0 IJ cp + ltpS x E r C YCJIOBHeM ltp ~ 0 Ha )]OCTaT04HO Y)]aJJeHHOH OT MarnHTHoH cpe)]bl rpaHHlle 3)]ecb

nOTeHIlHaJJ cps KaK ripe)JJ10)KeHO B pa60Te [12] Bbl4HCJIHeTCH no cpopMyJIe

fY Six Hrdt xY r (27)

r)]e cpS(xo) = ltpg - 3a11aHHOe 3Ha4eHHe B T04Ke Xo r O)]HaKo comaCHO

JIeMMe H3 [13] BeKTop jjS npe)]cTaBHM B BH)]e rpa11HeHTa B JII060M 06beMe me

7 jjS 0 H 7 x jjs O IlosToMY 3a11a4Y (26) CJIe)]yeT )]onOJIHHTb ypaBHeHHeM

7cps _jjs(x) x E r (28)

C YCJ10BHeM ltps (xo) ltpg H HaXO)]HTb pemeHHe STOro ypaBHeHHH C nOMOlllblO

CHCTeMbl aHaJJorn4HOH (12)

t (lif J7rNj 7rNidS = J(-jj~ (29) J=1 r r 1

r]le i 12 m MaTpHlla gtTOH CHCTeMbI HBJIHeTCH cHMMeTpWqHOH H nOJIOJKWshy

TeJIbHO onpe)]eJIeHHOH nosToMY )JJ1H pemeHuH (29) MOJKHO HCnOJIb30BaTb H3BecTshy

Hble scpcpeKTuBHble MeTO)]bI HanpuMep MeTO)] HenOJIHoro pa3JIO)KeHHH XOJIecshy

CKoro C COnpHJKeHHbIMH rpa11UeHTaMu [14] B OTJIH4He OT cpOpMyJIbl (27) B cpopshy

MyJIe (29) UCnOJIb3yeTcH )]ByMepHoe uHTerpHpoBaHHe no nosepxHocTH r 4TO lJaeT

60JIee 6bICTPOCxo)]HlllHecH npH6JIH)I(eHHH )JJ1H nOTeHUHaJJa cps CJIe)]yeT TaKJKe

OTMeTUTb 4TO )JJ1H BbI4UCJIeHUH cps no cpopMyJIe (27) 04eHb Ba)l(HbI HanpaBJIeHUH

uHTerpHpoBaHHH nOCKOJIbKY STa cpopMyJIa MOJKeT HaKaTIJIHBaTb omH6KH B03HHshy

KalOlllHe npH Bbl4HCJIeHHH BeKTopa jjs no Ky6aTypHbIM cpOpMyJIaM B npoTuBoshy

nOJIOJKHOCTb STOMY npollecc pemeHHH CHCTeMbl (29) He 3aBHCHT OT HanpaBJIeHUH

HHTerpupOBaHHH U He HaKanJIHBaeT omH6KH KOTopble B03HUKalOT npH BbI4UCJIeshy

HHH no Ky6aTYPHbiM cpopMyJIaM BeKTopa jjS

20

Pa3pa60TaHHbIH aJIrOpHTM peaJIH30BaH B paMKaX KOMnneKCa npOrpaMM

MSFE3D [IS H anp06HpOBaJIC~ npH paclIeTax TpexMepHblx MarHHTHblX nOJleH

HeCKOJlbKHX MarHHTHblX CHCTeM OnHllleM H3 HHX l1Be HaH60Jlee XapaKTepHble B KallCCTBe nepBoro npHMepa npHBeueM cneKTpoMeTpHlIecKHH MarHHT l1HnOJlbHOrO

THna He06xol1HMOCTb paclIeTa KOToporo B03HHKna npH nOl1rOTOBKe npoeKTa qmshy

3HlIeCKoro 3KcnepHMeHTa C nompH30BaHHoH MHllleHbJO B HT3lt1gt ( MocKBa) [16)

MarHHT HMeeT pa3Mepbl 3 4 xl 9 x 4 M3 ero 06ma~ nnomaub rpaHHUbl 68

M2 ero KOHqmrypaUHSI l1aHa B pa60Te [17] Y 3Toro MarmlTa TOJlbKO Ol1Ha nnocshy

KOCTb cHMMeTpHH oc06eHHocTbJO TaK)I(e ~BJ1~JOTC~ meml B BepxHeH lIaCTH Marshy

HHTonpoBoua TIPH BblllHcneHHH nOTeHUHaJIa ltps no nOBepxHocTH pa3ueIDIJOmeH

MarHHTHYJO H HeMarHHTHYJO cpeuy npH nOMomH cpopMynbI (27) nH60 B03HHKaJIH

3HallHTenbHbIe oWH6KH nH60 npouecc BblllHcneHH~ Tpe60BaJI CJlHWKOM 60JlbWOrO

BpeMeHH TI03TOMY B pa60Te [17] HHTerpHpoBaHHe no cpopMyne (27) npoBouHnOcb

6e3 YlIeTa meJleH B MarHHTOnpOBoue liTO npHBouHJlO K uonOJlHHTeJlbHbIM BblllHCJlHshy

TenbHbIM 3aTpaTaM Hcnonb30BaHHe aJIrOpHTMa onHcaHHoro B HaCTo~meH pa60Te

cpa3y )I(e uaeT xopOlllHe npH6JlH)I(eHH~ IJ)l~ nOTeHUHaJIa OT 06MOTKH TIPH 3TOM

um uOCTH)I(eHH~ He06xouHMOH TOllHOCTH Mouenb 06MOTKH uon)l(Ha 6blTb TaKOH

lIT06bI ypaBHeHH~

V iiS(]) = 0 V x iiS(x) 0 x E R3Os

BbmOJlH~JlHCb uOCTaTOllHO TOllHO

B KalleCTBe BTOpOro xapaKTepHoro npHMepa npHBeueM MarHHT cpH3HlIeCKoro

3KcnepHMeHTa L3 [18J KOTOPblH npeunOJlaraeTC~ HCnOJlb30BaTb B 3KcnepHMeHTe

ALICE (UEPH )l(eHeBa) [19] MarHHT L3 ~BJ1~eTC~ MarHHTOM COJleHOHuHoro

THna OH HMeeT pa3Mepbl 158 x 158 x 143 M3 KaK nOKa3aHO B pa60Tax

[2021 J HecMoTpSl Ha CnO)l(HYJO C60PHYJO KOHCTPYKUHJO 3Toro MarHHTa um pacshy

lIeTa C uocTaTollHoH TOllHOCTbJO ero MarHHTHoro non~ MO)l(HO HCnOJlb30BaTb 132

cHMMeTpHlIHYJO lIaCTb ero KOHcpmypaUHl1 Ha pHC4 npeucTaBJleHa 18 CHMMeshy

TPHlIH~ lIaCTb MouenH gtToro MarHHTa a B Ta6nHue 2 npHBeueHbl pe3YJlbTaTbi

BblllHcneHH~ nOTeHUHana 06MOTKH ltps Ha Tpex BJ10)l(eHHbIX ceTKax npH lICnOJlb30shy

BaHHH pa3Hbix THnOB 3JleMeHTOB H auanTHBHoro anropHTMa B Ta6nHlle lIepe3

centm2 0603HallaeTc~ npH6nH)I(eHHOe 3Ha1leHne nOTeHUHana HaHueHHoe Ha YKa3aHshy

HOM H Ha npeubluymeH ceTKax a OTHOCHTeJlbH~ 01llH6Ka BblllHCJl~eTCSI B Y3Jlax

npeubluymeH ceTKH KaK BHuHO H3 Ta6JlHUbi 2 Y)l(e npH caMOM rpy60M pa36HeHHH

cynep3JleMeHTOB rpaHHUbl no auanTHBHoMY anropHTMY nOJlYllaeTcSI nOTeHllHaJI C

TpeM~ BepHblMH 3HaKaMH TIPH 3TOM HCnOJlb3yeTcSI npHMepHo O)lHHaKOBoe lIHcno

3neMeHTOB BTOpOro H TpeTbero THnOB

ABTopbl 6JlarouapHbl ucp-MH BBPblJlbUOBy 3a nJlOllOTBOpHoe CTHMYJlHPYJOshy

mee COTpYl1HHlIeCTBO OH IOJ1IlaWeB 6JlarouapeH ucp-MH ACBouonbSlHoBY 3a

21

OT IHI f-Lo- MaI1lHTHasI nOCTOHHHaH J - 3a11aHHaH BeKTop-cpy-HKllHH nJIOTHOCTH

TOKa OTJIH4HaH OT HyJIH B HeKOTopOH 06JIaCTH Os COl bull B CJIyqae OTCYTCTBHH

MaI1lHTHOH Cpe)]bl pemeHHe 3a11a4H HMeeT BH)]

IloSTOMY npH BBe)]eHHH CKaJJHPHblX nOTeHIlHaJJOB IJ cp no cpopMyJIaM 7 IJ B MaI1lHTHOH cpe)]e H 7 cp - jj BHe MarnHTHOH cpe)]bl nOJIyqalOT [12] ]] cJIe)]yshy

lOUJyID KpaeByID 3a11a4y

7 f-L(I7IJI)7IJ 0 x E 0 7 7cp 0 x E 0 1 (26)

iimiddot (f-L(I7IJI)7IJ 7cp jjS) = 0 IJ cp + ltpS x E r C YCJIOBHeM ltp ~ 0 Ha )]OCTaT04HO Y)]aJJeHHOH OT MarnHTHoH cpe)]bl rpaHHlle 3)]ecb

nOTeHIlHaJJ cps KaK ripe)JJ10)KeHO B pa60Te [12] Bbl4HCJIHeTCH no cpopMyJIe

fY Six Hrdt xY r (27)

r)]e cpS(xo) = ltpg - 3a11aHHOe 3Ha4eHHe B T04Ke Xo r O)]HaKo comaCHO

JIeMMe H3 [13] BeKTop jjS npe)]cTaBHM B BH)]e rpa11HeHTa B JII060M 06beMe me

7 jjS 0 H 7 x jjs O IlosToMY 3a11a4Y (26) CJIe)]yeT )]onOJIHHTb ypaBHeHHeM

7cps _jjs(x) x E r (28)

C YCJ10BHeM ltps (xo) ltpg H HaXO)]HTb pemeHHe STOro ypaBHeHHH C nOMOlllblO

CHCTeMbl aHaJJorn4HOH (12)

t (lif J7rNj 7rNidS = J(-jj~ (29) J=1 r r 1

r]le i 12 m MaTpHlla gtTOH CHCTeMbI HBJIHeTCH cHMMeTpWqHOH H nOJIOJKWshy

TeJIbHO onpe)]eJIeHHOH nosToMY )JJ1H pemeHuH (29) MOJKHO HCnOJIb30BaTb H3BecTshy

Hble scpcpeKTuBHble MeTO)]bI HanpuMep MeTO)] HenOJIHoro pa3JIO)KeHHH XOJIecshy

CKoro C COnpHJKeHHbIMH rpa11UeHTaMu [14] B OTJIH4He OT cpOpMyJIbl (27) B cpopshy

MyJIe (29) UCnOJIb3yeTcH )]ByMepHoe uHTerpHpoBaHHe no nosepxHocTH r 4TO lJaeT

60JIee 6bICTPOCxo)]HlllHecH npH6JIH)I(eHHH )JJ1H nOTeHUHaJJa cps CJIe)]yeT TaKJKe

OTMeTUTb 4TO )JJ1H BbI4UCJIeHUH cps no cpopMyJIe (27) 04eHb Ba)l(HbI HanpaBJIeHUH

uHTerpHpoBaHHH nOCKOJIbKY STa cpopMyJIa MOJKeT HaKaTIJIHBaTb omH6KH B03HHshy

KalOlllHe npH Bbl4HCJIeHHH BeKTopa jjs no Ky6aTypHbIM cpOpMyJIaM B npoTuBoshy

nOJIOJKHOCTb STOMY npollecc pemeHHH CHCTeMbl (29) He 3aBHCHT OT HanpaBJIeHUH

HHTerpupOBaHHH U He HaKanJIHBaeT omH6KH KOTopble B03HUKalOT npH BbI4UCJIeshy

HHH no Ky6aTYPHbiM cpopMyJIaM BeKTopa jjS

20

Pa3pa60TaHHbIH aJIrOpHTM peaJIH30BaH B paMKaX KOMnneKCa npOrpaMM

MSFE3D [IS H anp06HpOBaJIC~ npH paclIeTax TpexMepHblx MarHHTHblX nOJleH

HeCKOJlbKHX MarHHTHblX CHCTeM OnHllleM H3 HHX l1Be HaH60Jlee XapaKTepHble B KallCCTBe nepBoro npHMepa npHBeueM cneKTpoMeTpHlIecKHH MarHHT l1HnOJlbHOrO

THna He06xol1HMOCTb paclIeTa KOToporo B03HHKna npH nOl1rOTOBKe npoeKTa qmshy

3HlIeCKoro 3KcnepHMeHTa C nompH30BaHHoH MHllleHbJO B HT3lt1gt ( MocKBa) [16)

MarHHT HMeeT pa3Mepbl 3 4 xl 9 x 4 M3 ero 06ma~ nnomaub rpaHHUbl 68

M2 ero KOHqmrypaUHSI l1aHa B pa60Te [17] Y 3Toro MarmlTa TOJlbKO Ol1Ha nnocshy

KOCTb cHMMeTpHH oc06eHHocTbJO TaK)I(e ~BJ1~JOTC~ meml B BepxHeH lIaCTH Marshy

HHTonpoBoua TIPH BblllHcneHHH nOTeHUHaJIa ltps no nOBepxHocTH pa3ueIDIJOmeH

MarHHTHYJO H HeMarHHTHYJO cpeuy npH nOMomH cpopMynbI (27) nH60 B03HHKaJIH

3HallHTenbHbIe oWH6KH nH60 npouecc BblllHcneHH~ Tpe60BaJI CJlHWKOM 60JlbWOrO

BpeMeHH TI03TOMY B pa60Te [17] HHTerpHpoBaHHe no cpopMyne (27) npoBouHnOcb

6e3 YlIeTa meJleH B MarHHTOnpOBoue liTO npHBouHJlO K uonOJlHHTeJlbHbIM BblllHCJlHshy

TenbHbIM 3aTpaTaM Hcnonb30BaHHe aJIrOpHTMa onHcaHHoro B HaCTo~meH pa60Te

cpa3y )I(e uaeT xopOlllHe npH6JlH)I(eHH~ IJ)l~ nOTeHUHaJIa OT 06MOTKH TIPH 3TOM

um uOCTH)I(eHH~ He06xouHMOH TOllHOCTH Mouenb 06MOTKH uon)l(Ha 6blTb TaKOH

lIT06bI ypaBHeHH~

V iiS(]) = 0 V x iiS(x) 0 x E R3Os

BbmOJlH~JlHCb uOCTaTOllHO TOllHO

B KalleCTBe BTOpOro xapaKTepHoro npHMepa npHBeueM MarHHT cpH3HlIeCKoro

3KcnepHMeHTa L3 [18J KOTOPblH npeunOJlaraeTC~ HCnOJlb30BaTb B 3KcnepHMeHTe

ALICE (UEPH )l(eHeBa) [19] MarHHT L3 ~BJ1~eTC~ MarHHTOM COJleHOHuHoro

THna OH HMeeT pa3Mepbl 158 x 158 x 143 M3 KaK nOKa3aHO B pa60Tax

[2021 J HecMoTpSl Ha CnO)l(HYJO C60PHYJO KOHCTPYKUHJO 3Toro MarHHTa um pacshy

lIeTa C uocTaTollHoH TOllHOCTbJO ero MarHHTHoro non~ MO)l(HO HCnOJlb30BaTb 132

cHMMeTpHlIHYJO lIaCTb ero KOHcpmypaUHl1 Ha pHC4 npeucTaBJleHa 18 CHMMeshy

TPHlIH~ lIaCTb MouenH gtToro MarHHTa a B Ta6nHue 2 npHBeueHbl pe3YJlbTaTbi

BblllHcneHH~ nOTeHUHana 06MOTKH ltps Ha Tpex BJ10)l(eHHbIX ceTKax npH lICnOJlb30shy

BaHHH pa3Hbix THnOB 3JleMeHTOB H auanTHBHoro anropHTMa B Ta6nHlle lIepe3

centm2 0603HallaeTc~ npH6nH)I(eHHOe 3Ha1leHne nOTeHUHana HaHueHHoe Ha YKa3aHshy

HOM H Ha npeubluymeH ceTKax a OTHOCHTeJlbH~ 01llH6Ka BblllHCJl~eTCSI B Y3Jlax

npeubluymeH ceTKH KaK BHuHO H3 Ta6JlHUbi 2 Y)l(e npH caMOM rpy60M pa36HeHHH

cynep3JleMeHTOB rpaHHUbl no auanTHBHoMY anropHTMY nOJlYllaeTcSI nOTeHllHaJI C

TpeM~ BepHblMH 3HaKaMH TIPH 3TOM HCnOJlb3yeTcSI npHMepHo O)lHHaKOBoe lIHcno

3neMeHTOB BTOpOro H TpeTbero THnOB

ABTopbl 6JlarouapHbl ucp-MH BBPblJlbUOBy 3a nJlOllOTBOpHoe CTHMYJlHPYJOshy

mee COTpYl1HHlIeCTBO OH IOJ1IlaWeB 6JlarouapeH ucp-MH ACBouonbSlHoBY 3a

21

Pa3pa60TaHHbIH aJIrOpHTM peaJIH30BaH B paMKaX KOMnneKCa npOrpaMM

MSFE3D [IS H anp06HpOBaJIC~ npH paclIeTax TpexMepHblx MarHHTHblX nOJleH

HeCKOJlbKHX MarHHTHblX CHCTeM OnHllleM H3 HHX l1Be HaH60Jlee XapaKTepHble B KallCCTBe nepBoro npHMepa npHBeueM cneKTpoMeTpHlIecKHH MarHHT l1HnOJlbHOrO

THna He06xol1HMOCTb paclIeTa KOToporo B03HHKna npH nOl1rOTOBKe npoeKTa qmshy

3HlIeCKoro 3KcnepHMeHTa C nompH30BaHHoH MHllleHbJO B HT3lt1gt ( MocKBa) [16)

MarHHT HMeeT pa3Mepbl 3 4 xl 9 x 4 M3 ero 06ma~ nnomaub rpaHHUbl 68

M2 ero KOHqmrypaUHSI l1aHa B pa60Te [17] Y 3Toro MarmlTa TOJlbKO Ol1Ha nnocshy

KOCTb cHMMeTpHH oc06eHHocTbJO TaK)I(e ~BJ1~JOTC~ meml B BepxHeH lIaCTH Marshy

HHTonpoBoua TIPH BblllHcneHHH nOTeHUHaJIa ltps no nOBepxHocTH pa3ueIDIJOmeH

MarHHTHYJO H HeMarHHTHYJO cpeuy npH nOMomH cpopMynbI (27) nH60 B03HHKaJIH

3HallHTenbHbIe oWH6KH nH60 npouecc BblllHcneHH~ Tpe60BaJI CJlHWKOM 60JlbWOrO

BpeMeHH TI03TOMY B pa60Te [17] HHTerpHpoBaHHe no cpopMyne (27) npoBouHnOcb

6e3 YlIeTa meJleH B MarHHTOnpOBoue liTO npHBouHJlO K uonOJlHHTeJlbHbIM BblllHCJlHshy

TenbHbIM 3aTpaTaM Hcnonb30BaHHe aJIrOpHTMa onHcaHHoro B HaCTo~meH pa60Te

cpa3y )I(e uaeT xopOlllHe npH6JlH)I(eHH~ IJ)l~ nOTeHUHaJIa OT 06MOTKH TIPH 3TOM

um uOCTH)I(eHH~ He06xouHMOH TOllHOCTH Mouenb 06MOTKH uon)l(Ha 6blTb TaKOH

lIT06bI ypaBHeHH~

V iiS(]) = 0 V x iiS(x) 0 x E R3Os

BbmOJlH~JlHCb uOCTaTOllHO TOllHO

B KalleCTBe BTOpOro xapaKTepHoro npHMepa npHBeueM MarHHT cpH3HlIeCKoro

3KcnepHMeHTa L3 [18J KOTOPblH npeunOJlaraeTC~ HCnOJlb30BaTb B 3KcnepHMeHTe

ALICE (UEPH )l(eHeBa) [19] MarHHT L3 ~BJ1~eTC~ MarHHTOM COJleHOHuHoro

THna OH HMeeT pa3Mepbl 158 x 158 x 143 M3 KaK nOKa3aHO B pa60Tax

[2021 J HecMoTpSl Ha CnO)l(HYJO C60PHYJO KOHCTPYKUHJO 3Toro MarHHTa um pacshy

lIeTa C uocTaTollHoH TOllHOCTbJO ero MarHHTHoro non~ MO)l(HO HCnOJlb30BaTb 132

cHMMeTpHlIHYJO lIaCTb ero KOHcpmypaUHl1 Ha pHC4 npeucTaBJleHa 18 CHMMeshy

TPHlIH~ lIaCTb MouenH gtToro MarHHTa a B Ta6nHue 2 npHBeueHbl pe3YJlbTaTbi

BblllHcneHH~ nOTeHUHana 06MOTKH ltps Ha Tpex BJ10)l(eHHbIX ceTKax npH lICnOJlb30shy

BaHHH pa3Hbix THnOB 3JleMeHTOB H auanTHBHoro anropHTMa B Ta6nHlle lIepe3

centm2 0603HallaeTc~ npH6nH)I(eHHOe 3Ha1leHne nOTeHUHana HaHueHHoe Ha YKa3aHshy

HOM H Ha npeubluymeH ceTKax a OTHOCHTeJlbH~ 01llH6Ka BblllHCJl~eTCSI B Y3Jlax

npeubluymeH ceTKH KaK BHuHO H3 Ta6JlHUbi 2 Y)l(e npH caMOM rpy60M pa36HeHHH

cynep3JleMeHTOB rpaHHUbl no auanTHBHoMY anropHTMY nOJlYllaeTcSI nOTeHllHaJI C

TpeM~ BepHblMH 3HaKaMH TIPH 3TOM HCnOJlb3yeTcSI npHMepHo O)lHHaKOBoe lIHcno

3neMeHTOB BTOpOro H TpeTbero THnOB

ABTopbl 6JlarouapHbl ucp-MH BBPblJlbUOBy 3a nJlOllOTBOpHoe CTHMYJlHPYJOshy

mee COTpYl1HHlIeCTBO OH IOJ1IlaWeB 6JlarouapeH ucp-MH ACBouonbSlHoBY 3a

21

1 I 1 j

0 0

1

0 0

1

-1

1 1 j

0 4P 0

~ 1 1

0 0

-1

FMc 1 0 - )3JIbI 3neMeHTa nepBoro FMc 2 0 - )3JIbJ 3J1eMeHTa BTOpOro

Tuna 0 - )3JIbI 3neMeHTa qeTBePToro Tuna bull - )3JIbl 3neMeHTa TpeTbero Tuna

Tuna

lll

1

0 0

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22

PIIC 4 18 CIiMMeTPllqHruI qaCTb MOLleJIH MarHHTa L3 1 - o6MoTKa 2 -MafHIITOnpOBOLl

23

nOMo~b B pa60Te BO BpeMSI nepBoM KOMaH)lIfpOBKM B UEPH ABTopbl Bblpa)l(alOT 6naronapHocTb PltlgtltlgtH 3a nOMep)l(KY (rpaHTbl 01-01-00726 M 99-01-01103)

TaWIHua 2

max I(centm2 - centm)centml maxcl

3neMeHTbi BTOpOro TMna

036841E-04 013678E-03

max I(centm2 centm)centml 074988E-04 maxcl 041298E-02 016093E-02

014891 E-04 056061E-03

ManTMBHbIM anropMTM

max I(centm2 - centm)centml 074814E-04 maxcl 041250E-02 016097E-02

0 18748E-04 047046E-03

JIIITEPATYPA

1 ERank IBabuska An expert system for the optimal mesh design in the hpshyversion of the finite element method Int j numer methods eng v24 pp 2087-2106 1987

2 LDemkowicz JTOden WRachowicz and OHardy Toward a universal h-p adaptive finite element strategy Part 1 Constrained approximation and data structure Compo meth appl mech eng v77 1989 pp79-112

3 Adaptive Finite and Boundary Element Methods (Ed CABrebbia and MHAliabadi) Wessex Institute of Technology UK 1992

4 MKGeorges MSShepard Automated adaptive two-dimensional system for the hp-version of the finite element method Int j num meth eng v 32 1991 pp867-893

5 CRIEmson JSimkin CWTrowbridge A status report on electromagnetic field computation IEEE Trans mag v30 N 4 1994 pp1533-1540

6 ltlgtCMlpne MeTon KOHeQHbIX 3neMeHTOB)lJUI 3J1JIMnTMQeCKMX 3al(aQ M MMP 1980

24

7 XraeBcKHM Krperep K3axapHac HeJlHHeMHble onepaTopHble ypaBHeHH5I H

onepaTopHble llHqqepeHUHaJlbHble ypaBHeHH5I M MHP 1978

8 HnMbICOBCKHX HHTepnOJl5lUHOHHbie Ky6aTYPHbre q0PMYJlbl M HaYKa

1981

9 rHMapLIYK BHAroWKOB BBe)leHHe B npoeKulloHHO-CeTOYHble MeTO)lbl M

HaYKa 1981

10 n)l(ACTpeTToH TeoplISl )JleKTpOMarHenoMa M rOCTeXH3)laT 1948

11 3AAMP5lH EnIKlIllKOB ABltlgte)lOpOB 6HXopOMCKJlM HAWeJlaeB

HnlOllHH OHlOJl)laWeB 1JllcneHHbre anropHTMbl paCyeTa MafHHTHblX CIIshy

CTeM YCKopHTendi 3ap5l)KeHHblX Llacnm ltlgtu3HKa )neMeHTapHblx yaCTIIU II

aToMHOro 51)lpa 1990 T21 Bblnl c 251-307

12 ISimkin CWTrowbridge Three dimensional non-linear electromagnetic

field computations using scalar potentials Proc lEE 1980 vo1127 PrB

N6 pp 368-374

13 EnJKH)lKOB MEIOJl)laWeBa HnIO)llIH OHIOn)laWeB MaTeMaTlfLlecKoe

MOllemlpOBamle npocTpaHcTBeHHoro MafHHTHOfO non51 cneKTpOMeTpl1 LlecKoro

MarHHTa Cn-40 OHSIH PI 1-94-160 ny6Ha 1994

14 IAMeijerink HAvan der Vorst An Iterative Solution for Linear Systems

of which the Coefficient Matrix is a Symmetric M-Matrix Math Comput

v 31 1977 pp 148-162

15 MElOJlllaWeBa OHIOn)1aWeB Ko~mneKc nporpaMM MSFE3D )1)151 paCyeTOB

npOCTpaHCTBeHHblX MarHHTOCTanlLleCKHX noneiL BepcHSI 12 OH5IH P II 94-202 ny6Ha 1994

16 HrArteKceeB BBPblJlbUOB Ii )lp HCCne)lOBaHlfe nonSlpH30BaHHbix napashy

MeTpoB B 6HHapHbIx peaKuHSlx pO)l()lemul cTpaHHbIx yaCTIlU 7rp 1-A(E) Ha YCKopHTeJle HT3ltJgt (npellJlO)KeHHe )KcnepHMeHTa) rnu PltJgt I1T3ltJgt 41-97

M OCKBa 1997

17 PHUaBJleTWHH EnJKHllKoB BBKymIKoB BBPblJlbUOB OHlOJl)lLlWeB

MEIOJ1)JameBa KOMnblOTepHaSl MO)1eJlb MarHHTa llJlSl npOeKTI )KcneplIMeHTa

c nOJlSlpH30BaHHoH MHweHblO B HT3ltlgt PacLleT OCHOBHblX BapllaHToB 6e3

nomlpH3YIOlllHX HaKOHelIHHKOB OH5IH P 11-98-351 ny6HI 1998

18 BAdeva et al The construction of the L3 experiment Nuclear instruments

and methods A v289 1990 pp35-102

25

19 ALICE Technical Proposal CERNfLHCC 95-71 1995 The Forward Muon Spectrometer of ALICE (Addendum to the ALICE technical proposal) CERNLHCC 96-32 1996

20 PGAkishin ASVodopianov IVPuzynin YuAShishov MBYuldashcva 01 Yu1dashev Computing Models of the L3 Magnet and Dipole Magnet for the ALICE Experiment ALICE 96-06 Int NoteMag 4 April 1996

21 I1rAKHllHiH ACBouonhSlHoB I1BI1y3hIHHH IOAWHIllOB MliIOJIuashyllIeBa OI1IOJ1LlallIeB MoueJIHpoBaHHe nonSi MIOOHHoro ueTeKTopa YCTaHoBKH ALICE XV COBemaHHe no YCKopHTeJISIM 3apSilKeHHhlx qaCTHU C60PHHK uoshyKJI3LlOB rnu PltIgt I1ltIgtB3 I1POTBHHO 1996 T 2 c182-185

I10JIyqeHo 19 anpeJI5I 2002 r

PIIC 4 18 CIiMMeTPllqHruI qaCTb MOLleJIH MarHHTa L3 1 - o6MoTKa 2 -MafHIITOnpOBOLl

23

nOMo~b B pa60Te BO BpeMSI nepBoM KOMaH)lIfpOBKM B UEPH ABTopbl Bblpa)l(alOT 6naronapHocTb PltlgtltlgtH 3a nOMep)l(KY (rpaHTbl 01-01-00726 M 99-01-01103)

TaWIHua 2

max I(centm2 - centm)centml maxcl

3neMeHTbi BTOpOro TMna

036841E-04 013678E-03

max I(centm2 centm)centml 074988E-04 maxcl 041298E-02 016093E-02

014891 E-04 056061E-03

ManTMBHbIM anropMTM

max I(centm2 - centm)centml 074814E-04 maxcl 041250E-02 016097E-02

0 18748E-04 047046E-03

JIIITEPATYPA

1 ERank IBabuska An expert system for the optimal mesh design in the hpshyversion of the finite element method Int j numer methods eng v24 pp 2087-2106 1987

2 LDemkowicz JTOden WRachowicz and OHardy Toward a universal h-p adaptive finite element strategy Part 1 Constrained approximation and data structure Compo meth appl mech eng v77 1989 pp79-112

3 Adaptive Finite and Boundary Element Methods (Ed CABrebbia and MHAliabadi) Wessex Institute of Technology UK 1992

4 MKGeorges MSShepard Automated adaptive two-dimensional system for the hp-version of the finite element method Int j num meth eng v 32 1991 pp867-893

5 CRIEmson JSimkin CWTrowbridge A status report on electromagnetic field computation IEEE Trans mag v30 N 4 1994 pp1533-1540

6 ltlgtCMlpne MeTon KOHeQHbIX 3neMeHTOB)lJUI 3J1JIMnTMQeCKMX 3al(aQ M MMP 1980

24

7 XraeBcKHM Krperep K3axapHac HeJlHHeMHble onepaTopHble ypaBHeHH5I H

onepaTopHble llHqqepeHUHaJlbHble ypaBHeHH5I M MHP 1978

8 HnMbICOBCKHX HHTepnOJl5lUHOHHbie Ky6aTYPHbre q0PMYJlbl M HaYKa

1981

9 rHMapLIYK BHAroWKOB BBe)leHHe B npoeKulloHHO-CeTOYHble MeTO)lbl M

HaYKa 1981

10 n)l(ACTpeTToH TeoplISl )JleKTpOMarHenoMa M rOCTeXH3)laT 1948

11 3AAMP5lH EnIKlIllKOB ABltlgte)lOpOB 6HXopOMCKJlM HAWeJlaeB

HnlOllHH OHlOJl)laWeB 1JllcneHHbre anropHTMbl paCyeTa MafHHTHblX CIIshy

CTeM YCKopHTendi 3ap5l)KeHHblX Llacnm ltlgtu3HKa )neMeHTapHblx yaCTIIU II

aToMHOro 51)lpa 1990 T21 Bblnl c 251-307

12 ISimkin CWTrowbridge Three dimensional non-linear electromagnetic

field computations using scalar potentials Proc lEE 1980 vo1127 PrB

N6 pp 368-374

13 EnJKH)lKOB MEIOJl)laWeBa HnIO)llIH OHIOn)laWeB MaTeMaTlfLlecKoe

MOllemlpOBamle npocTpaHcTBeHHoro MafHHTHOfO non51 cneKTpOMeTpl1 LlecKoro

MarHHTa Cn-40 OHSIH PI 1-94-160 ny6Ha 1994

14 IAMeijerink HAvan der Vorst An Iterative Solution for Linear Systems

of which the Coefficient Matrix is a Symmetric M-Matrix Math Comput

v 31 1977 pp 148-162

15 MElOJlllaWeBa OHIOn)1aWeB Ko~mneKc nporpaMM MSFE3D )1)151 paCyeTOB

npOCTpaHCTBeHHblX MarHHTOCTanlLleCKHX noneiL BepcHSI 12 OH5IH P II 94-202 ny6Ha 1994

16 HrArteKceeB BBPblJlbUOB Ii )lp HCCne)lOBaHlfe nonSlpH30BaHHbix napashy

MeTpoB B 6HHapHbIx peaKuHSlx pO)l()lemul cTpaHHbIx yaCTIlU 7rp 1-A(E) Ha YCKopHTeJle HT3ltJgt (npellJlO)KeHHe )KcnepHMeHTa) rnu PltJgt I1T3ltJgt 41-97

M OCKBa 1997

17 PHUaBJleTWHH EnJKHllKoB BBKymIKoB BBPblJlbUOB OHlOJl)lLlWeB

MEIOJ1)JameBa KOMnblOTepHaSl MO)1eJlb MarHHTa llJlSl npOeKTI )KcneplIMeHTa

c nOJlSlpH30BaHHoH MHweHblO B HT3ltlgt PacLleT OCHOBHblX BapllaHToB 6e3

nomlpH3YIOlllHX HaKOHelIHHKOB OH5IH P 11-98-351 ny6HI 1998

18 BAdeva et al The construction of the L3 experiment Nuclear instruments

and methods A v289 1990 pp35-102

25

19 ALICE Technical Proposal CERNfLHCC 95-71 1995 The Forward Muon Spectrometer of ALICE (Addendum to the ALICE technical proposal) CERNLHCC 96-32 1996

20 PGAkishin ASVodopianov IVPuzynin YuAShishov MBYuldashcva 01 Yu1dashev Computing Models of the L3 Magnet and Dipole Magnet for the ALICE Experiment ALICE 96-06 Int NoteMag 4 April 1996

21 I1rAKHllHiH ACBouonhSlHoB I1BI1y3hIHHH IOAWHIllOB MliIOJIuashyllIeBa OI1IOJ1LlallIeB MoueJIHpoBaHHe nonSi MIOOHHoro ueTeKTopa YCTaHoBKH ALICE XV COBemaHHe no YCKopHTeJISIM 3apSilKeHHhlx qaCTHU C60PHHK uoshyKJI3LlOB rnu PltIgt I1ltIgtB3 I1POTBHHO 1996 T 2 c182-185

I10JIyqeHo 19 anpeJI5I 2002 r

nOMo~b B pa60Te BO BpeMSI nepBoM KOMaH)lIfpOBKM B UEPH ABTopbl Bblpa)l(alOT 6naronapHocTb PltlgtltlgtH 3a nOMep)l(KY (rpaHTbl 01-01-00726 M 99-01-01103)

TaWIHua 2

max I(centm2 - centm)centml maxcl

3neMeHTbi BTOpOro TMna

036841E-04 013678E-03

max I(centm2 centm)centml 074988E-04 maxcl 041298E-02 016093E-02

014891 E-04 056061E-03

ManTMBHbIM anropMTM

max I(centm2 - centm)centml 074814E-04 maxcl 041250E-02 016097E-02

0 18748E-04 047046E-03

JIIITEPATYPA

1 ERank IBabuska An expert system for the optimal mesh design in the hpshyversion of the finite element method Int j numer methods eng v24 pp 2087-2106 1987

2 LDemkowicz JTOden WRachowicz and OHardy Toward a universal h-p adaptive finite element strategy Part 1 Constrained approximation and data structure Compo meth appl mech eng v77 1989 pp79-112

3 Adaptive Finite and Boundary Element Methods (Ed CABrebbia and MHAliabadi) Wessex Institute of Technology UK 1992

4 MKGeorges MSShepard Automated adaptive two-dimensional system for the hp-version of the finite element method Int j num meth eng v 32 1991 pp867-893

5 CRIEmson JSimkin CWTrowbridge A status report on electromagnetic field computation IEEE Trans mag v30 N 4 1994 pp1533-1540

6 ltlgtCMlpne MeTon KOHeQHbIX 3neMeHTOB)lJUI 3J1JIMnTMQeCKMX 3al(aQ M MMP 1980

24

7 XraeBcKHM Krperep K3axapHac HeJlHHeMHble onepaTopHble ypaBHeHH5I H

onepaTopHble llHqqepeHUHaJlbHble ypaBHeHH5I M MHP 1978

8 HnMbICOBCKHX HHTepnOJl5lUHOHHbie Ky6aTYPHbre q0PMYJlbl M HaYKa

1981

9 rHMapLIYK BHAroWKOB BBe)leHHe B npoeKulloHHO-CeTOYHble MeTO)lbl M

HaYKa 1981

10 n)l(ACTpeTToH TeoplISl )JleKTpOMarHenoMa M rOCTeXH3)laT 1948

11 3AAMP5lH EnIKlIllKOB ABltlgte)lOpOB 6HXopOMCKJlM HAWeJlaeB

HnlOllHH OHlOJl)laWeB 1JllcneHHbre anropHTMbl paCyeTa MafHHTHblX CIIshy

CTeM YCKopHTendi 3ap5l)KeHHblX Llacnm ltlgtu3HKa )neMeHTapHblx yaCTIIU II

aToMHOro 51)lpa 1990 T21 Bblnl c 251-307

12 ISimkin CWTrowbridge Three dimensional non-linear electromagnetic

field computations using scalar potentials Proc lEE 1980 vo1127 PrB

N6 pp 368-374

13 EnJKH)lKOB MEIOJl)laWeBa HnIO)llIH OHIOn)laWeB MaTeMaTlfLlecKoe

MOllemlpOBamle npocTpaHcTBeHHoro MafHHTHOfO non51 cneKTpOMeTpl1 LlecKoro

MarHHTa Cn-40 OHSIH PI 1-94-160 ny6Ha 1994

14 IAMeijerink HAvan der Vorst An Iterative Solution for Linear Systems

of which the Coefficient Matrix is a Symmetric M-Matrix Math Comput

v 31 1977 pp 148-162

15 MElOJlllaWeBa OHIOn)1aWeB Ko~mneKc nporpaMM MSFE3D )1)151 paCyeTOB

npOCTpaHCTBeHHblX MarHHTOCTanlLleCKHX noneiL BepcHSI 12 OH5IH P II 94-202 ny6Ha 1994

16 HrArteKceeB BBPblJlbUOB Ii )lp HCCne)lOBaHlfe nonSlpH30BaHHbix napashy

MeTpoB B 6HHapHbIx peaKuHSlx pO)l()lemul cTpaHHbIx yaCTIlU 7rp 1-A(E) Ha YCKopHTeJle HT3ltJgt (npellJlO)KeHHe )KcnepHMeHTa) rnu PltJgt I1T3ltJgt 41-97

M OCKBa 1997

17 PHUaBJleTWHH EnJKHllKoB BBKymIKoB BBPblJlbUOB OHlOJl)lLlWeB

MEIOJ1)JameBa KOMnblOTepHaSl MO)1eJlb MarHHTa llJlSl npOeKTI )KcneplIMeHTa

c nOJlSlpH30BaHHoH MHweHblO B HT3ltlgt PacLleT OCHOBHblX BapllaHToB 6e3

nomlpH3YIOlllHX HaKOHelIHHKOB OH5IH P 11-98-351 ny6HI 1998

18 BAdeva et al The construction of the L3 experiment Nuclear instruments

and methods A v289 1990 pp35-102

25

19 ALICE Technical Proposal CERNfLHCC 95-71 1995 The Forward Muon Spectrometer of ALICE (Addendum to the ALICE technical proposal) CERNLHCC 96-32 1996

20 PGAkishin ASVodopianov IVPuzynin YuAShishov MBYuldashcva 01 Yu1dashev Computing Models of the L3 Magnet and Dipole Magnet for the ALICE Experiment ALICE 96-06 Int NoteMag 4 April 1996

21 I1rAKHllHiH ACBouonhSlHoB I1BI1y3hIHHH IOAWHIllOB MliIOJIuashyllIeBa OI1IOJ1LlallIeB MoueJIHpoBaHHe nonSi MIOOHHoro ueTeKTopa YCTaHoBKH ALICE XV COBemaHHe no YCKopHTeJISIM 3apSilKeHHhlx qaCTHU C60PHHK uoshyKJI3LlOB rnu PltIgt I1ltIgtB3 I1POTBHHO 1996 T 2 c182-185

I10JIyqeHo 19 anpeJI5I 2002 r

7 XraeBcKHM Krperep K3axapHac HeJlHHeMHble onepaTopHble ypaBHeHH5I H

onepaTopHble llHqqepeHUHaJlbHble ypaBHeHH5I M MHP 1978

8 HnMbICOBCKHX HHTepnOJl5lUHOHHbie Ky6aTYPHbre q0PMYJlbl M HaYKa

1981

9 rHMapLIYK BHAroWKOB BBe)leHHe B npoeKulloHHO-CeTOYHble MeTO)lbl M

HaYKa 1981

10 n)l(ACTpeTToH TeoplISl )JleKTpOMarHenoMa M rOCTeXH3)laT 1948

11 3AAMP5lH EnIKlIllKOB ABltlgte)lOpOB 6HXopOMCKJlM HAWeJlaeB

HnlOllHH OHlOJl)laWeB 1JllcneHHbre anropHTMbl paCyeTa MafHHTHblX CIIshy

CTeM YCKopHTendi 3ap5l)KeHHblX Llacnm ltlgtu3HKa )neMeHTapHblx yaCTIIU II

aToMHOro 51)lpa 1990 T21 Bblnl c 251-307

12 ISimkin CWTrowbridge Three dimensional non-linear electromagnetic

field computations using scalar potentials Proc lEE 1980 vo1127 PrB

N6 pp 368-374

13 EnJKH)lKOB MEIOJl)laWeBa HnIO)llIH OHIOn)laWeB MaTeMaTlfLlecKoe

MOllemlpOBamle npocTpaHcTBeHHoro MafHHTHOfO non51 cneKTpOMeTpl1 LlecKoro

MarHHTa Cn-40 OHSIH PI 1-94-160 ny6Ha 1994

14 IAMeijerink HAvan der Vorst An Iterative Solution for Linear Systems

of which the Coefficient Matrix is a Symmetric M-Matrix Math Comput

v 31 1977 pp 148-162

15 MElOJlllaWeBa OHIOn)1aWeB Ko~mneKc nporpaMM MSFE3D )1)151 paCyeTOB

npOCTpaHCTBeHHblX MarHHTOCTanlLleCKHX noneiL BepcHSI 12 OH5IH P II 94-202 ny6Ha 1994

16 HrArteKceeB BBPblJlbUOB Ii )lp HCCne)lOBaHlfe nonSlpH30BaHHbix napashy

MeTpoB B 6HHapHbIx peaKuHSlx pO)l()lemul cTpaHHbIx yaCTIlU 7rp 1-A(E) Ha YCKopHTeJle HT3ltJgt (npellJlO)KeHHe )KcnepHMeHTa) rnu PltJgt I1T3ltJgt 41-97

M OCKBa 1997

17 PHUaBJleTWHH EnJKHllKoB BBKymIKoB BBPblJlbUOB OHlOJl)lLlWeB

MEIOJ1)JameBa KOMnblOTepHaSl MO)1eJlb MarHHTa llJlSl npOeKTI )KcneplIMeHTa

c nOJlSlpH30BaHHoH MHweHblO B HT3ltlgt PacLleT OCHOBHblX BapllaHToB 6e3

nomlpH3YIOlllHX HaKOHelIHHKOB OH5IH P 11-98-351 ny6HI 1998

18 BAdeva et al The construction of the L3 experiment Nuclear instruments

and methods A v289 1990 pp35-102

25

19 ALICE Technical Proposal CERNfLHCC 95-71 1995 The Forward Muon Spectrometer of ALICE (Addendum to the ALICE technical proposal) CERNLHCC 96-32 1996

20 PGAkishin ASVodopianov IVPuzynin YuAShishov MBYuldashcva 01 Yu1dashev Computing Models of the L3 Magnet and Dipole Magnet for the ALICE Experiment ALICE 96-06 Int NoteMag 4 April 1996

21 I1rAKHllHiH ACBouonhSlHoB I1BI1y3hIHHH IOAWHIllOB MliIOJIuashyllIeBa OI1IOJ1LlallIeB MoueJIHpoBaHHe nonSi MIOOHHoro ueTeKTopa YCTaHoBKH ALICE XV COBemaHHe no YCKopHTeJISIM 3apSilKeHHhlx qaCTHU C60PHHK uoshyKJI3LlOB rnu PltIgt I1ltIgtB3 I1POTBHHO 1996 T 2 c182-185

I10JIyqeHo 19 anpeJI5I 2002 r

19 ALICE Technical Proposal CERNfLHCC 95-71 1995 The Forward Muon Spectrometer of ALICE (Addendum to the ALICE technical proposal) CERNLHCC 96-32 1996

20 PGAkishin ASVodopianov IVPuzynin YuAShishov MBYuldashcva 01 Yu1dashev Computing Models of the L3 Magnet and Dipole Magnet for the ALICE Experiment ALICE 96-06 Int NoteMag 4 April 1996

21 I1rAKHllHiH ACBouonhSlHoB I1BI1y3hIHHH IOAWHIllOB MliIOJIuashyllIeBa OI1IOJ1LlallIeB MoueJIHpoBaHHe nonSi MIOOHHoro ueTeKTopa YCTaHoBKH ALICE XV COBemaHHe no YCKopHTeJISIM 3apSilKeHHhlx qaCTHU C60PHHK uoshyKJI3LlOB rnu PltIgt I1ltIgtB3 I1POTBHHO 1996 T 2 c182-185

I10JIyqeHo 19 anpeJI5I 2002 r

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Zhidkov E P Yuldashev 01 Yuldasheva M B Pll-2002-87 An Adaptive Algorithm for Calculating the Function on the Lipschitz Boundary of a 3D Solid with the Given Gradient and Its Application in the Magnetostatics

The present paper is devoted to elaboration of an effective method for finding the funcshytion $ on the --ipschitz boundary r of a three-dimensional solid if a sufficiently smooth vecshytor-function G 7$ is given and if in the point Xo E r it is known that $ (xo )=$ 0 is a given value The problem of calculating the function $ in the genera] case is reduced to solving a nonlinear equation On the basis of the theory of monotone operators under suitable condishytions for the function $ the existence of a unique generalized solution of the equation and convergence of the finite-element approximations to this solution have been proved The suggested adaptive algorithm on the basis of the finite element method allows one to obshytain the solutions with a given accuracy The developed method has important application in magnetostatics because it enables one to compute with needed accuracy the coil potential on a boundary between the regions with different magnetic properties The algorithm was approved for 3D magnetic field calculations for several magnetic systems including the dishypole type spectrometer magnet and the big solenoidal magnet of the physical experiment L3 (CERN Geneva)

The investigation has been performed at the Laboratory of Information Technologies lINR

Preprint of the Joint Institute for Nuclear Research Dubna 2002