Một số phương pháp hiệu chỉnh giải bài toán đặt không chỉnh

' - • 4^' BO GIAO DUG VA DAO TAO DAI HOC QUOC GIA HA N O I n^l/CJNG DAI HOC KHOA HOC Tl/NHIEN & NGUYEN VAN HUNG M O T S O PHUONG P H A P H I E U CHJNR GIAI BAI TO AN DAT KHONG CIliMH Chuy^n nganh Ma so : Toan hoc linh (can : 1.01.07 ^\ LUAN AN PHO^ TIEN SI KHOA HOC TOAN - LY NGUdl HUdNG DAN KHOA HOC: TIEN SI - PIIAM KY AN13 i^u-^rz-?.>^:(ss.-:- ••:':;^v:;^ u.y L^/4^ fri-ij^^K /rfa JtO yniir^ 3^ -.-4 ^'%-\'"^ '^•'•.'^i•^ ' • J - v - ' ^ ' j ' • N l l ^ •; ' j > i^ir, ^:T: * Ha nOi - 1996 MUC LUC L51 noi dau CHUONG I PHUONG PHAP COMPACT THU HEP CAI BIEN §1 Ma dau §2 Cac gia thi^'t cua bai toan §3 Thuat toan compact thu hep dang Robust §4 Tnrcmg hgp kh6ng nhat nghiem §5 Tnrcmg hgp ve phai va toan tir kh6ng biet chinh xac §6 Ap dung cho phirong trinh vi phan thuong 5 12 15 17 CHUDNG II BAI TOAN TUYEN TINH KHONG CHINH TREN COMPACT YEU §1 Ma dau §2 H6i tu y^'u kh6ng gian Hilbert §3 Danh gia tinh 6n dinh cua nghiem trSn compact y6u §4 Phuang phap khai tridn ky di chat cut §5 Phuang trinh tich phan dang tich chAp §6 Danh gia diam Vs phaang phap compact thu hep cua Gaponenko 23 25 27 29 31 36 CHUONG III M O T S O PHUONG PHAP LAP - HIEU CHINH §1 Phirang phap Gauss - Newton hi6u chinh (RGN) §2 Ki^m tra di^u kien B] cua Bakushinski §3 Phuang phap hieu chinh Gasse - Newton g'an dung §4 Phirang phap Seidel - Newton hieu chinh va bai toan phi tuyS'n cong huang I - Phuomg phap Seidel - Newton (SN) vd phuomg phap Seidel - Newton hieu chinh ( RSN) II - Syc hdi tu dia phuomg cua phuomg phap RSN III - Bdi todn Men tudn hodn cho phuomg trinh Duffing - Van derpol nil - Bdi todn bien tudn hodn doi v&iphuomg trinh Van derpol 40 41 46 52 Phan ket luan Phu luc 1- Bai toan - Lcfi giai 2- Thuat giai va chuong trinh 66 67 67 72 Tai lieu tham khao gg 52 54 57 65 PHXN Ud DAU # Ngay nay, ciing v6i vice sur dung ph6 biC'n may tinh, loan hoc cang dirge ung dung rong rai cac linh virc khoa hoc va ihurc ti6n Vice ap dung loan hoc mot each s^u sac da thiic day manh me sir phat trien cac phirang phap tinh toan Trong thirc te ta thirofng gap nhiJng bai toan ma dfr ki6n ban dau chi dugc biet gan dung, nhCmg thay ddi nho cua du* kien ban d'au c6 thd dSn dfi'n thay do! liiy y cua nghiem, do vi^c tun nghidm cua bai toan gap nhicu kho khan NhCmg bai toan khong on dinh nhu vay la m6t vi du ve bai toan dat khong chinh Khai niem bai toan dat khong chinh da dirge nha toan hoc Phap Hadamard J dira rfau tien cho Idrp phirang trinh vi phan [63,64] Thoat dau nguai ta cho rang bai toan kh6ng c6 y nghia toan hoc va thirc tiSn, n6n ft chu y de'n no Nhimg den cuo'i nhung nam 50 nguofi ta phat hien rang, nhicu bai toan ly thuyet va hau het Ccic bai toan ki thuat va ihirc 16' deu dan de'n bai loan dat khong chinh Tikhonov A N la ngirai c6 c6ng dau nghi^n cuti vah de nay[49,50] 6ng da d~e xuat khai niem bai toan dat khong chinh cho Icfp phirang trinh toan tu kh5ng gian T6p6 va cho dofi mot loat cong trinh xung quanh v^n dc I 0, tap K(r) = (x e Xo : fi[x] < r} la tap compact tuang doi X Phuang phap hieu chinh Tikhonov la lay nghiem g^an dung cua (0,1) la didm cue tieu cii phie'm ham tran: M "^ [x,y5 ] = p^ (Ax,y5 ) + a 0[x ] -> x e Xo Vai in6t s6' gia thie't nhat dinh c6 th^ chiing minh dugc rang i/ ! Xa= ArgminM''[ x,y5] X e Xo ii/ Tham s6' hieu chinh a = a (8) chon tu cac nguyen ly khong khc5rp, nguyen ly tua tO'i iru,v.v iii/Xa(6)^Xd ( - > ) - Ma r6ng nglu nhien bai toan la't dinh dat kh6ng chinh Ban luan an nghien citu mot so phuang phap hieu chinh giai bai toan dat khong chin Nhitng v^in d'e dugc quan tam luan an la: 1- Giai bai toan dat kh6ng chinh tren tap compact 2- Danh gia tfnh 6n dinh ye'u cua bai loan dat khOng chinh tren compact va compact yeu - Cac phuang phap lap hieu chinh giai bai toan dat kh6ng chinh Ban luan an gbm ba chirang, tai lieu tham khao va phan phu luc Chuomg I: Phuomg phap compact thu hep cat Men Chuang trinh bay phuang phap compact thu hep cai bien va danh gia dugc toe h6i tu cua phuang phap Chuang II: Bdi todn tuyen tinh khong chinh tren compact yeu Trong bai toan (0,1) xet truong hgp X,Y la cac khOng gian Hilbert, A la toan tur tuye'n tinh lien tuc vai khai tridn ki di da cho tru6c Trong chuang chung t6i trinh bay m6t thuat toan dang khai tri^n ki di chat cut giai phuang trinh (0,1) va danh gia tfnh on dinh ye'u cua nghiem Chuang I I I : Mot so phuomg phap lap hieu chinh Chuang trinh bay phuang phap lap hieu chinh Gauss-Newton gan dung va phuang phap Seidel - Newton hieu chinh Phan phu luc ; Trinh bay mot vi du so giai bdi todn Cauchy bang phuang phdp khai trien ky di chat cut Noi dung chfnh cua luan an da dugc cong bo bai bao dang tren cac tap chf cap truong, B5, qu6'c gia, qu6'c te' va dugc bao cao tai Xeminar toan hoe tfnh toan cua Dai Hoc Tong Hgp Ha Noi (Tien sy Pham Ky Anh chu tri) Hoi nghi khoa hoe 35 nam lap khoa Toan - Co - Tin DHTHHN hoc nam 1991, Hoi nghi khoa hoc khoa toan DHSPHN 1992, H5i nghi khoa hoc khoa Toan - Co - Tin hoc DHTHHN 1994, Hoi nghi quoc te' vt bai toan ngugc 1995 ( Tai Ho Chf Minh) CHUONGI PHUONG PHAP COMPACT THU HEP CAI B I C N $1 - Ma dau: Xet phuang trinh Ax = y • (LI) day A la toan tir phi tuye'n, X,, Y la cac khong gian vector t6p6 Gaponenko Yu.L [24,25] nam 1982 da de xua't phuang phap c6 ten la "compact thu hep" Ong da xay dung dugc cac tap Vg gbm huu han phan tir cho: V xg e Vg => II X5 - Xd II < diam V5 + -> ( -> 0) Arsenin V.Ia , ([5], nam 1989 ), xet bai toan (1.1), A : H ^ C[a,b], H la kh6ng gian Hilbert Thay vi biet ys e C[a,b] chi bie't m thi hien {y'sl "'1, han mOt nua y'5 thoa man dieu kien II / - y d ||cm/2) (1.2) Bang each sir dung ham Robust, Arsenin da dua mot phuang phap hieu chinh V6i ham M" [ x,y^5 ,.•• ,y'"8 ] = ^^ (Ax) + aC)(x), Arsenin V la da churng minh dugc cac ke't qua sau l/3!Xa-ArgminM"[x,y^8 y"\] 2/ a = a(5) : Xa(6) -> x* (5 -> ) Cung nhu phuang phap Tikhonov AN, phuang phap Robust kh6ng cho phep danh gia sai s6' cua nghiem g'an dung Y tuang cua Gaponenko va ky thuat cua Arsenin da dugc chung toi sir dung de xay dung thuat toan giai bai toan (LI) va da danh gia dugc tO'c dO hoi tu cua nghiem g^an dung 52 - Cac gia thiet cua bai toan Xet phuang trinh (1.1), A : X -> C[a,b], A la toan tir lien tuc, X la khdng gian Banach Goi Xi la mfen xac dinh cua phie'm ham on dinh D.[x] -6 Xi c: X va ta c6 OO Xi ==uK(n) (2.1) K(n) = (v e X] ,D.{v) r > m/2 < kj < m Gia su vai moi vi,V2 e K(n), A la loan tir thoa man dieu kien (2.3) II Avi-Av2|| la ham lien tuc, dan dieu tang theo cac bie'n, vai < t,r < +00, 4^(t,r) -> t ^ Goi S(n,h) e K(n) (Vn > 1) la (p(h) luai huu ban cua tap compact K(n), c6 nghia la: Vxe K(n), Xh e S (n,h) : ||xh - x|| < (p(h) Trong < (p(h), la ham dan dieu tang tren ( 0,1 ] va lim (p(h) - h->0 Xet phie'm ham Ibi b m b (l)6(y) = J Z P6 [ y's (t) - y(t)]dt Rg (t,y(t)) dt a i =1 a in day R5= E PsCy'sCO-x) s V (2 k) I sI < k I s I >k Trong d6 < K < C.6 , < C - Const -7- Ham (jjg (y) c6 tinh cha't sau (xem [5]) i/ T6n tai ygCt) e C[a,b] : (t)s(y5) = Inf {(^sCy) : y e C[a,b] ) ii/ V > : llyc-ysllc ^ V.5 iii/ G > : I(|)8(yi)-lkhiTo > yd(t) +Q6 5T iiiii/ T =To I < to < yd (t) - Q5 aR8(t,x) < m , V T , t e [a,b], > 5x S3 - Thuat toan Compact thu hep dang Robust Gia sir ta c6 he thiic b m (t)6(yd) = J Z P5(y'5(t)-yd(t))dt N o = > Q[Xd] n : r N < ^[Xd] 79 L:=L-1; Gotoxy(27,4); WritelnCGia tri L tim duoc = '.L); Gotoxy(25.22); Pause; Clrscr; if (L mod 2) then KI :=( L-3) div else KI : = ( L - ) d i v ; Gotoxy(30,4); WritelnCGia tri cua K= ', KI); Gotoxy(25,22); Pause; Clrscr; FIND_L:=K1; END; { WritelnC Ham tinh tong ')} FUNCTION SUM (x: Real; K:Integer): Real; Var : Integer; j Dau : Integer; Shll,Sh33 : Real; XI1 ,Ketqua : Real; Gt,Xl : Real; BEGIN Gt :=1; Dau := 1; Shll:=l/Shl; Sh33:=l/Sh3; XI : = X * p i ; Xll : - X l ; Ketqua := XI; Forj:=l ToKDo Begin Gt :=G t*(2*j)*(2*j + l); Dau := - Dau; XI :=X1 *sqr(xll); Ketqua :=: Ketqua + (Dau *xl)/Gt; End; SUM := Ketqua *(sqrt(2)*Shll/pi); END; -80 { ('ham tinh tich phan');} FUNCTION TICHPHAN (Y: AIT)- RealVar i : Byte; Result; Real; BEGIN Result := Y[0] + Y[100]; For i:= To 99 Do Begin Result := Result + * Y[i]; If ( i Mod 2) Then Result :=Result+ * Y[i]; End; TTCHPHAN := Resuit/(3*iOO); END; BEGIN { main program} Clrscr; Assign(Fkq,'DATAl PAS'); Rewrite(Fkq); Shl :=Sh(l); Sh3 := Sh(3); Sigma:= l/(pi * Shl* sqrt(2)); i:=l; h:= 1/100; Tiep := 'c'; While l^case(Tiep) = 'C begin Clrscr; REPEAT Gotoxy(25,10); WriteCHay nhap vao gia tri (>0) cua dental= '); Clreol; {$!-} Readln(dental); errorcode :=IOresult; If errorcode Then Writeln(Chr(7)); UNTIL (errorcode = 0) and (dental > 0.000001); K:-FIND_L(Dental); clrscr; Gotoxy(20,5); WRITEC XIN HAY VUI LONG DOI CHO MOT LAT '); 81F o r j : = T o 100 Do S[j] := SUMa*h,K); While ((Sigma/(2*i)) > Dental) Do Begin Forj :=:0To 100 Do Y[j] := sqrt(2)* S[j]*sin(i*pi*j*h); Ln[i] := TICHPHAN(Y); Sigma:= i/(pi* sh(i)*sqrt(2)); If abs(ln[i])>Dental Tlien Cn[i]:=Ln[i]/Sigma Else Cn[i]:=0; i:=i+l; End; NI : = i - l ; With Kqua Do Begin dell := dental; KK :=k; NN:=N1; F o r i : = l ToNl Do C[i]:=Cn[i]; End; Write(Fkq,Kqua); Clrscr; Gotoxy(25,22); pause; clrscr; gotoxy(25,22); WriteC Ban co muon thu voi dental khac khong(C/BC)?'); read In (Tiep); clrscr; end; close(Fkq); Gotoxy(25,2); WrileC KET QUA CIIAY CHUONG TRINH); Gotoxy(26,3 ); WriteC ****!f:*=fc******** 'v Gotoxy(l,5 ); WriteCDental K N C[l] C[2] C[3] C[4] C[5] C[6] Gotoxy(25,22); Writeln; hang := 8; Assign(Fkq;DATAl.PAS'); Reset(Fkq); C[7] C18]"); 82 While not Eof(Fkq) Do Begin read(Fkq,Kqua); With Kqua * begin Gotoxy(l,hang) ;Write(delt:0:4); Gotoxy(8,hang);Write(KK:2); Gotoxy(l Lhang);Write(NN:2); Cot:=15; for i := to NN Begin Gotoxy(cot,hang);Write(C[i]:6:4,"); Cot:=cot+8; End; End; hang :=hang+ 1; End; Close(Fkq); Writeln; Gotoxy(25,22); Pause; END 2.2- Trudng hgp 2: @' Thuat giai: Ta tha'y rang thuat giai tuong tu nhu trudng hgp.l @- Chirong trinh: {$N+,e+} ($M 65000,0,0} PROGRAM MAIN2.PAS; USES Crt,Myunit; Const N=100; Type AA= Array[1 8] of real; Arr = Array[ N] of Real; Ketqua = Record delt: real; BCK : Byte; NN : Byte; C :AA ; End; "83 Var Y.Ln,Cn,S : Arr ; i,j.Nl,k : Integer; h : Real ; Dental,sigma: Real ; Shl,Sh3 :Real '; errorcode,hang,cot: Byte ; Kqua :Ketqua; Tiep : Char; Fkq : File of Ketqua; { C Ham tim gia tri cua K)} FUNCTION FIND_L( Dental: Real): integer; Var K1,L : Integer; Denta2,x,r : Real; BEGIN Clrscr; Denta2 := (dental* shl * pi)/sqrt(2); L:=l; x:=l; r :=1; REPEAT r :=r * 3; X := X * (pi/L); L :=L+1; UNTIL ((X *(r +1) < denta2)); L:=L-1; Gotoxy(27,4); WritelnCGia tri L tim duoc = \L); Gotoxy(25,22); Pause; Clrscr; if (L mod 2) then KI :=(L-3)div2 else KI := ( L - 2) div 2; Gotoxy(30,4); WritelnCGia tri cua K = ', KI); Gotoxy(25,22); Pause; Clrscr; FIND_L:=K1; END; -84I (' Ham tinh tong ')) FUNCTION SUM (x: Real; K:Integer): Real; Var j : Integer; • Dau : Integer; ShIl,Sh33 :Real; Xll,Ketqua,t: Real; Gt,tt,Xl : Real; BEGIN Gt := 1; Dau : = ; Shll:=l/Shl; Sh33:=l/Sh3; t :=Shll+3*Sh33; tt : = ; I XI :=X*pi; X11:=X1; Ketqua := XI *t; Forj :=1 TokDo Begin Gt :=Gt*(2*j)*(2*j + l); Dau := - Dau; XI :=X1 *sqr(xll); tt := U * sqr(3); t :=Shll+tt*Sh33; Ketqua := Ketqua + (Dau * xl * t)/Gt; End; SUM := Ketqua * (sqrt(2)/pi); END; f j f | ' f • ( C ham tinh tich phan');) FUNCTION TICHPHAN (Y: Arr): Real; Var i : Byte; Result: Real; BEGIN Result :=Y[0]+Y[ 1001; For i:= To 99 Do Begin Result :=Result + * Y[i]; If ( i Mod 2) Then Result := Result + * Y[i]; End; TTCHPHAN := Result/(3*100); END: \ _- | j I -85 BEGIN I main program) Clrscr; Assign(Fkq;DATA2.PAS'); Rewrite(Fkq); Shl :=Sh(l); Sh3 := Sh(3); Sigma:= l/(pi * Shl* sqrt(2)); i :=1; h:== 1/100; Tiep := 'c'; While l^case(Tiep) = 'C begin Clrscr; REPEAT Gotoxy(25,10); Write(TTay nhap vao gia tri (>0) cua dentall= '); Clreol; ($M Readln(dental); errorcode :=IOresult; {$!+} If errorcode Then Writeln(Chr(7)); UNTIL (errorcode = 0) and (dental > 0.000001); K:=FIND_L(Dental); cb"scr; Gotoxy(20,5); WRITEC XIN HAY VUI LONG DOI CHO MOT LAT '); Forj:=OTo 100 Do SUl : - SUMG*h,K); While ((Sigma/(2*i)) > Dental) Do Begin Forj:=OTo 100 Do YLJ] :=sqrt(2)* S[j]*sin(i*pi*j*h); Ln[i] := TICHPHAN(Y); Sigma:= i/(pi* sh(i)*sqrt(2)); If abs(ln[i])>Dental Then Cn[i]:=Ln[il/Sigma Else Cn[i]:=0; i:=i+l; End; NI : = i - l ; With Kqua Do begin delt := dental; -86KK:=k; NN:=N1; For i := To NI Do C[i]:=Cn[i]; End; Write(Fkq,Kqua); clrscr; Gotoxy(25,22); pause; clrscr; gotoxy(25,22); WriteC Ban co muon thu voi dental khac khong(C/K)?'); readln(Tiep); clrscr; end; close(Fkq); Gotoxy(25,2 ); WriteC KET QUA CHAY CHUONG TRINH); Gotoxy(26,3 ); Gotoxy(l,5 ); WriteCDental K N C[ll C[2] C[3] C[4] C[5] Gotoxy(25,22); Writeln; hang := 8; Assign(Fkq,'DATA2.PAS'); Reset(Fkq); While not Eof(Fkq) Do Begin read(Fkq,Kqua); With Kqua begin Gotoxy(l ,hang) ;Write(delt:0:4); Gotoxy(8,hang);Write(KK:2); Gotoxy( 11 ,hang); Write(NN:2); Cot:=15; for i := to NN Begin Gotoxy(cot,hang);Write(C[i] :6:4,''); Cot:=cot+8; end; end; hang :=hang+ 1; end; close(Fkq); Gotoxy(25,22); pause; END C[6] C[7] C[81'); ' ~^ 87 * Cudi cung la listing Myunit tu lao thue hien cOng viec - Thu tuc chd( PAUSE) - Ham tinh Sh(x) Theo c6ng thiic X -X e -e Sh(x) = { Listing Myunit} {$n+) UNIT MYUNIT; INTERFACE USEs CRT; Procedure Pause; Function sh(x: Real): Extended; IMPLEMENTATION { Thu tuc cho } Procedure Pause; Var Ch: Char; Begin WritelnC Press any key to continune '); Ch := Readkey; Writeln; Writeln; End; { WritelnC ham tinh gia tri cua sh(x)')) Function SH(x:Real): Extended; Begin Sh := (Exp(x) - Exp(-x))/2; End; END .U •88- TAI LIEU THAM KHAO Nguygn Van Hiing Phuang phap compact thu hep cai bien va Cmg dung Thong tin khoa hoc Trufomg DHSPHN - 1993 So Tr 49 - 58 Nguyin Van Hung Phirang phap compact thu hep cai biSn tnrong hop phirong trinh toan tur loai I co w6 phai va toan tir khong biet chinh xac Thong bao khoa hoc Tnrong DHSPHN - 1995 S6 dac biet ky niem 20 nam lap tnrdng Tr 53 - 55 APCEHMH B.H., MBAHOB B.B pemeHHH HeKOTopHX HHTerpajn.HHx ynpaBJieHHft I pofla THna csepTKH jiecTojioM peryjwpHsanwH MM H M$ 1968 T }J c 310-32 APCEHMPI B.fl HeKoppeKTHO nocTaBJieHHHx sajajsnax ycnexH Marew HayK 1976 T if' c 89-101 APCEHMH B.H., KpHHCB A.B., IiyilKO-CMTHHKOB M.B npmeHeHHe podacTHHx MeTOflOB npH pemeHHH neKoppeKTHHx sasaq IBM H M$ 1989 T 29 Jfe c 653-661 BAKyUMHCKMH A.B 0(5 OflHOM ^HCjieHHOM MeTo;i;e pemeHHH HHTerpajEbHoro ypaBHeHHH ^pejuojiBMa I po;n;a IBM H M$ 1965 T )^ c 744-749 EAKyilMHCKMK A.B„ MsdpaHHHe BonpocH npHdjiUKenoro pemeHHH HeKoppeKTHHx sa^an M MSJI-BO MOCK yn-Ta 1968 BAKyilMHCMH A.B SaMe^aHHH o6 OJ;HOM KJiacce peryjinpHSHpycnpix ajiropHTMOB IBM H M^ 1973 T 13 i^ c 15961598 EAKyilMHCKHM A.B K ^pHH^H^y HTepaTHBHoM peryjinpHsannn IBM H m 1979 T 19 i^ c I040-I043 10 EAKyilMHCKMJ! A B , rOHHAPCKHH A.B ^HCjieHHHe weTOflH H npHjroweHHH M HeKoppeKTHUe saji^am MSJ-BO Miy 1989 -891 BAKyilMHCMK A.E., rOHqAPCKHK A.B HTepaTHBHHe meiom pemeHHH HCKoppeKTHHx saaaH M Hayna, 1989 12 EAKyilMHCKMH A.B K npodjiewe CXOBHMOCTH HTepaTHBHOperyjiHpH30BaHHoro MeTo^a rAycCA-HUlOTOHA FBM n m 1992 T.32 J^ c I503-I509 13 M.B , KB^IEB M»M HeKOTopue BonpocH ycTOHHHBOCTH HTepanHOHHbix MBTOflOB pemeHHH onepaTopHHx ypaBHeHHM B c(3"$yHKHHOHa7rbHHH anajiHs H ero npHJiOKeHWH" KasanB KasaH Yn-Ta, 1975 c 16-33 BOHKOB 14 EyillMAHOBA M.Bo KOHe^HOMepHUx npndjniKeHHHX K pemeHHio jfflHefiHoro onepaTopHOPo ypaBHeHHH I po^a Mss-BysoB MateMaTHKa 1977 !h c 11-77 15 BAMMKO r.Mo MeTOflH pemeHHH jranefiHHx neKoppeKTHo nocTaBJisHHHx saj^a-q B rnJCbdepTOBUx npocTpaHCTBax TapTy MsjI-BO AH CCCP 1964 I6o BACMH B B , TAHAHA B.IIo npHdjnaceHoe pemenne onepaTopHH: ypaBHeHHH nepBoro pojia MaTeM, sanncK-ypajiLC yn-Ta 1968 T 16 c 27-37 17 BACMH B B , TAHAHA Boll 0(5 ycTOH^BOCTH npoeKHHOHHUx MeTojUOB npH pemeHHH HeKoppeKTHHx 3aj];a¥.MM H M$ 1975 T.I5o J6 Ic c 19-29 18 BMHOKyPOB B.A norpemnocTH pemennn JiHHeftHUX onepaTopHHx ypaBHeHHHo IBM H m 1970 T.IO !f^ c 830-839o 19 BMHOKyPOB B,Ao, TAHOHEHKO lO.JI AnocTepnopHHe oneHKH pemeraia HeKoppeKTHHx o(5paTHUx 3a^a^ MR CCCP 1982 T.263 }f'- 277-280 TAnOHEHKO K).JI MeTOfl CTHTHBaBmKXOH KowmaKTOB jvm pemem HejmneHHKx ;];H$$epeHHHaji£HHx ypaBHennK BecTH WOCK yn-Ta, Cep 15 BH^HCjiHTejiHTejieHHH iviaTewaTHKa H KH(5epHeTHKa 1980 J^ A-Vbulk—.•• * ' ^^d -902 PAnOHEHKO K).JI MeTOfl j[HCKpeTHoffl $yHKHHH Ppmia ^JIH pemeHHH HSKOTopux KpaeBux sajian BBCTH, MOCK yn-Ta MaTewaTHKa H wexanHKa 1977 J6 I c - 2 PAnOHEHKO K).JI MeToji; corjiacoBanHOH annpoKCHMacHH BJIH pemeHHH nejiHHeHHHx onepaTopmix ypasHeHHfi IBM H m 1978 T I Jt c 767-769 PAnOHEHKO 10.JI MeToji; CTHrHBaBmnxcH KowinaKTOB JIJIH pemeHHH HejfflHeHHHx HeKoppeKTHUx sa^ian, IBM H M $ I I T I )f c 136-137 PAEOHEHKO K.JI 0(5 O^HOM Kjiacce snoJiHe peryjiHpHsyeMHx 0T0(5pa3ceHHfi IBM H m 1982 T.22 11? I c 3-9 PAIIOHEEIKO lO.JI HpHHUHH CTHTHBaBmnxCH KOwnaKTOB jum HejiHHefiHHx HCKoppeKTHHx 3aji;aH CH(5 MaTewi lypnaji 1982 T.23 if! c - PAHOHEHKO ID.Jl K Bonpocy oc5 ycTofi^HBocTH pemeHHH HHTBrpajiLHoro ypaBHeHHH I po^a na cjia(5oM KOMnaKTe IBM H M$ 1986 T.26 J£ c 970-980 PAHOHEHKO ED.JI HeKoppeKTHHe sana^H na cjia6ux KowmaKTHK Msfl-Bo MOCK yn-Ta 1989 IMJIflSOB C.$ MeTojm pemeHHH jraneHHUx HeKoppeKTHHx saflaq M Msji-BO MIY 1987 29 A B , flPOM A P , JIEOHOB A.C HeKOTopne GUeHKH CKOPOCTH CX0;[i;HM0CTH peryjiHpH30BaHHHx npH(5jIH3KeHHa fljiH ypaBHeHHH Tnna CBepTKH IHVI H M$ 1972 T I W- c 762-770 POHHAPCKMK 30 romiAPCKMH A B , JIEOHOB A.C , flPOJM A.P npHHiinne HeBHSKH npH pemeHHH HejraneHHHx HeKoppeKTOHHx 3a;5aH„ UAH CCCP 1974 To24 J6 c 499-500 -913 SflEPEB H E , CABEJIOBA T.M 06 oneHKe CKopocTH CXOJIHMOC peryjiHpH30BaHHHx pemeHHft ypaBHeHHH THna CBepTKH c norpemHOCTHMH B H^^pe H HpaBoft HaCTH IBM H M$ 1976 T I Jf^ c I I - I I I „ 32 MBAHOB B.K neKoppeKTno nocTasjieHHHx sa^^a^ax MaTeM C(5c 1963 T I J6 c 211-223 3 MBAHOB B.K HeKoppeKTHHe saHatm B TonoJiorn^ecraix npocTpaHCTBax CH(5 MaTeM jKypn 1969 T,IO„ ii c 10651074 34 MBAHOB B.K., TAHAHA B.H., BACMH B.B TeopHH jmneftHHx HeKoppeKTHHx 3ap.a^ H ee npHjiomenHH HSA-BO "HayKa" MOCK 1978 KOHTOPOBM^ JI.B ^ynnHonajiBHHH aHa.Ha3 H npHKjiaji;HaH MaTeMaTHKa YMH 1948 T.3o Jf c 89-184 36 KAHTOPOBM^ JI.B HeKOTopne jtajiBHemiiHe npHMenenHH MeToja HBBTOHa jyiH i^yHKHHOHajibHHx ypaBHennfi BecTHHK JHY 1957 T 68-103 KOJIMOPOPOB A.H., $OMHH C B OjiOMenTH TeopHH ^yHKHHfi H ^yHKiiHOHajiBHoro anajiHsa M HayKa, 1968 KYPHEJIL H.C npoeKiiHonno-HTepaTHBHHe MeTo^H pemeHHH onepaTopHHX ypaBnenHH K "HayKosa ^TMKa", 1968 c.245 39 JIABPEHTBEB M.M HCKOTopux HeKoppeKTHHx sajia^ax MaTeMaTHqecKOH $H3HKa HoBOCH(5epcK Msfl-Bo CO AH CCCP 1962 92 c JIABPEHTBEB M.M., AHMKOHOB B E , $A3HJI0B $.H Hpn^jmjcennoe pemenne ncKOTopHx nejiKnetaux onepaTopHHx ypaBneHnft JIAH CCCP I I T.200 if A c 770-772 JMCKOBEU O.A BapnaHHOHnHe MeTo^H pemenHH neycTOHMHBHx sajia^ MHHCK HayKa H TexnnKa, I I •92- I M03EP B EucTpo cxo^^HmnficH weTOfl HTepaiinfi H neJoraeHHHe J^II$epeH^HaJ^.HHe ypaBHeHHH YMH 1968 23 BHH,4(142) 179-238 M0P030B B.Ao Jlnnefinne H Hejmneflnue neKoppeKTHHe 3afla^H MTOPH HayKH H TexHHKH MaTewaTHHecKHH anajffls.T.II M BMHMTH 1973 c 129-178 44 M0P030B B.A„ PeryjinpHHe MBTOHU pemennn HeKoppeKTHHx sa^axio M Msfl-BO MOCK„ yn-Ta 1974 360 c 45 o HMPEHEEPP JI JIBKUHH no nejinnefinoMy ^ynKutHonajiLHOMy anajiHsy M Mnp 1977 46 OnoKlIEB BoMo 0(3pameHHe npHnnnna cscHMaiomHx OTodpaxenHH yWH 1978 T I BHH 4(190) c 169-198 47 OPTETA J I I , PHfflBOJIJIT B MTepHUHonnne MeTojtu pemenHH HejIHHeHHHX CHCTCM y p a B H e H H f l CO MHOrHMH HeH3BeCTHHMH M 1975 CABEJIOBA T.M HpoeKHHonnHe MCTOAH pemeHHH jnraeHHHx HeKoppeKTHHx ypaBneHHM IBM H M$ 1974 T I If c I027-I03I, TMXOHOB A.Ho 0(3 ycTofl^BOCTH odpaTHHx 3aj;a^ JIAH CCCP, 1943 T.39« ii c 195-198 50 TMXOHOB A.H pemenHH neKoppcKTHO nocTaBJiennHx 3aflaM H MeTOfle peryjiHpH3aii;HH JIAH CCCP 1963 T I I ih c 501-504 TMXOHOB A.H peryjiHpH3au,HH neKoppeKTHo nocTaBJiennHx sajiaM JIAH CCCP 1963 T I J^ c 49-52 52 TMXOHOB A.H pemenHH neOTHefinHx HHTerpajEbHHx ypaBneHHfi nepBoro po^a JIAH CCCP 1964 T 156 if c.12961299 93- THXOHOB A.H , TJIACKO B.B UvmiemimVi ueTom peryjiHpKsBixm B HejiMHefiHux sajia^ax IBM H m 1965 T Jf c 463-473 54 THXOHOB A.H APCEHMH B.fl saflBM M HayKa 1979 MeTojm pemeHKH HeKoppeKTHHx 55 $AM KM AHL npoeKUHOHHO-HTepaTWBHHx MGTojiax pemeHHH onepaTopHHx ypaBHeHHH IBM H M^ 1979 T I ]f> c.760765 56 5»AM KM AHL 06 O^HOM npHdjn^^eHHOM weTojie pemeHHH KsasajiHHefiHUx onepaTopHHx ypaBHeHHH MR CCCP 1980 To250 J^ c 291-295 57 $AM KM AHt npHfijiHKeHHoe pemeHHe KBasHjzHHeSHHx onepaTopHHx ypaBHeHHg UBTOJIOM HBK)T0Ha-3ett;i;ejrH I M H M