К.Ю. Богачёв - Практикум на ЭВМ. Методы приближения функций (1133845), страница 17
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.25,/D7 @H/1<55:ui k (x) = (x ; xi )k;1 k = 1 2 3 4 x 2 Sxi xi+1]vj l(y) = (y ; yj )l;1 l = 1 2 3 4 y 2 Syj yj+1]:"H/1<5+/.6Di 1(u) = u(xi) i 2(u) = di(u) i 3(u) = u(xi+1) i 4(u) = di+1 (u)j 1(v) = v(yj ) j 2(v) = dj (v) j 3(v) = v(yj+1) j 4(v) = dj+1(v)=37 076545/D di , dj +:97376;F-,; 0DC9.//D> >7-+3+> 5/-79:+6;<55, ,>.(I.19.4), (I.19.7), (I.19.8), (I.19.9), (I.19.12) (0 :+,673/7> ,6H4.7 076545/D di :+6H4.F-,; 0 972H6B-.-7 97E7/5; ,5,-7>D 65/78/DA H9.0/7/58)."+9>H6D (I.19.3) :973,-.06;F- ,+C+8 0D9.?7/5; 36; >.-95< A;1i 0 (2) 5;1Bj 0 (3).
78,-05-76B/+, 0D9.?7/5; (I.19.3) >+?/+ 2.:5,.-B 0 0537 (+C+2/.450hi = xi+1 ; xi )c1 i = f (xi)c2 i = dic3 i = h12 h3 (f (xi+1) ; f (xi)) ; 2di ; di+1ii3= ; 3 f (xi) ; 22 di + 33 f (xi+1) ; 12 di+1hihihihi 21c4 i = h2 di + di+1 ; h (f (xi+1 ) ; f (xi ))ii21= 3 f (xi) + 2 di ; 23 f (xi+1) + 12 di+1hihihihi.. !"# $B $Bx1. '&+-.- ')3 &)(-106673+0.-76B/+,ci (c1 i c2 i c3 i c4 i)t = A;1 (hi) (f (xi ) di f (xi+1) di+1)t=370 11000BBCCBB 0100 CCCB;1BCA (h) = B 3BB ; h3 ; h22 h33 ; h12 CCCB@ 2C1 ;2 1 Ah3 h2h3 h2;1;1;1.15> +C9.2+>, A;1i = A (xi+1 ; xi ), Bj = A (yj +1 ; yj ).
(9H=5>5 ,6+0.>5,41+I@@5<57/-D i = (i k )tk=1:::4 2 R4 :95C65?.FG78 @H/1<55 Pu = P i k ui k ,k=1H3+067-0+9;FG78 65/78/D> H,6+05;> 5/-79:+6;<55 i k (Pu) = i k (u), 0D45,6;F-,; 1.1 i = A;1(xi+1 ; xi)i(u), =37 i(u) = (i k (u))k=1:::4 2 R4 . +I@@54<57/-D j = (j l)tl=1:::4 2 R4 :95C65?.FG78 @H/1<55 Pv = P j lvj l , H3+067l=1-0+9;FG78 65/78/D> H,6+05;> 5/-79:+6;<55 j l(Pv) = j l(v), 0D45,6;F-,;1.1 j = A;1 (yj+1 ; yj )j (v), =37 j (v) = (j l(v))l=1:::4 2 R4 )..25, 0 :9+,-9./,-07 @H/1<58 { -7/2+9/DA :9+520737/58:wij kl(x y) = ui k (x)vj l(y) = (x ; xi)k;1(y ; yj )l;1 k l = 1 2 3 4:.25, 0 :9+,-9./,-07 @H/1<5+/.6+0 { -7/2+9/DA :9+520737/58:ij kl(f ) = i k (j l(f )) k l = 1 2 3 4:(11)+ -7+97>7 1 :95C65?.FG.; @H/1<5; 5>77- 053 (4):Pf =4 X4Xk=1 l=1ij klwij kl =4 X4Xk=1 l=1ij kl(x ; xi)k;1(y ; yj )l;1;1;1=37 1+I@@5<57/-D ij kl 0D45,6;F-,; 0 (5).
,56H A;1i = A (xi+1 ; xi ), Bj =A;1(yj+1 ; yj ) 5>77>:;ij = A;1(xi+1 ; xi )Fij (A;1(yj+1 ; yj ))t =37 ;ij = (ij kl)k l=1:::4 2 M4 (R), Fij = (ij kl(f ))k l=1:::4 2 M4(R)... !"# $B $Bx1. '&+-.- ')3 &)(-107&! 7!% { 6$& 5$6%# ,6H4.7 1H,+4/+8 5/-79:+6;<55 1HC547,15>5 >/+=+467/.>5 J9>5-. (,>.x I.19.3) >.-95<. Fij 5>77- 0530BBBBBBFij = BBBBBBB@@f (xi yj )@f (xi yj+1)f (xi yj )f(xi yj +1)@y@y@f (xi yj ) @ 2 f (xi yj ) @f (xi yj+1) @ 2 f (xi yj+1)@x@x@y@x@x@y yj+1)i+1 yj )f (xi+1 yj ) @f (x@yf (xi+1 yj+1) @f (xi+1@y@f (xi+1 yj ) @ 2 f (xi+1 yj ) @f (xi+1 yj+1) @ 2 f (xi+1 yj+1)@x@x@y@x@x@y1CCCCCCCCCCCCCA-.7.
0 1.?3+8 -+417 (xi yj ), i = 1 2 : : : n , j = 1 2 : : : m (CD-B >+?7-, 2.xi yj ) ,5,16F47/57> =9./54/DA) 3+6?/D CD-B 2.3./D 2/.47/5; f (xi yj ), @f (@x@f (xi yj ) , @ 2 f (xi yj ) .@y@x@y ,6H4.7 1H,+4/+8 5/-79:+6;<55 1HC547,15>5 >/+=+467/.>5 7,,76; (,>.x I.19.4), 1H,+4/+8 5/-79:+6;<55 1HC547,15>5 >/+=+467/.>5 >7-+3+> 15>D(,>. x I.19.5), 1H,+4/+8 5/-79:+6;<55 1HC547,15>5 >/+=+467/.>5 , 5,:+6B2+0./57> 9.23767//DA 9.2/+,-78 (,>.
x I.19.6), 5/-79:+6;<55 1HC547,15>5 ,:6.8/.>5(,>. x I.19.7) 0D9.?7/5; (I.19.7), (I.19.8), (I.19.9), (I.19.12) (3+:+6/7//D7 ,++-07-,-0HFG5>5 =9./54/D>5 H,6+05;>5, ,>. x I.19.8), >+?/+ 9.,,>.-950.-B 1.1,5,-7>D H9.0/7/58 36; +:973767/5; 1+I@@5<57/-+0 di (0 :790DA -97A >7-+3.A\,5,-7>." 5>77- 35.=+/.6B/HF >.-95<H). C+2/.45> 47972 G(x) (:95 xi := xi )5, ,++-07-,-07//+, G(y) (:95 xi := yi ), >.-95<H, 0D9.?.FGHF 97E7/57 ,5,-7>D(I.19.7), (I.19.8), (I.19.9) 565 (I.19.12) 47972 071-+9. F (x) = (f (xi))i=1 2 ::: n (,++-07-,-07//+, F (y) = (f (yj ))j=1 2 ::: m ), -.1, 4-+ D(x) = G(x) F (x) , D(y) = G(y) F (y) ,=37 D(x) = (di)i=1 2 ::: n :95 xi := xi , D(y) = (dj )j=1 2 ::: m :95 xi := yi .
.-95<.G(x) 5>77- 9.2>79 n n , G(y) { 9.2>79 m m , 5 H45-D0.7- ,++-07-,-0HFG57=9./54/D7 H,6+05;.7:79B >D >+?7> ,@+9>H659+0.-B .6=+95-> 0D45,67/5; @H/1<5+/.6+0ij kl(f ) 0 (11). H,-B F = (fij ) +C+2/.4.7- n m >.-95<H 2/.47/58 f (xi yj ).D45,65> (+35/ 9.2):n m >.-95<H F (x) = (fij(x)) G(x) F ,n m >.-95<H F (y) = (fij(y) ) F (G(y) )t ,n m >.-95<H F (xy) = (fij(xy) ) G(x)F (G(y) )t ... !"# $B $Bx1.
'&+-.- ')3 &)(-/.47/5; @H/1<5+/.6+0 Fij = (ij kl(f ))k l=1 2 3 4 0 (11):0BB fi jBB f (x)Fij = BBB i jBB fi+1 j@ (x)fi+1 jfi(yj)fi(xyj )y)fi(+1jxy)fi(+1jfi j+1)fi(xj+1fi+1 j+1x)fi(+1j +1fi(yj)+1fi(xyj+1)y)fi(+1j +1xy)fi(+1j +11081CCCCCCCCCA$)67; $; #& % 5; G*9./7/57 5/@+9>.<55 + @H/1<55 5 77 :95C65?7/55 H3+C/+ +,HG7,-05-B ,673HFG5> +C9.2+>.
A+3/D7 3.//D7 { 2/.47/5; @H/1<5+/.6+0 (ij kl(f )), i =1 2 : : : n , j = 1 2 : : : m , k l = 1 2 3 4, 9.2>7G.F-,; 0 3-A >79/+> >.,,507F = (fijk ), i = 1 2 : : : n , j = 1 2 : : : m , k = 1 2 3 4 9.2>79. (n m 4), =37fij 1 = ij 11(f ) = f (xi yj ) fij 2 = ij 12(f ) fij 3 = ij 21(f ) fij 4 = ij 22(f ):.-95<. Fij 0DE7 :+6H4.7-,; 52 I67>7/-+0 4-A >79/DA 071-+9+0 Fi j , Fi+1 j ,Fi j+1 , Fi+1 j+1 . +I@@5<57/-D :95C65?.FG78 @H/1<55 { (ij kl), i = 1 2 : : : n ,j = 1 2 : : : m , k l = 1 2 3 4, 0D45,6;F-,; +35/ 9.2 5 A9./;-,; 0 4-A >79/+>>.,,507 ; = (ijkl), i = 1 2 : : : n , j = 1 2 : : : m , k l = 1 2 3 4 9.2>79. (n m 4 4), =37 ijkl = ij kl .x 1.6.4.
-%#$; $5!;7;H,-B 1.?3D8 52 >7-+3+0 :95C65?7/5; :+ :797>7//D> x 5 y ;06;7-,;1H,+4/+-10.39.-54/+8 5/-79:+6;<578 (,>. x I.20), -.7. 00737/D -+4151(x) < a = x1 < 2(x) < x2 < 3(x) < x3 < : : : < xn;1 < n(x) < xn = b < n(x+1) 5)1(y) < c = y1 < 2(y) < y2 < 3(y) < y3 < : : : < ym;1 < m(y) < xm = d < m(y+15 5/-79:+659HFG.; @H/1<5; +3/+8 :797>7//+8 P (x)f ( P (y)f ) ,-9+5-,; -.x) ], i = 1 2 : : : n (S (y) (y) ],15> +C9.2+>, 4-+ /. 1.?3+> +-97217 Si(x) i(+1ii+1i = 1 2 : : : m ) +/.
;06;7-,; >/+=+467/+> Pi(x) ( Pj(y) ) ,-7:7/5 2, -.15>, 4-+9=Pi(x)(xi ) = f (xi)(12)x) ) = v (x) i = 1 2 : : : nPi(x)(i(x)) = vi(x) Pi(x)(i(+1i+1565, ,++-07-,-07//+,9=Pj(y) (yj ) = f (yj )j = 1 2 : : : m(13)Pj(y) (j(y) ) = vj(y) Pj(y) (j(y+1) ) = vj(y+1) .. !"# $B $Bx1. '&+-.- ')3 &)(-109=37 vi(x) , i = 1 2 : : : n+1, vj(y) , j = 1 2 : : : m+1 { ,0+C+3/D7 :.9.>7-9D, -+- 5655/+8 ,:+,+C 0DC+9. 1+-+9DA +:97376;7- >7-+3 1H,+4/+8 5/-79:+6;<55 10.39.-54/D>5 >/+=+467/.>5. >7-+37 5/-79:+6;<55 :.9.C+6547,15>5 ,:6.8/.>5I-5 :.9.>7-9D :+6H4.F-,; 1.1 97E7/57 ,5,-7>D 65/78/DA H9.0/7/58 (I.20.8).+=6.,/+ ,1.2.//+>H 0DE7 3+,-.-+4/+ 9.,,>+-97-B :95C65?7/57 /. :9;>+x) ] S (y) (y) ]. 95C65?.FG.; @H/1<5; (:+ 1.?3+8 52 :797H=+6B/517 Si(x) i(+1jj +1>7//DA) 0D45,6;7-,; :+ @+9>H67 (I.20.2), =37 1+I@@5<57/-D 2.3.F-,; (I.20.3).0737> 0,7 H4.,-0HFG57 0 :+,-./+017 2.3.45 65/78/+8 5/-79:+6;<55 -7/2+9/D>5 :9+520737/5;>5 +CO71-D.
.25,/D7 @H/1<55:x) ]ui k (x) = (x ; i(x))k;1 k = 1 2 3 x 2 Si(x) i(+1vj l (y) = (y ; j(y) )l;1 l = 1 2 3 y 2 Sj(y) j(y+1) ]:"H/1<5+/.6Dx) i 1(u) = vi(x) i 2(u) = u(xi) i 3(u) = vi(+1j 1(v) = vj(y) j 2(v) = v(yj ) j 3(v) = vj(y+1) :"+9>H6D (I.20.3) :973,-.06;F- ,+C+8 0D9.?7/5; 36; >.-95< A;1i 0 (2) 5;1Bj 0 (3). 78,-05-76B/+, (,>. 0D45,67/5; 0 x I.20.3) 0D9.?7/5; (I.20.3) >+?/+2.:5,.-B 0 0537 (+C+2/.450 hi = xi ; i , zi = i+1 ; xi , hi + zi = i+1 ; i )c1 i = vic2 i = ;vi h1 + h +1 z + f (xi) h1 + z1 ; vi+1 z1 ; h +1 ziiiiii 1i 1i 11111c3 i = vi h + z h ; f (xi ) h + z h + z + vi+1 h + z zii iiiiiii i673+0.-76B/+,ci (c1 i c2 i c3 i)t = A;1(hi zi ) (vi f (xi) vi+1 )t=3701100BB 1 11 CCC111BCA;1(h z) = BBB ; h + h + zh + z ; z ; h + z CCA@1 111111;h + z h + zh+z hh+z z(y) (y)(x) (x);1;1;1.15> +C9.2+>, A;1i = A (xi ; i i+1 ; xi ), Bj = A (yj ; j j +1 ; yj ).
(9H=5>5 ,6+0.>5, 1+I@@5<57/-D i = (i k )tk=1:::3 2 R3 :95C65?.FG78 @H/1<554Pi(x)u = P i k ui k , H3+067-0+9;FG78 65/78/D> H,6+05;> 5/-79:+6;<55 (12)k=1x) ; x ) (u), =37053. i k (P (x)u) = i k (u), 0D45,6;F-,; 1.1 i = A;1(xi ; i(x) i(+1i i.. !"# $B $Bx1. '&+-.- ')3 &)(-110i(u) = (i k (u))k=1:::3 2 R3 . +I@@5<57/-D j = (j l)tl=1:::3 2 R3 :95C65?.FG783@H/1<55 Pj(y) v = P j lvj l , H3+067-0+9;FG78 65/78/D> H,6+05;> 5/-79:+6;l=1<55 (12) 053. j l(Pj(y)v) = j l(v), 0D45,6;F-,; 1.1 j = A;1 (yj ; j(y) j(y+1) ;yj )j (v), =37 j (v) = (j l(v))l=1:::3 2 R3 )..25, 0 :9+,-9./,-07 @H/1<58 { -7/2+9/DA :9+520737/58:wij kl(x y) = ui k (x)vj l (y) = (x ; i(x))k;1(y ; j(y) )l;1 k l = 1 2 3:.25, 0 :9+,-9./,-07 @H/1<5+/.6+0 { -7/2+9/DA :9+520737/58:ij kl(f ) = i k (j l(f )) k l = 1 2 3:+ -7+97>7 1 :95C65?.FG.; @H/1<5; 5>77- 053 (4):3 X33 X3XXPf =ij klwij kl =ij kl(x ; i(x) )k;1(y ; j(y) )l;1k=1 l=1(14)k=1 l=1(x) (x);1=37 1+I@@5<57/-D ij kl 0D45,6;F-,; 0 (5).
,56H A;1i = A (xi ; i i+1 ; xi ),Bj;1 = A;1(yj ; j(y) j(y+1) ; yj ) 5>77>:x) ; x )F (A;1 (y ; (y) (y) ; y ))t ;ij = A;1 (xi ; i(x) i(+1i ijjjjj +1=37 ;ij = (ij kl)k l=1:::3 2 M3 (R), Fij = (ij kl(f ))k l=1:::3 2 M3(R).&! 7!% { 6$& 5$6%#A+3/D>5 3.//D>5 .6=+95->. ;06;F-,; 2/.47/5; f (xi yj ) = i 2(j 2(f )),i = 1 2 : : : n , j = 1 2 : : : m , 5 -7 565 5/D7 19.70D7 H,6+05; (,>.
x I.20.4).6; :+,-9+7/5; 5/-79:+659HFG78 @H/1<55 -97CH7-,; 0D45,65-B 0,7 +,-.6B/D72/.47/5; @H/1<5+/.6+0 ij kl(f ) 0 (14).C+2/.45> 0 (12) V (x) = (vi(x))i=1 2 ::: n+1 2 Rn+1 , F (x) = (f (xi))i=1 2 ::: n 2 Rn ,5 0 (13) V (y) = (vj(y) )j=1 2 ::: m+1 2 Rm+1 , F (y) = (f (yj ))j=1 2 ::: m 2 Rm . 5,-7>. (I.20.8), 3+:+6/7//.; 19.70D>5 H,6+05;>5, :973,-.06;7- ,+C+8 ,0;2B >7?3H>7?3H 071-+9.>5 V (x) 5 F (x) (:95 xi := xi , := (x) ), . -.1?7 >7?3H 071-+9.>5 V (y) 5 F (y) (:95 xi := yi , := (y) ). C+2/.45> 47972 G(x) 5, ,++-07-,-07//+,G(y) , >.-95<H, 0D9.?.FGHF 97E7/57 ,5,-7>D (I.20.8) 47972 071-+9. F (x) (,++-07-,-07//+, F (y) ), -.1, 4-+ V (x) = G(x)F (x) , V (y) = G(y) F (y) . .-95<.
G(x) 5>779.2>79 (n + 1) n , G(y) { 9.2>79 (m + 1) m , 5 H45-D0.7- ,++-07-,-0HFG57=9./54/D7 H,6+05;.7:79B >D >+?7> ,@+9>H659+0.-B .6=+95-> 0D45,67/5; @H/1<5+/.6+0ij kl(f ) 0 (14). H,-B F = (fij ) +C+2/.4.7- n m >.-95<H 2/.47/58 f (xi yj ).D45,65> (+35/ 9.2):(n + 1) m >.-95<H F (x) = (fij(x)) G(x) F ,n (m + 1) >.-95<H F (y) = (fij(y) ) F (G(y) )t ,.. !"# $B $Bx2.
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