К.Ю. Богачёв - Практикум на ЭВМ. Методы приближения функций (1133845), страница 14
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;06;F-,; 30H>; /73+,-.FG5>5 H9.0/7/5;>5, 1+-+9D7 36; ,+A9./7/5; -97A35.=+/.6B/+,-5 ,5,-7>D (8) /.3+ ,376.-B ,++-07-,-07//+ :790D>5 :+,673/5> H9.0/7/5;>5 I-+8 ,5,-7>D.x 20.4.3.\%&" )$&!%;J-+- >7-+3 5,:+6B2H7-,;, 1+=3. /51.1+8 3+:+6/5-76B/+8 5/@+9>.<55 + :95C65?.7>+8 @H/1<55 f /7-.+=3. :+-97CH7> +- :95C65?.FG78 @H/1<55 Pf(Pf )00(x1 ) = 0(Pf )00(xn) = 0-.7.P100(x1 ) = 0 Pn00(xn ) = 0:2 ,++-/+E7/58 (9) :+6H4.7> 30.
3+:+6/5-76B/DA H9.0/7/5;Pi00(xi ) = 0 i = 1 n:J-5 9.07/,-0. ;06;F-,; 30H>; /73+,-.FG5>5 H9.0/7/5;>5, 1+-+9D7 36; ,+A9./7/5; -97A35.=+/.6B/+,-5 ,5,-7>D (8) /.3+ ,376.-B ,++-07-,-07//+ :790D>5 :+,673/5> H9.0/7/5;>5 I-+8 ,5,-7>D.,65 H :95C65?.7>+8 @H/1<55 f 00(x1 ) 6= 0 565 f 00 (xn) 6= 0, -+ :9+520+6 0 0DC+97 =9./54/DA H,6+058 :950+35- 1 :+0DE7//+8 :+=97E/+,-5 0 :95=9./54/DAH26.A... !"# # $#x20. '&+-.- &+*x 20.4.4.!% \ %;88 &+,6!" % 5$)$&6!J-+- >7-+3 5,:+6B2H7-,;, 1+=3.
/51.1+8 3+:+6/5-76B/+8 5/@+9>.<55 + :95C65?.7>+8 @H/1<55 f /7-.+=3. :+-97CH7> +- :95C65?.FG78 @H/1<55 PfP1 P2 Pn;1 Pn(10)-.7. >/+=+467/ P1 P2 5/-79:+659H7- f /. S1 3] :+ -+41.> x1 x2 : P1(x1 ) =f (x1 ) P1(x2 ) = f (x2) 5 P1 (3) = v3 , . >/+=+467/ Pn;1 Pn 5/-79:+659H7- f /.Sn;1 n+1] :+ -+41.> xn;1 xn : Pn(xn;1 ) = f (xn;1 ) Pn(xn) = f (xn) 5 Pn(n;1) =vn;1 . 673+0.-76B/+, >/+=+467/ P1 P2 ,-9+5-,; 1.1 5/-79:+6;<5+//D8 >/+=+467/ 0-+9+8 ,-7:7/5, :95/5>.FG58 0 -+41.A x1 x2 3 2/.47/5; f (x1) f (x2) v3 Y>/+=+467/ Pn Pn;1 ,-9+5-,; 1.1 5/-79:+6;<5+//D8 >/+=+467/ 0-+9+8 ,-7:7/5, :95/5>.FG58 0 -+41.A n;1 xn;1 xn 2/.47/5; vn;1 f (xn;1) f (xn)..15> +C9.2+>, 0>7,-+ n >/+=+467/+0 Pi i = 1 2 : : : n 5 n + 1 :.9.>7-9+0vi i = 1 2 : : : n + 1 5>77> n ; 2 >/+=+467/.
P1 P2 P3 P4 : : : Pn;2 Pn;1 Pn5 n ; 3 :.9.>7-9. v3 v4 : : : vn;1 . >7-+37 :95C65?7/5; :.9.C+6547,15>5 ,:6.8/.>5 0+2>+?/D 30. :+3A+3..1. 2 H,6+05; Pf 2 C (1) (Sa b]) :+6H4.7>Pi0;1(i) = Pi0(i)i = 3 4 : : : n ; 1{ n;3 H9.0/7/5; +-/+,5-76B/+ n;3 /75207,-/DA v3 v4 : : : vn;1 . 9.0/7/5; 36;i = 4 5 : : : n ; 2 ,+0:.3.F- , 0D:5,.//D>5 9./77 H9.0/7/5;>5 (8), H9.0/7/5;36;P10 (3) P20 (3) = P30 (3) Pn0 ;2(n;1) = Pn0 ;1(n;1) Pn0 (n;1)52>7/;-,; 0 ,56H 52>7/50E7=+,; 053.
>/+=+467/+0 P1 P2 , Pn;1 Pn ..). D:5E7> /+0+7 :790+7 H9.0/7/57 ,5,-7>D (8). + 5/-79:+6;<5+//+8 @+9>H67 BF-+/.P1 (x) = a1 1 + (x ; x1 )a2 1 + (x ; x1 )(x ; x2 )a3 1=37a1 1 = f (x1) a2 1 = f (x1 Y x2 ) a3 1 = 13+I-+>H;x1!v3 ; f (x2) ; f (x Y x ) :1 23 ; x2P10 (3) = a2 1 + (23 ; x1 ; x2)a3 1 =!2v3 ; x1 ; x23 ; f (x 2 ); f (x1Y x2) == f (x1Y x2 ) + ; x313 ; x2!11x2 f (x Y x ):= ; x + ; x (v3 ; f (x2 )) ; 3 ;;x 1 23..13231 !"# # $#x20.
'&+-.- &+* &+,89D9.?7/57 36; P30 (3) :+6H4.7> 52 (6) :95 i = 3:!!1111P30 (3) = ;v3 x ; + ; ; v4 ; x ; ; +3 3 4 3 ! 4 3 4 31 + 1+ f (x3 ) x ;3 4 ; x3 :3673+0.-76B/+, H9.0/7/57 5>77- 053!!111111v3 ; x + ; x + x ; + ; + v4 ; x ; ; =343! 4 31 3 2 3 ! 3 4 31 + 13 ; x2 f (x Y x ):+= f (x2 ) ;1 x + ;1 x + f (x3 ) x ;3 4 ; x3 3 ; x1 1 231323C). D:5E7> /+0+7 :+,673/77 H9.0/7/57 ,5,-7>D (8). + 5/-79:+6;<5+//+8@+9>H67 BF-+/.Pn(x) = a1 n + (x ; n;1)a2 n + (x ; n;1)(x ; xn;1 )a3 n=371 (f (x Y x ) ; a ) :) ; vn;1 a =a1 n = vn;1 a2 n = f (xxn;1;3nn;1 n2nxn ; n;1n;1 n;1+I-+>HPn0 (n;1) = a2 n + (n;1 ; xn;1 )a3 n = a2 n ; xxn;1;;n;1 (f (xn;1 Y xn) ; a2 n) =nn;1xn ; 2n;1 a ; xn;1 ; n;1 f (x Y x ) == xn;1x+ ;2nn;1 nx;nn;1nn;1!11(f (xn;1 ) ; vn;1 ) ; xxn;1;;n;1 f (xn;1 Y xn):= x ; + x ;n;1n;1nn;1nn;1D9.?7/57 36; Pn0 ;2(n;1) :+6H4.7> 52 (7) :95 i = n ; 1:!110Pn;2(n;1) = vn;2 x ; ; ; +n;2n;2n;1n;2!11+ vn;1n;1 ; xn;2 + n;1 ; n;2 ;!; f (xn;2) x ;1 + ;1 xn;2n;2n;1n;2673+0.-76B/+, H9.0/7/57 5>77- 053!11vn;2 x ; ; ; +n;2n;2n;1n;2..
!"# # $#x20. '&+-.- &+*1!1+vn;1+++n;1 ; xn;2 n;1 ; n;2 xn;1 ; n;1 ! xn ; n;1 =11++= f (xn;2)xn;2 ; n;2 n;1 ; xn;2!1xn;1 ; n;1 f (x Y x ):1;+f (xn;1) x ; + x ; xn ; n;1 n;1 nn;1n;1nn;12. H37> 2.3.0.-B P1 P2 : : : Pn 1.1 9./BE7 /.C+9+> v1 v2 : : : vn+1 . D:+6/7/5; H,6+05; (10) 3+CO7>,;, :+-97C+0.0 /7:979D0/+,-5 (Pf )00 0 -+41.A 2 5n , -.7.P100(2) = P200(2) Pn00;1(n) = Pn00(n)(11)(H,6+05; (10) 5 (11) I1050.67/-/D 36; >/+=+467/+0 0-+9+8 ,-7:7/5, :+,1+6B1H 0,56H (4) H P1 5 P2 , Pn;1 5 Pn ,+0:.3.F- 0,7 :9+520+3/D7 +- /H670+=+ 3+ 0-+9+=+:+9;31. 016F45-76B/+).++-/+E7/5; (11) 3.F- 30. /73+,-.FG5A 36; 2.>D1./5; ,5,-7>D (8) H9.0/7/5;.2 :973,-.067/5; >/+=+467/.
Pi 0 0537 (2) /.A+35>, 4-+ ,++-/+E7/5; (11)I1050.67/-/Dc3 1 = c3 2 c3 n;1 = c3 n(12)x 20.4.5.190 &+,5!!"&16! % 5$)$&6!J-+- >7-+3 5,:+6B2H7-,;, 1+=3. /51.1+8 3+:+6/5-76B/+8 5/@+9>.<55 + :95C65?.7>+8 @H/1<55 f /7-.0737> 3+:+6/5-76B/D7 -+415 x0 < x1 5 xn+1 > xn 5 2/.47/5; :95C65?.7>+8 @H/1<55 f (x0) 5 f (xn+1) 0 I-5A -+41.A. +=3. :+6.=.7> 1 = x0 5n+1 = xn+1v1 = f (1) vn+1 = f (n+1):J-5 9.07/,-0. ;06;F-,; 30H>; /73+,-.FG5>5 H9.0/7/5;>5, 1+-+9D7 36; ,+A9./7/5; -97A35.=+/.6B/+,-5 ,5,-7>D (8) /.3+ ,376.-B ,++-07-,-07//+ :790D>5 :+,673/5> H9.0/7/5;>5 I-+8 ,5,-7>D.x 20.4.6.G$5!;7; % 5$)$&6!J-+- >7-+3 5,:+6B2H7-,;, 1+=3. /51.1+8 3+:+6/5-76B/+8 5/@+9>.<55 + :95C65?.7>+8 @H/1<55 f /7-.
1.47,-07 :95C65?7//+=+ 2/.47/5; f 0(x1) f 0(xn) 65C+ f 00(x1 ) f 00(xn) C79H-,;2/.47/5; Q0k (x1 ) Rk0 (xn) 65C+ Q00k (x1 ) Rk00(xn), =37 Qk (x) { 5/-79:+6;<5+//D8>/+=+467/ .=9./?. ,-7:7/5 k ; 1, :+,-9+7//D8 :+ -+41.> x1 x2 : : : xk , Rk (x){ 5/-79:+6;<5+//D8 >/+=+467/ .=9./?.
,-7:7/5 k ;1, :+,-9+7//D8 :+ -+41.>xn;k+1 xn;k+2 : : : xn , k 2. +6H47//D7 2/.47/5; f 0 565 f 00 0 -+41.A x1 5xn 5,:+6B2HF-,; 1.1 H1.2D0.6+,B 0DE7 (,>. :+,-9+7/57 =9./54/DA H,6+058 :+2/.47/5;> :790+8 565 0-+9+8 :9+520+3/+8 @H/1<55 0 =9./54/DA H26.A)... !"# # $#91 II. ! "#$% &'$()*$ +, $$!,/-79:+6;<5; @H/1<58 f : [ Rn ! R1 H16.3D0.7-,; 0 +CGHF ,A7>H2.3.45 65/78/+8 5/-79:+6;<55. + I-. ,A7>. -97CH7- +C9.G7/5; >.-95<D (I.1.3)5 :+-+>H 36; +3/+>79/+=+ ,6H4.; 0DE7 CD6 :9736+?7/ 9;3 >7-+3+0, 0 1+-+9DAI-. >.-95<. /7 C+677, 47> -97A35.=+/.6B/..6; >/+=+>79/DA +C6.,-78 :+,-9+7/57 A+9+E+ +C9.-5>+8 >.-95<D (I.1.3):9+520+35-,;, 0 +,/+0/+>, 30H>; >7-+3.>5:1) 5/-79:+659+0./57> -7/2+9/D>5 :9+520737/5;>5,2) :95C65?7/57> >7-+3+> 1+/74/DA I67>7/-+0.1.
" ,, xJ-+- >7-+3 5/-79:+6;<55 :+20+6;7- /.5C+677 :9+,-+ ,07,-5 2.3.4H 5/-97:+6;<55 @H/1<55 >/+=5A :797>7//DA, 2.3.//+8 0 :.9.66767:5:737, 1 2.3.475/-79:+6;<55 @H/1<58 +3/+8 :797>7//+8.x 1.1.-% 6# $5!;7 6$& 5$6%#; $ $$H,-B 2.3./. @H/1<5; f : Z = X Y ! R1 , f 2 F (Z R1) { /71+-+9+>H @H/1<5+/.6B/+>H :9+,-9./,-0H, /.C+9D @H/1<58 u1 : : : un , ui : X ! R1 ,ui 2 F (X R1) 5 v1 : : : vm , vi : Y ! R1 , vi 2 F (Y R1) 5 /.C+9D 65/78/DA@H/1<5+/.6+0 1 : : : n /. :9+,-9./,-07 F (X R1) 5 1 : : : m /. :9+,-9./,-07 F (Y R1). 97CH7-,; :+,-9+5-B .::9+1,5>.<5F Pf @H/1<55 f 053.Pf =n XmXi=1 j =1i j wi j-.1HF, 4-+CD CD65 0D:+6/7/D 65/78/D7 H,6+05; 5/-79:+6;<55i j (Pf ) = i j (f ) i = 1 : : : n j = 1 : : : m..(1) !"# $B $Bx1.
'&+-.- ')3 &)(-=3792wi j = ui vj -.7. wi j (x y) = ui(x)vj (y)i j = i j -.7. i j (f ) = i(j (fx(y))) = j (i(fy (x)))fx(y) = f (x y) 2 F (Y R1) fy (x) = f (x y) 2 F (X R1)f 2 F (Z R1) i = 1 : : : n j = 1 : : : m x 2 X y 2 Y:C+2/.45>(X)nnU = huiii=1 =iui : 2 Rn V = hvj imj=1 i=1U = hiini=1 M = hj inj=1 W = U V = fw = u v : u 2 U v 2 V g :$ 1. 1.
u1 : : : un , 1 : : : nA = (i(uj )) i j=1 ::: n32. v1 : : : vm , (2)1 : : : mB = (i(vj )) i j=1 ::: m(3).1. : wij = ui vj , W nm .2. ; .3. . Pf =n XmXi=1 j =1W = U Vi j wi j,(4) ;1; = (i j )(i=1 ::: n j=1 ::: m) = A;1 (i j (f ))(i=1 ::: n j=1 ::: m) B t :(5)6!"%. ,;158 I67>7/- w 2 W = U V :973,-.06;7-,; 0 0537w=n XmXi=1 j =1i j wi j78,-05-76B/+, w 2 U V 7,-B w = u v , -.7. w(x y) = u(x)v(y). .1 1.1u1 : : : un ,+,-.06;F- C.25, U 5 v1 : : : vm ,+,-.06;F- C.25, V , -+u(x) =..nXi=1iui(x)v(y) =mXj =1j vj (y): !"# $B $Bx1. '&+-.- ')3 &)(-+I-+>Hw(x y) =n XmXi=1 j =193ij ui(x)vj (y):9+0795>, 4-+ I-+ :973,-.067/57 735/,-07//+ (I-+ 5 CH37- +2/.4.-B, 4-+ wij =ui vj +C9.2HF- C.25, U V ).
.,,>+-95>0n m1 n mXXXXi j k (ui)l (vj ) =kl(w) = (k l )(w) = (k l ) @i j ui vj A =i=1 j =1i=1 j =1n Xmn XmXX=i j akiblj =akii j blj = (A;B t )kli=1 j =1i=1 j =1+ H,6+05F >.-95<D A 5 B +C9.-5>D, ,673+0.-76B/+; = A;1(kl (w))k=1 ::: n l=1 ::: m(B t);1(6)-,F3. 0D-71.7-, 4-+ 1+I@@5<57/-D ij +3/+2/.4/+ +:97376;F-,; @H/1<578w , -.7.
wij { C.25, W .2 (1) 5 (6) 0D-71.F- 0-+9+7 5 -97-B7 H-079?37/5; -7+97>D.7+97>. 3+1.2./..x 1.2.7 ! $ 5$7<7/5> -9H3+7>1+,-B 0D45,67/58 :+ @+9>H67 (5).1) :+>+GBF, /.:95>79, LU {9.26+?7/5; >D >+?7> 0D45,65-B A;1 , 2.-9.-+8 O(n3) .95@>7-547,15A +:79.<58.2) :+>+GBF, /.:95>79, LU {9.26+?7/5; >D >+?7> 0D45,65-B B ;1 , 2.-9.-+8 O(m3) .95@>7-547,15A +:79.<58.3) .-95<. T = A;1(kl (f )) 0D45,6;7-,; :+ +:973767/5F :9+520737/5; >.-95< , 2.-9.-+8 O(n2m) .95@>7-547,15A +:79.<58.4) .-95<.
; = T (B t );1 0D45,6;7-,; :+ +:973767/5F :9+520737/5; >.-95<, 2.-9.-+8 O(nm2) .95@>7-547,15A +:79.<58..15> +C9.2+>, +CG77 1+6547,-0+ .95@>7-547,15A +:79.<58, -97CH7>DA 36;:9+0737/5; 0D45,67/58 :+ @+9>H67 (5), 9.0/+ O(n3 + n2m + nm2 + m3 ).F 1. 7:+,973,-07//+7 +:973767/57 1+I@@5<57/-+0 ; = (i j ) 52,5,-7>Dn XmXi=1 j =1k (ui)l (vj )ij = kl (f )k = 1 2 : : : n l = 1 2 : : : m52 nm 65/78/DA H9.0/7/58 +-/+,5-76B/+ nm /75207,-/DA ij -97CH7- 2 (nm)33,6+?7/58 5 ,-+6B1+ ?7 H>/+?7/58 :+ >7-+3H .H,,.. .:95>79, :95 n = m I-+3.7- O(n6) .95@>7-547,15A +:79.<58, 0 -+ 097>;, 1.1 @+9>H67 (5) O(n3)... !"# $B $Bx1.
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