К.Ю. Богачёв - Практикум на ЭВМ. Методы приближения функций (1133845), страница 13
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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 3. +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).x20. x 20.1.>=; H,-B 2.3./D -+415 a = x1 < x2 < : : : < xn = b 5 2/.47/5;f (x1 ) f (x2) : : : f (xn). 0737> 3+:+6/5-76B/D7 -+415 1 2 : : : n+1 -.1, 4-+CD1 < a = x1 < 2 < x2 < 3 < x3 < : : : < xn;1 < n < xn = b < n+1:.:95>79, >+?/+ :+6+?5-Bi = xi;12+ xi i = 2 3 : : : n:/-79:+659HFG.; @H/1<5; Pf ,-9+5-,; -.15> +C9.2+>, 4-+ /. 1.?3+> +-97217Si i+1], i = 1 : : : n +/. ;06;7-,; >/+=+467/+> Pi ,-7:7/5 2, -.15>, 4-+9=Pi(xi) = f (xi)i = 1 : : : n(1)Pi(i) = viPi(i+1) = vi+1 =37 vi , i = 1 2 : : : n + 1 { ,0+C+3/D7 :.9.>7-9D, -+- 565 5/+8 ,:+,+C 0DC+9.1+-+9DA +:97376;7- >7-+3 1H,+4/+8 5/-79:+6;<55 :.9.C+6547,15>5 >/+=+467/.>5.
+6H47//.; @H/1<5; Pf ,+0:.3.7- , f 0 -+41.A xi , i = 1 : : : n 5 36;6FC+=+ /.C+9. :.9.>7-9+0 vi Pf 2 C (Sa b])... !"# # $#x20. '&+-.- &+*x 20.2.81 &+,!)$ %&!; E 7% )!+I@@5<57/-D >/+=+467/. Pi , 2.:5,.//+=+ 0 @+9>7PiPi(x) = a1 i + a2 i(x ; i) + a3 i(x ; i)(x ; xi )>+=H- CD-B 0D45,67/D :+ 5/-79:+6;<5+//+8 @+9>H67 BF-+/. (,>. +:973767/57 (4.1) 9.23767//DA 9.2/+,-78):vif (xi)n.nf (xi) ; vixi ; invi+1 ; f (xi ) .i+1 ; xi.vi+1 ; f (xi) ; f (xi) ; vii+1 ; i i+1 ; xixi ; i1!vi+1-+6C<D 0D45,6;F-,; ,670.
/.:9.0+, I67>7/-D ,-+6C<. 0D45,6;F-,; ,/52H0079A, ,-97615 H1.2D0.F- ;4781H :.>;-5, 0 1+-+9HF :+>7G.7-,; 972H6B-.-.-,F3. :+6H4.7>a1 i = vi) ; via2 i = f (xxi;i i !v1i+1 ; f (xi ) f (xi ) ; via3 i = ; ; x ; x ; i+1ii+1iii6; :973,-.067/5; 0 0537Pi(x) = c1 i + c2 i(x ; i) + c3 i(x ; i)2(2)2.:5E7>(x ; i)(x ; xi) = (x ; i)(x ; i + i ; xi ) = (x ; i)2 ; (xi ; i)(x ; i)+-1H3. :+6H4.7>c1 i = a1 i = vic2 ic3 i!f(xxi ) ; vii ; i vi+1 ; f (xi ) f (xi ) ; vi= a2 i ; (xi ; i)a3 i =; i+1 ; i i+1 ; xi ; xi ; ix;ii!v1i+1 ; f (xi ) f (xi ) ; vi= a3 i = ; ; x ; x ; i+1ii+1iii(3)..
!"# # $#x20. '&+-.- &+*x 20.3. &+,82$5!;7; 5$>! 5!/-79:+6;<5; :.9.C+6547,15>5 ,:6.8/.>5 5,:+6B2H7-,; 36; :+6H47/5;/.5C+677 =6.31+8 :95C65?.FG78 @H/1<55 Pf 0 9.,,>.-950.7>+> 16.,,71H,+4/+-10.39.-54/DA >/+=+467/+0..9.>7-9D vi , i = 1 2 : : : n + 1 0 +CG78 ,A7>7 5/-79:+6;<55 1H,+4/+10.39.-54/D>5 @H/1<5;>5 0DC59.F-,; -.1, 4-+CD :95C65?.FG.; @H/1<5;Pf CD6. ,:6.8/+> 2-=+ :+9;31. -.7. Pf 2 C (1) (Sa b]) 565Pi0;1 (i) = Pi0(i)i = 2 3 : : : n:(4)2 :973,-.067/5; >/+=+467/. Pi 0 0537 (2) /.A+35>Pi0;1(i) = c2 i;1 + 2(i ; i;1)c3 i;1Pi0(i) = c2 i:(5) :+>+GBF (3) 0D45,65>. -.1?7!f(xxi ) ; vii ; i vi+1 ; f (xi ) f (xi ) ; vic2 i = x ; ; ; ; x ; x ; =ii i i!i+1 i i+1 i= 1 + xi ; i f (xi ) ; vi ; xi ; i vi+1 ; f (xi) i+1 ; i xi ; i! i+1 ; i i+1 ; xi1 + 1xi ; i= ;vi x ;;v+i+1(i i i+1 ; ii+1 ; i )(i+1 ; xi )!11xi ; i+ f (xi ) x ; + ; + ( ; )( ; x )iii+1i+1!i i i+1 i !1 ; 11 + 1;v+= ;vi x ;i+1i i i+1 ; ii+1 ; xi i+1 ; i!11+ f (xi )x ; + ;x ii+1i!vi+1 ; f (xi ) f (xi ) ; vic3 i = ; ; x ; x ; =i+1iii i+1 i!!11111= ; vi x ; + vi+1 ; x ; f (xi) x ; + ; x :i+1iiii+1iiii+1i15ic2 i + 2(i+1 ; i)c3 i = !!1111= ;vi x ; + ; ; vi+1 ; x ; ; +ii+1ii+1i i i+1 i !1 + 1+ f (xi) x ; ;x +i..ii+1i !"# # $#x20.
'&+-.- &+*83 &+,!!1111+ 2 vi+ vi+1; f (xi) xi ; i + i+1 ; xi =xi ; ii+1 ; xi!!1111= vixi ;i ; i+1 ; i + vi!+1 i+1 ; xi + i+1 ; i ;; f (xi) x ;1 + 1; xiii+1i :+>+>+GBF I-5A 9.07/,-0 5 (5) /.A+35>11!11!Pi (i) = ;vi x ; + ; ; vi+1 ; x ; ; +i ii+1ii+1ii+1i!11+ f (xi) x ; + ; x 0i5ii+1i!11 + 1P (i) = vi;1 x ; ; ;1+ vii;1 i;1ii;1! i ; xi;1 i ; i;1; f (xi;1) x ;1 + ;1x0i;1i;1i;1i!!;(7)i;1+3,-.06;; (6) 5 (7) 0 (4), :+6H4.7> 36; 0,7A i = 2 3 : : : n(6)!11 + 1 + 1 + 1;vi;1 +xi;1 ; i;1 i ; i;1 i ; xi;1 i ;! i;1 xi ; i i+1 ; i vi ++ 1; x ; 1; vi+1 =i+1 ! ii+1 i!1111f (xi;1) + x ; + ; x f (xi)= x ; + ;xi;1i;1ii;1iii+1i(8)J-+ ,5,-7>.
52 n ; 1 65/78/DA H9.0/7/58 +-/+,5-76B/+ n + 1 /75207,-/DAv1 v2 : : : vn+1 . 6; 77 2.>D1./5; -97CH7-,; 7G7 30. H9.0/7/5;, 1+-+9D7 ,-9+;-,;, 5,A+3; 52 -+8 565 5/+8 3+:+6/5-76B/+8 5/@+9>.<55 + @H/1<55 f (,>./5?7).5,-7>. (8) ;06;7-,; ,5,-7>+8 H9.0/7/58 , -97A35.=+/.6B/+8 >.-95<78 ,(,-9+=5>) 35.=+/.6B/D> :97+C6.3./57> 5 77 >+?/+ 97E.-B >7-+3+> .H,,.
C720DC+9. =6.0/+=+ I67>7/-..,/+0/D7 %% 5$>!@<= 7 Pf :1) Pf ;06;7-,; /76+1.6B/+8 1H,+4/+->/+=+467//+8 @H/1<578, 2/.47/5; 1+-+9+8 2.05,;- +- :+0737/5; @H/1<55 f 0 1.?3+8 -+417 x1 x2 : : : xn .12) 7-+3 :95C65?7/5; ;06;7-,; 65/78/D>: P (f + g) = Pf + Pg , P (f ) =Pf ... !"# # $#x20. '&+-.- &+* &+,84+=97E/+,-B .::9+1,5>.<55 :.9.C+6547,15>5 ,:6.8/.>5 (0 ,56H /76+1.6B/+,-5 ,:6.8/.) 2.05,5- +- 0DC+9.
3+:+6/5-76B/DA H9.0/7/58 36; 2.>D1./5; ,5,-7>D (8) 5 ,+,-.06;7- +- O(h1) 3+ O(h3), =37 h +:97376;7-,; 9.07/,-0+> (19.5).x 20.4.5$#! #<= )$&!%,:+6B2HF-,; /7,1+6B1+ >7-+3+0 :+,-9+7/5; 30HA 3+:+6/5-76B/DA H9.0/7/58 36; 2.>D1./5; ,5,-7>D (8), 1+-+9D7 0DC59.F-,; 0 2.05,5>+,-5 +- -+=+,1.1.; 3+:+6/5-76B/.; 5/@+9>.<5; 5207,-/. + :95C65?.7>+8 @H/1<55.x 20.4.1.5$#! #<= )$& !% 56%& 6; 5$% 5$6%# 7 % )$&6!H,-B 5207,-/D 2/.47/5; f 0(x1 ) 5 f 0(xn) :9+520+3/+8 f 0 @H/1<55 f 0 =9./54/DA H26.A x1 5 xn . +2>+?/D 30.
I1050.67/-/DA :+3A+3..1. 0. 3+:+6/5-76B/DA H9.0/7/5; :+6H4.F-,; 52 H,6+058P10 (x1 ) = f 0(x1 )Pn0 (xn) = f 0(xn ):D45,65>Pi0(xi ) = c2 i + 2(xi ; i)c3 i = !!1111= ;vixi; i + i+1 ; i ; !vi+1 i+1 ; xi ; i+1 ; i +1 + 1+ f (xi)xi ; i i+1 ; xi +!!1111xi ; i+ 2 ; vi x ; + vi+1 ; x ; f (xi) x ; + ; xi+1iiii+1iiii+1i+,1+6B1Hx i ; i1 = 1 ; 1 i+1 ; i i+1 ; xi i+1 ; xi! i+1 ; ixi ; i1 + 1i+1 ; i= xi ; i= 1 i+1 ; i xi ; i i+1 ; xii+1 ; i (i+1 ; xi)(xi ; i) i+1 ; xi-+!!11110Pi (xi ) = ;vi x ; + ; ; vi+1 ; x ; ; +ii+1ii+1i i i+1 i !1 + 1+ f (xi) x ;+i i i+1 ; xi..
!"# # $#x20. '&+-.- &+*+ 2vi+1+11!;;2f (xi) 1 =!i ; xi i ; i! i ; xi i ; i1 ; 11 ; 1= vi+ vi+i ; i xi ; ii ; xi i ; i!11+ f (xi) x ; ; ; xiiii+ 2vi185 &+,1+1+1+1+1+1+1+1+10. 3+:+6/5-76B/DA H9.0/7/5; :+6H4.F-,; 52 H,6+058Pi0(xi ) = f 0(xi )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.2. DC59.7> i = x1 , n+1 = xn .
+=3. 0 ,5,-7>H H9.0/7/58 (8) v1 5 vn+1 /70+83H-, . P1 5 Pn ,-9+;-,; :+ 5/-79:+6;<5+//+8 @+9>H67 BF-+/. , 19.-/D>5H26.>5: >/+=+467/P1(x) = c1 1 + (x ; x1 )c2 1 + (x ; x1 )2 c3 1,-9+5-,; 52 H,6+058P1(x1 ) = f (x1 ) P10 (x1 ) = f 0(x1 ) P1 (2) = v24-+ 3.7-c1 1 = f (x1) c2 1 = f (x1 ) c3 1 = 102./.6+=54/+, >/+=+467/;x1!v2 ; f (x1) ; f 0(x ) Y12 ; x1Pn(x) = c1 n + (x ; n)c2 n + (x ; n)2c3 n,-9+5-,; 52 H,6+058Pn(n) = vn P1(xn) = f (xn) P10 (xn) = f 0(xn )4-+ 3.7-!f(x1f(xn ) ; vnn ) ; vn00c1 n = vn c2 n = 2 x ; ; f (xn ) c3 n = x ; f (xn) ; x ; :nnnnnn2 H,6+05; Pf 2 C (1) (Sa b]) :+6H4.7>Pi0;1 (i) = Pi0(i)..i = 2 3 : : : n: !"# # $#x20.
'&+-.- &+* &+,86{ n ; 1 H9.0/7/5; +-/+,5-76B/+ n ; 1 /75207,-/DA v2 v3 : : : vn . 9.0/7/5; 36;i = 3 4 : : : n ; 1 ,+0:.3.F- , 0D:5,.//D>5 9./77 H9.0/7/5;>5 (8), H9.0/7/5;36;P10 (2) = P20 (2) Pn0 ;1(n) = Pn0 (n)52>7/;-,; 0 ,56H 52>7/50E7=+,; 053. >/+=+467/+0 P1 5 Pn;1 .1. D:5E7> /+0+7 :790+7 H9.0/7/57 ,5,-7>D (8). >77>!v;f(x)21P10 (2) = c2 1 + 2(2 ; x1 ) = f 0(x1 ) + 2 ; x ; f 0(x1 )21= 2 v2;;f (xx1) ; f 0(x1 )210D9.?7/57 36; P (2) :+6H4.7> 52 (6) :95 i = 2:!!11110P2 (2) = ;v2 x ; + ; ; v3 ; x ; ; +2 2 3 2 ! 3 2 3 21 + 1+ f (x2 ) x ;2 3 ; x2 :2673+0.-76B/+, H9.0/7/57 5>77- 053!!11112v2 ; x + x ; + ; + v3 ; x ; ; =212232323 2!211= f (x1) ; x + f (x2 ) x ; + ; x + f 0(x1 ):2122322. D:5E7> /+0+7 :+,673/77 H9.0/7/57 ,5,-7>D (8).
>77>) ; vn ; f 0(x )Pn0 (n) = c2 n = 2 f (xxn;nn n0D9.?7/57 36; Pn0 ;1(n) :+6H4.7> 52 (7) :95 i = n :!!11110Pn;1(n) = vn;1 x ; ; ; + vn ; x + ; ;n;1n;1nn;1nn;1nn;1!11; f (xn;1) xn;1 ; n;1 + n ; xn;1673+0.-76B/+, H9.0/7/57 5>77- 053!!11112vn;1 x ; ; ; + vn ; x + ; + x ; =n;1n;1nn;1nn;1!n n;1 n n11++ f (xn) 2 ; f 0(xn):= f (xn;1)xn;1 ; n;1 n ; xn;1xn ; n02..
!"# # $#x20. '&+-.- &+* &+,87x20.4.2. 5$#! #<= )$& !% 56%& 6; %$ 5$6%# 7 % )$&6!H,-B 5207,-/D 2/.47/5; f 00(x1 ) 5 f 00(xn) :9+520+3/+8 f 00 @H/1<55 f 0 =9./54/DA H26.A x1 5 xn . +=3. :+-97CH7> +- :95C65?.FG78 @H/1<55 Pf(Pf )00(x1 ) = f 00(x1 )(Pf )00(xn) = f 00(xn)-.7.P100(x1 ) = f 00(x1 ) Pn00(xn ) = f 00(xn):2 (2) /.A+35>Pi00(xi ) = 2c3 i =!!21111=i+1 ; i vi xi ; i + vi+1 i+1 ; xi ; f (xi) xi ; i + i+1 ; xi :(9)+=3. 3+:+6/5-76B/D7 H9.0/7/5;Pi00(xi ) = f 00(xi ) i = 1 n:J-5 9.07/,-0.
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