К.Ю. Богачёв - Практикум на ЭВМ. Методы приближения функций (1133845), страница 9
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5/78/.; /72.05,5>+,-B >/+=+467/+0 Tb0 Tb1 : : : Tbn;1 3+-6!"%1.2./. 0 67>>7 13.1.+1.?7>, 4-+ 0,;158 >/+=+467/ Pn;1 2 Pn;1 >+?7- CD-B :973,-.067/ 0 053765/78/+8 1+>C5/.<55 >/+=+467/+0 Tb0 Tb1 : : : Tbn;1 .:973765> :+ -+41.> (13.7) 35,197-/+7 \,1.6;9/+7" :9+520737/57 (13.1) 5 :+,-9+5> 9.26+?7/57 (13.10) :+ >/+=+467/.> 7CDE70. Tb0 Tb1 : : : Tbn;1 36; @H/1<55 f = Pn;1 .
,56H 67>>D 13.3 Pn;1(xi ) = Pf (xi), i = 1 2 : : : n . +,1+6B1HPn;1 5 Pf ;06;F-,; >/+=+467/.>5 ,-7:7/5 /7 0DE7 n ; 1 5 ,+0:.3.F- 0 n-+41.A, -+ Pn;1 Pf . -.1, 0,;158 >/+=+467/ Pn;1 2 Pn;1 :973,-.067/ 0 0537 65/78/+8 1+>C5/.<55 >/+=+467/+0 Tb0 Tb1 : : : Tbn;1 (:9547> 1+I@@5<57/-D9.26+?7/5; 0D45,67/D /.>5 ;0/+, ,>. (13.10)).7>>.
3+1.2./.... !"# # $#x17. ' 1<@ ('( 2. 1 0# #:Tb0 Tb1 : : : Tbn;157 Sa b]-ZbZbbi(y)Tbj (y)b2T22qq T0 (y )dy = 2 ij i + j 6= 0dy = :a (b ; y )(y ; a)a (b ; y )(y ; a)6!"%. +1.?7> 0/.4.67, 4-+ >/+=+467/D T0 T1 : : : Tn;1 /. +-97217 S;1 1] H3+067-0+9;F- ,++-/+E7/5;>:Z1 T 2(x)Z1 Ti(x)Tj (x)22p1 ; x2 dx = 2 ij i + j 6= 0 p10 ; x2 dx = :(1);1;1>77>:Z1 Ti (x)Tj (x)Z1 1p1 ; x2 dx = p1 ; x2 cos(i arccos x) cos(j arccos x) dx:;1;1376.7> 237,B 2.>7/H x = cos , 2 S0 ]:Z0Z1 Ti (x)Tj (x)p1 ; x2 dx = sin1 cos(i) cos(j)(; sin ) d =;1Z= cos(i) cos(j) d =0=37ZZ11=cos((i ; j )) d +cos((i + j )) d =2020= 21 (I (i ; j ) + I (i + j ))98 1>>sin(k)=0k=60=< k0= k 0:I (k) = cos(k) d = > Z >:1d=k=000Z673+0.-76B/+,8122Z1 Ti (x)Tj (x)11p1 ; x2 dx = 2 i j + 2 i ;j = <: 2 ij ii =+jj=6=0 0;1376.7> 237,B 2.>7/H :797>7//DA (11.4): x = x(y), x : Sa b] ! S;1 1], x(y) =2y ; (b + a) :b;aZbZ1 Ti(x)Tj (x)p1 ; x2 dx = r 21y;(b+a) 2 Tbi(y)Tbj (y) b ;2 a dya 1;;1b;a..
!"# # $#x17. ' 1<@ ('(.1 1.1r581 q(b ; a)2 ; (2y ; (b + a))2 == 2y;(b+a) 2 b ; a11 ; b;aq1= b ; a (b ; a ; (2y ; (b + a)))(b ; a + 2y ; (b + a)) =qq= b ;1 a (2b ; 2y)(2y ; 2a) = b ;2 a (b ; y)(y ; a)-+Z1 Ti (x)Tj (x)Zb Tbi (y)Tbj (y)p1 ; x2 dx = q(b ; y)(y ; a) dya;12 (1) :+6H4.7> -97CH7>D8 972H6B-.-.7>>. 3+1.2./..x 17.2.-% 6# ! $5!;70737> ,1.6;9/+7 :9+520737/57ZbSu v] =/+9>H5 :9+,-9./,-0+ ,56Haq u(y)v(y) dy(b ; y)(y ; a)kuk= Su u]1=2L2 (Sa b]) = f f : kf k < 1g:Zbdy<1a (b ; y )(y ; a)5>77- >7,-+ 016F47/57 C (Sa b]) L2 (Sa b]).+ >7-+3H /.5>7/BE5A 10.39.-+0 36; @H/1<55 f 2 L2 (Sa b]) -97CH7-,; :+,-9+5-B :95C65?7/57 Pf 2 Pn;1 -.1+7, 4-+kf ; Pf k = g2Pinf;1 kf ; gk :qn ,56H 67>>D 1 >+?/+ 5,1.-B Pf 0 0537Pf =nX;1j =0j Tbj :(2);1673+0.-76B/+, /.3+ /.8-5 fj gnj=0-.157, 4-+kf ; jX j Tbj kn;1=0..= g2Pinf;1 kf ; gk :n !"# # $#x17. ' 1<@ ('(59 701653+0+> :9+,-9./,-07 L2 Sa b] >5/5>H> kf ; Pf k 97.652H7-,;D b En;1/.
@H/1<55 Pf , ;06;FG78,; :9+71<578 f /. :+3:9+,-9./,-0+ Pn;1 = Tj j=0 . 673+n on;10.-76B/+, I67>7/- f ; Pf +9-+=+/.67/ :+3:9+,-9./,-0H Pn;1 . .1 1.1 Tbj j=0{ C.25, Pn;1 , -+ H,6+057 +9-+=+/.6B/+,-5 Pn;1 I1050.67/-/+ +9-+=+/.6B/+,-5Tbi , i = 0 1 : : : n ; 1:hif ; Pf Tbi = 0 i = 0 1 : : : n ; 1-.7.hi h iPf Tbi = f Tbi i = 0 1 : : : n ; 1:(3),6+05; (3) :973,-.06;F- ,+C+8 65/78/D7 H,6+05; 5/-79:+6;<55, 1+-+9D>3+6?/. H3+067-0+9;-B :95C65?.FG.; @H/1<5; (2).2 (3) 5 (2) /.A+35>:nX;1 hih iTbj Tbi j = f Tbi i = 0 1 : : : n ; 1:(4)j =0 ,56H 67>>D 2hb bih iTj Ti = ij Tbi Tbi i j = 0 1 : : : n ; 1:+I-+>H 52 (4) :+6H4.7>h bif Tii = h b b i i = 0 1 : : : n ; 1Ti Ti-.7.h ih i0 = 1 f Tb0 i = 2 f Tbi x 17.3.i = 1 2 : : : n ; 1(5)&! E 7% $6!@;6; 0D45,67/5; 1+I@@5<57/-+0 9.26+?7/5; (5) -97CH7-,; 0D45,6;-B 5/-7=9.6Dh b i Zb f (y)Tbi(y)dyf Ti = q(b;y)(y;a)a;a,0737> -+415 a = y0 < y1 < : : : < yN = b (/.:95>79, yj = a + jh , h = b Nj = 0 1 : : : N , N > 0, 4.,-+ N n ).
+=3.y;1 Z +1bh b i NXq f (y)Ti(y) dyf Ti =(b ; y)(y ; a)j =0 yj(6)j.. !"# # $#x17. ' 1<@ ('(6095C6525> @H/1<5Fg(y) = f (y)Tbi(y)(7)/. +-97217 Syj yj+1] 65/78/+8 @H/1<578 { 5/-79:+6;<5+//D> >/+=+467/+> .=9./?. L(2j) :790+8 ,-7:7/5, :+,-9+7//D> :+ -+41.> yj 5 yj+1 :g(yj ) :L(2j)(y) = g(yj ) + (y ; yj )g(yj Y yj+1) = g(yj ) + (y ; yj ) g(yyj+1) ;j +1 ; yj95 I-+> 0 ,56H 67>>D 15.1 ,:9.073650+ /79.07/,-0+kg ; L(2j)kC(y y +1]) 18 (yj+1 ; yj )2 kg00kC(y y +1]):673+0.-76B/+, y +1yZ +1(j ) Z q g(y)dy ; q L2dy (b ; y)(y ; a) y (b ; y)(y ; a)yyZ +1(8) 18 (yj+1 ; yj )2 kg00kC(y y +1]) q(b ; y1)(y ; a) dy =y1= (yj+1 ; yj )2 kg00kC (y y +1])Cj (a b)8=37yZ +11Cj (a b) = qdy:(b ; y)(y ; a)yyZ +1dy ,673HFG5> ,:++I-+>H >D >+?7> :95C6525-B 5/-7=9.6 q g(y)(b ; y)(y ; a)y,+C+>:yZ +1yZ +1(j )g(y)qdy q L2 (y)dy(b;y)(y;a)(b;y)(y;a)yy, +E5C1+8 (8).
+3,-.06;; I-+ 0 (6), /.A+35>y(j );1 Z +1h b i NXq L2 (y)dy(9)f Ti (b ; y)(y ; a)j =0 y, +E5C1+8h i NXy(j );1;1 Z +1 NX1 (y ; y )2 kg00kL2 (y) f Tbi ;qdyj +1jC (y y +1 ]) Cj (a b) (b ; y)(y ; a) j=0 8j =0 yN ;1 18 kg00kC(a b]) h2 X Cj (a b) =j =012b= 8 C (a b) h k(f Ti)00kC (a b])(10)jjjjjjjjjjjjjjjjjjjjjjjjjjjj.. !"# # $#x17. ' 1<@ ('(=3761Zb1dy:(b ; y)(y ; a)h = j=0max(yj+1 ; yj ) C (a b) = q1 ::: N ;1aD45,65>(j )Z +1 g(yj ) + (y ; yj )g(yj Y yj+1)L2 (y )qqdy =dy =(b;y)(y;a)(b ; y)(y ; a)yyyZ +1yZ +1dy= (g(yj ) ; yj g(yj Y yj+1)) q+ g(yj Y yj+1) q y dy(b ; y)(y ; a)(b ; y)(y ; a)yy(11).1 1.1ZZ b;a dxZ dxdy2qpp1 ; x2 = arcsin x==b;a 1 ; x2(b ; y)(y ; a) y= b+a + b;a x2yZ +1yjjjjjjjj2-+aj =yZ +1jyj.1 1.12y +1dy2y;(b+a) :q= arcsinb ; a y(b ; y)(y ; a)j(12)jZ b+2 a + b;2 a x b;2 a dx Z b + a dxydyqp==b;a p1 ; x22 1 ; x2 +(b ; y)(y ; a) y= b+a + b;a x222Z b ; a x dxb+ab ; a ; p1 ; x2 p=+arcsinx+2 1 ; x222-+y +1vyZ +1!u2ubj = q y dy= b + a aj ; b ; a t1 ; 2y ; (b + a) : (13)22b;a(b ; y)(y ; a)yyZjjjj+3,-.06;; (12) 5 (13) 0 (11), :+6H4.7>yZ +1(j )Lq 2 (y)dy = (g(yj ) ; yj g(yj Y yj+1))aj + g(yj Y yj+1)bj =(b ; y)(y ; a)y; g(yj) == aj g(yj ) + (bj ; aj yj )g(yj Y yj+1)) = aj g(yj ) + (bj ; aj yj ) g(yyj+1) ;j +1 yj!aj yj g(y ) + bj ; aj yj g(y ) = aj yj+1 ; bj g(y ) + bj ; aj yj g(y )= aj ; ybj ; ;jjyy ; y j+1y ;yy ; y j+1jjj +1..jj +1jj +1jj +1j !"# # $#x17.
' 1<@ ('(C+2/.45>62cj = ayj yj+1;;ybj dj = ybj ; ;aj yyj :j +1+=3.jj +1(14)jyZ +1(j )Lq 2 (y) dy = cj g(yj ) + dj g(yj+1):(b ; y)(y ; a)y+3,-.06;; I-+ 0 (9), /.A+35>;1NX;1Nh b i NXXf Ti (cj g(yj ) + dj g(yj+1)) = cj g(yj ) + dj;1g(yj ) =jjj =0= c0g(y0) +NX;1j =1j =0j =1(cj + dj;1)g(yj ) + dN ;1g(yN ) ,56H (7) 5>77>NX;1h bif Ti c0f (y0)Tbi(y0) + (cj + dj;1)f (yj )Tbi (yj ) + dN ;1f (yN )Tbi(yN )j =1C+2/.45>u0 = c0f (y0) uj = (cj + dj;1)f (yj ) j = 1 2 : : : N ; 1 uN = dN ;1f (yN ) (15)(+->7-5>, 4-+ uj /7 2.05,;- +- i ).
+=3.Nh bi Xf Ti uj Tbi (yj )(16)j =0, +E5C1+8 (,>. (10))Nh b i X 1 f Ti ; uj Tbi (yj ) 8 C (a b) h2 k(f Tbi)00 kC (a b])j =0x 17.4.(17)!)$ %&!; E 7% $6!@;/.4.67 , :+>+GBF @+9>H6 (12), (13), (14), (15) 0D45,6;F-,; 0,7 1+I@@5<57/-D uj , j = 0 1 : : : N . J-+ +,HG7,-06;7-,; 2. 65/78/+7 :+ N 45,6+ .95@>7-547,15A +:79.<58..-7> , :+>+GBF @+9>H6D (16) 0D45,6;F-,; 1+I@@5<57/-D 9.26+?7/5; (5):NNXX0 = 1 uj Tb0(yj ) i = 2 uj Tbi(yj )j =0..j =0i = 1 2 : : : n ; 1(18) !"# # $#x17. ' 1<@ ('(63H37> 0D45,6;-B I-5 ,H>>D ./.6+=54/+ 1+I@@5<57/-.> 9.26+?7/5; :+ >/+=+467/.> 7CDE70...:5E7> 0D9.?7/5; (18) 36; i 0 0537 -.C65<D0 = 1 ( u0Tb0 (y1) + u1Tb0 (y2) + : : : + uN Tb0 (yN ) )1 = 2 ( u0Tb1 (y1) + u1Tb1 (y2) + : : : + uN Tb1 (yN ) )(19)2 = 2 ( u0Tb2 (y1) + u1Tb2 (y2) + : : : + uN Tb2 (yN ) )...............n;1 = 2 ( u0Tbn;1 (y1) + u1Tbn;1(y2) + : : : + uN Tbn;1(yN ) )+6B2H;,B 9711H97/-/D>5 @+9>H6.>5 (13.12), CH37> 0D45,6;-B 0 ,H>>.A (19) ,-+6C<D ,670.
/.:9.0+, . 1.?3D8 ,-+6C7< { ,079AH 0/52.C+2/.45>gi j = uj Tbi (yj ) zj = 2 2yj ;b ;(ba+ a) i = 0 : : : n ; 1 j = 0 1 : : : N:+ @+9>H6.> (13.12) 36; 0,7A j = 0 1 : : : N 0D45,6;F-,;g0 j = uj g1 j = 12 zj uj = 21 zj g0 j (20)gi j = zj gi;1 j ; gi;2 j i = 2 : : : n ; 1:J-5 @+9>H6D :+20+6;F- 0D45,6;-B I67>7/-D -.C65<D (19) :+ ,-+6C<.> (,670. /.:9.0+), :95 I-+> 1.?3D8 ,-+6C7< 2.:+6/;7-,; ,079AH 0/52. .1 -+6B1++47973/+8 ,-+6C7< -.C65<D 0D45,67/, +/ :95C.06;7-,; 1 ,H>>7 :973D3HG5A,-+6C<+0:i := i + gij i = 0 : : : n ; 1:(21)+,67 0D45,67/58 :+ @+9>H6.> (20), (21) 36; 0,7A j = 0 1 : : : N 0D45,6;F-,; 1+I@@5<57/-D i :0 := 1 0 i := 2 i i = 1 2 : : : n ; 1:(22)+,67 -+=+, 1.1 1+I@@5<57/-D i 0D45,67/D, 2/.47/57 :95C65?.FG7=+n;1>/+=+467/.
0 -+417 Pf (y) = P iTbi(y) 0D45,6;7-,; , 5,:+6B2+0./57> @+9i=0>H6 (13.12)Tb0 (y) = 1 Tb1(y) = z=2 Pf (y) = 0Tb0 (y) + 1 Tb1(y)(23)Tbi (y) = z Tbi;1 (y) ; Tbi;2(y) Pf (y) := Pf (y) + i Tbi(y)i = 2 : : : n ; 1:=37 z = 2 2y ;b ;(b a+ a) ... !"# # $#x18. '&+-.- 4+@ 70.,x 17.5.647 !% $ 5$76; 0,7A j = 0 1 : : : N /.> -97CH7-,; :9+5207,-5 0D45,67/5; :+ @+9>H6.> (12), (13), (14), (15) 5 0D45,65-B 0,7 1+I@@5<57/-D uj . J-+ +,HG7,-06;7-,;2. 65/78/+7 :+ N 45,6+ .95@>7-547,15A +:79.<58: O(N ).6; 0,7A j = 0 1 : : : N /.> -97CH7-,; :9+5207,-5 0D45,67/5;1) + @+9>H6.> (20).
. I-+ -97CH7-,; n + O(1) >H6B-5:651.-50/DA 5 ,-+6B1+ ?7 .335-50/DA +:79.<58.2) + @+9>H6.> (21). . I-+ -97CH7-,; n + O(1) .335-50/DA +:79.<58.CG77 1+6547,-0+ +:79.<58 36; 0D45,67/58 :+ @+9>H6.> (20) 5 (21) 36; 0,7Aj = 0 1 : : : N : (N + 1)n + O(N ) >H6B-5:651.-50/DA 5 2(N + 1)n + O(N ) .335-50/DA +:79.<58.6; 0D45,67/5; 1+I@@5<57/-+0 i :+ @+9>H6.> (22) -97CH7-,; 7G7 n + O(1)>H6B-5:651.-50/DA +:79.<58.673+0.-76B/+, ,H>>.9/+7 1+6547,-0+ +:79.<58, /7+CA+35>DA 36; 0D45,67/5; 1+I@@5<57/-+0 i , 9.0/+ Nn + O(N + n) >H6B-5:651.-50/D> 5 2Nn +O(N + n) .335-50/D> +:79.<5;>..
0D45,67/57 2/.47/5; :95C65?.FG7=+ >/+=+467/. Pf 0 -+417 :+ @+9>H6.> (23) -97CH7-,; 2n + O(1) >H6B-5:651.-50/DA 5 ,-+6B1+ ?7 .335-50/DA+:79.<58.F 1..,-+ 0DC59.F- N = n + 1 5 -+415y0 = a yj = a +2 b + b ;2 a cos (2j2n; 1) j = 1 2 : : : n yn+1 = b(-.7. -+415 y1 y2 : : : yn ;06;F-,; /H6;>5 >/+=+467/. 7CDE70. Tbn /. +-97217Sa b]). 95 -.1+> 0DC+97 0D45,67/5; :+ @+9>H6.> (12), (13), (14), (15) H:9+G.F-,;.x18.
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