К.Ю. Богачёв - Практикум на ЭВМ. Методы приближения функций (1133845), страница 7
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!"# # $#x13. )+ & 4*+ *3<(40J-5 @+9>H6D :+20+6;F- 0D45,6;-B I67>7/-D -.C65<D (11) :+ ,-+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:(14)+,67 0D45,67/58 :+ @+9>H6.> (13), (14) 36; 0,7A j = 1 2 : : : n 0D45,6;F-,; 1+I@@5<57/-D i :0 := n1 0 i := n2 i i = 1 2 : : : n ; 1:(15)+,67 -+=+, 1.1 1+I@@5<57/-D i 0D45,67/D, 2/.47/57 :95C65?.FG7=+n;1>/+=+467/. 0 -+417 Pf (x) = P iTbi(x) 0D45,6;7-,; , 5,:+6B2+0./57> @+9i=0>H6 (12)Tb0(x) = 1 Tb1 (x) = z=2Tbi(x) = z Tbi;1(x) ; Tbi;2 (x)=37 z = 2 2x ; (b + a) .b;ax 13.3.Pf (x) = 0 Tb0(x) + 1Tb1 (x)Pf (x) := Pf (x) + iTbi(x)i = 2 : : : n ; 1:(16)7 !% $ 5$76; 0,7A j = 1 2 : : : n /.> -97CH7-,; :9+5207,-5 0D45,67/5;1) + @+9>H6.> (13).
. I-+ -97CH7-,; n + O(1) >H6B-5:651.-50/DA 5 ,-+6B1+ ?7 .335-50/DA +:79.<58.2) + @+9>H6.> (14). . I-+ -97CH7-,; n + O(1) .335-50/DA +:79.<58.CG77 1+6547,-0+ +:79.<58 36; 0D45,67/58 :+ @+9>H6.> (13) 5 (14) 36; 0,7Aj = 1 2 : : : n : n2 +O(n) >H6B-5:651.-50/DA 5 2n2 +O(n) .335-50/DA +:79.<58.6; 0D45,67/5; 1+I@@5<57/-+0 i :+ @+9>H6.> (15) -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/+ n2 + O(n) >H6B-5:651.-50/D> 5 2n2 + O(n) .335-50/D> +:79.<5;>..
0D45,67/57 2/.47/5; :95C65?.FG7=+ >/+=+467/. Pf 0 -+417 :+ @+9>H6.> (16) -97CH7-,; 2n + O(1) >H6B-5:651.-50/DA 5 ,-+6B1+ ?7 .335-50/DA+:79.<58... !"# # $#x13. )+ & 4*+ *3<(x 13.4.41%;6" $6!@; 5 )! B>&C% $5!;795 :+,-9+7/55 >/+=+467/. Pf /7 -97C+0.6+,B, 4-+CD +/ ,+0:.3.6 , @H/1<578 f 0 1.15A-CD -+ /5 CD6+ -+41.A (-97C+0.6+,B 65EB 0D:+6/7/57 65/78/DAH,6+058 5/-79:+6;<55 (4)).
3/.1+, +1.2D0.7-,;, 4-+ :+,-9+7//D8 0DE7 :95C65?.FG58 >/+=+467/ Pf >+?/+ 9.,,>.-950.-B 1.1 5/-79:+6;<5+//D8 >/+=+467/. 3. # (10) (7) - %! !, ..(Pf )(xm) = f (xm ) m = 1 2 : : : n(17), , (&, 0# Tn .b. D45,65> >/+=+467/ (10) 0 -+41.A xm , m = 1 2 : : : n :6!"%Db ETj f(Pf )(xm) = D b b En Tbj (xm ) =j =0 Tj Tj nnX;1nX= D b 1b ETbj (xi )f (xi)Tbj (xm ) =j =0 Tj Tj n i=1nnX;1X= f (xi) D b 1b E Tbj (xi)Tbj (xm )i=1j =0 Tj Tj nnX;1.1+> +C9.2+>, 36; 3+1.2.-76B,-0. (17) (. 2/.45-, 5 67>>D) /.> 3+,-.-+4/+:+1.2.-B, 4-+nX;1D b 1b E Tbj (xi)Tbj (xm ) = imj =0 Tj Tj n-.7.;12 Tb (x )Tb (x ) = 1 + nXimn j=1 n j i j m565nX;1(18)Tbj (xi )Tbj (xm ) = n2 im + 12 :j =0>77>! !n;1bTj (xi)Tbj (xm ) = X cos j (2i ; 1) cos j (2m ; 1) :2n2nj =0j =0nX;1..
!"# # $#x13. )+ & 4*+ *3<(+,:+6B2H7>,; @+9>H6+8 cos cos = 1 (cos( ; ) + cos( + )):2! nX!n;1;1nP;1X1(i;m)1(i+m;1)Tbj (xi)Tbj (xm ) = 2 cos j n+ 2 cos j=nj =0j =0j =0= S (i ; m) + S (i + m ; 1)=37 ! nX ( )()!nX;1;1jk1ijk;ijk1S (k) = 2 cos n = 4exp n + exp n=j =0j =00( )1nX;11A=exp ijk= 4 @1 +nj =;(n;1)0( )1nX1= @1 ; exp fikg +exp ijk A =4nj =;(n;1)0() 2X( )!j 1n;1 1ik(n;1)ik A= 4 @1 ; (;1)k + exp ;expnnj =0+,1+6B1H8 2n>2Xn;1< q ; 1 :95 q 6= 1jq => q;1: 2nj =0:95 q = 1-+ :95 k 6= 042(19)01()BBCC 1 1ik(n;1)expf2ikg;1kBCC = 1 ; (;1)k ()S (k) = 4 B1 ; (;1) + exp ; n@A 4exp ik;1n( )-.1 1.1 exp f2ikg = 1 36; 0,7A k .
95 k = 0 q = exp ik = 1 5nS (0) = 1 (1 ; 1 + 2n) = n-.1,8><S (k) = >:421 1 ; (;1)k :95 k 6= 04n:95 k = 02+3,-.06;; I-+ 2/.47/57 0 (19), :+6H4.7> :95 i 6= mnX;1Tbj (xi)Tbj (xm ) = S (i ; m) + S (i + m ; 1) = 14 1 ; (;1)i;m +j =0+ 1 1 ; (;1)i+m;1 = 1 + 1 = 144 4 2.. !"# # $#x14. 043*-4*+- && .-(:+,1+6B1H 45,6. i ; m 5 i + m ; 1 5>7F- 9.2/HF 47-/+,-B).
95 i = mnX;1j =0Tbj (xi )Tbj (xm ) = S (i ; m) + S (i + m ; 1) = n2 + 41 1 ; (;1)2i;1 = n2 + 21 :7> ,.>D> @+9>H6. (18) 5 (0>7,-7 , /78 67>>.) 3+1.2./..,7 9.,,>+-97//D7 0DE7 .::9+1,5>.<55 Pf @H/1<55 f 5>765 053: /. +-97217 Sa b] @H/1<5; f :95C65?.7-,; >/+=+467/+> Pf = Lm ,-7:7/5 m .
9.0/5>I-+- ,:+,+C ,+ ,673HFG5>: 9.23765> +-792+1 Sa b] /. k 4.,-78 5 /. 1.?3+8 52/5A :95C6525> >/+=+467/+> Ln ,-7:7/5 n . .1.; .::9+1,5>.<5; /.2D0.7-,;1H,+4/+->/+=+467//+8.14. &- x<7/15 :+=97E/+,-5 0,7A 9.,,>+-97//DA 9./77 >7-+3+0 5>7F- 053kf ; Pf kC const(n) distC (f Pn )(=37 Pf { .::9+1,5>.<5; >/+=+467/+> ,-7:7/5 n ; 1, Pn { :9+,-9./,-0+>/+=+467/+0 ,-7:7/5 n ; 1). + -7+97>7 ?71,+/. (,>.
-7+97>H 12.3) :95 n >r+1!r !b;ab;arkf ; Pf kC const(r) n ; 1 ! f Y 2(n ; 1 ; r) :;1;1( )673+0.-76B/+, +<7/1. :+=97E/+,-5!r !b;ab;akf ; Pf kC const(n) const(r) n ; 1 ! f (r)Y 2(n ; 1 ; r) :076547/57 -+4/+,-5 :95 @51,59+0.//+8 076545/7 >+3H6; /7:979D0/+,-5@H/1<55 f (r) >+?7- CD-B 3+,-5=/H-+ 65C+ H076547/57> n , 65C+ H>7/BE7/57> b ; a Y 9.2C57/57 +-9721. Sa b] /.
k 4.,-78 3.7- -+- ?7 I@@71-, 4-+ 5 5,:+6B2+0./57 >/+=+467/+0 ,-7:7/5 kn . 975>HG7,-0. 9.2C57/5; +-9721. :7973H076547/57> ,-7:7/5 >/+=+467/.:1) 97CH7-,; 97E.-B (0 +CG78 ,A7>7 2.3.45 65/78/+8 5/-79:+6;<55) k ,5,-7>9.2>79. n 0>7,-+ +3/+8 9.2>79. kn .2) 9+,-+> ,-7:7/5 >/+=+467/. n 9.,-7- +CH,6+067//+,-B C.25,., . :95 9.2C57/55 +-9721. +/. /7 >7/;7-,;, -.1 1.1 9.2>79/+,-B C.25,. n (,-7:7/B>/+=+467/. /. 1.?3+> 52 +-9721+0 9.2C57/5;) /7 52>7/;7-,; , 9+,-+> 1+6547,-0. +-9721+0 9.2C57/5; k ... !"# # $#x15.
0x44*-+,- '&+-.-15. - H,-B /. +-97217 Sa b] 2.3./D -+415 a = x1 < x2 < : : : < xn = b 5 2/.47/5;f (x1 ) f (x2) : : : f (xn) /71+-+9+8 @H/1<55 f 2 C (Sa b]). 97CH7-,; :+,-9+5-B:95C65?7/57 @H/1<55 f 1H,+4/+-65/78/+8 @H/1<578 I2f , ,+0:.3.FG78 , f0 -+41.A xi , i = 1 2 : : : n . 9H=5>5 ,6+0.>5, -97CH7-,; :+,-9+5-B @H/1<5FI2f -.1HF, 4-+ 36; 0,7A i = 1 2 : : : n ; 1 /. +-97217 Sxi xi+1] @H/1<5; I2 f;06;7-,; 65/78/+8 @H/1<578, 2/.47/5; 1+-+9+8 0 -+41.A xi 5 xi+1 ,+0:.3.F,+ 2/.47/5;>5 f (xi) 5 f (xi+1) @H/1<55 f 0 I-5A -+41.A.+ 5/-79:+6;<5+//+8 @+9>H67 BF-+/. 36; 0,7A x 2 Sxi xi+1], i =1 2 : : : n ; 1; f (xi) (1)I2f (x) = f (xi) + (x ; xi )f (xiY xi+1) = f (xi ) + (x ; xi ) f (xxi+1) ;i+1 xi 1. f 2 C 2 (Sa b]) , #- - kf ; I f kC a b 18 h kf kC a b2(])200(])h = i=1max(xi+1 ; xi ):2 ::: n;1.
+ -7+97>7 7.2 :+6H4.7>00kf ; Pf kC(x x +1]) 2!1 x2maxj(x;x)(x;x(2)ii+1 )jkf kC (x x +1 ]) :x x +1]"H/1<5; g(x) = (x ; xi )(x ; xi+1 ) = x2 ; (xi + xi+1 )x + xixi+1 0 36; 0,7Ax 2 Sxi xi+1] 5 5>77- /H6B :9+520+3/+8 (xi + xi+1 )=2, :95/.367?.G58 +-9721HSxi xi+1]. +,1+6B1H g(xi) = g(xi+1) = 0, -+ xi + xi+1 xi+1 ; xi xi ; xi+1 1 = = (x ; x )2 :max jg(x)j = gx2x x +1 ]222 4 i+1 i+3,-.06;; I-+ 0 (2), :+6H4.7>kf ; Pf kC(x x +1]) 81 (xi+1 ; xi)2 kf 00kC(x x +1]):673+0.-76B/+,1 h2 kf 00kkf ; I2f kC(a b]) = i=1maxkf;PfkC (a b]) :C(x x +1 ]) 2 ::: n;187>>. 3+1.2./..5$#!.
C+2/.45> 47972 S2 65/78/+7 :9+,-9./,-0+ /7:979D0/DA 6+>./DA 65/58 /. +-97217 Sx1 xn] = Sa b] , 526+>.>5 x2 < x3 < : : : < xn;1 .6!"%iiiiiiiiiiii..ii !"# # $#x16. ' 1<@ ('(452 2. fC (Sa b]) , # - distC (f S2 ) kf ; I2f kC (a b]) 2distC (f S2):6!"%. ,56H 735/,-07//+,-5 5/-79:+6;<5+//+=+ >/+=+467/. /.1.?3+> 52 +-9721+0 Sxi xi+1], i = 1 2 : : : n ; 1 :+6H4.7> 36; 0,;1+8 f 2 S2I2f = f(3)47053/+, 4-+kf ; I2f kC(a b]) ginf2S2 kf ; gkC(a b]) = distC (f S2): ,56H 65/78/+,-5 I2f /.
Sxi xi+1 ]kI2f kC(a b]) = i=1maxj(I2f )(xi)j = i=1maxjf (xi)j kf kC(a b])2 ::: n2 ::: n-.7. 36; 0,;1+8 f 2 C (Sa b])kI f kC a b kf kC a b :2(])(])(4)>77> 36; 0,;1+8 g 2 S2kf ; I2f kC(a b]) = k(f ; g) + (g ; I2f )kC(a b]):,:+6B2H; (3) 5 65/78/+,-B I2 , :9+3+6?.7>kf ; I2f kC(a b]) = k(f ; g) ; I2(f ; g)kC(a b]) k(f ; g)kC(a b]) + kI2(f ; g)kC(a b]): :+>+GBF (4) /.A+35>kf ; I2f kC(a b]) k(f ; g)kC(a b]) + k(f ; g)kC(a b]) = 2k(f ; g)kC(a b])36; 0,;1+8 g 2 S2 . 673+0.-76B/+,kf ; I2f kC(a b]) ginf2S k(f ; g)kC(a b]) = 2distC (f S2):7>>. 3+1.2./..x216.
- (, :973D3HG7> 9.23767 >D :+,-9+565 1H,+4/+-65/78/HF @H/1<5F, ,+0:.3.FGHF , @H/1<578 f 0 2.3.//DA -+41.A. 3/.1+, 7,65 77 2/.47/5; 5207,-/D ,/71+-+9+8 :+=97E/+,-BF, -+ I-+ >+?7- 3.-B :95C65?7/57 :6+A+=+ 1.47,-0.. +:5,D0.7>+> /5?7 .6=+95->7 C652+,-B @H/1<55 5 77 :95C65?7/5; 52>79;7-,;0 5/-7=9.6B/+8 /+9>7, 5, ,673+0.-76B/+, :95C65?.FG.; @H/1<5; CH37- >7/774H0,-05-76B/+8 1 0DC9+,.> 0 2/.47/5;A :95C65?.7>+8 @H/1<55... !"# # $#x16. ' 1<@ ('(x 16.1.46-% 6# ! $5!;7H,-B -97CH7-,; :+,-9+5-B :95C65?7/57 @H/1<55 f 1H,+4/+-65/78/+8@H/1<578 L2 f 2 S2 -.1+7, 4-+kf ; L f kL2 a b ! min2-.7.(kf ; L f kL2 a b2=37kgkL2 a b(])(= (g g)1=2L2(a b])])])= ginfkf ; gkL2(a b])2S2Zb (u v)L2(a b]) = u(x)v(x) dx:a0737> C.25, :9+,-9./,-0.
S2Hj 2 S2 Hj (xi) = ij i = 1 2 : : : n j = 1 2 : : : n:,;1.; g 2 S2 9.,16.3D0.7-,; :+ I-+>H C.25,H ,673HFG5> +C9.2+>g(x) =(1)nXj =1g(xj )Hj (x):(2)(3)78,-05-76B/+, 0 ,56H (2) 5>77> 36; gb(x) = P g(xj )Hj (x): gb(xi) = g(xi),j =1bi = 1 2 : : : n . 673+0.-76B/+, g(x) = I2g { 1H,+4/+-65/78/.; 5/-79:+659HFG.; @H/1<5;. + 3+1.2.//+>H 0 67>>7 15.2 36; g 2 S2 0D:+6/7/+ (15.3).673+0.-76B/+, g = gb ..15> +C9.2+>, 2.3.4. (1) >+?7- CD-B ,@+9>H659+0./. 0 ,673HFG7> 0537:/.8-5 (j )j=1 2 ::: n -.157, 4-+ @H/1<5;n(L2f )(x) =nXj =1j Hj (x)(4)H3+067-0+9;7- ,++-/+E7/5Fn X= ginfkf ; gkL2(a b]):f ; j Hj 2S2j =1L2 (a b])(5)207,-/+, 4-+ 0 701653+0+> :9+,-9./,-07 (0 /.E7> ,6H4.7 { L2 (Sa b])) >5/5>H> (5) 97.652H7-,; /.
@H/1<55 (4), ;06;FG78,; :9+71<578 @H/1<55 f /.:+3:9+,-9./,-0+ S2 . 673+0.-76B/+, I67>7/- f ; L2 f +9-+=+/.67/ :+3:9+,-9./,-0H S2 . +,1+6B1H fHigni=1 ,+,-.06;F- C.25, S2 , -+ H,6+057 +9-+=+/.6B/+,-5S2 I1050.67/-/+ H,6+05F +9-+=+/.6B/+,-5 0,7> Hi , i = 1 : : : n :(f ; L2f Hi)L2 (a b]) = 0 36; 0,7A i = 1 : : : n.. !"# # $#x16. ' 1<@ ('(47-.7.(L2f Hi)L2 (a b]) = (f Hi)L2 (a b]) i = 1 : : : n:(6),6+05; (6) :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; (4).2 (6) 5 (4) /.A+35>:nX565j =1n ZbXj =1 a(Hj Hi)L2(a b]) j = (f Hi)L2 (a b]) i = 1 : : : nZbHj (x)Hi(x) dx j = f (x)Hi(x) dx i = 1 : : : n:(7)a+- ?7 972H6B-.- (7) >+?7- CD-B :+6H47/ 52 H,6+05;2n XF () = f ; j Hj j =1L2 (a b])01nnXX= @f ; j Hj f ; j Hj Aj =1j =1L2 (a b])! min:H-7> 35@@797/<59+0./5; @H/1<55 F ():0 011nnX@F = @ @ @f ; Xj Hj A f ; j Hj A+@i@j =1j =1L2 (a b])0011nnX@ @f ; X+ @f ; j Hj @j Hj AA=j =1j =1L2 (a b])01nX= @;2Hi f ; j Hj Aj =1L2 (a b])5 :959.0/50./5; 1 /H6F :9+520+3/+8 @F 0 -+417 >5/5>H>.:@i@F = 0 i = 1 : : : n:@ix 16.2.&! $7& & 6# !$5!;7D45,65> >.-95<H ,5,-7>D (7).
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