К.Ю. Богачёв - Практикум на ЭВМ. Методы приближения функций (1133845), страница 6
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Tn+1(x) :+6H4.7-,; 52 ,-.9E7=+ 467/. >/+=+467/. Tn(x) H>/+?7/57> /. 2x . 673+0.-76B/+, ,-.9E58 467/>/+=+467/. Tn(x) :95 n > 0 7,-B 2n;1x .+,1+6B1H 36; 0,;15A 5 cos cos = 21 (cos( + ) + cos( ; )), -+ :956FC+> cos((n + 1)) = 2 cos cos(n) ; cos((n ; 1)):+6+?50 = arccos x , :+6H45> :95 x 2 S;1 1]cos((n + 1) arccos x) = 2x cos(n arccos x) ; cos((n ; 1) arccos x)-.7. @H/1<5; 'n(x) = cos(n arccos x) H3+067-0+9;7- ,++-/+E7/5;> (2) 36; 0,7An > 0 5 x 2 S;1 1].
.1 1.1 '0(x) = cos(0 arccos x) = 1, '1(x) = cos(1 arccos x) =1 :95 x 2 S;1 1], -.7. +/. H3+067-0+9;7- (1), -+ Tn(x) 'n(x):Tn(x) = cos(n arccos x) n 0 x 2 S;1 1]:(3)2 :973,-.067/5; (3) 0D-71.F- ,673HFG57 ,0+8,-0. >/+=+467/+0 7CDE70.:1)jTn(x)j 1 :95 x 2 S;1 1]:2) +415 I1,-97>H>. >/+=+467/. Tn(x) /. S;1 1], -.7. -+415 xbm , jTn(xbm )j = 17,-B xbm = cos mn m = 0 1 : : : n:9547>Tn (xbm ) = cos m = (;1)m :3) H65 >/+=+467/. Tn(x) /. S;1 1] 7,-B!(2m;1) m = 1 2 : : : n:xm = cos2n+,1+6B1H /.837/D n /H678 >/+=+467/.
Tn(x) ,-7:7/5 n , -+ 39H=5A /H678 H/7=+ /7- 5Tn(x) = 2n;1(x ; x1) : : : (x ; xn ):5$#!.. !"# # $#x11. 4*+3 *3<(32. 0737>5$#!T n(x) = 2n1;1 Tn (x) = xn + : : : 1. Pn(x) = xn + : : : { n # %-, 1, 1 :maxjP(x)jmaxjT(x)j=nnx2;1 1]x2;1 1]2n;1. 973:+6+?5> :9+-50/+7, -.7. ,HG7,-0H7- -.1+8 >/+=+467/,-7:7/5 n ,+ ,-.9E5> 1+I@@5<57/-+>, 9.0/D> 1, 4-+ jPn(x)j <1 36; 0,7A x 2 S;1 1]. .,>+-95> Q (x) = T (x) ; P (x) { >/+=+467/nnn;12n;1nn,-7:7/5 /7 0DE7 n ; 1, :+,1+6B1H T n(x) = x + : : : , Pn(x) = x + : : : .
D45,65>2/.1 >/+=+467/. Qn;1 0 -+41.A xbm , m = 0 1 : : : n :1msign Qn;1(xbm ) = sign(T n(xbm ) ; Pn(xbm )) = sign (;1) 2n;1 == sign (;1)m n1;1 = (;1)m2(:+,1+6B1H jPn(xbm )j < 2n1;1 ). +34791/7>, 4-+ Qn;1(xbm ) 6= 0, m = 0 1 : : : n ..15> +C9.2+>, /. +-97217 Sxbm xbm+1 ], m = 0 1 : : : n ; 1 @H/1<5; Qn;1(x)>7/;7- 2/.1 5 :+I-+>H 5>77- 1+97/B, :95/.367?.G58 5/-790.6H (xbm xbm+1 ),m = 0 1 : : : n ; 1. 673+0.-76B/+, >/+=+467/ Qn;1 ,-7:7/5 /7 0DE7 n ; 1 5>77n 9.2654/DA 1+9/78 5 :+-+>H Qn;1 0. 3/.1+, :+ 3+1.2.//+>H Qn;1 (xbm ) 6= 0,m = 0 1 : : : n . +6H47//+7 :9+-50+97457 3+1.2D0.7- 67>>H.5$#!. /+=+467/ T n (x) /.2D0.7-,; >/+=+467/+> ,-7:7/5 n , /.
+-97217 S;1 1].H,-B Sa b] 7,-B 2.3.//D8 +-972+1.5$#!. 0737> +-+C9.?7/5;y : S;1 1] ! Sa b] y(x) = (a + b) ;2 (a ; b)x (4)2y;(b+a)x : Sa b] ! S;1 1] x(y) = b ; a5 0# Sa b]!bTn(y) = Tn(x(y)) = Tn 2y ; (b + a) :b;a6!"%Pn(x) = xn + : : :H65 >/+=+467/.
Tbn(y) (0,7 :95/.367?.- +-9721H Sa b]) 7,-Bym = y(xm) = a +2 b + b ;2 a cos (2m2n; 1) m = 1 2 : : : n:..(5) !"# # $#x12. ).- &4<' '&+-.33 2 n2n;1b-.9E58 1+I@@5<57/- Tn(y) 7,-B2n;1 = 2 n .b;a(b ; a)5$#!. +6+?5>!a)n T (y) = (b ; a)n T 2y ; (b + a) :eTn(y) = (b ;2n;1n222n;1 nb;a 2. 1 Ten (y ) , , ..
Pn (y ) = y n + : : : n #%, 1,a) :en(y)j = (b ;2n;1maxjP(y)jmaxjTny2a b]y2a b]26!"%. -079?37/57 0D-71.7- 52 67>>D 1 :H-7> 2.>7/D :797>7//+8.n12. ( x5/5>5259H7> :+=97E/+,-B 5/-79:+6;<55 @H/1<55 f /. +-97217 Sa b] , :+>+GBF 5/-79:+6;<5+//+=+ >/+=+467/. .=9./?. Ln ,-7:7/5 n ; 1 2. ,47- 0DC+9. H26+0 5/-79:+6;<55 x1 : : : xn .+ -7+97>7 7.2(n)ff (x) ; Ln(x) = n!( ) (x ; x1 ) : : : (x ; xn)(1)=37 x1 : : : xn x 2 Sa b], 2 Sa b].
673+0.-76B/+, (n) (2)kf ; LnkC(a b]) fn! k(x ; x1) : : : (x ; xn)kC(a b])C (a b])=37kgkC(a b]) = xmaxjg(x)j:2a b]5/5>5259H7> :9.0HF 4.,-B, 0DC9.0 >/+=+467/ Pn(x) = (x ; x1 ) : : : (x ; xn )/.5>7/77 H16+/;FG5>,; +- /H6; /. +-97217 Sa b], -.7. , >5/5>.6B/+8 /+9>+8kPnkC(a b]) . +2B>7>n 2x ; (b + a) !(b;a)Pn(x) = (x ; x1 ) : : : (x ; xn) = Ten(x) = 22n;1 Tn:b;a2 I-+=+ 9.07/,-0.
0D-71.7-, 4-+ -+415 x1 : : : xn ;06;F-,; /H6;>5 >/+=+467/.Ten(x) /. +-97217 Sa b]. ,56H (11.5)xm = ym = a +2 b + b ;2 a cos (2m2n; 1) m = 1 2 : : : n:(3).. !"# # $#x12. ).- &4<34' '&+-.+ 67>>7 11.2 :95 -.1+> 0DC+97 H26+0kPnkC a b((b ; a)]) =22n;1n5 +<7/1. (2) 5>77- 053nkf ; LnkC a b n1! f n C a b (b2;n a) :(( )])(])(4)2 ;1+6H47//.; +<7/1. (4) /7H6H4E.7>..n78,-05-76B/+, 0+2B>7> f (x) = anxn + : : : + a0 = P am xm { >/+=+467/m=0,-7:7/5 n . +=3. f n( ) = ann! 5 +<7/1.
:+=97E/+,-5 (1) :95>7- 053f (x) ; Ln (x) = an(x ; x1 ) : : : (x ; xn ) = anPn(x)=37 Pn(x) { >/+=+467/ (/. +-97217 Sa b]) ,+ ,-.9E5> 1+I@@5<57/-+>, 9.0/D> 1.+ 67>>7 11.2 :+6H4.7>kf ; LnkC a b = janjkPnkC a b janj (b2;n a)n(])(])2 ;1 a)n = 1 f (n) C (a b]) (b ;n!22n;14-+ 5 +2/.4.7- /7H6H4E.7>+,-B +<7/15 (4).-.1, 36; >5/5>52.<55 :+=97E/+,-5 0 1.47,-07 H26+0 x1 : : : xn 5/-79:+6;<55 /.3+ C9.-B /H65 (3) >/+=+467/. 7CDE70. Ten(x)..1+8 0DC+9 H6H4E.7- -.1?7 1.47,-07//+7 :+0737/57 :+=97E/+,-5 :95n ! 1.1-$$ (C72 3+1.2.-76B,-0.).
.,,>+-95> @H/1<5F f (x) =1 + 25x2 5 +-972+1 Sa b] = S;1 1]. 6; 9.0/+>79/+ 9.,:973767//DA /. +-97217 Sa b] -+471x1 : : : xn (-.7. xi = a + (i ; 1)h , h = nb ;; a1 , i = 1 : : : n ) :+=97E/+,-B 5/-79:+6;<55 kf ; LnkC (;1 1]) ! 1 :95 n ! 1 . 6; -+471 x1 : : : xn { /H678 (3)>/+=+467/. 7CDE70. Ten(x) :+=97E/+,-B 5/-79:+6;<55 kf ; LnkC (;1 1]) ! 0:95 n ! 1 .+4/HF A.9.1-795,-51H :+0737/5; :+=97E/+,-5 5/-79:+6;<55 :+ /H6;> >/+=+467/. 7CDE70. Ten(x) 3.F- ,673HFG57 H-079?37/5;.n= Q x ; xjj =1 xi ; xj$ 1.
( ). ' %-! (& li (x)2 ln n + 1 ...xm = ym(3) #-j 6=i !"# # $#x13. )+ & 4*+ *3<(;35$ 2. # fLn & f (& Ln n 1 , (3) 0#, Pn;1;dist(f Pn;1 ) kf ; LnkC (a b]) 2 + 2 ln n distC (f Pn;1){ n ; 1 ,distC (f Pn;1) = P ;1infkf ; Pn;1kC(a b]):2P ;1nnf 2 C (r) Sa b] n > r + 1 , !r b ; a !r 6(3e)b;a(r)distC (f Pn;1) 1 + r n ; 1 ! f Y 2(n ; 1 ; r) :$ 3 (@). !(gY h) = supfjg(x) ; g(y)j : x y 2 Sa b] jx ; yj h g ..
.,,>+-95> @H/1<5F f (x) = jxj , 2 (0 1) /. +-97217 S;1 1].+=3. !(f Y h) = h . + -7+97>7 3 :95 r = 0, n > 1 :+6H4.7>!2= 6 :distC (f Pn;1) 6! f Y2(n ; 1)(n ; 1)+ -7+97>7 2 2 6kf ; LnkC(;1 1]) 2 + ln n (n ; 1) :q.:95>79, 36; f (x) = jxj 2 6kf ; LnkC(;1 1]) 2 + ln n (n ; 1)1=2 nln1=n2 :95 n ! 1:-$$x13. (H,-B n { 2.3.//+7 <76+7 45,6+, Sa b] { 2.3.//D8 +-972+1, x1 : : : xn { 9.2654/D7 -+415, :95/.367?.G57 +-9721H Sa b]. 0737> > :9+,-9./,-07 C (Sa b])35,197-/+7 \,1.6;9/+7" :9+520737/57: 36; 0,;15A f g 2 C (Sa b]) :+6+?5>hf gin = mX f (xm)g(xm):n=1..(1) !"# # $#x13.
)+ & 4*+ *3<(x 13.1.36-% 6# ! $5!;7.,,>+-95> 2.3.4H :+,-9+7/5; 9.26+?7/5;Pf (x) =nX;1i=0i Tbi(x)(2)@H/1<55 f 2 C (Sa b]) :+ >/+=+467/.> 7CDE70. /. +-97217 Sa b]:!2x;(b+a)Tbi (x) = Ti i = 0 1 : : : n ; 1b;a-.1, 4-+CD CD65 0D:+6/7/D 65/78/D7 H,6+05; 5/-79:+6;<55:Db E Db ETi Pf n = Ti f n i = 0 1 : : : n ; 1:D Ei(f ) = Tbi f n 1. 1Tb0 Tb1 : : : Tbn;1(3)(4) , .. 973:+6+?5>, 4-+ >/+=+467/D Tb0 Tb1 : : : Tbn;1 65/78/+n;12.05,5>D, -.7. ,HG7,-0H7- 2 Rn , 6= 0 -.1+7, 4-+ Pn;1(x) = P iTbi (x) = 0i=036; 0,7A x 2 Sa b]. .1 1.1 deg Tbi = i , -+ ,-.9E58 1+I@@5<57/- >/+=+467/.n;1Pn;1(x) = P i Tbi(x) 9.07/ ,-.9E7>H 1+I@@5<57/-H >/+=+467/.
Tbj (x) (=37 j =i=0maxfj : j 6= 0g ), H>/+?7//+>H /. j . +,1+6B1H >/+=+467/ Pn;1(x) 0, -+7=+ ,-.9E58 1+I@@5<57/- 9.07/ 0. 673+0.-76B/+, j = 0, 4-+ :9+-50+9745-+>H, 4-+ j = maxfj : j 6= 0g . +6H47//+7 :9+-50+97457 3+1.2D0.7- :790+7H-079?37/57 67>>D.D E973:+6+?5>, 4-+ @H/1<5+/.6D i(f ) = Tbi f n 65/78/+ 2.05,5>D, -.7.n;1,HG7,-0H7- 2 Rn , 6= 0 -.1+7, 4-+ = P ii = 0. 9H=5>5 ,6+0.>5,6!"%i=0D b E nP;1 b (f ) = i Ti f n =iTi f = 0 36; 0,;1+8 f 2 C (Sa b]). +2B>7>i=0i=0nn x ; xiQf (x) = lk (x) = x ; x . +,1+6B1H lk (xj ) = kj , -+i=1 kii6=k*nX+ X;1n nX;1n nX;1Xb0 = (lk ) =iTi lk =iTbi(xj )lk (xj ) =i Tbi(xj )kj =nP;1i=0nj =1 i=0=nX;1i=0j =1 i=0iTbi (xk ) 36; 0,7A k = 1 2 : : : n:n;1673+0.-76B/+, >/+=+467/ Pn;1(x) = P iTbi(x) ,-7:7/5 /7 0DE7 n ; 1 5>77i=0nP;1n /H678 5 :+-+>H Pn;1(x) 0, -.7.
i Tbi(x) = 0 36; 0,7A x . ,56H 3+i=01.2.//+8 0DE7 65/78/+8 /72.05,5>+,-5 >/+=+467/+0 Tb0 Tb1 : : : Tbn;1 :+6H4.7>.. !"# # $#x13. )+ & 4*+ *3<(37 = 0. +6H47//+7 :9+-50+97457 , :973:+6+?7/57> 6= 0 3+1.2D0.7- 0-+9+7H-079?37/57 67>>D.p2n .$ 1. ( ). ' 0#Tb0 Tb1 : : : Tbn;1 #2 67>>D 1 5 -7+97>D 1.1 0D-71.7-, 4-+ 7,65 >.-95<.DEA = (aij ) aij = Tbi Tbj n i j = 0 1 : : : n ; 1(5)+C9.-5>., -+ 2.3.4. 65/78/+8 5/-79:+6;<55 :+ >/+=+467/.> 7CDE70. (3) 1+9971-/.
5 77 97E7/57 5>77- 053 (2), =37 1+I@@5<57/-D 9.26+?7/5; = A;1DETbi f n i=0 1 ::: n;1 :(6)+3C797> -+415 x1 : : : xn -.1, 4-+CD >.-95<. A CD6. 67=1+ +C9.-5>.. 2. xm = ym = a +2 b + b ;2 a cos (2m2n; 1) m = 1 2 : : : nTbn(7)n . Db b EDb b E nDb b ETi Tj n = 0 i 6= jTj Tj n = 2 j 6= 0T0 T0 n = n:!2x;(b+a)b36; 0,7A x 26!"%. .1 1.1 Tj (x) = cos j arccosb;aSa b], -+! !nnDb b E XX(2m;1)(2m;1)Ti Tj n = Tbi(xm )Tbj (xm ) = cos i 2ncos j 2n:m=1m=1 0#+,:+6B2H7>,; @+9>H6+8 cos cos = 1 (cos( ; ) + cos( + )):2!!nnDb b EXX1(2m;1)1(2m;1)Ti Tj n = 2 cos (i ; j ) 2n+ 2 cos (i + j ) 2n=m=1m=1= F (i ; j ) + F (i + j )..(8) !"# # $#x13. )+ & 4*+ *3<(=3738!nXF (k) = 21 cos k (2m2;n 1) =m=1 ()()!n 1X1k(2m;1)k(2m;1)= 2 2 exp i 2n+ exp ;i 2n=m=1 ()()!nX1ik(2m;1)ik(2(;(m;1));1)=exp+ exp=4 m=12n2n()nXik(2m;1)1exp==42nm=;(n;1)() X ( )!mn1ikexp ik== 4 exp ; 2nnm=;(n;1)() ( )!;n X ( )!m2n1ikik= 4 exp ; 2n exp nexp iknm=1+,1+6B1H-+ :95 k 6= 08 2n< q ;1X m >q = > q q ; 1 :95 q 6= 1: 2nm=1:95 q = 12n)( )ikik)g ; 1 = 0; 2n exp f;ikg exp ikn exp (f2ikexpn ;1( )-.1 1.1 exp f2ikg = 1 36; 0,7A k .
95 k = 0 q = exp ik = 1 5nF (k) = 41 exp-.1,(F (0) = 14 1 1 2n = n2 :8>< 0 :95 k 6= 0F (k) = > n: 2 :95 k = 0+3,-.06;; I-+ 2/.47/57 0 (8), :+6H4.7>Db b ETi Tj n = 0 + 0 = 0 :95 i 6= jDb bE nTi Ti n = 2 + 0 = n2 :95 i 6= 0Db b E n nT0 T0 n = 2 + 2 = n.. !"# # $#x13. )+ & 4*+ *3<(397>>. 3+1.2./..2 67>>D 2 :+6H4.7>, 4-+ :95 0DC+97 -+471 x1 : : : xn 1.1 0 (7) >.-95<. A0 (5) 7,-B nnA = diag n 2 : : : 2(9)5 5,1+>+7 9.26+?7/57 (2), (6) 5>77- 053Pf (x) =nX;1i=0x 13.2.iTbi (x)Db ETi fi = D b b En i = 0 1 : : : n ; 1:Ti Ti n(10)!)$ 5$; $6!@;.:5E7> 0D9.?7/5; (10) 36; i 0 0537 -.C65<D0 = n1 ( Tb0 (x1)f (x1 ) + Tb0 (x2 )f (x2) + : : : + Tb0(xn )f (xn) )1 = n2 ( Tb1 (x1)f (x1 ) + Tb1 (x2 )f (x2) + : : : + Tb1(xn )f (xn) )2 = n2 ( Tb2 (x1)f (x1 ) + Tb2 (x2 )f (x2) + : : : + Tb2(xn )f (xn) ) (11)...............n;1 = n2 ( Tbn;1(x1 )f (x1 ) + Tbn;1(x2 )f (x2 ) + : : : + Tbn;1(xn)f (xn) )2 9711H97/-/DA @+9>H6 (11.1), (11.2) 36; >/+=+467/+0 7CDE70.
Tn /.+-97217 S;1 1] 2.>7/+8 :797>7//DA (11.4) :+6H4.F-,;! 9711H97/-/D7 @+9>H6D2x;(b+a)36; >/+=+467/+0 7CDE70. Tbn(x) = Tn/. +-97217 Sa b]:b;aTb0 (x) = 1 Tb1(x) = 2x ;b ;(b a+ a)(12)bTn+1(x) = 2 2x ; (b + a) Tbn (x) ; Tbn;1(x) :95 n > 0:b;a+6B2H;,B I-5>5 @+9>H6.>5, CH37> 0D45,6;-B 0 ,H>>.A (11) ,-+6C<D ,670./.:9.0+, . 1.?3D8 ,-+6C7< { ,079AH 0/52.C+2/.45>gi j = Tbi(xj )f (xj ) zj = 2 cos (2jn; 1) i = 0 : : : n ; 1 j = 1 : : : n:+ @+9>H6.> (12) 36; 0,7A j = 1 2 : : : n 0D45,6;F-,;g0 j = f (xj ) g1 j = 21 zj f (xj ) = 12 zj g0 j (13)gi j = zj gi;1 j ; gi;2 j i = 2 : : : n ; 1:..
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