二、低阶插值型求导公式 1两点公式 n=1f f(k=Lk )+E,(xk) k=0, 1 X-x X-x L,(x)=fo +f1 Xo -X 1 1 1 L1(x)=f6 E1(x) f2( (xk-x)k=0,1,≠k 2 若令h=x1-x,则
二、低阶插值型求导公式 n = 1时 ( ) ( ) ( ) k 1 k 1 k f ¢ x = L¢ x + E x 1.两点公式 k = 0,1 ( ) 1 L x ( ) 1 L¢ x ( ) 2 ( ) ( ) (2) 1 k k j x x f E x = - x k = 0,1, j ¹ k 若令h = x1 - x0 ,则 0 1 1 0 x x x x f - - = 1 0 0 1 x x x x f - - + 1 0 1 0 1 0 1 1 x x f x x f - + - =
f(o)=lr(o+e,xo (f1-f0) h f2() h f(x1)=L1(x1)+E1(x1) =(1-f6)+f2(5 h 2 --(5) (4)(5)式称为带余项的两点求导公式由于E=o(h) f(x0)≈f(x1)≈,(-f0) 精度1阶 h
( ) ( ) ( ) 0 1 0 1 0 f ¢ x = L¢ x + E x ( ) 1 1 0 f f h = - ( ) 2 (2) f x h - ( ) ( ) ( ) 1 1 1 1 1 f ¢ x = L¢ x + E x ( ) 1 1 0 f f h = - ( ) 2 (2) f x h + --------(4) --------(5) ( ) 1 ( ) ( ) 0 1 1 0 f f h f ¢ x » f ¢ x » - (4)(5)式称为带余项的两点求导公式 即 精度1阶 由于E = o(h)
2.三点公式 n=2时(xk)=L2(x)+E2(xk)k=0,1,2 x-Xo(x L2(x)=fo (x-x1)(x-x2)c(x-x0)(x-x2)c( X1 2 1 x-x1)+ xx+x-x x-xo+(x I2(x)=f0 f1 +f2 (x-x)(x-x2)(x1-x0)(x1-x2)“2(x2-x)( X -x (3) E2(rk) ∏I(xk-x) k 若x,x1x2为等距节点,即h=x1-x=x2-x1,则
2.三点公式 n = 2时 ( ) ( ) ( ) k 2 k 2 k f ¢ x = L¢ x + E x k = 0,1,2 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) 2 0 2 1 0 1 2 1 0 1 2 0 2 1 0 1 0 2 1 2 2 0 x x x x x x x x f x x x x x x x x f x x x x x x x x L x f - - - - + - - - - + - - - - = ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 2 0 2 1 0 1 2 1 0 1 2 0 2 1 0 1 0 2 1 2 2 0 x x x x x x x x f x x x x x x x x f x x x x x x x x L x f - - - + - + - - - + - + - - - + - ¢ = Õ ¹ = = - 2 0 (3) 2 ( ) 3! ( ) ( ) j k j k k j x x f E x x 若 x0 , x1 , x2为等距节点，即h = x1 - x0 = x2 - x1，则
3h 2h h 1 (x0)=f012+f1 2h h 2h22h (-36+4f1-f2) h h-h h 2h h +J2 2h22h (-f6+f2 2h 3h L2(x2)=f612+f112+2 2h 2h 2h(0-4f1+3/2) E、(x3)f(3(Ey h xo-x,lxo-x )=f((2) 3! 0 3 E2(x) f( 3! (x1-x)(x1-x2) f(2) (3) E 3!
2 0 0 2 1 2 2 2 2 2 2 3 ( ) h h f h h f h h L x f - + - - + - ¢ = ( )( ) 3! ( ) ( ) 0 1 0 2 (3) 2 0 x x x x f E x = - - x ( 3 4 ) 2 1 0 1 2 f f f h = - + - 2 1 0 2 1 2 2 2 2 2 ( ) h h f h h h f h h L x f + - - + - ¢ = ( ) 2 1 0 2 f f h = - + 2 2 0 2 1 2 2 2 2 2 3 2 ( ) h h f h h f h h L x f + - ¢ = + ( 4 3 ) 2 1 0 1 2 f f f h = - + ( ) 3 (3) 2 f x h = ( )( ) 3! ( ) ( ) 1 0 1 2 (3) 2 1 x x x x f E x = - - x ( ) 6 (3) 2 f x h = - ( )( ) 3! ( ) ( ) 2 0 2 1 (3) 2 2 x x x x f E x = - - x ( ) 3 (3) 2 f x h =