) 中图学技术大荸学系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS n=22(x)(x-x-51+(x-6x-5 (x x (一)(一3)2(-5)一3)"√2(一)(誓一)2 sin500≈L,(5z )≈0.76543 18 cos R2(x)= 3! (x-2)x-)(x-2) <cOS&.< 2 0.00044<R2 5|<0.00077 sin50°=0.7660444. 18 2次插值的实际误差≈000061 高次插值通常优于 低次插值
数 学 系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS n = 2 2 3 ( )( ) ( )( ) 2 1 ( )( ) ( )( ) 2 1 ( )( ) ( )( ) ( ) 3 6 3 4 6 4 4 6 4 3 6 3 6 4 6 3 4 3 2 − − − − + − − − − + − − − − = x x x x x x L x ) 18 5 sin 50 (2 0 L 0.76543 2 3 cos 2 1 ); 3 )( 4 )( 6 ( 3 ! cos ( ) 2 − − − − = x x R x x x x 0.00077 18 5 0.00044 2 R sin 50 = 0.7660444… 2次插值的实际误差 0.00061 高次插值通常优于 低次插值
) 中图学技术大荸学系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS 误差R,(x)=f(x)-Ln(x) 解 f(x1)=Ln2(x),i=0,…n . R n (x1)=0,i=0, R,(x)=k(x(x-xo).(x-xn) 求k(x)=? Va,k(a=? 设v(t)=f(1)-Ln()-k(a)(t-x)…(t-xn) 易知v(x)=0,i=0,… n and y(a)=0
数 学 系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS 误差 R (x) f (x) L (x) n = − n 解： ( ) ( )( ) ( ) ( ) 0 , 0, ( ) ( ) , 0, n 0 n n i i n i R x k x x x x x R x i n f x L x i n = − − = = = = 求 k(x) = ? ( ) 0, 0, and ( ) 0 ( ) ( ) ( ) ( )( ) ( ) , ( ) ? 0 = = = = − − − − = x i n a t f t L t k a t x t x a k a i n n 设 易知
) 中图学技术大荸学系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS y(t)有n+2个零点 ∴彐 (n+1) ()=0 (n+1 c(n+1) (2)-k(a)(n+1) k(a)= f(n() (n+1)! 由a的任意性 R,(x)= fm(2) (x-x0)…(x-xn) (n+1)
数 学 系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS (t) 有n+2个零点 ( 1)! ( ) ( ) ( ) ( ) ( )( 1)! , ( ) 0 ( 1) ( 1) ( 1) ( 1) + = = − + = + + + + n f k a f k a n n n n n ( ) ( ) ( 1)! ( ) ( ) 0 ( 1) n n n x x x x n f R x − − + = + 由a的任意性
) 中图学技术大荸学系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS 事后误差估计 给定气任取n+1个构造Ln(x) 如 i=0. → 另取i=1,…,n+1 则 (n+ f(x-L(x)= 1(x-x0)…(x-x,) (n+1) (n+1) f(x)-L,(x)= (x-x1)…(x-xn+1) n+
数 学 系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS 事后误差估计 给定 1 0 + = n i i x 任取n+1个构造 L (x) n ( ) ~ 1, , 1 0, , ( ) i n L x i n L x n n = + = 如： 另取 则 ( ) ( ) ( 1)! ( ) ( ) ~ ( ) ( ) ( ) ( 1)! ( ) ( ) ( ) 1 1 2 ( 1) 0 1 ( 1) + + + − − + − = − − + − = n n n n n n x x x x n f f x L x x x x x n f f x L x
) 中图学技术大荸学系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS 近似f(1)≈f(2) f(x-Ln,(x)x- f(x-L(x)x-x n+1 X-x X-x →f(x) n+Ln(x)+ n+1 n+1 X-x →f(x)-Ln(x)≈ X X n+1
数 学 系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS 近似 ( ) ( ) 2 ( 1) 1 ( 1) + + n n f f 则 ( ( )) ~ ( ) ( ) ( ) ( ) ~ ( ) ( ) ( ) ~ ( ) ( ) ( ) 0 1 0 1 0 0 0 1 1 1 0 L x L x x x x x f x L x L x x x x x L x x x x x f x x x x x f x L x f x L x n n n n n n n n n n n n − − − − − − + − − − − − − + + + + +