2.分段线性插值的误差估计 由第二节定理1可知n次 Lagrange插值多项式的余项为 R(x)=/(x)-P(x)=(), n+ n+1(X 那么分段线性插值n(x)的余项为 R(x)=f(x)-L1(x)=f(x)-l(x) f"( (x-xk)(x-xk+1)5,x∈[xx+1且5与x有关 R1(x)|≤ maxI f"(x)l max (x X(X-X k+1 a<x<b a≤x<b k ≤·M2-h Mh( ) ( 1)! ( ) 1 ( 1) x n f n n + + + = w x 由第二节定理1可知,n次Lagrange插值多项式的余项为 R (x) f (x) P (x) n = - n 那么分段线性插值L1 (x)的余项为 ( ) ( ) ( ) 1 1 R x = f x - L x ( ) ( ) ( ) 1 f x L x k = - ( )( ) 2 ( ) - - +1 ¢¢ = k k x x x x f x x , x Î[xk , xk +1 ],且x与x有关 |R1 (x)| max| ( )| max|( )( )| 2 1 +1 £ £ £ £ £ × ¢¢ × - k - k k a x b a x b f x x x x x 2 2 4 1 2 1 £ × M × h 2 2 8 1 = M h 2. 分段线性插值的误差估计