分段线性插值
分段线性插值
分段线性插值，就是将插值点用折线段连接起来逼近f(x)。设已知节点 a = x 0 < x 1 < ⋅ ⋅ ⋅ < x n = b a=x_0<x_1<···<x_n=b a=x0<x1<⋅⋅⋅<xn=b上的函数值 f 0 , f 1 , . . . , f n f_0,f_1,...,f_n f0,f1,...,fn，记 h k = x k + 1 − x k , h = m a x h k h_k=x_{k+1}-x_k,h=\\max h_k hk=xk+1−xk,h=maxhk。
称 I h ( x ) I_h(x) Ih(x)为分段线性插值函数，如果满足：
(1) 记 I h ( x ) ∈ [ a , b ] I_h(x)∈[a,b] Ih(x)∈[a,b];
(2) I h ( x k ) = f k ( k = 0 , 1 , . . . , n ) I_h(x_k)=f_k(k=0,1,...,n) Ih(xk)=fk(k=0,1,...,n);
(3) I h ( x ) I_h(x) Ih(x)在每个区间 [ x k , x k + 1 ] [x_k,x_{k+1}] [xk,xk+1]上是线性函数，
则由定义， I h ( x ) I_h(x) Ih(x)在每个小区间 [ x k , x k + 1 ] [x_k,x_{k+1}] [xk,xk+1]上可表示为
I h ( x ) = x − x k + 1 x k − x k + 1 f k + x − x k x k + 1 − x k f k + 1 ( x k ≤ x ≤ x k + 1 ) I_h(x)=\frac{x-x_{k+1}}{x_k-x_{k+1}}f_k+\frac{x-x_k}{x_{k+1}-x_k}f_{k+1} \ \ \ (x_k≤x≤x_{k+1}) Ih(x)=xk−xk+1x−xk+1fk+xk+1−xkx−xkfk+1 (xk≤x≤xk+1)
若用插值基函数表示，则I_h(x)在整个区间[a,b]上可表示为
I h ( x ) = ∑ j = 0 n f j l j ( x ) I_h(x)=\sum_{j=0}^nf_jl_j(x) Ih(x)=j=0∑nfjlj(x)
其中基函数 l j ( x ) l_j(x) lj(x)满足条件 l j ( x k ) = δ j k ( j , k = 0 , 1 , . . . , n ) l_j(x_k)=δ_jk(j,k=0,1,...,n) lj(xk)=δjk(j,k=0,1,...,n)。