拉格朗日插值 定义 Ln(x)=∑i=0nli(x)f(xi)其中li(x)=∏0≤j≤n,i≠jx−xjxi−xjL_n(x)=\sum_{i=0}^{n}l_i(x)f(x_i) \\ 其中l_i(x)=\prod_{0\le j \le n,i\ne j }{\frac{x-x_j}{x_i-x_j}} Ln(x)=i=0∑nli(x)f(xi)其中li(x)=0≤j≤n,i=j∏xi−xjx−xj 误差 设Ln(x)L_n(x)Ln(x)是[a,b][a,b][a,b]上过{(xi,f(xi)),i=0,1...,n}\{(x_i,f(x_i)),i=0,1...,n\}{(xi,f(xi)),i=0,1...,n}的nnn次插值多项式,xi∈[a,b],xix_i\in[a,b],x_ixi∈[a,b],xi互不相同，当f∈Cn+1[a,b]f\in C^{n+1}[a,b]f∈Cn+1[a,b]时，误差 Rn(x)=f(n+1)(ξ)(n+1)!(x−x0)(x−x1)...(x−xn),其中ξ∈[a,b]R_n(x)=\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_0)(x-x_1)...(x-x_n),其中\xi\in[a,b] Rn(x)=(n+1)!f(n+1)(ξ)(x−x0)(x−x1)...(x−xn),其中ξ∈[a,b] 牛顿插值 待更新
