插值方法 - Lagrange插值多项式

利用基函数，假设插值多项式的形式： $${\mathit{L}}_{\mathit{n}}\left(\mathit{x}\right)=\underset{\mathit{i}=0}{\overset{\mathit{n}}{\sum }}\mathit{a}{}_{\mathit{i}}\mathit{l}{}_{\mathit{i}}\left(\mathit{x}\right)$$
记 $$\mathit{w}{}_{\mathit{n}+1}\left(\mathit{x}\right)=\underset{\mathit{i}=0}{\overset{\mathit{n}}{\prod }}(\mathit{x}-\mathit{x}{}_{\mathit{i}})$$
得Lagrange插值多项式的完整形式： $${\mathit{L}}_{\mathit{n}}\left(\mathit{x}\right)=\underset{\mathit{i}=0}{\overset{\mathit{n}}{\sum }}\mathit{f}{}_{\mathit{i}}\left(\mathit{x}\right)\mathit{l}{}_{\mathit{i}}\left(\mathit{x}\right)$$其中 $$\mathit{l}{}_{\mathit{i}}\left(\mathit{x}\right)=\frac{\mathit{w}{}_{\mathit{n}+1}\left(\mathit{x}\right)}{(\mathit{x}-\mathit{x}{}_{\mathit{i}})\mathit{w}{}^{,}{}_{\mathit{n}+1}\left(\mathit{x}\right)}$$