# 拉格朗日插值法

## 定义

${\displaystyle (x_{0},y_{0}),\ldots ,(x_{k},y_{k})}$

${\displaystyle L(x):=\sum _{j=0}^{k}y_{j}\ell _{j}(x)}$

${\displaystyle \ell _{j}(x):=\prod _{i=0,\,i\neq j}^{k}{\frac {x-x_{i}}{x_{j}-x_{i}}}={\frac {(x-x_{0})}{(x_{j}-x_{0})}}\cdots {\frac {(x-x_{j-1})}{(x_{j}-x_{j-1})}}{\frac {(x-x_{j+1})}{(x_{j}-x_{j+1})}}\cdots {\frac {(x-x_{k})}{(x_{j}-x_{k})}}.}$[3]

## 范例

• ${\displaystyle f(4)=10}$
• ${\displaystyle f(5)=5.25}$
• ${\displaystyle f(6)=1}$

${\displaystyle \ell _{0}(x)={\frac {(x-5)}{(4-5)}}\cdot {\frac {(x-6)}{(4-6)}}}$
${\displaystyle \ell _{1}(x)={\frac {(x-4)}{(5-4)}}\cdot {\frac {(x-6)}{(5-6)}}}$
${\displaystyle \ell _{2}(x)={\frac {(x-4)}{(6-4)}}\cdot {\frac {(x-5)}{(6-5)}}}$

${\displaystyle p(x)=f(4)\ell _{0}(x)+f(5)\ell _{1}(x)+f(6)\ell _{2}(x)}$
${\displaystyle .\,\,\,\,\,\,\,\,\,\,=10\cdot {\frac {(x-5)(x-6)}{(4-5)(4-6)}}+5.25\cdot {\frac {(x-4)(x-6)}{(5-4)(5-6)}}+1\cdot {\frac {(x-4)(x-5)}{(6-4)(6-5)}}}$
${\displaystyle .\,\,\,\,\,\,\,\,\,\,={\frac {1}{4}}(x^{2}-28x+136)}$

## 证明

### 存在性

${\displaystyle L(x_{j})=\sum _{i=0}^{k}y_{i}\ell _{i}(x_{j})=0+0+\cdots +y_{j}+\cdots +0=y_{j}}$

${\displaystyle (x-x_{0})\cdots (x-x_{j-1})(x-x_{j+1})\cdots (x-x_{k})}$

${\displaystyle \ell _{j}(x):=\prod _{i=0,\,i\neq j}^{k}{\frac {x-x_{i}}{x_{j}-x_{i}}}={\frac {(x-x_{0})}{(x_{j}-x_{0})}}\cdots {\frac {(x-x_{j-1})}{(x_{j}-x_{j-1})}}{\frac {(x-x_{j+1})}{(x_{j}-x_{j+1})}}\cdots {\frac {(x-x_{k})}{(x_{j}-x_{k})}}}$

## 几何性质

${\displaystyle P=\lambda _{0}\ell _{0}+\lambda _{1}\ell _{1}+\cdots +\lambda _{n}\ell _{n}=0}$

${\displaystyle \lambda _{0}=\lambda _{1}=\cdots =\lambda _{n}=0}$

## 重心拉格朗日插值法

${\displaystyle \ell (x)=(x-x_{0})(x-x_{1})\cdots (x-x_{k})}$

${\displaystyle \ell _{j}(x)={\frac {\ell (x)}{x-x_{j}}}{\frac {1}{\prod _{i=0,i\neq j}^{k}(x_{j}-x_{i})}}}$

${\displaystyle w_{j}={\frac {1}{\prod _{i=0,i\neq j}^{k}(x_{j}-x_{i})}}}$

${\displaystyle \ell _{j}(x)=\ell (x){\frac {w_{j}}{x-x_{j}}}}$

${\displaystyle L(x)=\ell (x)\sum _{j=0}^{k}{\frac {w_{j}}{x-x_{j}}}y_{j}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)}$

${\displaystyle \forall x,\,g(x)=\ell (x)\sum _{j=0}^{k}{\frac {w_{j}}{x-x_{j}}}}$

${\displaystyle L(x)={\frac {\sum _{j=0}^{k}{\frac {w_{j}}{x-x_{j}}}y_{j}}{\sum _{j=0}^{k}{\frac {w_{j}}{x-x_{j}}}}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)}$[7]

## 参考文献

### 引用

1. ^ E. Waring. Problems Concerning Interpolations. Philosophical Transactions of the Royal Society of London. 1779, 69: 59–67.
2. ^ （英文）E. Meijering. A chronology of interpolation: From ancient astronomy to modern signal and image processing,. Proceedings of the IEEE: 323.
3. ^ （英文）Julius Orion Smith III. Lagrange_Interpolation. Center for Computer Research in Music and Acoustics (CCRMA), Stanford University. [2009-12-22]. （原始内容存档于2009-06-28）.
4. ^ 冯有前，《数值分析》，第63页
5. ^ 李庆扬，《数值分析》第4版，第31页
6. ^ 冯有前，《数值分析》，第64页
7. Jean-Paul Berrut, Lloyd N. Trefethen. Barycentric Lagrange Interpolation (PDF). SIAM Review. 2004, 46 (3): 501–517. doi:10.1137/S0036144502417715.[永久失效链接]
8. ^ 王兆清，李淑萍，唐炳涛. 一维重心型插值：公式、算法和应用. 山东建筑大学学报. 2007, 22 (5): 447–453.
9. ^ NICHOLAS J. HIGHAM. The numerical stability of barycentric Lagrange Interpolation (PDF). IMA Journal of Numerical Analysis. 2004, 24 (4): 547–556.