# 多项式插值

## 定义

${\displaystyle p(x_{i})=y_{i}{\mbox{ , }}i=0,\ldots ,n.}$

${\displaystyle L_{n}:\mathbb {K} ^{n+1}\to \Pi _{n}}$

## 构建插值多项式

${\displaystyle p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{2}x^{2}+a_{1}x+a_{0}.\qquad (1)}$

p 对数据点进行插值其含义为

${\displaystyle p(x_{i})=y_{i}\qquad {\mbox{for all }}i\in \left\{0,1,\dots ,n\right\}.}$

${\displaystyle {\begin{bmatrix}x_{0}^{n}&x_{0}^{n-1}&x_{0}^{n-2}&\ldots &x_{0}&1\\x_{1}^{n}&x_{1}^{n-1}&x_{1}^{n-2}&\ldots &x_{1}&1\\\vdots &\vdots &\vdots &&\vdots &\vdots \\x_{n}^{n}&x_{n}^{n-1}&x_{n}^{n-2}&\ldots &x_{n}&1\end{bmatrix}}{\begin{bmatrix}a_{n}\\a_{n-1}\\\vdots \\a_{0}\end{bmatrix}}={\begin{bmatrix}y_{0}\\y_{1}\\\vdots \\y_{n}\end{bmatrix}}.}$

## 插值误差

n 阶多项式在节点 x0、...、xn 对函数 f 插值的误差为：

${\displaystyle f(x)-p_{n}(x)=f[x_{0},\ldots ,x_{n},x]\prod _{i=0}^{n}(x-x_{i})}$

${\displaystyle f[x_{0},\ldots ,x_{n},x]}$

${\displaystyle f(x)-p_{n}(x)={\frac {f^{(n+1)}(\xi )}{(n+1)!}}\prod _{i=0}^{n}(x-x_{i})}$

## 勒贝格常数

${\displaystyle \|f-X(f)\|\leq (L+1)\|f-p^{*}\|.}$

${\displaystyle L\geq {\frac {2}{\pi }}\log(n+1)+C\quad {\mbox{for some constant }}C.}$

## 收敛性

${\displaystyle \lim _{n\to \infty }X_{n}f=f,}$

## 参考文献

• Kendell A. Atkinson (1988). An Introduction to Numerical Analysis (2nd ed.), Chapter 3. John Wiley and Sons. ISBN 0-471-50023-2
• L. Brutman (1997), Lebesgue functions for polynomial interpolation — a survey, Ann. Numer. Math. 4, 111–127.
• M.J.D. Powell (1981). Approximation Theory and Method, Chapter 4. Cambridge University Press. ISBN 0-521-29514-9.
• Michelle Schatzman (2002). Numerical Analysis: A Mathematical Introduction, Chapter 4. Clarendon Press, Oxford. ISBN 0-19-850279-6.
• Endre Süli and David Mayers (2003). An Introduction to Numerical Analysis, Chapter 6. Cambridge University Press. ISBN 0-521-00794-1.