19 - Error Bound for Gauss Quadrature

Error Bound for Gauss Quadrature

Let $f^{2n+2}$ be continuous on $(a,b)$ then

$${\int }_a^{b}w(x)f(x)dx=\sum _{j=0}^n{w}_{j}f(x_j)+\frac{f^{(2n+2)}(\eta )}{(2n+2)!}{\int }_a^{b}w(x)w(x)\prod _{j=0}^n(x-x_j{)}^{2}dx$$

for some $\eta \in (a,b)$

Proof

Using Hermite interpolating polynomial $H_{2n+1}$ to $f$ on $x_0,x_1,\cdots ,x_n$ then
Error for Hermite interpolation is defined by

$$f(x)-H_{2n+1}(x)=\frac{1}{(2n+2)!}{f}^{(2n+2)}(\eta (x))\prod _{j=0}^n(x-x_j{)}^2$$

for some $\eta =\eta (x)\in (a,b)$

As $H_{2n+1}\in {\Pi }_{2n+1}$ then

$${\int }_a^{b}w(x)H_{2n+1}(x)dx=\sum _{j=0}^n{w}_j{H}_{2n+1}(x_j)=\sum _{j=0}^n{w}_{j}f(x_j)$$

as Gauss quadrature is exact for polynomials of that degree and then by interpolation

Hence

$$\begin{array}{rl}{\int }_a^{b}w(x)f(x)dx-\sum _{j=0}^n{w}_{j}f(x_j)& ={\int }_a^{b}w(x)[f(x)-H_{2n+1}(x)]dx\\ & =\frac{1}{(2n+2)!}{\int }_a^b{f}^{(2n+2)}(\eta (x))w(x)\prod _{j=0}^n(x-x_j{)}^{2}dx\end{array}$$

hence result follows from integral mean value theorem as

$$w(x)\prod _{j=0}^n(x-x_j{)}^2\ge 0$$