18 - Exactness of Gaussian Quadrature

Exactness of Gaussian Quadrature

Let $x_0<x_1<\cdots <x_n$ be the roots of $n+1$-st degree orthogonal polynomial ${\varphi }_{n+1}$ under

$$⟨g,h⟩={\int }_a^{b}w(x)g(x)h(x)dx$$

Then Quadrature Formula with weights is exact if $f\in {\Pi }_{2n+1}$

$$w_j={\int }_a^{b}w(x)L_{n,j}(x)dx$$

Proof

Let $p\in {\Pi }_{2n+1}$ then

By Division Algorithm

$$p(x)=q(x){\varphi }_{n+1}(x)+r(x)\text{ with }q,r\in {\Pi }_n$$

So

$${\int }_a^{b}w(x)p(x)dx={\int }_a^{b}w(x)q(x){\varphi }_{n+1}(x)dx+{\int }_a^{b}w(x)r(x)dx=\sum _{j=0}^n{w}_{j}r(x_j)$$

as first integral is zero by Roots of orthogonal polynomials and other is exact as $r\in {\Pi }_n$
Then

$$p(x_j)=q(x_j){\varphi }_{n+1}(x_j)+r(x_j)=r(x_j)\text{ for }j=0,1,\cdots ,n$$

as $x_j$ are roots of ${\varphi }_{n+1}$

Hence

$${\int }_a^{b}w(x)p(x)dx=\sum _{j=0}^n{w}_{j}p(x_j)$$

where

$$w_j={\int }_a^{b}w(x)L_{n,j}(x)dx\text{ if }p\in {\Pi }_{2n+1}$$