07 - Numerical Quadrature

Quadrature

Quadrature is equivalent to numerical integration

Roots of Orthogonal Polynomials lemma

Let $\{{\varphi }_0,{\varphi }_1,\cdots ,{\varphi }_n,\cdots \}$ be orthogonal polynomials for inner product

$$⟨f,g⟩={\int }_a^{b}w(x)f(x)g(x)dx$$

Then for each $k=0,1,\cdots$

$${\varphi }_k\text{ has }k\text{ distinct roots in interval }(a,b)$$

Proof

As

$${\varphi }_0(x)\equiv \text{constant}\ne 0$$

then holds trivially for $k=0$

Suppose for $k\ge 1$ then

$$⟨{\varphi }_k,{\varphi }_0⟩={\int }_a^{b}w(x){\varphi }_k(x){\varphi }_0(x)dx=0$$

and as ${\varphi }_0$ is constant then implies

$${\int }_a^{b}w(x){\varphi }_k(x)dx=0\text{ with }w(x)>0,x\in (a,b)$$

Thus

$${\varphi }_k(x)\text{ changes sign in }(a,b)$$

Hence at least one root in $(a,b)$

Suppose there exist $1\le \ell \le k$ points $a<r_1<r_2<\cdots <r_{\ell }<b$ where ${\varphi }_k$ changes sign
Then

$$q(x)=\prod _{j=1}^{\ell }(x-r_j)\cdot \text{ sign of }{\varphi }_k\text{ on }(r_{\ell },b)$$

has the same sign as ${\varphi }_k$ on $(a,b)$

Hence

$$⟨{\varphi }_k,q⟩={\int }_a^{b}w(x){\varphi }_k(x)q(x)dx>0$$

As $q$ is of degree $\ell$ then it must be degree $\ge k$ so $\ell \ge k$
However ${\varphi }_k$ is of exact degree $k$ hence number of distinct roots $\ell$ must be $\le k$
Thus

$$\ell =k$$

Hence ${\varphi }_k$ has $k$ distinct roots in $(a,b)$

Optimal Choice of Quadrature Points

Quadrature points $x_0,x_1,\cdots ,x_n$ should be chosen to match exactness of quadrature rule

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

for polynomials of degree as high as possible

Quadrature Weights from Lagrange Interpolation

Let $x_{0}k,x_1,\cdots ,x_n$ be the quadrature points then

Lagrange Interpolating Polynomials defined as

$$p_n=\sum _{j=0}^{n}f(x_j)L_{n,j}\in {\Pi }_n$$

is unique

So $f\in {\Pi }_n⟹p_n\equiv f$ independent of interpolation points and

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

where

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

Gauss Quadrature

Let inner product be denoted by

$$⟨g,h⟩={\int }_a^{b}w(x)g(x)h(x)dx$$

with $w$ being a non-negative weight function

$(n+1)-$point Gauss quadrature approximates

$${\int }_a^{b}w(x)f(x)dx\text{ by }{\int }_a^{b}w(x)p_n(x)dx$$

where $p_n$ is the polynomial interpolant to $f$ at $\{x_i{\}}_{i=0}^n$
with $x_i$ being the distinct roots of $(n+1)$th orthogonal polynomial

Types of Quadrature Rules

Gauss-Legendre Quadrature

$$w(x)\equiv 1,(a,b)=(-1,1)$$

Gauss-Chebyshev Quadrature

$$w(x)\equiv (1-x^2{)}^{-\frac{1}{2}},(a,b)=(-1,1)$$

Gauss-Laguerre Quadrature

$$w(x)\equiv e^{-x},(a,b)=(0,\infty )$$

[!example] Gauss-Hermite Quadrature

$$w(x)\equiv e^{-x^2},(a,b)=(-\infty ,\infty )$$