Gaussian quadrature

Builds on Quadrature rule Degree of exactness of a quadrature rule Orthogonal polynomials

Definition

Let $(a, b)$ be an interval, bounded or not, and let $w$ be a weight function on $(a, b)$ . This means that $w$ is integrable, $w > 0$ almost everywhere, and every moment $\int_a^b x^k w(x)\,dx$ is finite. Let $n \ge 1$ . Consider a quadrature rule $\int_a^b f(x)\, w(x)\,dx \approx \sum_{i=1}^{n} w_i f(x_i)$ with distinct nodes $x_1, \dots, x_n$ in $(a, b)$ and real weights $w_1, \dots, w_n$ . It is an $n$ -point Gaussian quadrature rule for $w$ if it is an equality for every polynomial $f$ of degree at most $2n - 1$ .

Remarks

For each $w$ and $n$ such a rule exists and is unique. Its nodes are the zeros of the degree $n$ orthogonal polynomial $p_n$ for the inner product $\langle f, g \rangle = \int_a^b f g\, w\,dx$ , and its weights are positive. No $n$ -point rule is exact for all polynomials of degree $2n$ : the polynomial $\prod_{i}(x - x_i)^2$ has a positive integral but vanishes at every node. So $2n - 1$ is the largest possible degree of exactness.

Example: $w = 1$ on $(-1, 1)$ and $n = 1$ . The rule $\int_{-1}^{1} f \approx 2 f(0)$ is exact for $f = 1$ (both sides are $2$ ) and for $f = x$ (both sides are $0$ ), so it is the one-point Gaussian rule, the midpoint rule.

Non-example: the trapezoidal rule $\int_{-1}^{1} f \approx f(-1) + f(1)$ has two nodes but fails for $f = x^2$ , since the left side is $2/3$ and the right side is $2$ . So it is not exact up to degree $3$ and is not a Gaussian rule.

The rule for $w = 1$ on $(-1, 1)$ is Gauss–Legendre quadrature.

Reading path

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Degree of exactness of a quadrature rule concept
Orthogonal polynomials concept
Quadrature rule concept
Gaussian quadrature definition

Tag	`SZ6XCSQA5H`
URI	`m2://SZ6XCSQA5H`
Kind	definition
Writing	W0 stub
MSC	65D32, 41A55, 33C45
Defines	Gaussian quadrature
Uses	Quadrature rule, Degree of exactness of a quadrature rule, Orthogonal polynomials

Gaussian quadrature

Definition

Remarks

Reading path

Further reading

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