Gauss–Legendre quadrature

Builds on Gaussian quadrature Legendre polynomials

Definition

Let $n \ge 1$ and let $P_n$ be the Legendre polynomial of degree $n$ . Let $x_1, \dots, x_n$ be the zeros of $P_n$ ; they are distinct and lie in $(-1, 1)$ . The $n$ -point Gauss–Legendre quadrature rule is $\int_{-1}^{1} f(x)\,dx \approx \sum_{i=1}^{n} w_i f(x_i), \qquad w_i = \frac{2}{(1 - x_i^2)\, P_n'(x_i)^2}.$ It is the $n$ -point Gaussian quadrature rule for the weight $w = 1$ on $(-1, 1)$ .

Remarks

The rule is exact for every polynomial of degree at most $2n - 1$ . For an interval $[a, b]$ one uses $\int_a^b g(t)\,dt = \frac{b - a}{2} \int_{-1}^{1} g\big(\frac{b - a}{2} x + \frac{a + b}{2}\big)\,dx$ .

Example: $n = 2$ . Here $P_2(x) = \frac12 (3x^2 - 1)$ , with zeros $\pm 1/\sqrt{3}$ , and $P_2'(x) = 3x$ . The weights are $2 / \big((2/3) \cdot 3\big) = 1$ . For $f(x) = x^2$ the rule gives $2/3$ , and for $f(x) = x^3$ it gives $0$ , both exact.

Non-example: for $n = 2$ and $f(x) = x^4$ the rule gives $2/9$ while the integral is $2/5$ . This is expected, since $4 > 2n - 1 = 3$ .

Reading path

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Tag	`5ZA6VRDNB9`
URI	`m2://5ZA6VRDNB9`
Kind	definition
Writing	W0 stub
MSC	65D32, 41A55, 33C45
Defines	Gauss–Legendre quadrature
Uses	Gaussian quadrature, Legendre polynomials

Gauss–Legendre quadrature

Definition

Remarks

Reading path

Further reading

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