Fungrim home page

Gaussian quadrature

Table of contents: Gauss-Legendre quadrature

Gauss-Legendre quadrature

Related topics: Legendre polynomials

0745ee
zeroszCPn ⁣(z)={xn,k:k{1,2,n}}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} P_{n}\!\left(z\right) = \left\{ x_{n,k} : k \in \{1, 2, \ldots n\} \right\}
ea4754
wn,k=2(1(xn,k)2)([ddtPn ⁣(t)]t=xn,k)2w_{n,k} = \frac{2}{\left(1 - {\left(x_{n,k}\right)}^{2}\right) {\left(\left[ \frac{d}{d t}\, P_{n}\!\left(t\right) \right]_{t = x_{n,k}}\right)}^{2}}
47b181
11f ⁣(t)dtk=1nwn,kf ⁣(xn,k)64M15(1ρ2)ρ2n   where M=suptEρf ⁣(t)\left|\int_{-1}^{1} f\!\left(t\right) \, dt - \sum_{k=1}^{n} w_{n,k} f\!\left(x_{n,k}\right)\right| \le \frac{64 M}{15 \left(1 - {\rho}^{-2}\right) {\rho}^{2 n}}\; \text{ where } M = \mathop{\operatorname{sup}}\limits_{t \in \mathcal{E}_{\rho}} \left|f\!\left(t\right)\right|
545987
abf ⁣(t)dtba2k=1nwn,kf ⁣(ba2xn,k+a+b2)ba264M15(1ρ2)ρ2n   where M=suptEρf ⁣(ba2t+a+b2)\left|\int_{a}^{b} f\!\left(t\right) \, dt - \frac{b - a}{2} \sum_{k=1}^{n} w_{n,k} f\!\left(\frac{b - a}{2} x_{n,k} + \frac{a + b}{2}\right)\right| \le \frac{\left|b - a\right|}{2} \frac{64 M}{15 \left(1 - {\rho}^{-2}\right) {\rho}^{2 n}}\; \text{ where } M = \mathop{\operatorname{sup}}\limits_{t \in \mathcal{E}_{\rho}} \left|f\!\left(\frac{b - a}{2} t + \frac{a + b}{2}\right)\right|

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-07-15 23:42:41.550119 UTC