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单词 Gaussian quadrature
释义

Gaussian quadrature

中文百科

高斯求积

以德国数学家卡尔·弗里德里希·高斯所命名的一种数值积分中的求积规则。当我们要求解某个函数的积分\int_{-1}^{1}f(x) dx ，其数值解可以由\sum_{i=1}^n w_i f(x_i)近似，其中w_i, i = 1 ... n为权重。高斯求积仅仅当函数f(x)可以由在区间[-1, 1]的多项式近似时才能获得准确的近似解，这种方法并不适合函数具有奇异点的情况。于是乎，我们可以把函数f(x)写作f(x) = W(x)g(x)，其中g(x)是近似多项式，W(x)是已知的权重函数，这样我们就有

\int_{-1}^1 f(x) dx = \int_{-1}^1 W(x)g(x)dx \approx \sum_{i=1}^n w_i' g(x_i) 。

常用的权重函数有

W(x) = (1 - x^2)^{-1/2} （高斯切比雪夫）

以及

W(x) = e^{-x^2} （高斯埃米特）。
英语百科

Gaussian quadrature 高斯求积

Comparison between 2-point Gaussian and trapezoidal quadrature. The blue line is the polynomial , whose integral in [-1, 1] is 2/3. The trapezoidal rule returns the integral of the orange dashed line, equal to . The 2-point Gaussian quadrature rule returns the integral of the black dashed curve, equal to . Such a result is exact since the green region has the same area as the red regions.
Graphs of Legendre polynomials (up to n = 5)

In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. (See numerical integration for more on quadrature rules.) An n-point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the points xi and weights wi for i = 1, ..., n. The domain of integration for such a rule is conventionally taken as [−1, 1], so the rule is stated as
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更新时间：2025/6/22 0:18:24