Bivariate Gaussian Quadrature

版本 1.0.2 (2.6 KB) 作者: zhang
A function implementing first, second and third order Gaussian qudrature in 2d cases with a test and illustration script attached.
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更新时间 2024/4/10

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For quadrature order over 3, symbolic toolbox is required to calculate the Gauss points and weights using Legendre Polynomials.
function quadrature = gaussianQuadrature2D(func, xmin, xmax, ymin, ymax, order)
%GAUSSIANQUADRATURE2D Perform a 2D Gaussian quadrature over a specified range.
% QUADRATURE = GAUSSIANQUADRATURE2D(FUNC, XMIN, XMAX, YMIN, YMAX, ORDER)
% performs a 2D Gaussian quadrature numerical integration on a bivariate
% function FUNC of order ORDER, over the range [XMIN, XMAX] and [YMIN, YMAX].
% Algebra accuracy of ORDER-k Gaussian quadrature is (2*k - 1).
%
% Inputs:
% FUNC - a function handle to the integrand, supposed to be bivariate
% XMIN, XMAX - the lower and upper bounds of the x range
% YMIN, YMAX - the lower and upper bounds of the y range
% ORDER - the order of Gaussian quadrature, determining the number of
% sample points and weights
%
% Outputs:
% QUADRATURE - the result of the numerical integration
A simple demo and test script:
% test_gauss_quad.m
clear
clc
% example 1
n = 10;
[xmin, xmax, ymin, ymax] = deal(-1.5, 2, -1, 0.5);
result = zeros(n, 5);
for k = 1:n
func = @(x, y) (x + 0.5).^k .* (y - 0.3).^k + 1;
% Gaussian quadrature with 1st/2nd/5th order
gaussQuad1 = gaussianQuadrature2D(func, xmin, xmax, ymin, ymax, 1);
gaussQuad2 = gaussianQuadrature2D(func, xmin, xmax, ymin, ymax, 2);
gaussQuad3 = gaussianQuadrature2D(func, xmin, xmax, ymin, ymax, 5);
% reference solution using built-in `integral2` function
normalQuad = integral2(func, xmin, xmax, ymin, ymax);
% logging result
result(k, :) = [k, gaussQuad1, gaussQuad2, gaussQuad3, normalQuad];
end
% show result:
% k-th order gaussian handles (2k-1) or lower polynomials accurately.
result = array2table(result); ...
result.Properties.VariableNames = ...
{'Poly.Order', 'Gauss-1', 'Gauss-2', 'Gauss-5', 'Ref.Sol.'};
disp(result);
result:
Poly.Order Gauss-1 Gauss-2 Gauss-5 Ref.Sol.
__________ _______ _______ _______ ________
1 3.0844 3.0844 3.0844 3.0844
2 6.1433 9.3231 9.3231 9.3231
3 4.8815 -1.5406 -1.5406 -1.5406
4 5.402 17.027 19.903 19.903
5 5.1873 -15.115 -27.349 -27.349
6 5.2759 40.493 83.539 83.539
7 5.2393 -55.736 -189.11 -189.11
8 5.2544 110.78 504.8 504.8
9 5.2482 -177.37 -1309.3 -1309.3
10 5.2507 321.27 3534.2 3536.7

引用格式

zhang (2024). Bivariate Gaussian Quadrature (https://www.mathworks.com/matlabcentral/fileexchange/163096-bivariate-gaussian-quadrature), MATLAB Central File Exchange. 检索时间: .

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版本 已发布 发行说明
1.0.2

Arbitrary order is supported
1.0.1

arbitrary range of x, y
1.0.0