create a gauss legendre in matlab

To create a Gauss-Legendre quadrature in MATLAB, you can use the built-in function legendre() to obtain the Legendre polynomial coefficients, and then solve for the roots of the polynomial to obtain the quadrature points. Here's an example code snippet that shows how to use Gauss-Legendre quadrature to integrate a function over a given interval:

main.m
% Define the function to be integrated
f = @(x) 1./(1+x.^2);

% Define the interval of integration
a = -1;
b = 1;

% Set the number of quadrature points
n = 4;

% Obtain the Legendre polynomial coefficients
p = legendre(n);

% Solve for the roots of the polynomial to obtain quadrature points
x = roots(p);

% Compute the quadrature weights
w = 2./((1-x.^2).*polyval(polyder(p),x).^2);

% Evaluate the integral using the quadrature rule
I = sum(w.*f((b-a)/2*x + (b+a)/2))*(b-a)/2;
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In this example code, we define the function f(x) to be integrated, and the interval of integration [a,b]. We also set the number of quadrature points n. We then obtain the Legendre polynomial coefficients using legendre(n), and solve for the roots of the polynomial to obtain the quadrature points. We use these points to compute the quadrature weights w, and then evaluate the integral using the quadrature rule sum(w.*f((b-a)/2*x + (b+a)/2))*(b-a)/2.

Note that this implementation assumes that the interval of integration is [-1,1]. To use Gauss-Legendre quadrature over a different interval [a,b], you need to transform the quadrature points and weights accordingly.