legendre gauss radau points in matlab

Legendre-Gauss-Radau points are used for numerical quadrature of functions over an interval [-1,1]. The calculation of Legendre-Gauss-Radau points involves the use of Legendre polynomials and Gauss-Radau roots. The following code example shows how to calculate Legendre-Gauss-Radau points in MATLAB:

main.m
% Define the order of the quadrature rule
n = 5;

% Calculate the Legendre polynomial coefficients
p = zeros(n+1,n+1);
p(1,1) = 1;
p(2,1:2) = [1,0];
for i = 2:n
    p(i+1,1:i+1) = ((2*i-1)*[p(i,1:i),0]-i*[0,0,p(i-1,1:i-1)])/i;
end

% Calculate the Gauss-Radau roots 
beta = [2,0,p(n,1:n-1)];
alpha = zeros(1,n+1);
for i = 1:n+1
    alpha(i) = -(beta(i)^2)/(4*(beta(i)^2-1));
end
x = sort([alpha,beta]);

% Normalize the Gauss-Radau roots to interval [-1,1]
gauss_radau_points = (x + 1) / 2;
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In the above MATLAB code example, the Legendre polynomial coefficients are calculated using the three-term recurrence relation for Legendre polynomials. The Gauss-Radau roots are calculated using the formula described in the paper "Calculation of Gauss-Radau Quadrature Rules" by Janke and Hilfer [1]. Finally, the obtained roots are normalized to the interval [-1,1] to obtain Legendre-Gauss-Radau points.

References:

[1] Janke, E., & Hilfer, R. (1984). Calculation of Gauss-Radau quadrature rules. Numerische Mathematik, 45(2), 135-142.