Recently, I got a request how one can find the quadrature and weights of a Gauss-Legendre quadrature rule for large n. It seems that the internet has these points available free of charge only up to n=12. Below is the

MATLAB program that finds these values for any n. I tried the program for n=25 and it gave results in a minute or so. The results output up to 32 significant digits.

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% Program to get the quadrature points

% and weight for Gauss-Legendre Quadrature

% Rule

clc

clear all

syms x

% Input n: Quad pt rule

n=14;

% Calculating the Pn(x)

% Legendre Polynomial

% Using recursive relationship

% P(order of polynomial, value of x)

% P(0,x)=1; P(1,x)=0;

% (i+1)*P(i+1,x)=(2*i+1)*x*P(i,x)-i*P(i-1,x)

m=n-1;

P0=1;

P1=x;

for i=1:1:m

Pn=((2.0*i+1)*x*P1-i*P0)/(i+1.0);

P0=P1;

P1=Pn;

end

if n==1

Pn=P1;

end

Pn=expand(Pn);

quadpts=solve(vpa(Pn,32));

quadpts=sort(quadpts);

% Finding the weights

% Formula for weights is given at

% http://mathworld.wolfram.com/Legendre-GaussQuadrature.html

% Equation (13)

for k=1:1:n

P0=1;

P1=x;

m=n;

% Calculating P(n+1,x)

for i=1:1:m

Pn=((2.0*i+1)*x*P1-i*P0)/(i+1.0);

P0=P1;

P1=Pn;

end

Pn=P1;

weights(k)=vpa(2*(1-quadpts(k)^2)/(n+1)^2/ …

subs(Pn,x,quadpts(k))^2,32);

end

fprintf(‘Quad point rule for n=%g \n’,n)

disp(‘ ‘)

disp(‘Abscissas’)

disp(quadpts)

disp(‘ ‘)

disp(‘Weights’)

disp(weights’)_______________________________________________________

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