Gauss-Laguerre quadrature rule

% matlab script to derive the 2-point Gauss-Laguerre quadrature rule
% and use it on an example

% inelegantly set up equations from method of undetermined coefficients
% and solve

clear all 
close all
format long

eq1 = 'w1*1 + w2*1 = 1';

eq2 = 'w1*x1 + w2*x2 = 1';

eq3 = 'w1*x1^2 + w2*x2^2 = 2';

eq4 = 'w1*x1^3 + w2*x2^3 = 6';

[w1,w2,x1,x2] = solve(eq1,eq2,eq3,eq4)
pause

% note: there are two solutions
% we pick the one where x1 < x2

[x1,index] = min(double(x1))
w1 = double(w1(index))

x2 = double(x2(index))
w2 = double(w2(index))

% define the integrand

f = @(x) exp(x).*log(1+exp(-x));

% use Gauss-Laguerre quadrature to approximate it

glq = w1*feval(f,x1) + w2*feval(f,x2)

% now let's find the exact answer and see what the error is ...

syms x
I = double(int(log(1+exp(-x)),0,Inf))

error = I - glq
pause

% or we can estimate the neglected tail
% trying integral(f,0,Inf) or integral(f,0,realmax)
% won't work!

f = @(x) log(1+exp(-x));
Q10 = integral(f,0,10)
Q20 = integral(f,0,20)
Q30 = integral(f,0,30)
% from here we see that we have convergence to 8 decimal places

% we can also perform a change of variable to make the interval finite
F = @(t) log(1+t)./t;
integral(F,0,1)

% check for problems at t=0
ezplot(F,0,1)
integral(F,realmin,1)