function [tp,yp,abserr] = colloc1(n);
%
%   function [t,y,abserr] = colloc1(n);
%
% Solves the linear BVP from page 3.61
% using collocation with polynomials.
%

Np = 1000;
h  = 1.0 / n;
t  = linspace(h,1.0,n);
tp = linspace(0.0,1.0,Np);

% Determine the right-hand side vector

g = (400 * (cos(pi*t)).^2 + 2*pi^2 * cos(2*pi*t))';
g(n,1) = 0.0;

% Determine the matrix from the evaluation of y''

A1 = zeros(n,n);

for l=2:n
    for k=1:n-1
        A1(k,l) = l*(l-1)*(t(k))^(l-2);
    end;
end;

% Determine the matrix from the evaluation of y

A2 = zeros(n,n);

for k=1:n-1
    for l=1:n
        A2(k,l) = (t(k))^l;
    end;
end;

for l=1:n
    A2(n,l) = 1.0;
end;

% Solve the linear system

A = A1 - 400.0 * A2;

x = (A \ g)';

% Compute the exact solution and the absolute error

yexact = -(cos(pi*tp)).^2 + (exp(20*(tp-1)) + exp(-20*tp))/(1+exp(-20));
yp     = 0.0 * tp + x(n);

for k=n-1:-1:1
    yp = yp .* tp + x(k);
end;

yp = yp .* tp;
abserr = norm(yp-yexact,inf);