function [tp,yp] = colloc3(x0);
%
%   function [t,y] = colloc3(x0);
%
% Solves the modified Euler buckling problem
% from page 3.65 with the collocation method.
% The Newton iteration starts at the column
% vector x0.
%

global LAMBDA;

Np     = 1000;
errtol = 1.0e-12;

n  = length(x0);
h  = pi / (n+1);
t  = linspace(h,pi-h,n);
tp = linspace(0,pi,Np);

% Create the collocation matrix from y'', and the auxiliary matrix B

A = zeros(n,n);
B = zeros(n,n);

for l=1:n
    for k=1:n
        A(l,k) = -k*k*cos(k*t(l));
        B(l,k) =      cos(k*t(l));
    end;
end;

% Start the Newton iteration

it   = 0;
x    = x0;
delx = x;

while norm(delx)>errtol
    it   = it + 1;    
    F    = A*x + LAMBDA * sin(B*x);
    DF   = A + LAMBDA * (diag(cos(B*x)) * B);
    delx = DF \ (-F);
    x    = x + delx;
end;

fprintf('\n');
fprintf('Converged in %d iterations!\n\n',it);

yp     = 0.0 * tp;

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