function [x,sol]=spec_colloc_solve(N)

% compute numerical solution to u''+u=0 with u(-1)=u(1)=1 using spectral
% collocation
% exact solution is u(x)=cos(x)/cos(1)

% set up grid points and weights
x=(cos((0:N)*pi/N))'; 
w=(-1).^(0:N); w(1)=1/2; w(end)=w(end)/2;

% set up differentiation matrix
D=zeros(N+1);
for j=0:N
    for k=[0:j-1,j+1:N]
        D(j+1,k+1)=w(k+1)/w(j+1)/(x(j+1)-x(k+1));
    end
end
for j=0:N
    D(j+1,j+1)=-sum(D(j+1,[1:j,j+2:N+1]));
end
% matrix representing operator
L=D*D+eye(N+1);

% apply boundary conditions u(-1)=1 and u(1)=1
L(1,:)=[1, zeros(1,N)];
L(N+1,:)=[zeros(1,N), 1];
rhs=zeros(N+1,1); rhs(1)=1; rhs(N+1)=1; 

% solve linear system
sol=L\rhs;

% compute norm of error
err=norm(sol-cos(x)/cos(1),Inf)
plot(x,sol)