function nonlinearBvp()

%   u''(t) = -sin(t)     t in (0,2*pi)
%
%         u(0) = alpha
%      u(2*pi) = beta
% 

   maxiter = 200;
         N = 200;
         t = 1:1/80:2*pi; t=t';  N = length(t)
%         t = linspace(0,2*pi,N)';
%         u = 0.7*cos(t) + 0.5*sin(t);      %  initial guess
         u = 0.7 + sin(t/2);     
         h = t(2) - t(1);
         
         f = zeros(N,1);
         alpha = 0.7;
          beta = 0.7;
         
         tol = 10e-12;
         iter = 0;
         du = 1;
          I = 2:N-1;

   while ( norm(du,inf)> tol )
   
                f(1) = (alpha - 2*u(1) + u(2))/(h^2) + sin(u(1));
                f(I) = (u(I-1) - 2*u(I) + u(I+1))/(h^2) + sin(u(I));
                f(N) = (u(N-1) - 2*u(N) + beta)/(h^2) + sin(u(N));
                
                 up = diag(ones(1,N),1);
                low = diag(ones(1,N),-1);
                  J = ( diag(( -2 + (h^2)*cos(u) )) + up(2:end,2:end) + low(2:end,2:end) )/(h^2);
                  
            du = -J\f;
             u = u + du;
             
          iter = iter+1;
          if iter==maxiter, disp('exceeded maximum iterations'), break;  end
          
          plot(t,u)
          pause
       
   end
   
   iter