function fish_test
% FISH TEST
%
% This is an example of how to use Tim's fast Poisson solver (P2D.m)
% to solve Poisson's equation in Matlab. You will have to use Matlab's
% reshape command.
%
% My codes require that h be a power of 1/2. In this example h=1/64.
%
% You will need these codes, which I have posted on the course web page.
%
% P2D.m (solver)
% lapmf.m  (matrix-free negative Laplacian)
% 
% Both of these codes expect you to reshape to a linear array and
% then you have to undo it when the solver returns.
%
% The first step is to set up the grid. I will use N=63 interior grid
% points in each direction. Note that the last command makes x a column
% vector.
%
n=63; h=1/(n+1); x=1:n; x=x'*h;
%
% Now I will invent a solution u(x,y) = sin(pi x)*sin(pi*y)*rand(n,n).
%
sinx=sin(pi*x); u=sinx*sinx'; ru=rand(n,n); u=u.*ru;
%
% So the right side is f(x,y) = -u_xx -u_yy
%
% It's now time to do the reshape. Turn u from a 2D grid function into
% a linear array u1.
%
u1=reshape(u,n*n,1);
%
% Now apply the 2D Laplacian in matrix-free form.
%
rhs=lapmf(u1);
%
% Feed it to the solver
%
sol=P2D(rhs);
%
% Use reshape again to map to the physical grid.
%
uout=reshape(sol,n,n);
%
% Check the results.
%
max(max(uout-u))