% check WKB eigenvalue accuracy (against numerical shooting method via ODE45)
% barnett 4/18/08

k = @(x) 2-x.^2;
q = @(x) k(x).^2;
b = 1;                        % end of interval
n=1:30;                       % which eigenvalues to check
l_wkb = (3*pi*n/5).^2;        % WKB formula from worksheet
l = []; figure;
for j=1:numel(n)
  l(j) = fzero(@(l) shooting(l, q, b), l_wkb(j));
  fprintf('l_WKB = %g\t l_EXACT = %g\t relative error = %g\n', ...
          l_wkb(j), l(j), (l_wkb(j)-l(j))/l(j));
  [yend t y] =  shooting(l(j), q, b);                % get the eigenfunction
  subplot(5,6,j); plot(t, y(:,1), '-'); hline(0); drawnow;
end
figure; loglog(1./sqrt(l), (l_wkb-l)./l, '+'); xlabel('eps'); ylabel('rel err');
title('rel error vs eps = 1/l^{1/2}: appears to be O(eps^2)');