%
%             alias.m
%
%
%% Code used to generate the figures and ranks for Case ii: n = 101, p = 1000 for Example 10.7.
%  This can be easily modified for Case i: p = n = 5.
%

  clear all
%  close all

  n = 101;
  p = 1000;
%  n = 5;
%  p = 5;
  h = 1/(n-1);
  t = 0:h:1;
  tol = 1e-10;


  [I,T] = meshgrid(1:p,t);
  A = sin(2*pi*I.*T);
  rank(A,tol)

%
% RandSVD on A (using an oversampling of 10)
%

  k = 65; 
  tic
  [U1,S1,V1] = randsvd(A,k,10);
  toc

%
% RandSVD on A^T (note that the left and right singular vectors change places
%

  tic
  [V2,S2,U2] = randsvd(A',k,10);
  toc

%
% Compute the SVD of A for comparison to the randomized results.
%

  S = svd(A);

%
% Plot the aliasing results, singular values and comparison of the singular values.
%

  figure(1)
%  plot(t,A(:,1),'b--',t,A(:,2),'-k',t,A(:,3),'b-.',t,0*t,'k-','linewidth',3)
  plot(t,A(:,1),'-k',t,A(:,101),'--r',t,A(:,201),'-.b',t,A(:,901),':k','linewidth',3)
  hold on
  plot(t,0*t,'k-','linewidth',3)
  hold off
  set(gca,'Fontsize',[22]);
  xlabel('Time (s)')
  ylabel('Column Entries of A')
%  legend('A(:,1)','A(:,2)','A(:,3)','Location','South')
  legend('A(:,1)','A(:,101)','A(:,201)','A(:,901)','Location','NorthEast')

  figure(2)
  plot(t,A(:,1),'--b',t,A(:,51),'-r','linewidth',3)
  hold on
  plot(t,0*t,'k-','linewidth',3)
  hold off
  set(gca,'Fontsize',[22]);
  xlabel('Time (s)')
  ylabel('Column Entries of A')
  legend('A(:,1)','A(:,51)','Location','South')


%  figure(4)
%  plot(Q(:,1),'x','linewidth',3)
%  hold on
%  plot(V(:,1),'o','linewidth',3)
%  hold off
%  set(gca,'Fontsize',[22]);
%  legend('Q(:,1)','Vs(:,1)','Location','South')
%  ylabel('First Basis Function for I(q)')

  figure(3)
  semilogy(1:k,S(1:k), 'xr', 1:k,diag(S1), 'ob', 'linewidth',3)
  axis([0 k 1e-13 1e+2])
  set(gca,'Fontsize',[22]);
  legend('Deterministic Algorithm','Randomized Algorithm','Location','East')
  xlabel('Index')
  ylabel('Singular Values')


  Sdiff = abs(S(1:k) - diag(S1));
  figure(4)
  semilogy(Sdiff,'bx','linewidth',4)
  axis([0 k 1e-14 1e-12])
  set(gca,'Fontsize',[22]);
  xlabel('Index')
  ylabel('Absolute Difference in Singular Values')


%  figure,
%  semilogy(1:k,S(1:k), 'k-', 1:k, diag(S1), 'r:', 1:k, diag(S2), 'b--', 'LineWidth',3);
%  xlabel('Index')
%  ylabel('Singular Values')
%  legend('True', 'RandSVD A', 'RandSVD A^T');
%  set(gca, 'FontSize', 18);