visualize the norm of the difference of the solution in every iteration with the final solution to see the convergence:
clear all;
% very large matrix
dim = 2000;
vec = -1*ones(dim-1,1);
K = 4*eye(dim) + diag(vec,1)+diag(vec,-1);
K(1,1) = 100; % change Matrix to get an illconditioned matrix
% compute Condition matrix
cond(K)
%define righthandside
f = ones(dim,1);
%set maximal iteration and Toleranz
maxiter = 1000;
tol = 10e-6;
%set initial vector
u_0 = zeros(dim,1);
% run your implemented cg method
[u_cg, iter] = cg(K,f,maxiter,tol,u_0);
% compute norm of the difference of the solution in every iteration with the final solution
norm_diff_cg = zeros(iter, 1);
for i = 1:iter
norm_diff_cg(i) = norm(u_cg(1:i) - u_cg);
end
% run your implemented pcg
[u_pcg, iter_pcg] = p_cg(K,f,maxiter,tol,u_0);
% compute norm of the difference of the solution in every iteration with the final solution
norm_diff_pcg = zeros(iter_pcg, 1);
for i = 1:iter_pcg
norm_diff_pcg(i) = norm(u_pcg(1:i) - u_pcg);
end
% visualize the norm of the difference
figure;
semilogy(1:iter, norm_diff_cg);
xlabel('Iteration');
ylabel('Norm of the Difference');
title('Norm of the Difference for CG');
figure;
semilogy(1:iter_pcg, norm_diff_pcg);
xlabel('Iteration');
ylabel('Norm of the Difference');
title('Norm of the Difference for PCG');
This code will generate two plots, one for the norm of the difference of the solution in every iteration with the final solution for the CG method, and the other for the PCG method. You can compare the two plots to see how the convergence rate differs between the two methods.
