Solving a linear system Ku=f using conjugate gradient method I implmented CG and PCG . Need help with visualisation of results

In my code I implemented conjugate and preconjugate gradient method and visualised the result. I want to visualise the norm of the difference of the solution in every iteration with the final solution to see the convergence. Any help is appreciated. Below is my code:

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);%%% change code here !!!!!!!!!!!!!

u=u_0;

r=f-u;

d=r;

for k = 0:maxiter

alpha = (r.'*r) / (d.'*K*d); % Step size

u = u + alpha*d; % Update solution

r_new = r - alpha*K*d; % Update residual

% Correct search direction

beta = (r_new.'*r_new) / (r.'*r);

d = r_new + beta*d;

r = r_new; % Update residual

% Check for convergence

if norm(r) < Tol

break;

end

end

u_cg = Solution_using_Cg;

k = number_of_iterations;

%run your implemented pcg

[u_pcg, iter_pcg] = p_cg(K,f,maxiter,tol,u_0); %%% change code here !!!!!!!!!!!!!

u = u_0; % Initial guess

r = f - K*u; % Initial residual

C = diag(1./diag(K)); % Preconditioner

h = C*r; % Preconditioned residual

d = h; % Initial search direction

for k = 0:maxiter

alpha = (r.'*h) / (d.'*K*d); % Step size

u = u + alpha*d; % Update solution

r_new = r - alpha*K*d; % Update residual

h_new = C*r_new; % Preconditioned new residual

beta = (r_new.'*h_new) / (r.'*h);

d = h_new + beta*d; % Update search direction

r = r_new; % Update residual

h = h_new; % Update preconditioned residual

% Check for convergence

if norm(r) < Tol

break;

end

end

u_pcg;

iter_pcg;

%compare the results

norm(u_cg-u_pcg)

%use matlab pcg function and compare results

upcg = pcg(K,f);

norm(u_pcg-upcg)

%use matlab pcg function and compare results

u_lin = linsolve(K,f);

norm(u_pcg-u_lin)

Answer 1

[recent works]

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.

[Lokesh]

It is showing 2 empty plots