Successive Over-Relaxation ... what is wrong? Improvements?

[Joza]

Below is an implementation I have written of the Successive Over-Relaxation
Method (for pedagogical purposes). I must use a starting guess of zeros and
b in Ax=b is all ones. A is sparse. 

I've run Octave's CGS and it solved the problem in about 1 second ... my
code has been running for well over 5 minutes and nothing ...

I'm sure there is something I'm missing here. Any ideas? Feel free to run
the code yourself ... w is the relaxation parameter ( try 1.35) and tol is
the convergence tolerance. 

This is the second major algorithm I've had trouble with recently. Must be
something up with mu approach! Anyway, I'd really like to hear all
critiques!
************************************************************************************************************************************************

function [x, iters] = SOR(w, tol)

r = sparse([1 1], [1 2], [2 -1], 1, 1000);
A = toeplitz(r);
b = ones(1000,1);
x = zeros(1000,1);

iters = 0;
converged = 0;

while !converged
        
        norm_x = 0.0;
        norm_delta_x = 0.0;
        for i=1:1:1000;
                sum = 0.0;
                
                if i==1
                        for j=1:2
                                sum = sum + A(i,j)*x(j);
                        end
                
                elseif i==1000
                        for j=999:1000
                                sum = sum + A(i,j)*x(j);
                        end
                
                else
                        for j=i-1:i+1
                                sum = sum + A(i,j)*x(j);
                        end
                end

                delta_x = w*(b(i) - sum)/A(i,i);
                x(i) = x(i) + delta_x
                iters = iters + 1;

                norm_x = max( norm_x, abs(x(i)) );
                norm_delta_x = max(norm_delta_x, abs(delta_x) )
        end
        
        if norm_delta_x <= tol + 4.0*(10^-16)*norm_x
                converged = 1;
        end
end




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