How to solve linear equation for 3 unkowns using SOR in GPU

I am trying to find the motion (u,v,w) for each pixel in my frame by solving the linear system Ax=b

A is a 3 by 3 ... each element for example A (1,1) is equivalent to the frame size ... let's say ( 512 X 512)

A = [ A B C ; D E F ; G H I]

x is unknown need to be estimated ( u,v,w) ... u , v and w should be ( 512 X 512)

b is a 3 by 1 matrix and each element is equivalent to the frame size as well

b= [ J K L]

I am trying to solve this system using the SOR iteration method so I have this provided function

problems:

-In this code I have to use omega = 1 otherwise I got the wrong answer totaly !

-How can I solve this system using the SOR iteration method?

-How can I improve it to use GPU to speed up?

Any assistance will be so appreciated

function [u,v,w,error]=SORstep(A,B,C,D,E,F,G,H,I,J,K,L,omega,du_temp,dv_temp,dw_temp) % du_temp, dv_temp, dw_temp previouse estimation 
u=((1.0-omega).*du_temp)+(omega./A).*(J-(B.*dv_temp+C.*dw_temp));
v=((1.0-omega).*dv_temp)+(omega./E).*(K-(D.*u+F.*dw_temp));
w=((1.0-omega).*dw_temp)+(omega./I).*(L-(G.*u+H.*v));
error = norm([u;v;w]-[du_temp;dv_temp;dw_temp])./norm([u;v;w]);
end 

Answer 1

[Matt J]

Not sure why you would be using an iterative method when an analytical solution is available.

A=gpuArray.rand(3,3,512^2);
b=gpuArray.rand(3,1,512^2);

x=pagefun(@mldivide, A,b);

u=reshape(x(1,:),512,512);
v=reshape(x(2,:),512,512);
w=reshape(x(3,:),512,512);

[Seereen]

I need iteration because it is very big system! ... I have to solve 262144 equations .. right?

so I think iterative method help in this case

[Matt J]

The language of your post is confusing. You should not say your A matrix is 3x3 when it is in fact 262144 x 262144...

Answer 2

[Matt J]

If you have the Parallel Computing Toolbox, you can make all the variables in your SORstep function

A,B,C,D,E,F,G,H,I,J,K,L,omega,du_temp,dv_temp,dw_temp

into gpuArrays. Then, all the matrix arithmetic done in your SORstep function will be done on the GPU.