How to accelerate matrix calculation with matlab?

I have 40800 linear sparse equations that have the following forms:

   F(1)=2*x(1)+x(2)-10;
   F(2)=x(9)+3*x(10)-7;......

   F(40800)=x(40200)-x(1)+....-10;

I want to convert these equations to matrices by evaluating them at the columns of speye(N);

   N=40200; %the number of variables
   E=speye(N,N+1);
   columns=cell(1,N);
   beq=-evaluate_equations(E(:,N+1));
   for i=1:N
     columns{i}=evaluate_equations(E(:,i))+ beq;
   end
   Aeq=cell2mat(columns);
   full(Aeq),
   full(beq),
function F=evaluate_equations(x)
   F(1)=2*x(1)+x(2)-10;
   F(2)=x(9)+3*x(10)-7;......

   F(40800)=x(40200)-x(1)+....-10;
   F=F(:); %column vector

But Matlab is very slow. It can not build the matrices. However i use 8 workers to accelerate the computing. Have you an idea to overcome this problem? Thanks.

Answer 1

[Walter Roberson]

If they are sparse linear equations, could you not code them as a matrix multiplication in the first place? If you did that you could also skip the "for i" loop and do it as a single matrix multiplication.

[imed NASRI]

I don't understand what do you mean by single matrix multiplication? Can you explain more please?

[Walter Roberson]

A = [2 1 0 0 0 0 0 0 0 0 -10;     %predetermined
     0 0 0 0 0 0 0 0 1 3  -7]

x = [E(1:size(A,2)-1, :); ones(1, size(E,2))];  %column-oriented!

A * x

This is for general E. Considering that your particular E is spyeye(), the answer probably just comes out as A.

Thus, you would code them in matrix form instead of in equation form.

Are you generating the equations somehow, perhaps using the symbolic toolbox? If so then the symbolic toolbox can be used to convert into matrix form.

[imed NASRI]

I generate my equations using a matlab code. Equations are generated in a text file using fprintf. Then i copy them into an other m file to generate the desired matrices Aeq and beq into an other text file(the code that I have posted above)

[imed NASRI]

Aeq and beq are unknown. I have to generate them from equations:

    F(1)=2*x(1)+x(2)-10;
   F(2)=x(9)+3*x(10)-7;......
   F(40800)=x(40200)-x(1)+....-10;

such that we have the following matrix form:

    Aeq*x=beq

So, I would like to get my generated matrice Aeq and beq and save them to a text file because i will call them after in the ga() solver