System of linear equations

u1, u2, v1, v2, w1, w2 are each 5 by 5 matrix. Following pair of linear equations hold at each grid, for example

At (1,1 ): c1*u1(1,1) + c2*v1(1,1) = w1(1,1)
and        c1*u2(1,1) + c2*v2(1,1) = w2(1,1) 
At (1,2 ): c1*u1(1,2) + c2*v1(1,2) = w1(1,2)
and        c1*u2(1,2) + c2*v2(1,2) = w2(1,2)

...

At (5,5): c1*u1(5,5) + c2*v1(5,5) = w1(5,5)
and       c1*u2(5,5) + c2*v2(5,5) = w2(5,5).

How can I solve for c1 and c2 (each in 5 by 5 matrix form)?

[Walter Roberson]

Could you confirm that you need this solved in Simulink? If so then would using a MATLAB Function Block be acceptable, or do you need to make it out of more basic blocks ?

[Sk Zeeshan Ali]

Hi, I am just looking for the solution in matlab. Giving the tag Simulink was unintentional.

Answer 1

[Walter Roberson]

c1*u1(1,1) + c2*v1(1,1) = w1(1,1)
How can I solve for c1 and c2 (each in 5 by 5 matrix form)? 

If c1 and c2 are 5 x 5, then the implication is that at each location (J,K) in the grid, c1*u1(J,K) + c2*v1(J,K) = w1(J,K) but c1 and c2 are 5 x 5, so the left side would be 5 x 5 and the right side would be a consistent scalar. That would imply that

c1 = (w1(J,K) - c2*v1(J,K))./u1(J,K)
c1 = w1(J,K)./u1(J,K) - c2*(v1(J,K)./u1(J,K))

and that has to hold for all J,K, with c1 and c2 being the same 5 x 5 matrices for each J,K location.

And at the same time,

c1 = w2(J,K)./u2(J,K) - c2*(v2(J,K)./u2(J,K))

This leads to

c1 = (v2.*w1 - v1.*w2) ./ (u1.*v2 - u2.*v1)
c2 = (u1.*w2 - u2.*w1) ./ (u1.*v2 - u2.*v1)

But are you sure that is the equation system you want??

[Sk Zeeshan Ali]

The procedure is indeed correct if we apply Cramer's rule. However, i am looking for some quick matlab operator to solve this problem. Because instead of c1 and c2, if we have more number of similar constants, then it would be difficult to apply Cramer's rule. Can we solve this problem using \ operator or similar thing? Any suggestions, much appreciated!

[Walter Roberson]

c12 = [u1(:), v1(:); u2(:), v2(:)] \ [w1(:); w2(:)];
c1 = reshape(c12(1:25), 5, 5);
c2 = reshape(c12(26:end), 5, 5);