Solving under-determined matrix equations

I need to Solve : A1*x+B1*y=c; A2*x+B2*y=d in Matlab where A1, A2, B1 and B2 are m-by-n complex matrices with m<n and rank=m. x, y, c and d are n-by-1 vectors. Hence the system is under-determined. I have seen solution techniques for solving system of equations in the form Ax=b, but how can I apply that to my case? Please let me know..

Thanks in advance S.Paul

[Youssef Khmou]

hi, are you sure that x,y,c and are nx1? just to make sure

[Subhra]

You are right. c and d are m x 1. only x and y are n x 1.

Answer 1

[Youssef Khmou]

Hi Subhra,

try to check the lengths you assigned to vectors c,d ; Under-determined system means you have more variables and less equations , in your example c and d must be mx1 vectors to get under-determined system : In this case the matrix A is not square and other techniques must be used : like

1)The Moore-Penrose pseudoinverse matrix or

2)The Least squares algorithm , TLS,..etc

Try this example :

 m=4;
 n=6; % m<n

 A1=randn(m,n)+j*randn(m,n);
 B1=randn(m,n)+j*randn(m,n);
 A2=randn(m,n)+j*randn(m,n);
 B2=randn(m,n)+j*randn(m,n);
 rank(A1) % rank(A1)=rank(A2)=rank(B1)=rank(B2)

 c=rand(m,1);
 d=rand(m,1);

 % Matrices concatenation  to get  AX=B

 A=[A1 B1;A2 B2]; % size{A} is 2mx2n and rank{A} is 2m .

 B=[c;d]; % size{B}is 2mx1

 % Solution
 %  Moore-Penrose pseudoinverse of matrix since A is not square :
 X=pinv(A)*B;
 error=X*A-B;

 % LS
 X2=pinv(A'*A)*A'*B;

 % Now get x and y :
 %x=X(1:4);  y=X(5:end);

[Subhra]

Hi Youssef,

Thanks a lot for your quick reply. That was really a simple idea to convert to AX=B form! Should have got that. One point though. I think, as A is 2m x 2n and rank is 2m, its right pseudoinverse exists, not the left. i.e. A*pinv(A)=I, but pinv(A)*A is not identity matrix. SO, the solution X can be modified as X=A'*pinv(A*A'). However, I have tried this and also your solutions for X. But neither of these is working. Although they are giving very small residual A*X-B, but the final field solution is erratic. My m =31, n=63.

[Youssef Khmou]

hi,

ok, did you try the function "linsolve" ?

   X=linsolve(A,B); %

[Subhra]

Nope. Ok I will see.

[Subhra]

No! Its not working either. My problem is ill-conditioned.