Solve overdetermined system of matrices

I have an overdetermined system of the form Ax = B, where A, x, and B are all matrices. My system looks like this:

A1 * x = B1
A2 * x = B2
    ...
AN * x = BN

I am trying to find the x which best fits my system. I initially attempted this using lsqonneg function, but this did not work because x, A, and B are matrices. I then manually wrote a least squares function:

    function [difference] = least_squares(x)
        difference = zeros(m,n);

        for i = 1 : number of equations
           difference = difference + (A{i}*x - B{i}).^2;
        end
    end

    eqn = @least_squares;
    x = fmincon(eqn,x0);

This solves the problem of matrix inputs. However, the fmincon and fminsearch functions still expect difference to be a scalar. Any suggestions for solving this problem are greatly appreciated!

Answer 1

[Star Strider]

I don’t understand that ‘A, x, and B are all matrices.’

I can’t run your code, but if you’re doing a least-squares cost function, see if adding the sum function does what you want. This assumes ‘A’ is a matrix and ‘x’ and ‘B’ are vectors:

difference = difference + sum((A{i}*x - B{i}).^2);

It should give you a scalar value, and if all goes well, is the cost function that does what you want.

If they’re actually all matrices and my assumption from your code is in error, so that the result of the calculation in your ‘difference’ assignment is a matrix rather than a vector, consider using the norm function.

[Kelly]

Hi Star,

Yes, A, B, and x are all matrices. A is 3x4, x is 4x4, and B is 3x4. How would you suggest using norm?

Regards, Kelly

[Torsten]

A=vertcat(A1,A2,A3,A4);
B=vertcat(B1,B2,B3,B4);
X=A\B;

Best wishes

Torsten.

[Matt J]

I second Torsten's suggestion. Since there are no constraints in play, A\B should work just fine here. If there are linear constraints (e.g., positivity) there are ways to solve the problem with LSQLIN.

[Kelly]

I implemented what Torsten suggested. It sounds like I was making an easy problem much more difficult than it needed to be. Thank you Torsten and Matt the suggestion and explanations!