Linear least squares overdetermined system

Hello all,

I was hoping for some help solving the following;

k1*x = s1

k2*y = s2

x+y = s3

Whereby k1,k2 are scalar multipliers and x, y ,s_i are 448x448x10 matrices.

AKA I'm solving 448x448x10 sets of 3 equations.

I'm not sure the syntax or correct method to enter into matlab.

?linlsq vs solve vs other?

Thank you!

[Bjorn Gustavsson]

Did I understand your question right?

Answer 1

[Bjorn Gustavsson]

If I understand your problem correct you want to solve a reasonably large number of very small linear systems?

In that case you have a problem of the type:

A = [k1,0;0,k2;1 1];
A*[x;y] = [s1;s2;s3];

where you want to solve for x and y. Since you have a large number of so small equations to solve, why not calculate the least-square estimator explicitly?

A_dagger = inv(A'*A)*A';

The general advice is not to do this, but you have one 3x2 matrix to "invert" and on the order of 2e6 equations to solve.

With an explicit inverse, A_dagger, you can write the all the solutions for x and y explicitly. If k1 and k2 are numerically known you're set, if you have access to the symbolic toolbox it too lets you solve for A_dagger. This ought to be the most efficient way to solve the problem the way I've understood it.

HTH

[Bjorn Gustavsson]

You can one-line this if you can manage to reshape this properly:

XY = A\[s1(:),s2(:),s3(:)]';

This will give you XY as a [2 x (448*448*10)] array, from which you'd get all x from the first row and y from the second. You'd have to reshape them to your prefered 448x448x10 after that. I'd still go with the above solution.