Solving large overdetermined system of equations

I am trying to solve an overdetermenind system of equations (45 equations with 18 unknowns) in the form of A*X = B; where: A = [8.19 0.3; 12.39 0.86; 16.15 1.68; 17.7 2.16; 24.6 8.33]; A is 5 rows, 2 columns

B = [0.85 3.8 0.225 0.175 0.05 0.05 0 1.919 0.165; 1.5 6.825 0.45 0.4 0.175 0.125 0.025 2.907 0.315; 3 7.625 0.9 1.05 0.475 0.225 0.1 3.823 0.544; 3.05 7.9 1.1 1.8 1.225 0.25 0.45 4.153 0.881; 5.1 12.95 1.975 3.65 3.075 0.25 1.075 4.045 1.217]; B is 5 rows, 9 columns

X = [a1 b1 c1 d1 e1 f1 g1 h1 i1;a2 b2 c2 d2 e2 f2 g2 h2 i2]; X is 2 rows, 9 columns

I need to solve X with the following constraints: sum (a1 to i1) = 1 & sum (a2 to i2) = 1 & all elements in X shall be higher than or equal to zero (non negative).

Answer 1

[John D'Errico]

I almost always need to laugh out loud, when someone says they have a "large" system. That is not even close to large, by factors of thousands at least in the number of variables. If you said a million unknowns, yep, that is large. :)

But you seem confused even at that. First, you say 45x18. Then you say 5x2. Neither is large.

Actually, what you seem to have is a 5x2 system of equations, with 9 distinct right hand sides. So you have 9separate problems, each of which shares the same matrix A, so the same left hand side. You don't have 45 equations. Just 5 equations, solved 9 times, all essentially in one call.

And, wanting to do this?

X = [a1 b1 c1 d1 e1 f1 g1 h1 i1;a2 b2 c2 d2 e2 f2 g2 h2 i2]; X is 2 rows, 9 columns

A godawful bad idea.

Anyway, the way to solve ANY linear system of equations of the form A*X=B, large or small, is

X = A\B;

Note that this solves your problem, all 9 right hand sides at once. The result will be a 2x9 array.

[Mohamed Abdelhameed]

Thank you John for your valuable reply. I already tried to solve my problem using X = A\B. But the resulting 2x9 array contains some negative elements, additionally the summation of each row is not equal to 1. My question is: is there any way to add these two constraints (non negative elements & each row summation = 1) in the solution? Many thanks.

[Torsten]

Use "lsqlin".

Best wishes

Torsten.

[John D'Errico]

If you have the optimization toolbox, then use LSQLIN. Note that LSQLIN cannot solve all problems at once. So you will need to loop over the columns of B, creating one column of the result at each time through. But this is easy.

LSQLIN allows nonnegativity constraints, so you will provide LOWER bounds of [0 0] for each element.

LB = [0 0];
UB = [];

since you have no upper bounds. And of course, since there are no linear inequality constraints:

A = [];
b = [];

As well, LSQLIN allows you to specify the sum of the elements, as a linear equality constraint. So you would specify

Aeq = [1 1];
beq = 1;

to be passed in.

If you don't have the optimization toolbox (why would anyone not have that toolbox?) then this problem becomes more difficult, but not impossibly so.