Ax = B wrong solution when A contains a row of zeros

[eladr07]

Hi,

I am using MathNet.Numerics v4.7.0.

I try to solve Ax = B (for x) when A is a 4 by 6 matrix with the following value array (note that the first row is all zeros - indices 0,7,13,19):

[0]: 0
[1]: 1.06261732
[2]: 0
[3]: 0.00038268
[4]: 0
[5]: 1.063
[6]: 0
[7]: 0
[8]: 3.335926901
[9]: 0.020869227
[10]: 0.104069263
[11]: 3.4609
[12]: 0
[13]: 0
[14]: 0
[15]: 1.0538
[16]: 0
[17]: 1.0538
[18]: 0
[19]: 0
[20]: 0
[21]: 0.000663424
[22]: 0.933717888
[23]: 0.9344

And B is a 6-vector with the following values (note that first element is non-zero):

[0]: -147339.22693314467
[1]: 90907.703152322269
[2]: -71858.548209015382
[3]: 67094.2200240139
[4]: 61199.2777493189
[5]: 0

And A.Solve(B) returns a solution which is obviously incorrect.

What am I missing?

Thanks!

[cdrnet]

Solve is supposed to return the solution which minimizes the squared errors i.e. |Ax - b|. The returned vector looks ok to me. Is there another solution with a lower least-squares error?

Note that the error is not necessarily zero. In your particular problem there exists no solution with zero error, since there is no solution for the equation 0 * t = -147339.22693314467 corresponding to the first row of your 6x4 matrix and the first element of your 6-vector.

[eladr07]

I understand now, thanks for your explanation.

Is there any way to determine how "good" is the solution? Currently I'm looking at determining it from a business POV and then looking at

var error = A
                .Multiply(x)
                .Subtract(B)
                .Norm(2.0);

[cdrnet]

For a start, you could check whether the rank of the augmented matrix A|b is greater than the rank of A. If so, the system is inconsistent and there exists no solution - like in your example, where the former is 5 and the latter 4.

if(A.Append(b.ToColumnMatrix()).Rank() > A.Rank())
{
    // inconsistent, i.e. there is no solution
}