C++ armadillo not correctly solving poorly conditioned matrix

Question

I have a relatively simple question regarding the linear solver built into Armadillo. I am a relative newcomer to C++ but have experience coding in other languages. I am solving a fluid flow problem by successive linearization, using the armadillo function Solve(A,b) to get the solution at each iteration.

The issue that I am running into is that my matrix is very ill-conditioned. The determinant is on the order of 10^-20 and the condition number is 75000. I know these are terrible conditions but it's what I've got. Does anyone know if it is possible to specify the precision in my A matrix and in the solve function to something beyond double (long double perhaps)? I know that there are double matrix classes in Armadillo but I haven't found any documentation for higher levels of precision.

To approach this from another angle, I wrote some code in Mathematica and the LinearSolve worked very well and the program converged to the correct answer. My reasoning is that Mathematica variables have higher precision which can handle the higher levels of rounding error.

If anyone has any insight on this, please let me know. I know there are other ways to approach a poorly conditioned matrix (like preconditioning and pivoting), but my work is more in the physics than in the actual numerical solution so I'm trying to steer clear of that.

EDIT: I just limited the precision in the Mathematica version to 15 decimal places and the program still converges. This leads me to believe it is NOT a variable precision question but rather an issue with the method.

Answer 1

As you said "your work is more in the physics": rather than trying to increase the accuracy, I would use the Moore-Penrose Pseudo-Inverse, which in Armadillo can be obtained by the function pinv . You should then experience a bit with the parameter tolerance to set it to a reasonable level.

The geometrical interpretation is as follows: bad condition numbers are due to the fact that the row/column-vectors are linearly dependent. In physics, such linearly dependencies usually have an origin which at least needs to be interpreted. The pseudoinverse first projects the matrix onto a lower dimensional space in which the vectors are "less linearly dependent" by dropping all singular vectors with singular values smaller than the parameter tolerance . The reulting matrix has a better condition number such that the standard inverse can be constructed with less problems.