I am developing an application in C# language — an electric simulator that uses the node-voltage method in the AC frequency domain.
I need to solve large systems of linear equations (over $\mathbb C$), one for every frequency step. So, for example, when the program calculates a voltage-frequency curve for an electric node, in a $14$ node electric circuit, with $400$ frequency points, it has to solve a $14$-variable linear equation matrix, $400$ times.
Currently I have the solver implemented with dense matrix format, LU factorization. then Gaussian elimination. I took these algorithms from the book Numeric Methods from Ward Cheney & David Kincaid.
The example described above, to solve $400$ times, a $14$-variable matrix, takes in my machine, approximately $500$ milliseconds, measured with Visual Studio 2019. Very slow. I need something quicker.
But in practice, the matrices are very sparse, or in other words, low density, so the ideal thing would be to use sparse matrices!
What algorithms I have available to solve quickly linear equations with sparse matrixes?, (remember to be algorithmically solved with programming language C#).
Also I DO NOT need to have ALL variables calculated, only a few of interest, this could speed up the calculation process a little further, I guess..
What algorithms I have available to solve quickly linear equations with sparse matrixes?
Thank you very much.