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The most famous free codes that I know is "GSL" - The GNU Scientific Library. The unix version is free but the windows version starts charging (if you get on emule, you may still get free windows codes). I have used both of them and they are really great. GSL can solve a lot of mathematics problems and it certainly has a matrix solver. All matrix functions start with gsl_matrix.
When you start using it, you better find some examples first as it took me some time to get used to it.
Most larger scale sparse matrix code is exploded in application direction because it was find out that if the code is optimized to the known structure of the problem it runs much faster. Take for example spice which generate only 4-6 entries per row but could have very dense rows (global nodes). If you apply special strategies to the structure the code runs much faster. Markowitz ordering is the best practice for the last 20 years used in spice. Ken Kundert wrote the sparse kernel Sparse 1.4 for spectre. It works good today up to 100k. The run time is about n^(1.8-2). It is estimated that newer algorithm outperform this because they operate at n^(1.2-1.7). That gives big run time advantages in the range 500k-10M for spice applications. They use a combination of direct elemination, ordering and iterative elemination.
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