Some open-source libraries for linear system computations

Free libraries are readily available for the majority of linear system problems encountered in scientific computing, A Google search will show scores of them, unfortunately without an external objective evaluation of the libraries. A library should provide:
  1. Interface(s) to C/C++ and Fortran
  2. Have at least a simple and an expert interface, where the basic one chooses most of the parameters and settings for you.
  3. A test suite to verify an installation on your machine was correct
  4. Numerically reliability, and when that is not possible, a warning and estimate of the potential size of computational error
  5. Thorough documentation
Following are a few that I personally recommend, and they usually handle 80% of the cases I've encountered. The list is not comprehensive, and many unlisted ones also have excellent quality satisfying the above metrics.

These are primarily intended for cases where the system is too large for Matlab to handle, or where Matlab cannot be used. In general, for small to midsize applications Matlab is the best and easiest to use. Increasingly Matlab is able to handle large scale systems, so it can even solve problems with millions of unknowns (circa 2012).

References

The most readily available text is Matrix Computations, by Golub and van Loan. Converting one of the algorithm statements in the book to a Matlab function can almost be done without changing the text. It is definitely targeted towards practical implementations. For sparse linear system solvers, Alan George and Joseph Liu's Computer Solution of Large Sparse Positive Definite Systems, (Prentice Hall, 1981) lays out the foundations and terminology still in use. Unfortunately it is now expensive ($148 for a used copy on Amazon, $360 for a new one!). Tim Davis's Direct Methods for Sparse Linear Systems is more current and provides a really good and brief overview of the title topic. Duff, Reid and Erisman have the early and definitive book Direct Methods for Sparse Matrices

SIAM ...

... is not just a colonial name for a country in Asia. Indiana University is an institutional member of SIAM, which means students can join it for free. Long before other computer science societies recognized scientific computing as a field, SIAM was advocating it. Their main revenue is from publishing books and journals, but the books are usually much cheaper than what textbook publishers demand. Maybe not as cheap as the elapsed copyright books from Dover Publications, but closer to it than Prentice-Hall in prices. Also, many of the authors make no money on the books and instead have their royalties fed into a pool for sponsoring students to attend SIAM conferences. Take advantage of IU's institutional membership, and not just for this one book.

For a good mathematical reference on the methods used in modern eigenvalue solvers, see The Matrix Eigenvalue Problem: GR and Krylov Subspace Methods by Watkins, another SIAM publication.

Libraries

The most widely used library for the dense systems is LAPACK. If you need to find LU solvers or eigenvalues of dense matrices that exceed the memory capability of a single machine, use ScaLAPACK. As the name suggests it scales up, to to thousands of processors if necessary. Those provide full decompositions (e.g., all of the eigenvalues and eigenvectors) of dense matrices and some quasi-sparse ones like triangular or banded systems. Both of those rely upon good implementations of the BLAS, one of the reasons for emphasizing them in P573.

The site that hosts LAPACK is netlib.org also hosts a large collection of publically available libraries, books, papers, and databases in scientific computing. Some are antique but even those are reliable and classic, in the English lit sense that they "have withstood the test of time".

Warning: a compatibility library (AKA a "reference implementation") of the BLAS is also on netlib.org, but it is not a high-performance one. It was designed as a temporary measure when the BLAS were first proposed, with the idea that it would be supplanted by people implementing the AFPI in more efficient versions. Instead use a BLAS library supplied by your compiler vendor (Sun, Intel, PGI) and if one is not available try ATLAS, FLAME, or similar projects.

Tim Davis's book is also associated with a library UMFPACK, which implements a "multifrontal" method for LU factorization. The other major method for sparse LU uses "supernodes", and SuperLU is an excellent and well-maintained library for it.

Other Junk Stuff

All of the above references primarily just provide keywords for doing a Google search. Try doing a search with just the word "flame" and see how many pages it takes before the link for the scientific library shows up. Don't forget the second fundamental rule of scientific computing: never write code if instead it can be scavenged on the Web (but give a citation when you do that). Use the Google, that's what it is for.