Edit: I was advised to replace the question with a more specific one.Coming from a very theoretical background, I'm pretty ignorant about what practical matrix solvers exist. (I have been, and will continue to scour the web for information, but I figured I would get direct and concise answers here as well.)Currently, what are the most important matrix solvers? (implemented in any software)I have a rudimentary programming background, and I'm familiar with classical theoretical methods for solving matrix systems. However, I'm well aware computer scientists have been at work finding ingenious ways to solve larger systems more quickly, especially for specialized classes of matrices.I'm looking at applications of matrix solvers in industry, in particular for use in simulation programs.
Wanting to learn about matrix solvers
matrices;reference request;linear solver
The best high-level overview that I know of is Trefethen and Bau. If I had to boil it down to a list, it would be (somewhat in pedagogical order):Dense $QR$ factorizationDense symmetric/Hermitian Eigenvalue Decomposition (EVD)Dense Singular Value Decomposition (SVD)The Conjugate Gradient Method (CG)Generalize Minimum Residual method (GMRES)Sparse Cholesky and $LU$ factorizationFast methods, such as multigrid, the Fast Fourier Transform (FFT), and the Fast Multipole Method (FMM)