IterativeSolvers.jl
IterativeSolvers.jl is a Julia package that provides efficient iterative algorithms for solving large linear systems, eigenproblems, and singular value problems. Most of the methods can be used matrix-free.
For bug reports, feature requests and questions please submit an issue. If you're interested in contributing, please see the Contributing guide.
For more information on future methods have a look at the package roadmap on deterministic methods, for randomized algorithms check here.
What method should I use for linear systems?
When solving linear systems $Ax = b$ for a square matrix $A$ there are quite some options. The typical choices are listed below:
| Method | When to use it |
|---|---|
| Conjugate Gradients | Best choice for symmetric, positive-definite matrices |
| MINRES | For symmetric, indefinite matrices |
| GMRES | For nonsymmetric matrices when a good preconditioner is available |
| IDR(s) | For nonsymmetric, strongly indefinite problems without a good preconditioner |
| BiCGStab(l) | Otherwise for nonsymmetric problems |
We also offer Chebyshev iteration as an alternative to Conjugate Gradients when bounds on the spectrum are known.
Stationary methods like Jacobi, Gauss-Seidel, SOR and SSOR can be used as smoothers to reduce high-frequency components in the error in just a few iterations.
When solving least-squares problems we currently offer just LSMR and LSQR.
Eigenproblems and SVD
For the Singular Value Decomposition we offer SVDL, which is the Golub-Kahan-Lanczos procedure.
For eigenvalue problems we have at this point just the Power Method and some convenience wrappers to do shift-and-invert.
Randomized algorithms
Randomized algorithms have gotten some traction lately. Some of the methods mentioned in [Halko2011] have been implemented as well, although their quality is generally poor. Also note that many classical methods such as the subspace iteration, BiCG and recent methods like IDR(s) are also "randomized" in some sense.
Halko, Nathan, Per-Gunnar Martinsson, and Joel A. Tropp. "Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions." SIAM review 53.2 (2011): 217-288.