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.
What method should I use for linear systems?
Square 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.
Non-square systems: least squares
When solving least-squares problems we currently offer LSMR and LSQR. LSMR generally converges more quickly than 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.