petsc-3.10.5 2019-03-28
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# Summary of Sparse Linear Solvers Available from PETSc

### Requests and contributions welcome

|  | Algorithm | Matrix types | External Package* | Parallel | Complex
--- | --- | --- | --- | --- | --- | ---
* External Package If this column is filled in you must install the external package indicated to use this feature. See instructions ** A "block Jacobi" style SOR with lits local smoothing followed by update of ghost points from other processors, see MatSOR() *** Use PCType of PCSOR
Preconditioners |  | Jacobi | aij, baij, sbaij, dense | --- | X | X
|  | point block Jacobi | baij, bs = 2,3,4,5 |  | X | X
|  | SOR | seqdense, seqaij, seqsbaij, mpiaij ** | --- |  | X
|  | point block SOR*** | seqbaij, bs = 2,3,4,5 |  |  | X
|  | block Jacobi | aij,baij,sbaij | --- | X | X
|  | additive Schwarz | aij,baij,sbaij | --- | X | X
| Incomplete factorizations | ILU(k) | seqaij,seqbaij | --- |  | X
|  | ICC(k) | seqaij,seqbaij | --- |  | X
|  | ILU dt | aij | pilut/hypre (LLNL) | X |
| Matrix-free | infrastructure |  |  | X | X
| Multigrid | infrastructure |  |  | X | X
|  | geometric (DMDA for structured grid) |  |  | X | X
|  | smoothed aggregation | aij |  | X | X
|  | structured geometric | hyprestruct | PFMG from hypre | X |
|  | classical algebraic | aij | BoomerAMG/hypre (LLNL) | X |
|  |  | aij | ML/Trilinos (SNL) | X |
| Physics-based splitting | relaxation and Schur-complement | aij, baij, nest |  | X | X
|  | least squares commutator | schurcomplement |  | X | X
| Approximate inverses |  | aij | Parasails/hypre (LLNL) | X |
|  |  | aij | SPAI | X |
| Substructuring | balancing Neumann-Neumann | is | --- | X | X
|  | BDDC | is | (petsc-master only) | X | X
Direct solvers | LU | LU | seqaij,seqbaij | --- |  | X
|  | LU | seqaij | MATLAB |  | X
|  |  | aij | PaStiX (INRIA) | X | X
|  |  | aij | SuperLU (BNL) SuperLU Sequential LU solver / SuperLU_DIST Parallel LU solver | X | X
|  |  | aij | MUMPS | X | X
|  |  | seqaij | ESSL (IBM) |  |
|  |  | seqaij | UMFPACK Part of SuiteSparse |  | X
|  |  | seqaij | KLU Part of SuiteSparse |  | X
|  |  | seqaij | LUSOL |  |
|  |  | dense | X | X
| Cholesky | Cholesky | seqaij,seqsbaij | --- |  | X
|  |  | sbaij | PaStiX (INRIA) | X | X
|  |  | sbaij | MUMPS | X | X
|  |  | seqaij, seqsbaij | CHOLMOD Part of SuiteSparse |  | X
|  |  | dense | X | X
| QR |  | matlab | MATLAB |  |
| XXt and XYt |  | aij | --- | X |
Krylov methods |  | Richardson |  | --- | X | X
|  | Chebyshev |  | --- | X | X
|  | conjugate gradients |  | --- | X | X
|  | GMRES |  | --- | X | X
|  | Bi-CG-stab |  | --- | X | X
|  | transpose free QMR |  | --- | X | X
|  | conjugate residuals |  | --- | X | X
|  | conjugate gradient squared |  | --- | X | X
|  | bi-conjugate gradient |  | --- | X | X
|  | MINRES |  | --- | X | X
|  | flexible GMRES |  | --- | X | X
|  | LSQR(supports rectangular matrices) |  | --- | X | X
|  | CG for least squares(supports rectangular matrices) |  | --- | X | X
|  | SYMMLQ |  | --- | X | X
|  | LGMRES |  | --- | X | X
|  | GCR |  | --- | X | X
|  | Conjugate gradient on the normal equations |  | --- | X | X

This page was inspired by Jack Dongarra's freely available software for linear algebra list