Summary of Sparse Linear Solvers Available In PETSc#
Preconditioners#
Algorithm 
Associated Type 
Matrix Types 
External Packages 
Parallel 
Complex 


Generic 
Jacobi 
— 
X 
X 

Point Block Jacobi 
— 
X 
X 

Variable Point Block Jacobi 
— 
X 
X 

Block Jacobi 
— 
X 
X 

SOR 
— 
X 
X 

Point Block SOR 

— 
X 
X 

Kaczmarz 
— 
X 
X 

Additive Schwarz 
— 
X 
X 

Vanka/overlapping patches 
— 
X 
X 

Deflation 
All 
— 
X 
X 

Incomplete 
ILU 
— 
X 

ILU with drop tolerance 
X 

Euclid/hypre ( 
X 

ICholesky 
— 
X 

Algebraic recursive multilevel 
X 

Matrix Free 
Infrastructure 
All 
— 
X 
X 

Multigrid 
Infrastructure 
All 
— 
X 
X 

Geometric 
All 
— 
X 
X 

Smoothed Aggregation 
— 
X 
X 

Smoothed Aggregation (ML) 
X 
X 

Structured Geometric 
X 

Classical Algebraic 
X 

Multigroup MG 
— 
X 
X 

Domain Decomposition 
X 
X 

Hierarchical matrices 
\(\mathcal H^2\) 
X 

Physicsbased Splitting 
Relaxation & Schur Complement 
— 
X 
X 

Galerkin composition 
Any 
— 
X 
X 

Additive/multiplicative 
Any 
— 
X 
X 

Least Squares Commutator 
— 
X 
X 

Parallel transformation 
Redistribution 
— 
X 
X 

Telescoping communicator 
— 
X 
X 

Distribute for MPI 
— 
X 
X 

Approximate Inverse 
AIV 
X 

Substructuring 
Balancing NeumannNeumann 
— 
X 
X 

Balancing Domain Decomposition 
— 
X 
X 

2level Schwarz wire basket 
— 
X 
X 
Direct Solvers#
Algorithm 
Associated Type 
Matrix Types 
External Packages 
Parallel 
Complex 


Direct LU 
LU 
— 
X 

X 

X 
X 

X 
X 

X 
X 

X 

X 

X 

X 
X 

X 
X 

Direct Cholesky 
Cholesky 
— 
X 

X 
X 

X 
X 

X 

X 
X 

X 

Direct SVD 
Singular value decomposition 
Any 
— 
X 
X 

Direct QR 
QR 

XXt and XYt 
— 
X 
Krylov Methods#
Algorithm 
Associated Type 
External Packages 
Parallel 
Complex 

Richardson 
— 
X 
X 

Chebyshev 
— 
X 
X 

GMRES 
— 
X 
X 

Flexible GMRES 
— 
X 
X 

LGMRES 
— 
X 
X 

Deflated GMRES 
— 
X 

Twostage with least squares residual minimization 
— 
X 
X 

Conjugate Gradient 
— 
X 
X 

Conjugate Gradient Squared 
— 
X 
X 

Conjugate Gradient for Least Squares 
— 
X 
X 

Conjugate Gradient on Normal Equations 
— 
X 
X 

Nash Conjugate Gradient with trust region constraint 
— 
X 
X 

Conjugate Gradient with trust region constraint 
— 
X 
X 

Gould et al Conjugate Gradient with trust region constraint 
— 
X 
X 

Steinhaug Conjugate Gradient with trust region constraint 
— 
X 
X 

Left Conjugate Direction 
— 
X 
X 

BiConjugate Gradient 
— 
X 
X 

Stabilized BiConjugate Gradient 
— 
X 
X 

Improved Stabilized BiConjugate Gradient 
— 
X 
X 

Transposefree QMR 
— 
X 
X 

Tony Chan QMR 
— 
X 
X 

QMR BiCGStab 
— 
X 
X 

Flexible Conjugate Gradients 
— 
X 
X 

Flexible stabilized BiConjugate Gradients 
— 
X 
X 

Flexible stabilized BiConjugate Gradients with fewer reductions 
— 
X 
X 

Stabilized BiConjugate Gradients with length \(\ell\) recurrence 
— 
X 
X 

Conjugate Residual 
— 
X 
X 

Generalized Conjugate Residual 
— 
X 
X 

Generalized Conjugate Residual (with inner normalization and deflated restarts) 
X 
X 

Minimum Residual 
— 
X 
X 

LSQR 
— 
X 
X 

SYMMLQ 
— 
X 
X 

FETIDP (reduction to dualprimal subproblem) 
— 
X 
X 

Gropp’s overlapped reduction Conjugate Gradient 
— 
X 
X 

Pipelined Conjugate Gradient 
— 
X 
X 

Pipelined Conjugate Gradient with residual replacement 
— 
X 
X 

Pipelined depth \(\ell\) Conjugate Gradient 
— 
X 
X 

Pipelined predictandrecompute Conjugate Gradient 
— 
X 
X 

Pipelined Conjugate Gradient over iteration pairs 
— 
X 
X 

Pipelined flexible Conjugate Gradient 
— 
X 
X 

Pipelined stabilized BiConjugate Gradients 
— 
X 
X 

Pipelined Conjugate Residual 
— 
X 
X 

Pipelined flexible GMRES 
— 
X 
X 

Pipelined Generalized Conjugate Residual 
— 
X 
X 

Pipelined GMRES 
— 
X 
X 