Summary of Sparse Linear Solvers Available In PETSc

Preconditioners

	Algorithm	Associated Type	Matrix Types	External Packages	Parallel	Complex
Generic	Jacobi	PCJACOBI	MATAIJ, MATBAIJ, MATSBAIJ, MATDENSE	—	X	X
	Point Block Jacobi	PCPBJACOBI	MATAIJ, MATBAIJ, MATSBAIJ, MATKAIJ, MATMPISELL, MATIS	—	X	X
	Variable Point Block Jacobi	PCPBJACOBI	MATAIJ, MATBAIJ, MATSBAIJ	—	X	X
	Block Jacobi	PCBJACOBI	MATAIJ, MATBAIJ, MATSBAIJ	—	X	X
	SOR	PCSOR	MATAIJ, MATSEQDENSE, MATSEQSBAIJ	—	X	X
	Point Block SOR		MATSEQBAIJ (only for bs = 2,3,4,5)	—	X	X
	Kaczmarz	PCKACZMARZ	MATAIJ	—	X	X
	Additive Schwarz	PCASM	MATAIJ, MATBAIJ, MATSBAIJ	—	X	X
	Vanka/overlapping patches	PCPATCH	MATAIJ	—	X	X
	Deflation	PCDEFLATION	All	—	X	X
Incomplete	ILU	PCILU	MATSEQAIJ, MATSEQBAIJ	—		X
	ILU with drop tolerance	PCILU	MATSEQAIJ	SuperLU Sequential ILU solver		X
		PCILU	MATAIJ	Euclid/hypre (PCHYPRE)	X
	ICholesky	PCICC	MATSEQAIJ, MATSEQBAIJ, MATSEQSBAIJ	—		X
	Algebraic recursive multilevel	PCPARMS	MATSEQAIJ	pARMS		X
Matrix Free	Infrastructure	PCSHELL	All	—	X	X
Multigrid	Infrastructure	PCMG	All	—	X	X
	Geometric		All	—	X	X
	Smoothed Aggregation	PCGAMG	MATAIJ	—	X	X
	Smoothed Aggregation (ML)	PCML	MATAIJ	ML/Trilinos	X	X
	Structured Geometric	PCPFMG, PCSYSPFMG, PCSMG	MATHYPRESTRUCT	hypre	X
	Classical Algebraic	PCHYPRE, PCAMGX	MATAIJ	BoomerAMG/hypre, AmgX	X
	Multi-group MG	PCHMG	MATAIJ	—	X	X
	Domain Decomposition	PCHPDDM	MATAIJ, MATBAIJ, MATSBAIJ, MATIS	HPDDM	X	X
Hierarchical matrices	\(\mathcal H^2\)	PCH2OPUS	MATHTOOL, MATH2OPUS	H2OPUS	X
Physics-based Splitting	Relaxation & Schur Complement	PCFIELDSPLIT	MATAIJ, MATBAIJ, MATNEST	—	X	X
	Galerkin composition	PCGALERKIN	Any	—	X	X
	Additive/multiplicative	PCCOMPOSITE	Any	—	X	X
	Least Squares Commutator	PCLSC	MATSCHURCOMPLEMENT	—	X	X
Parallel transformation	Redistribution	PCREDISTRIBUTE	MATAIJ	—	X	X
	Telescoping communicator	PCTELESCOPE	MATAIJ	—	X	X
	Distribute for MPI	PCMPI	MATAIJ	—	X	X
Approximate Inverse	AIV	PCHYPRE, PCSPAI	MATAIJ	Parasails/hypre, SPAI	X
Substructuring	Balancing Neumann-Neumann	PCNN	MATIS	—	X	X
	Balancing Domain Decomposition	PCBDDC	MATIS	—	X	X
	2-level Schwarz wire basket	PCEXOTIC	MATAIJ	—	X	X

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Direct Solvers

	Algorithm	Associated Type	Matrix Types	External Packages	Parallel	Complex
Direct LU	LU	PCLU	MATSEQAIJ, MATSEQBAIJ	—		X
			MATSEQAIJ	MATLAB		X
			MATAIJ	PaStiX	X	X
			MATAIJ	SuperLU	X	X
			MATAIJ, MATBAIJ	MUMPS	X	X
			MATSEQAIJ	ESSL
			MATSEQAIJ	UMPFPACK (SuiteSparse)		X
			MATSEQAIJ	KLU (SuiteSparse)		X
			MATSEQAIJ	LUSOL
			MATSEQAIJ, MATSEQBAIJ	MKL PARDISO		X
			MATMPIAIJ, MATMPIBAIJ	MKL CPardiso	X	X
			MATDENSE	Elemental	X	X
Direct Cholesky	Cholesky	PCCHOLESKY	MATSEQAIJ, MATSEQSBAIJ	—		X
			MATSBAIJ	PaStiX	X	X
			MATSBAIJ	MUMPS	X	X
			MATSEQAIJ, MATSEQSBAIJ	CHOLMOD (SuiteSparse)		X
			MATDENSE	Elemental	X	X
			MATSEQSBAIJ	MKL PARDISO
			MATMPIAIJ, MATMPIBAIJ	MKL CPardiso	X
Direct SVD	Singular value decomposition	PCSVD	Any	—	X	X
Direct QR	QR	PCQR	MATSEQAIJ	SuiteSparse QR
	XXt and XYt		MATAIJ	—	X

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Krylov Methods

Algorithm	Associated Type	External Packages	Parallel	Complex
Richardson	KSPRICHARDSON	—	X	X
Chebyshev	KSPCHEBYSHEV	—	X	X
GMRES	KSPGMRES	—	X	X
Flexible GMRES	KSPFGMRES	—	X	X
LGMRES	KSPLGMRES	—	X	X
Deflated GMRES	KSPDGMRES	—	X
Two-stage with least squares residual minimization	KSPTSIRM	—	X	X
Conjugate Gradient	KSPCG	—	X	X
Conjugate Gradient Squared	KSPCGS	—	X	X
Conjugate Gradient for Least Squares	KSPCGLS	—	X	X
Conjugate Gradient on Normal Equations	KSPCGNE	—	X	X
Nash Conjugate Gradient with trust region constraint	KSPNASH	—	X	X
Conjugate Gradient with trust region constraint	KSPSTCG	—	X	X
Gould et al Conjugate Gradient with trust region constraint	KSPGLTR	—	X	X
Steinhaug Conjugate Gradient with trust region constraint	KSPQCG	—	X	X
Left Conjugate Direction	KSPLCD	—	X	X
Bi-Conjugate Gradient	KSPBICG	—	X	X
Stabilized Bi-Conjugate Gradient	KSPBCGS	—	X	X
Improved Stabilized Bi-Conjugate Gradient	KSPIBCGS	—	X	X
Transpose-free QMR	KSPTFQMR	—	X	X
Tony Chan QMR	KSPTCQMR	—	X	X
QMR BiCGStab	KSPQMRCGS	—	X	X
Flexible Conjugate Gradients	KSPFCG	—	X	X
Flexible stabilized Bi-Conjugate Gradients	KSPFBCGS	—	X	X
Flexible stabilized Bi-Conjugate Gradients with fewer reductions	KSPFBCGSR	—	X	X
Stabilized Bi-Conjugate Gradients with length \(\ell\) recurrence	KSPBCGSL	—	X	X
Conjugate Residual	KSPCR	—	X	X
Generalized Conjugate Residual	KSPGCR	—	X	X
Generalized Conjugate Residual (with inner normalization and deflated restarts)	KSPHPDDM	HPDDM	X	X
Minimum Residual	KSPMINRES	—	X	X
LSQR	KSPLSQR	—	X	X
SYMMLQ	KSPSYMMLQ	—	X	X
FETI-DP (reduction to dual-primal sub-problem)	KSPFETIDP	—	X	X
Gropp’s overlapped reduction Conjugate Gradient	KSPGROPPCG	—	X	X
Pipelined Conjugate Gradient	KSPPIPECG	—	X	X
Pipelined Conjugate Gradient with residual replacement	KSPPIPECGRR	—	X	X
Pipelined depth \(\ell\) Conjugate Gradient	KSPPIPELCG	—	X	X
Pipelined predict-and-recompute Conjugate Gradient	KSPPIPEPRCG	—	X	X
Pipelined Conjugate Gradient over iteration pairs	KSPPIPECG2	—	X	X
Pipelined flexible Conjugate Gradient	KSPPIPEFCG	—	X	X
Pipelined stabilized Bi-Conjugate Gradients	KSPPIPEBCGS	—	X	X
Pipelined Conjugate Residual	KSPPIPECR	—	X	X
Pipelined flexible GMRES	KSPPIPEFGMRES	—	X	X
Pipelined Generalized Conjugate Residual	KSPPIPEGCR	—	X	X
Pipelined GMRES	KSPPGMRES	—	X	X