petsc-3.14.5 2021-03-03
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# KSPSolve

Solves linear system.

### Synopsis

    #include "petscksp.h"
    #include "petscmat.h"
    PetscErrorCode KSPSolve(KSP ksp,Vec b,Vec x)

Collective on ksp

### Parameters

| ksp | - iterative context obtained from KSPCreate()
| b | - the right hand side vector
| x | - the solution (this may be the same vector as b, then b will be overwritten with answer)

### Options Database Keys

| -ksp_view_eigenvalues | - compute preconditioned operators eigenvalues
| -ksp_view_eigenvalues_explicit | - compute the eigenvalues by forming the dense operator and using LAPACK
| -ksp_view_mat binary | - save matrix to the default binary viewer
| -ksp_view_pmat binary | - save matrix used to build preconditioner to the default binary viewer
| -ksp_view_rhs binary | - save right hand side vector to the default binary viewer
| -ksp_view_solution binary | - save computed solution vector to the default binary viewer (can be read later with src/ksp/tutorials/ex10.c for testing solvers)
| -ksp_view_mat_explicit | - for matrix-free operators, computes the matrix entries and views them
| -ksp_view_preconditioned_operator_explicit | - computes the product of the preconditioner and matrix as an explicit matrix and views it
| -ksp_converged_reason | - print reason for converged or diverged, also prints number of iterations
| -ksp_view_final_residual | - print 2-norm of true linear system residual at the end of the solution process
| -ksp_view | - print the ksp data structure at the end of the system solution

### Notes

If one uses KSPSetDM() then x or b need not be passed. Use KSPGetSolution() to access the solution in this case.

The operator is specified with KSPSetOperators().

Call KSPGetConvergedReason() to determine if the solver converged or failed and why. The number of iterations can be obtained from KSPGetIterationNumber().

If you provide a matrix that has a MatSetNullSpace() and MatSetTransposeNullSpace() this will use that information to solve singular systems in the least squares sense with a norm minimizing solution.

                      A x = b   where b = b_p + b_t where b_t is not in the range of A (and hence by the fundamental theorem of linear algebra is in the nullspace(A') see MatSetNullSpace()

       KSP first removes b_t producing the linear system  A x = b_p (which has multiple solutions) and solves this to find the ||x|| minimizing solution (and hence

       it finds the solution x orthogonal to the nullspace(A). The algorithm is simply in each iteration of the Krylov method we remove the nullspace(A) from the search

       direction thus the solution which is a linear combination of the search directions has no component in the nullspace(A).

       We recommend always using GMRES for such singular systems.

       If nullspace(A) = nullspace(A') (note symmetric matrices always satisfy this property) then both left and right preconditioning will work

       If nullspace(A) != nullspace(A') then left preconditioning will work but right preconditioning may not work (or it may).

Developer Note: The reason we cannot always solve nullspace(A) != nullspace(A') systems with right preconditioning is because we need to remove at each iteration the nullspace(AB) from the search direction. While we know the nullspace(A) the nullspace(AB) equals B^-1 times the nullspace(A) but except for trivial preconditioners such as diagonal scaling we cannot apply the inverse of the preconditioner to a vector and thus cannot compute the nullspace(AB).

If using a direct method (e.g., via the KSP solver KSPPREONLY and a preconditioner such as PCLU/PCILU), then its=1. See KSPSetTolerances() and KSPConvergedDefault() for more details.

### Understanding Convergence

The routines KSPMonitorSet(), KSPComputeEigenvalues(), and KSPComputeEigenvaluesExplicitly() provide information on additional options to monitor convergence and print eigenvalue information.

### See Also

KSPCreate(), KSPSetUp(), KSPDestroy(), KSPSetTolerances(), KSPConvergedDefault(),
KSPSolveTranspose(), KSPGetIterationNumber(), MatNullSpaceCreate(), MatSetNullSpace(), MatSetTransposeNullSpace(), KSP, KSPConvergedReasonView()

### Level

beginner

### Location

src/ksp/ksp/interface/itfunc.c

### Examples

src/dm/impls/stag/tutorials/ex1.c.html
src/dm/impls/stag/tutorials/ex2.c.html
src/dm/impls/stag/tutorials/ex3.c.html
src/dm/impls/stag/tutorials/ex4.c.html
src/ksp/pc/tutorials/ex1.c.html
src/ksp/pc/tutorials/ex2.c.html
src/ksp/pc/tutorials/ex3.c.html
src/ksp/ksp/tutorials/ex1.c.html
src/ksp/ksp/tutorials/ex2.c.html
src/ksp/ksp/tutorials/ex3.c.html
src/ksp/ksp/tutorials/ex4.c.html

### Implementations

KSPSolve_BCGS in src/ksp/ksp/impls/bcgs/bcgs.c
KSPSolve_FBCGS in src/ksp/ksp/impls/bcgs/fbcgs/fbcgs.c
KSPSolve_FBCGSR in src/ksp/ksp/impls/bcgs/fbcgsr/fbcgsr.c
KSPSolve_PIPEBCGS in src/ksp/ksp/impls/bcgs/pipebcgs/pipebcgs.c
KSPSolve_BCGSL in src/ksp/ksp/impls/bcgsl/bcgsl.c
KSPSolve_BiCG in src/ksp/ksp/impls/bicg/bicg.c
KSPSolve_CG in src/ksp/ksp/impls/cg/cg.c
KSPSolve_CG_SingleReduction in src/ksp/ksp/impls/cg/cg.c
KSPSolve_CGLS in src/ksp/ksp/impls/cg/cgls.c
KSPSolve_CGNE in src/ksp/ksp/impls/cg/cgne/cgne.c
KSPSolve_GROPPCG in src/ksp/ksp/impls/cg/groppcg/groppcg.c
KSPSolve_PIPECG in src/ksp/ksp/impls/cg/pipecg/pipecg.c
KSPSolve_PIPECG2 in src/ksp/ksp/impls/cg/pipecg2/pipecg2.c
KSPSolve_PIPECGRR in src/ksp/ksp/impls/cg/pipecgrr/pipecgrr.c
KSPSolve_InnerLoop_PIPELCG in src/ksp/ksp/impls/cg/pipelcg/pipelcg.c
KSPSolve_ReInitData_PIPELCG in src/ksp/ksp/impls/cg/pipelcg/pipelcg.c
KSPSolve_PIPELCG in src/ksp/ksp/impls/cg/pipelcg/pipelcg.c
KSPSolve_PIPEPRCG in src/ksp/ksp/impls/cg/pipeprcg/pipeprcg.c
KSPSolve_CGS in src/ksp/ksp/impls/cgs/cgs.c
KSPSolve_Chebyshev in src/ksp/ksp/impls/cheby/cheby.c
KSPSolve_CR in src/ksp/ksp/impls/cr/cr.c
KSPSolve_PIPECR in src/ksp/ksp/impls/cr/pipecr/pipecr.c
KSPSolve_FCG in src/ksp/ksp/impls/fcg/fcg.c
KSPSolve_PIPEFCG_cycle in src/ksp/ksp/impls/fcg/pipefcg/pipefcg.c
KSPSolve_PIPEFCG in src/ksp/ksp/impls/fcg/pipefcg/pipefcg.c
KSPSolve_FETIDP in src/ksp/ksp/impls/fetidp/fetidp.c
KSPSolve_GCR_cycle in src/ksp/ksp/impls/gcr/gcr.c
KSPSolve_GCR in src/ksp/ksp/impls/gcr/gcr.c
KSPSolve_PIPEGCR_cycle in src/ksp/ksp/impls/gcr/pipegcr/pipegcr.c
KSPSolve_PIPEGCR in src/ksp/ksp/impls/gcr/pipegcr/pipegcr.c
KSPSolve_AGMRES in src/ksp/ksp/impls/gmres/agmres/agmres.c
KSPSolve_DGMRES in src/ksp/ksp/impls/gmres/dgmres/dgmres.c
KSPSolve_FGMRES in src/ksp/ksp/impls/gmres/fgmres/fgmres.c
KSPSolve_GMRES in src/ksp/ksp/impls/gmres/gmres.c
KSPSolve_LGMRES in src/ksp/ksp/impls/gmres/lgmres/lgmres.c
KSPSolve_PGMRES in src/ksp/ksp/impls/gmres/pgmres/pgmres.c
KSPSolve_PIPEFGMRES in src/ksp/ksp/impls/gmres/pipefgmres/pipefgmres.c
KSPSolve_HPDDM in src/ksp/ksp/impls/hpddm/hpddm.cxx
KSPSolve_IBCGS in src/ksp/ksp/impls/ibcgs/ibcgs.c
KSPSolve_LCD in src/ksp/ksp/impls/lcd/lcd.c
KSPSolve_LSQR in src/ksp/ksp/impls/lsqr/lsqr.c
KSPSolve_MINRES in src/ksp/ksp/impls/minres/minres.c
KSPSolve_PREONLY in src/ksp/ksp/impls/preonly/preonly.c
KSPSolve_QCG in src/ksp/ksp/impls/qcg/qcg.c
KSPSolve_Richardson in src/ksp/ksp/impls/rich/rich.c
KSPSolve_SYMMLQ in src/ksp/ksp/impls/symmlq/symmlq.c
KSPSolve_TCQMR in src/ksp/ksp/impls/tcqmr/tcqmr.c
KSPSolve_TFQMR in src/ksp/ksp/impls/tfqmr/tfqmr.c
KSPSolve_TSIRM in src/ksp/ksp/impls/tsirm/tsirm.c
Index of all KSP routines
Table of Contents for all manual pages
Index of all manual pages