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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML3.2 EN">
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<TITLE>KSPSolve</TITLE>
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<BODY BGCOLOR="FFFFFF">
<div id="version" align=right><b>petsc-3.10.3 2018-12-18</b></div>
<div id="bugreport" align=right><a href="mailto:petsc-maint@mcs.anl.gov?subject=Typo or Error in Documentation &body=Please describe the typo or error in the documentation: petsc-3.10.3 v3.10.3 docs/manualpages/KSP/KSPSolve.html "><small>Report Typos and Errors</small></a></div>
<A NAME="KSPSolve"><H1>KSPSolve</H1></A>
Solves linear system.
<H3><FONT COLOR="#CC3333">Synopsis</FONT></H3>
<PRE>
#include "petscksp.h"
<A HREF="../Sys/PetscErrorCode.html#PetscErrorCode">PetscErrorCode</A> <A HREF="../KSP/KSPSolve.html#KSPSolve">KSPSolve</A>(<A HREF="../KSP/KSP.html#KSP">KSP</A> ksp,<A HREF="../Vec/Vec.html#Vec">Vec</A> b,<A HREF="../Vec/Vec.html#Vec">Vec</A> x)
</PRE>
Collective on <A HREF="../KSP/KSP.html#KSP">KSP</A>
<P>
<H3><FONT COLOR="#CC3333">Parameter</FONT></H3>
<TABLE border="0" cellpadding="0" cellspacing="0">
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>ksp </B></TD><TD>- iterative context obtained from <A HREF="../KSP/KSPCreate.html#KSPCreate">KSPCreate</A>()
</TD></TR>
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>b </B></TD><TD>- the right hand side vector
</TD></TR>
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>x </B></TD><TD>- the solution (this may be the same vector as b, then b will be overwritten with answer)
</TD></TR></TABLE>
<P>
<H3><FONT COLOR="#CC3333">Options Database Keys</FONT></H3>
<TABLE border="0" cellpadding="0" cellspacing="0">
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>-ksp_compute_eigenvalues </B></TD><TD>- compute preconditioned operators eigenvalues
</TD></TR>
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>-ksp_plot_eigenvalues </B></TD><TD>- plot the computed eigenvalues in an X-window
</TD></TR>
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>-ksp_plot_eigencontours </B></TD><TD>- plot the computed eigenvalues in an X-window with contours
</TD></TR>
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>-ksp_compute_eigenvalues_explicitly </B></TD><TD>- compute the eigenvalues by forming the dense operator and using LAPACK
</TD></TR>
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>-ksp_plot_eigenvalues_explicitly </B></TD><TD>- plot the explicitly computing eigenvalues
</TD></TR>
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>-ksp_view_mat binary </B></TD><TD>- save matrix to the default binary viewer
</TD></TR>
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>-ksp_view_pmat binary </B></TD><TD>- save matrix used to build preconditioner to the default binary viewer
</TD></TR>
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>-ksp_view_rhs binary </B></TD><TD>- save right hand side vector to the default binary viewer
</TD></TR>
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>-ksp_view_solution binary </B></TD><TD>- save computed solution vector to the default binary viewer
(can be read later with src/ksp/examples/tutorials/ex10.c for testing solvers)
</TD></TR>
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>-ksp_view_mat_explicit </B></TD><TD>- for matrix-free operators, computes the matrix entries and views them
</TD></TR>
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>-ksp_view_preconditioned_operator_explicit </B></TD><TD>- computes the product of the preconditioner and matrix as an explicit matrix and views it
</TD></TR>
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>-ksp_converged_reason </B></TD><TD>- print reason for converged or diverged, also prints number of iterations
</TD></TR>
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>-ksp_final_residual </B></TD><TD>- print 2-norm of true linear system residual at the end of the solution process
</TD></TR>
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>-ksp_view </B></TD><TD>- print the ksp data structure at the end of the system solution
</TD></TR></TABLE>
<P>
<H3><FONT COLOR="#CC3333">Notes</FONT></H3>
<P>
If one uses <A HREF="../KSP/KSPSetDM.html#KSPSetDM">KSPSetDM</A>() then x or b need not be passed. Use <A HREF="../KSP/KSPGetSolution.html#KSPGetSolution">KSPGetSolution</A>() to access the solution in this case.
<P>
The operator is specified with <A HREF="../KSP/KSPSetOperators.html#KSPSetOperators">KSPSetOperators</A>().
<P>
Call <A HREF="../KSP/KSPGetConvergedReason.html#KSPGetConvergedReason">KSPGetConvergedReason</A>() to determine if the solver converged or failed and
why. The number of iterations can be obtained from <A HREF="../KSP/KSPGetIterationNumber.html#KSPGetIterationNumber">KSPGetIterationNumber</A>().
<P>
If you provide a matrix that has a <A HREF="../Mat/MatSetNullSpace.html#MatSetNullSpace">MatSetNullSpace</A>() and <A HREF="../Mat/MatSetTransposeNullSpace.html#MatSetTransposeNullSpace">MatSetTransposeNullSpace</A>() this will use that information to solve singular systems
in the least squares sense with a norm minimizing solution.
<pre>
</pre>
<pre>
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 <A HREF="../Mat/MatSetNullSpace.html#MatSetNullSpace">MatSetNullSpace</A>()
</pre>
<pre>
</pre>
<pre>
<A HREF="../KSP/KSP.html#KSP">KSP</A> 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
</pre>
<pre>
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
</pre>
<pre>
direction thus the solution which is a linear combination of the search directions has no component in the nullspace(A).
</pre>
<pre>
</pre>
<pre>
We recommend always using GMRES for such singular systems.
</pre>
<pre>
If nullspace(A) = nullspace(A') (note symmetric matrices always satisfy this property) then both left and right preconditioning will work
</pre>
<pre>
If nullspace(A) != nullspace(A') then left preconditioning will work but right preconditioning may not work (or it may).
</pre>
<P>
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).
<P>
<P>
If using a direct method (e.g., via the <A HREF="../KSP/KSP.html#KSP">KSP</A> solver
<A HREF="../KSP/KSPPREONLY.html#KSPPREONLY">KSPPREONLY</A> and a preconditioner such as <A HREF="../PC/PCLU.html#PCLU">PCLU</A>/<A HREF="../PC/PCILU.html#PCILU">PCILU</A>),
then its=1. See <A HREF="../KSP/KSPSetTolerances.html#KSPSetTolerances">KSPSetTolerances</A>() and <A HREF="../KSP/KSPConvergedDefault.html#KSPConvergedDefault">KSPConvergedDefault</A>()
for more details.
<P>
<H3><FONT COLOR="#CC3333">Understanding Convergence</FONT></H3>
The routines <A HREF="../KSP/KSPMonitorSet.html#KSPMonitorSet">KSPMonitorSet</A>(), <A HREF="../KSP/KSPComputeEigenvalues.html#KSPComputeEigenvalues">KSPComputeEigenvalues</A>(), and
<A HREF="../KSP/KSPComputeEigenvaluesExplicitly.html#KSPComputeEigenvaluesExplicitly">KSPComputeEigenvaluesExplicitly</A>() provide information on additional
options to monitor convergence and print eigenvalue information.
<P>
<P>
<H3><FONT COLOR="#CC3333">Keywords</FONT></H3>
solve, linear system
<BR>
<P>
<H3><FONT COLOR="#CC3333">See Also</FONT></H3>
<A HREF="../KSP/KSPCreate.html#KSPCreate">KSPCreate</A>(), <A HREF="../KSP/KSPSetUp.html#KSPSetUp">KSPSetUp</A>(), <A HREF="../KSP/KSPDestroy.html#KSPDestroy">KSPDestroy</A>(), <A HREF="../KSP/KSPSetTolerances.html#KSPSetTolerances">KSPSetTolerances</A>(), <A HREF="../KSP/KSPConvergedDefault.html#KSPConvergedDefault">KSPConvergedDefault</A>(),
<BR><A HREF="../KSP/KSPSolveTranspose.html#KSPSolveTranspose">KSPSolveTranspose</A>(), <A HREF="../KSP/KSPGetIterationNumber.html#KSPGetIterationNumber">KSPGetIterationNumber</A>(), <A HREF="../Mat/MatNullSpaceCreate.html#MatNullSpaceCreate">MatNullSpaceCreate</A>(), <A HREF="../Mat/MatSetNullSpace.html#MatSetNullSpace">MatSetNullSpace</A>(), <A HREF="../Mat/MatSetTransposeNullSpace.html#MatSetTransposeNullSpace">MatSetTransposeNullSpace</A>(), <A HREF="../KSP/KSP.html#KSP">KSP</A>
<P><B></B><H3><FONT COLOR="#CC3333">Level</FONT></H3>beginner<BR>
<H3><FONT COLOR="#CC3333">Location</FONT></H3>
</B><A HREF="../../../src/ksp/ksp/interface/itfunc.c.html#KSPSolve">src/ksp/ksp/interface/itfunc.c</A>
<A HREF="../../../src/ksp/pc/examples/tutorials/ex1.c.html">src/ksp/pc/examples/tutorials/ex1.c.html</A><BR>
<A HREF="../../../src/ksp/pc/examples/tutorials/ex2.c.html">src/ksp/pc/examples/tutorials/ex2.c.html</A><BR>
<A HREF="../../../src/ksp/pc/examples/tutorials/ex3.c.html">src/ksp/pc/examples/tutorials/ex3.c.html</A><BR>
<A HREF="../../../src/ksp/ksp/examples/tutorials/ex1.c.html">src/ksp/ksp/examples/tutorials/ex1.c.html</A><BR>
<A HREF="../../../src/ksp/ksp/examples/tutorials/ex2.c.html">src/ksp/ksp/examples/tutorials/ex2.c.html</A><BR>
<A HREF="../../../src/ksp/ksp/examples/tutorials/ex3.c.html">src/ksp/ksp/examples/tutorials/ex3.c.html</A><BR>
<A HREF="../../../src/ksp/ksp/examples/tutorials/ex5.c.html">src/ksp/ksp/examples/tutorials/ex5.c.html</A><BR>
<A HREF="../../../src/ksp/ksp/examples/tutorials/ex6.c.html">src/ksp/ksp/examples/tutorials/ex6.c.html</A><BR>
<A HREF="../../../src/ksp/ksp/examples/tutorials/ex7.c.html">src/ksp/ksp/examples/tutorials/ex7.c.html</A><BR>
<A HREF="../../../src/ksp/ksp/examples/tutorials/ex8.c.html">src/ksp/ksp/examples/tutorials/ex8.c.html</A><BR>
<P><H3><FONT COLOR="CC3333">Implementations</FONT></H3><A HREF="../../../src/ksp/ksp/impls/bcgs/bcgs.c.html#KSPSolve_BCGS">KSPSolve_BCGS in src/ksp/ksp/impls/bcgs/bcgs.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/bcgs/fbcgs/fbcgs.c.html#KSPSolve_FBCGS">KSPSolve_FBCGS in src/ksp/ksp/impls/bcgs/fbcgs/fbcgs.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/bcgs/fbcgsr/fbcgsr.c.html#KSPSolve_FBCGSR">KSPSolve_FBCGSR in src/ksp/ksp/impls/bcgs/fbcgsr/fbcgsr.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/bcgs/pipebcgs/pipebcgs.c.html#KSPSolve_PIPEBCGS">KSPSolve_PIPEBCGS in src/ksp/ksp/impls/bcgs/pipebcgs/pipebcgs.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/bcgsl/bcgsl.c.html#KSPSolve_BCGSL">KSPSolve_BCGSL in src/ksp/ksp/impls/bcgsl/bcgsl.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/bicg/bicg.c.html#KSPSolve_BiCG">KSPSolve_BiCG in src/ksp/ksp/impls/bicg/bicg.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/cg/cg.c.html#KSPSolve_CG">KSPSolve_CG in src/ksp/ksp/impls/cg/cg.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/cg/cg.c.html#KSPSolve_CG_SingleReduction">KSPSolve_CG_SingleReduction in src/ksp/ksp/impls/cg/cg.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/cg/cgls.c.html#KSPSolve_CGLS">KSPSolve_CGLS in src/ksp/ksp/impls/cg/cgls.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/cg/cgne/cgne.c.html#KSPSolve_CGNE">KSPSolve_CGNE in src/ksp/ksp/impls/cg/cgne/cgne.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/cg/groppcg/groppcg.c.html#KSPSolve_GROPPCG">KSPSolve_GROPPCG in src/ksp/ksp/impls/cg/groppcg/groppcg.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/cg/pipecg/pipecg.c.html#KSPSolve_PIPECG">KSPSolve_PIPECG in src/ksp/ksp/impls/cg/pipecg/pipecg.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/cg/pipecgrr/pipecgrr.c.html#KSPSolve_PIPECGRR">KSPSolve_PIPECGRR in src/ksp/ksp/impls/cg/pipecgrr/pipecgrr.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/cg/pipelcg/pipelcg.c.html#KSPSolve_InnerLoop_PIPELCG">KSPSolve_InnerLoop_PIPELCG in src/ksp/ksp/impls/cg/pipelcg/pipelcg.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/cg/pipelcg/pipelcg.c.html#KSPSolve_ReInitData_PIPELCG">KSPSolve_ReInitData_PIPELCG in src/ksp/ksp/impls/cg/pipelcg/pipelcg.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/cg/pipelcg/pipelcg.c.html#KSPSolve_PIPELCG">KSPSolve_PIPELCG in src/ksp/ksp/impls/cg/pipelcg/pipelcg.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/cgs/cgs.c.html#KSPSolve_CGS">KSPSolve_CGS in src/ksp/ksp/impls/cgs/cgs.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/cheby/cheby.c.html#KSPSolve_Chebyshev">KSPSolve_Chebyshev in src/ksp/ksp/impls/cheby/cheby.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/cr/cr.c.html#KSPSolve_CR">KSPSolve_CR in src/ksp/ksp/impls/cr/cr.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/cr/pipecr/pipecr.c.html#KSPSolve_PIPECR">KSPSolve_PIPECR in src/ksp/ksp/impls/cr/pipecr/pipecr.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/fcg/fcg.c.html#KSPSolve_FCG">KSPSolve_FCG in src/ksp/ksp/impls/fcg/fcg.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/fcg/pipefcg/pipefcg.c.html#KSPSolve_PIPEFCG_cycle">KSPSolve_PIPEFCG_cycle in src/ksp/ksp/impls/fcg/pipefcg/pipefcg.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/fcg/pipefcg/pipefcg.c.html#KSPSolve_PIPEFCG">KSPSolve_PIPEFCG in src/ksp/ksp/impls/fcg/pipefcg/pipefcg.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/fetidp/fetidp.c.html#KSPSolve_FETIDP">KSPSolve_FETIDP in src/ksp/ksp/impls/fetidp/fetidp.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/gcr/gcr.c.html#KSPSolve_GCR_cycle">KSPSolve_GCR_cycle in src/ksp/ksp/impls/gcr/gcr.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/gcr/gcr.c.html#KSPSolve_GCR">KSPSolve_GCR in src/ksp/ksp/impls/gcr/gcr.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/gcr/pipegcr/pipegcr.c.html#KSPSolve_PIPEGCR_cycle">KSPSolve_PIPEGCR_cycle in src/ksp/ksp/impls/gcr/pipegcr/pipegcr.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/gcr/pipegcr/pipegcr.c.html#KSPSolve_PIPEGCR">KSPSolve_PIPEGCR in src/ksp/ksp/impls/gcr/pipegcr/pipegcr.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/gmres/agmres/agmres.c.html#KSPSolve_AGMRES">KSPSolve_AGMRES in src/ksp/ksp/impls/gmres/agmres/agmres.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/gmres/dgmres/dgmres.c.html#KSPSolve_DGMRES">KSPSolve_DGMRES in src/ksp/ksp/impls/gmres/dgmres/dgmres.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/gmres/fgmres/fgmres.c.html#KSPSolve_FGMRES">KSPSolve_FGMRES in src/ksp/ksp/impls/gmres/fgmres/fgmres.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/gmres/gmres.c.html#KSPSolve_GMRES">KSPSolve_GMRES in src/ksp/ksp/impls/gmres/gmres.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/gmres/lgmres/lgmres.c.html#KSPSolve_LGMRES">KSPSolve_LGMRES in src/ksp/ksp/impls/gmres/lgmres/lgmres.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/gmres/pgmres/pgmres.c.html#KSPSolve_PGMRES">KSPSolve_PGMRES in src/ksp/ksp/impls/gmres/pgmres/pgmres.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/gmres/pipefgmres/pipefgmres.c.html#KSPSolve_PIPEFGMRES">KSPSolve_PIPEFGMRES in src/ksp/ksp/impls/gmres/pipefgmres/pipefgmres.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/ibcgs/ibcgs.c.html#KSPSolve_IBCGS">KSPSolve_IBCGS in src/ksp/ksp/impls/ibcgs/ibcgs.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/lcd/lcd.c.html#KSPSolve_LCD">KSPSolve_LCD in src/ksp/ksp/impls/lcd/lcd.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/lsqr/lsqr.c.html#KSPSolve_LSQR">KSPSolve_LSQR in src/ksp/ksp/impls/lsqr/lsqr.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/minres/minres.c.html#KSPSolve_MINRES">KSPSolve_MINRES in src/ksp/ksp/impls/minres/minres.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/preonly/preonly.c.html#KSPSolve_PREONLY">KSPSolve_PREONLY in src/ksp/ksp/impls/preonly/preonly.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/qcg/qcg.c.html#KSPSolve_QCG">KSPSolve_QCG in src/ksp/ksp/impls/qcg/qcg.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/rich/rich.c.html#KSPSolve_Richardson">KSPSolve_Richardson in src/ksp/ksp/impls/rich/rich.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/symmlq/symmlq.c.html#KSPSolve_SYMMLQ">KSPSolve_SYMMLQ in src/ksp/ksp/impls/symmlq/symmlq.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/tcqmr/tcqmr.c.html#KSPSolve_TCQMR">KSPSolve_TCQMR in src/ksp/ksp/impls/tcqmr/tcqmr.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/tfqmr/tfqmr.c.html#KSPSolve_TFQMR">KSPSolve_TFQMR in src/ksp/ksp/impls/tfqmr/tfqmr.c</A><BR>
<A HREF="../../../src/ksp/ksp/impls/tsirm/tsirm.c.html#KSPSolve_TSIRM">KSPSolve_TSIRM in src/ksp/ksp/impls/tsirm/tsirm.c</A><BR>
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