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<section class="tex2jax_ignore mathjax_ignore" id="advanced-features-of-matrices-and-solvers">
<span id="ch-advanced"></span><h1>Advanced Features of Matrices and Solvers<a class="headerlink" href="#advanced-features-of-matrices-and-solvers" title="Link to this heading">#</a></h1>
<p>This chapter introduces additional features of the PETSc matrices and
solvers.</p>
<section id="extracting-submatrices">
<span id="sec-matsub"></span><h2>Extracting Submatrices<a class="headerlink" href="#extracting-submatrices" title="Link to this heading">#</a></h2>
<p>One can extract a (parallel) submatrix from a given (parallel) using</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../manualpages/Mat/MatCreateSubMatrix.html">MatCreateSubMatrix</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">A</span><span class="p">,</span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="w"> </span><span class="n">rows</span><span class="p">,</span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/MatReuse.html">MatReuse</a></span><span class="w"> </span><span class="n">call</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="o">*</span><span class="n">B</span><span class="p">);</span>
</pre></div>
</div>
<p>This extracts the <code class="docutils notranslate"><span class="pre">rows</span></code> and <code class="docutils notranslate"><span class="pre">cols</span></code> of the matrix <code class="docutils notranslate"><span class="pre">A</span></code> into
<code class="docutils notranslate"><span class="pre">B</span></code>. If call is <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MatReuse.html">MAT_INITIAL_MATRIX</a></span></code> it will create the matrix
<code class="docutils notranslate"><span class="pre">B</span></code>. If call is <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MatReuse.html">MAT_REUSE_MATRIX</a></span></code> it will reuse the <code class="docutils notranslate"><span class="pre">B</span></code> created
with a previous call. This function is used internally by <code class="docutils notranslate"><span class="pre"><a href="../manualpages/PC/PCFIELDSPLIT.html">PCFIELDSPLIT</a></span></code>.</p>
<p>One can also extract one or more submatrices per MPI process with</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../manualpages/Mat/MatCreateSubMatrices.html">MatCreateSubMatrices</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">A</span><span class="p">,</span><span class="n"><a href="../manualpages/Sys/PetscInt.html">PetscInt</a></span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="w"> </span><span class="n">rows</span><span class="p">[],</span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="w"> </span><span class="n">cols</span><span class="p">[],</span><span class="n"><a href="../manualpages/Mat/MatReuse.html">MatReuse</a></span><span class="w"> </span><span class="n">call</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="o">*</span><span class="n">B</span><span class="p">[]);</span>
</pre></div>
</div>
<p>This extracts n (zero or more) matrices with the <code class="docutils notranslate"><span class="pre">rows[k]</span></code> and <code class="docutils notranslate"><span class="pre">cols[k]</span></code> of the matrix <code class="docutils notranslate"><span class="pre">A</span></code> into an array of
sequential matrices <code class="docutils notranslate"><span class="pre">B[k]</span></code> on this process. If call is <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MatReuse.html">MAT_INITIAL_MATRIX</a></span></code> it will create the array of matrices
<code class="docutils notranslate"><span class="pre">B</span></code>. If call is <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MatReuse.html">MAT_REUSE_MATRIX</a></span></code> it will reuse the <code class="docutils notranslate"><span class="pre">B</span></code> created
with a previous call. The <code class="docutils notranslate"><span class="pre"><a href="../manualpages/IS/IS.html">IS</a></span></code> arguments are sequential. The array of matrices should be destroyed with <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MatDestroySubMatrices.html">MatDestroySubMatrices</a>()</span></code>.
This function is used by <code class="docutils notranslate"><span class="pre"><a href="../manualpages/PC/PCBJACOBI.html">PCBJACOBI</a></span></code> and <code class="docutils notranslate"><span class="pre"><a href="../manualpages/PC/PCASM.html">PCASM</a></span></code>.</p>
<p>Each submatrix may be parallel, existing on a <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Sys/MPI_Comm.html">MPI_Comm</a></span></code> associated with each pair of <code class="docutils notranslate"><span class="pre"><a href="../manualpages/IS/IS.html">IS</a></span></code> <code class="docutils notranslate"><span class="pre">rows[k]</span></code> and <code class="docutils notranslate"><span class="pre">cols[k]</span></code>,
using</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../manualpages/Mat/MatCreateSubMatricesMPI.html">MatCreateSubMatricesMPI</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">A</span><span class="p">,</span><span class="n"><a href="../manualpages/Sys/PetscInt.html">PetscInt</a></span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="w"> </span><span class="n">rows</span><span class="p">[],</span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="w"> </span><span class="n">cols</span><span class="p">[],</span><span class="n"><a href="../manualpages/Mat/MatReuse.html">MatReuse</a></span><span class="w"> </span><span class="n">call</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="o">*</span><span class="n">B</span><span class="p">[]);</span>
</pre></div>
</div>
<p>Finally this version has a specialization</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../manualpages/Mat/MatGetMultiProcBlock.html">MatGetMultiProcBlock</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">A</span><span class="p">,</span><span class="w"> </span><span class="n"><a href="../manualpages/Sys/MPI_Comm.html">MPI_Comm</a></span><span class="w"> </span><span class="n">subComm</span><span class="p">,</span><span class="w"> </span><span class="n"><a href="../manualpages/Mat/MatReuse.html">MatReuse</a></span><span class="w"> </span><span class="n">scall</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="o">*</span><span class="n">subMat</span><span class="p">);</span>
</pre></div>
</div>
<p>where collections of non-overlapping MPI processes share a single parallel matrix on their sub-communicator.
This function is used by <code class="docutils notranslate"><span class="pre"><a href="../manualpages/PC/PCBJACOBI.html">PCBJACOBI</a></span></code> and <code class="docutils notranslate"><span class="pre"><a href="../manualpages/PC/PCASM.html">PCASM</a></span></code>.</p>
<p>The routine</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../manualpages/Mat/MatCreateRedundantMatrix.html">MatCreateRedundantMatrix</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">A</span><span class="p">,</span><span class="n"><a href="../manualpages/Sys/PetscInt.html">PetscInt</a></span><span class="w"> </span><span class="n">nsubcomm</span><span class="p">,</span><span class="n"><a href="../manualpages/Sys/MPI_Comm.html">MPI_Comm</a></span><span class="w"> </span><span class="n">subcomm</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/MatReuse.html">MatReuse</a></span><span class="w"> </span><span class="n">reuse</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="o">*</span><span class="n">matredundant</span><span class="p">);</span>
</pre></div>
</div>
<p>where <code class="docutils notranslate"><span class="pre">nsubcomm</span></code> copies of the entire matrix are stored, one on each <code class="docutils notranslate"><span class="pre">subcomm</span></code>. The routine <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Sys/PetscSubcommCreate.html">PetscSubcommCreate</a>()</span></code> and its
<code class="docutils notranslate"><span class="pre"><a href="../manualpages/Sys/PetscSubcomm.html">PetscSubcomm</a></span></code> object may, but need not be, used to construct the <code class="docutils notranslate"><span class="pre">subcomm</span></code>.</p>
<p>The routine</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../manualpages/Mat/MatMPIAdjToSeq.html">MatMPIAdjToSeq</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">A</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="o">*</span><span class="n">B</span><span class="p">);</span>
</pre></div>
</div>
<p>is a specialization that duplicates an entire <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MATMPIADJ.html">MATMPIADJ</a></span></code> matrix on each MPI process.</p>
</section>
<section id="matrix-factorization">
<span id="sec-matfactor"></span><h2>Matrix Factorization<a class="headerlink" href="#matrix-factorization" title="Link to this heading">#</a></h2>
<p>Normally, PETSc users will access the matrix solvers through the <code class="docutils notranslate"><span class="pre"><a href="../manualpages/KSP/KSP.html">KSP</a></span></code>
interface, as discussed in <a class="reference internal" href="ksp.html#ch-ksp"><span class="std std-ref">KSP: Linear System Solvers</span></a>, but the
underlying factorization and triangular solve routines are also directly
accessible to the user.</p>
<p>The ILU, LU, ICC, Cholesky, and QR matrix factorizations are split into two or three
stages depending on the user’s needs. The first stage is to calculate an
ordering for the matrix. The ordering generally is done to reduce fill
in a sparse factorization; it does not make much sense for a dense
matrix.</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../manualpages/MatGraphOperations/MatGetOrdering.html">MatGetOrdering</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">matrix</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/MatOrderingType.html">MatOrderingType</a></span><span class="w"> </span><span class="n">type</span><span class="p">,</span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="o">*</span><span class="w"> </span><span class="n">rowperm</span><span class="p">,</span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="o">*</span><span class="w"> </span><span class="n">colperm</span><span class="p">);</span>
</pre></div>
</div>
<p>The currently available alternatives for the ordering <code class="docutils notranslate"><span class="pre">type</span></code> are</p>
<ul class="simple">
<li><p><code class="docutils notranslate"><span class="pre">MATORDERINGNATURAL</span></code> - Natural</p></li>
<li><p><code class="docutils notranslate"><span class="pre">MATORDERINGND</span></code> - Nested Dissection</p></li>
<li><p><code class="docutils notranslate"><span class="pre">MATORDERING1WD</span></code> - One-way Dissection</p></li>
<li><p><code class="docutils notranslate"><span class="pre">MATORDERINGRCM</span></code> - Reverse Cuthill-McKee</p></li>
<li><p><code class="docutils notranslate"><span class="pre">MATORDERINGQMD</span></code> - Quotient Minimum Degree</p></li>
</ul>
<p>These orderings can also be set through the options database.</p>
<p>Certain matrix formats may support only a subset of these. All of
these orderings are symmetric at the moment; ordering routines that are
not symmetric may be added. Currently we support orderings only for
sequential matrices.</p>
<p>Users can add their own ordering routines by providing a function with
the calling sequence</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="kt">int</span><span class="w"> </span><span class="nf">reorder</span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">A</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/MatOrderingType.html">MatOrderingType</a></span><span class="w"> </span><span class="n">type</span><span class="p">,</span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="o">*</span><span class="w"> </span><span class="n">rowperm</span><span class="p">,</span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="o">*</span><span class="w"> </span><span class="n">colperm</span><span class="p">);</span>
</pre></div>
</div>
<p>Here <code class="docutils notranslate"><span class="pre">A</span></code> is the matrix for which we wish to generate a new ordering,
<code class="docutils notranslate"><span class="pre">type</span></code> may be ignored and <code class="docutils notranslate"><span class="pre">rowperm</span></code> and <code class="docutils notranslate"><span class="pre">colperm</span></code> are the row and
column permutations generated by the ordering routine. The user
registers the ordering routine with the command</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../manualpages/MatGraphOperations/MatOrderingRegister.html">MatOrderingRegister</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/MatOrderingType.html">MatOrderingType</a></span><span class="w"> </span><span class="n">ordname</span><span class="p">,</span><span class="kt">char</span><span class="w"> </span><span class="o">*</span><span class="n">path</span><span class="p">,</span><span class="kt">char</span><span class="w"> </span><span class="o">*</span><span class="n">sname</span><span class="p">,</span><span class="n"><a href="../manualpages/Sys/PetscErrorCode.html">PetscErrorCode</a></span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="n">reorder</span><span class="p">)(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/MatOrderingType.html">MatOrderingType</a></span><span class="p">,</span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="o">*</span><span class="p">,</span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="o">*</span><span class="p">)));</span>
</pre></div>
</div>
<p>The input argument <code class="docutils notranslate"><span class="pre">ordname</span></code> is a string of the user’s choice,
either an ordering defined in <code class="docutils notranslate"><span class="pre">petscmat.h</span></code> or the name
of a new ordering introduced by the user. See the code in
<code class="docutils notranslate"><span class="pre">src/mat/impls/order/sorder.c</span></code> and other files in that
directory for examples on how the reordering routines may be written.</p>
<p>Once the reordering routine has been registered, it can be selected for
use at runtime with the command line option
<code class="docutils notranslate"><span class="pre">-pc_factor_mat_ordering_type</span></code> <code class="docutils notranslate"><span class="pre">ordname</span></code>. If reordering from the API, the
user should provide the <code class="docutils notranslate"><span class="pre">ordname</span></code> as the second input argument of
<code class="docutils notranslate"><span class="pre"><a href="../manualpages/MatGraphOperations/MatGetOrdering.html">MatGetOrdering</a>()</span></code>.</p>
<p>PETSc matrices interface to a variety of external factorization/solver packages via the <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MatSolverType.html">MatSolverType</a></span></code> which can be
<code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MATSOLVERSUPERLU_DIST.html">MATSOLVERSUPERLU_DIST</a></span></code>, <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MATSOLVERMUMPS.html">MATSOLVERMUMPS</a></span></code>, <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MATSOLVERPASTIX.html">MATSOLVERPASTIX</a></span></code>, <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MATSOLVERMKL_PARDISO.html">MATSOLVERMKL_PARDISO</a></span></code>, <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MATSOLVERMKL_CPARDISO.html">MATSOLVERMKL_CPARDISO</a></span></code>,
<code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MATSOLVERUMFPACK.html">MATSOLVERUMFPACK</a></span></code>, <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MATSOLVERCHOLMOD.html">MATSOLVERCHOLMOD</a></span></code>, <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MATSOLVERKLU.html">MATSOLVERKLU</a></span></code>,
<code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MATSOLVERCUSPARSE.html">MATSOLVERCUSPARSE</a></span></code>, and <code class="docutils notranslate"><span class="pre">MATSOLVERCUDA</span></code>.
The last three of which can run on GPUs, while <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MATSOLVERSUPERLU_DIST.html">MATSOLVERSUPERLU_DIST</a></span></code> can partially run on GPUs.
See <a class="reference internal" href="../overview/linear_solve_table.html#doc-linsolve"><span class="std std-ref">Summary of Sparse Linear Solvers Available In PETSc</span></a> for a table of the factorization based solvers in PETSc.</p>
<p>Most of these packages compute their own orderings and cannot use ones provided so calls to the following routines with those
packages can pass NULL as the <code class="docutils notranslate"><span class="pre"><a href="../manualpages/IS/IS.html">IS</a></span></code> permutations.</p>
<p>The following routines perform incomplete and complete, in-place, symbolic, and
numerical factorizations for symmetric and nonsymmetric matrices:</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../manualpages/Mat/MatICCFactor.html">MatICCFactor</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">matrix</span><span class="p">,</span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="w"> </span><span class="n">permutation</span><span class="p">,</span><span class="k">const</span><span class="w"> </span><span class="n"><a href="../manualpages/Mat/MatFactorInfo.html">MatFactorInfo</a></span><span class="w"> </span><span class="o">*</span><span class="n">info</span><span class="p">);</span>
<span class="n"><a href="../manualpages/Mat/MatCholeskyFactor.html">MatCholeskyFactor</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">matrix</span><span class="p">,</span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="w"> </span><span class="n">permutation</span><span class="p">,</span><span class="k">const</span><span class="w"> </span><span class="n"><a href="../manualpages/Mat/MatFactorInfo.html">MatFactorInfo</a></span><span class="w"> </span><span class="o">*</span><span class="n">info</span><span class="p">);</span>
<span class="n"><a href="../manualpages/Mat/MatILUFactor.html">MatILUFactor</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">matrix</span><span class="p">,</span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="w"> </span><span class="n">rowpermutation</span><span class="p">,</span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="w"> </span><span class="n">columnpermutation</span><span class="p">,</span><span class="k">const</span><span class="w"> </span><span class="n"><a href="../manualpages/Mat/MatFactorInfo.html">MatFactorInfo</a></span><span class="w"> </span><span class="o">*</span><span class="n">info</span><span class="p">);</span>
<span class="n"><a href="../manualpages/Mat/MatLUFactor.html">MatLUFactor</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">matrix</span><span class="p">,</span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="w"> </span><span class="n">rowpermutation</span><span class="p">,</span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="w"> </span><span class="n">columnpermutation</span><span class="p">,</span><span class="k">const</span><span class="w"> </span><span class="n"><a href="../manualpages/Mat/MatFactorInfo.html">MatFactorInfo</a></span><span class="w"> </span><span class="o">*</span><span class="n">info</span><span class="p">);</span>
<span class="n"><a href="../manualpages/Mat/MatQRFactor.html">MatQRFactor</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">matrix</span><span class="p">,</span><span class="w"> </span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="w"> </span><span class="n">columnpermutation</span><span class="p">,</span><span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n"><a href="../manualpages/Mat/MatFactorInfo.html">MatFactorInfo</a></span><span class="w"> </span><span class="o">*</span><span class="n">info</span><span class="p">);</span>
</pre></div>
</div>
<p>The argument <code class="docutils notranslate"><span class="pre">info->fill</span> <span class="pre">></span> <span class="pre">1</span></code> is the predicted fill expected in the
factored matrix, as a ratio of the original fill. For example,
<code class="docutils notranslate"><span class="pre">info->fill</span> <span class="pre">=</span> <span class="pre">2.0</span></code> would indicate that one expects the factored matrix
to have twice as many nonzeros as the original.</p>
<p>For sparse matrices it is very unlikely that the factorization is
actually done in-place. More likely, new space is allocated for the
factored matrix and the old space deallocated, but to the user it
appears in-place because the factored matrix replaces the unfactored
matrix.</p>
<p>The two factorization stages can also be performed separately, by using
the preferred out-of-place mode, first one obtains that matrix object that will
hold the factor using</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../manualpages/Mat/MatGetFactor.html">MatGetFactor</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">matrix</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/MatSolverType.html">MatSolverType</a></span><span class="w"> </span><span class="n">package</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/MatFactorType.html">MatFactorType</a></span><span class="w"> </span><span class="n">ftype</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="o">*</span><span class="n">factor</span><span class="p">);</span>
</pre></div>
</div>
<p>and then performs the factorization</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../manualpages/Mat/MatICCFactorSymbolic.html">MatICCFactorSymbolic</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">factor</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">matrix</span><span class="p">,</span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="w"> </span><span class="n">perm</span><span class="p">,</span><span class="k">const</span><span class="w"> </span><span class="n"><a href="../manualpages/Mat/MatFactorInfo.html">MatFactorInfo</a></span><span class="w"> </span><span class="o">*</span><span class="n">info</span><span class="p">);</span>
<span class="n"><a href="../manualpages/Mat/MatCholeskyFactorSymbolic.html">MatCholeskyFactorSymbolic</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">factor</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">matrix</span><span class="p">,</span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="w"> </span><span class="n">perm</span><span class="p">,</span><span class="k">const</span><span class="w"> </span><span class="n"><a href="../manualpages/Mat/MatFactorInfo.html">MatFactorInfo</a></span><span class="w"> </span><span class="o">*</span><span class="n">info</span><span class="p">);</span>
<span class="n"><a href="../manualpages/Mat/MatCholeskyFactorNumeric.html">MatCholeskyFactorNumeric</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">factor</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">matrix</span><span class="p">,</span><span class="k">const</span><span class="w"> </span><span class="n"><a href="../manualpages/Mat/MatFactorInfo.html">MatFactorInfo</a></span><span class="p">);</span>
<span class="n"><a href="../manualpages/Mat/MatILUFactorSymbolic.html">MatILUFactorSymbolic</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">factor</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">matrix</span><span class="p">,</span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="w"> </span><span class="n">rowperm</span><span class="p">,</span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="w"> </span><span class="n">colperm</span><span class="p">,</span><span class="k">const</span><span class="w"> </span><span class="n"><a href="../manualpages/Mat/MatFactorInfo.html">MatFactorInfo</a></span><span class="w"> </span><span class="o">*</span><span class="n">info</span><span class="p">);</span>
<span class="n"><a href="../manualpages/Mat/MatLUFactorSymbolic.html">MatLUFactorSymbolic</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">factor</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">matrix</span><span class="p">,</span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="w"> </span><span class="n">rowperm</span><span class="p">,</span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="w"> </span><span class="n">colperm</span><span class="p">,</span><span class="k">const</span><span class="w"> </span><span class="n"><a href="../manualpages/Mat/MatFactorInfo.html">MatFactorInfo</a></span><span class="w"> </span><span class="o">*</span><span class="n">info</span><span class="p">);</span>
<span class="n"><a href="../manualpages/Mat/MatLUFactorNumeric.html">MatLUFactorNumeric</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">factor</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">matrix</span><span class="p">,</span><span class="k">const</span><span class="w"> </span><span class="n"><a href="../manualpages/Mat/MatFactorInfo.html">MatFactorInfo</a></span><span class="w"> </span><span class="o">*</span><span class="n">info</span><span class="p">);</span>
<span class="n"><a href="../manualpages/Mat/MatQRFactorSymbolic.html">MatQRFactorSymbolic</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">factor</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">matrix</span><span class="p">,</span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="w"> </span><span class="n">perm</span><span class="p">,</span><span class="k">const</span><span class="w"> </span><span class="n"><a href="../manualpages/Mat/MatFactorInfo.html">MatFactorInfo</a></span><span class="w"> </span><span class="o">*</span><span class="n">info</span><span class="p">);</span>
<span class="n"><a href="../manualpages/Mat/MatQRFactorNumeric.html">MatQRFactorNumeric</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">factor</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">matrix</span><span class="p">,</span><span class="k">const</span><span class="w"> </span><span class="n"><a href="../manualpages/Mat/MatFactorInfo.html">MatFactorInfo</a></span><span class="w"> </span><span class="o">*</span><span class="n">info</span><span class="p">);</span>
</pre></div>
</div>
<p>In this case, the contents of the matrix <code class="docutils notranslate"><span class="pre">factor</span></code> is undefined between
the symbolic and numeric factorization stages. It is possible to reuse
the symbolic factorization. For the second and succeeding
factorizations, one simply calls the numerical factorization with a new
input <code class="docutils notranslate"><span class="pre">matrix</span></code> and the <em>same</em> factored <code class="docutils notranslate"><span class="pre">factor</span></code> matrix. It is
<em>essential</em> that the new input matrix have exactly the same nonzero
structure as the original factored matrix. (The numerical factorization
merely overwrites the numerical values in the factored matrix and does
not disturb the symbolic portion, thus enabling reuse of the symbolic
phase.) In general, calling <code class="docutils notranslate"><span class="pre">XXXFactorSymbolic</span></code> with a dense matrix
will do nothing except allocate the new matrix; the <code class="docutils notranslate"><span class="pre">XXXFactorNumeric</span></code>
routines will do all of the work.</p>
<p>Why provide the plain <code class="docutils notranslate"><span class="pre">XXXFactor</span></code> routines when one could simply call
the two-stage routines? The answer is that if one desires in-place
factorization of a sparse matrix, the intermediate stage between the
symbolic and numeric phases cannot be stored in a <code class="docutils notranslate"><span class="pre">factor</span></code> matrix, and
it does not make sense to store the intermediate values inside the
original matrix that is being transformed. We originally made the
combined factor routines do either in-place or out-of-place
factorization, but then decided that this approach was not needed and
could easily lead to confusion.</p>
<p>We do not provide our own sparse matrix factorization with pivoting
for numerical stability. This is because trying to both reduce fill and
do pivoting can become quite complicated. Instead, we provide a poor
stepchild substitute. After one has obtained a reordering, with
<code class="docutils notranslate"><span class="pre"><a href="../manualpages/MatGraphOperations/MatGetOrdering.html">MatGetOrdering</a>(<a href="../manualpages/Mat/Mat.html">Mat</a></span> <span class="pre">A,MatOrdering</span> <span class="pre">type,<a href="../manualpages/IS/IS.html">IS</a></span> <span class="pre">*row,<a href="../manualpages/IS/IS.html">IS</a></span> <span class="pre">*col)</span></code> one may call</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../manualpages/Mat/MatReorderForNonzeroDiagonal.html">MatReorderForNonzeroDiagonal</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">A</span><span class="p">,</span><span class="n"><a href="../manualpages/Sys/PetscReal.html">PetscReal</a></span><span class="w"> </span><span class="n">tol</span><span class="p">,</span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="w"> </span><span class="n">row</span><span class="p">,</span><span class="w"> </span><span class="n"><a href="../manualpages/IS/IS.html">IS</a></span><span class="w"> </span><span class="n">col</span><span class="p">);</span>
</pre></div>
</div>
<p>which will try to reorder the columns to ensure that no values along the
diagonal are smaller than <code class="docutils notranslate"><span class="pre">tol</span></code> in a absolute value. If small values
are detected and corrected for, a nonsymmetric permutation of the rows
and columns will result. This is not guaranteed to work, but may help if
one was simply unlucky in the original ordering. When using the <code class="docutils notranslate"><span class="pre"><a href="../manualpages/KSP/KSP.html">KSP</a></span></code>
solver interface the option <code class="docutils notranslate"><span class="pre">-pc_factor_nonzeros_along_diagonal</span> <span class="pre"><tol></span></code>
may be used. Here, <code class="docutils notranslate"><span class="pre">tol</span></code> is an optional tolerance to decide if a value
is nonzero; by default it is <code class="docutils notranslate"><span class="pre">1.e-10</span></code>.</p>
<p>The external <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MatSolverType.html">MatSolverType</a></span></code>’s <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MATSOLVERSUPERLU_DIST.html">MATSOLVERSUPERLU_DIST</a></span></code> and <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MATSOLVERMUMPS.html">MATSOLVERMUMPS</a></span></code>
do manage numerical pivoting internal to their API.</p>
<p>The external factorization packages each provide a wide number of options to chose from,
details on these may be found by consulting the manual page for the solver package, such as,
<code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MATSOLVERSUPERLU_DIST.html">MATSOLVERSUPERLU_DIST</a></span></code>. Most of the options can be easily set via the options database
even when the factorization solvers are accessed via <code class="docutils notranslate"><span class="pre"><a href="../manualpages/KSP/KSP.html">KSP</a></span></code>.</p>
<p>Once a matrix has been factored, it is natural to solve linear systems.
The following four routines enable this process:</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../manualpages/Mat/MatSolve.html">MatSolve</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">A</span><span class="p">,</span><span class="n"><a href="../manualpages/Vec/Vec.html">Vec</a></span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n"><a href="../manualpages/Vec/Vec.html">Vec</a></span><span class="w"> </span><span class="n">y</span><span class="p">);</span>
<span class="n"><a href="../manualpages/Mat/MatSolveTranspose.html">MatSolveTranspose</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">A</span><span class="p">,</span><span class="w"> </span><span class="n"><a href="../manualpages/Vec/Vec.html">Vec</a></span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n"><a href="../manualpages/Vec/Vec.html">Vec</a></span><span class="w"> </span><span class="n">y</span><span class="p">);</span>
<span class="n"><a href="../manualpages/Mat/MatSolveAdd.html">MatSolveAdd</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">A</span><span class="p">,</span><span class="n"><a href="../manualpages/Vec/Vec.html">Vec</a></span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n"><a href="../manualpages/Vec/Vec.html">Vec</a></span><span class="w"> </span><span class="n">y</span><span class="p">,</span><span class="w"> </span><span class="n"><a href="../manualpages/Vec/Vec.html">Vec</a></span><span class="w"> </span><span class="n">w</span><span class="p">);</span>
<span class="n"><a href="../manualpages/Mat/MatSolveTransposeAdd.html">MatSolveTransposeAdd</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">A</span><span class="p">,</span><span class="w"> </span><span class="n"><a href="../manualpages/Vec/Vec.html">Vec</a></span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n"><a href="../manualpages/Vec/Vec.html">Vec</a></span><span class="w"> </span><span class="n">y</span><span class="p">,</span><span class="w"> </span><span class="n"><a href="../manualpages/Vec/Vec.html">Vec</a></span><span class="w"> </span><span class="n">w</span><span class="p">);</span>
</pre></div>
</div>
<p>matrix <code class="docutils notranslate"><span class="pre">A</span></code> of these routines must have been obtained from a
factorization routine; otherwise, an error will be generated. In
general, the user should use the <code class="docutils notranslate"><span class="pre"><a href="../manualpages/KSP/KSP.html">KSP</a></span></code> solvers introduced in the next
chapter rather than using these factorization and solve routines
directly.</p>
<p>Some of the factorizations also support solves with multiple right-hand sides stored in a <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/Mat.html">Mat</a></span></code> using</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../manualpages/Mat/MatMatSolve.html">MatMatSolve</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">A</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">B</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">X</span><span class="p">);</span>
</pre></div>
</div>
<p>and</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../manualpages/Mat/MatMatSolveTranspose.html">MatMatSolveTranspose</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">A</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">B</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">X</span><span class="p">);</span>
</pre></div>
</div>
<p>Finally, <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MATSOLVERMUMPS.html">MATSOLVERMUMPS</a></span></code>, provides access to Schur complements obtained after partial factorizations as well
as the inertia of a matrix via <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MatGetInertia.html">MatGetInertia</a>()</span></code>.</p>
</section>
<section id="matrix-matrix-products">
<span id="sec-matmatproduct"></span><h2>Matrix-Matrix Products<a class="headerlink" href="#matrix-matrix-products" title="Link to this heading">#</a></h2>
<p>PETSc matrices provide code for computing various matrix-matrix products. This section will introduce the two sets of routines
available. For now consult <code class="docutils notranslate"><span class="pre">MatCreateProduct()</span></code> and <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MatMatMult.html">MatMatMult</a>()</span></code>.</p>
</section>
<section id="creating-pc-s-directly">
<h2>Creating PC’s Directly<a class="headerlink" href="#creating-pc-s-directly" title="Link to this heading">#</a></h2>
<p>Users obtain their preconditioner contexts from the <code class="docutils notranslate"><span class="pre"><a href="../manualpages/KSP/KSP.html">KSP</a></span></code>
context with the command <code class="docutils notranslate"><span class="pre"><a href="../manualpages/KSP/KSPGetPC.html">KSPGetPC</a>()</span></code>. It is possible to create,
manipulate, and destroy <code class="docutils notranslate"><span class="pre"><a href="../manualpages/PC/PC.html">PC</a></span></code> contexts directly, although this
capability should rarely be needed. To create a <code class="docutils notranslate"><span class="pre"><a href="../manualpages/PC/PC.html">PC</a></span></code> context, one uses
the command</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../manualpages/PC/PCCreate.html">PCCreate</a></span><span class="p">(</span><span class="n"><a href="../manualpages/Sys/MPI_Comm.html">MPI_Comm</a></span><span class="w"> </span><span class="n">comm</span><span class="p">,</span><span class="n"><a href="../manualpages/PC/PC.html">PC</a></span><span class="w"> </span><span class="o">*</span><span class="n">pc</span><span class="p">);</span>
</pre></div>
</div>
<p>The routine</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../manualpages/PC/PCSetType.html">PCSetType</a></span><span class="p">(</span><span class="n"><a href="../manualpages/PC/PC.html">PC</a></span><span class="w"> </span><span class="n">pc</span><span class="p">,</span><span class="n"><a href="../manualpages/PC/PCType.html">PCType</a></span><span class="w"> </span><span class="n">method</span><span class="p">);</span>
</pre></div>
</div>
<p>sets the preconditioner method to be used. The routine</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../manualpages/PC/PCSetOperators.html">PCSetOperators</a></span><span class="p">(</span><span class="n"><a href="../manualpages/PC/PC.html">PC</a></span><span class="w"> </span><span class="n">pc</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">Amat</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="n">Pmat</span><span class="p">);</span>
</pre></div>
</div>
<p>set the matrices that are to be used with the preconditioner. The
routine</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../manualpages/PC/PCGetOperators.html">PCGetOperators</a></span><span class="p">(</span><span class="n"><a href="../manualpages/PC/PC.html">PC</a></span><span class="w"> </span><span class="n">pc</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="o">*</span><span class="n">Amat</span><span class="p">,</span><span class="n"><a href="../manualpages/Mat/Mat.html">Mat</a></span><span class="w"> </span><span class="o">*</span><span class="n">Pmat</span><span class="p">);</span>
</pre></div>
</div>
<p>returns the values set with <code class="docutils notranslate"><span class="pre"><a href="../manualpages/PC/PCSetOperators.html">PCSetOperators</a>()</span></code>.</p>
<p>The preconditioners in PETSc can be used in several ways. The two most
basic routines simply apply the preconditioner or its transpose and are
given, respectively, by</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../manualpages/PC/PCApply.html">PCApply</a></span><span class="p">(</span><span class="n"><a href="../manualpages/PC/PC.html">PC</a></span><span class="w"> </span><span class="n">pc</span><span class="p">,</span><span class="n"><a href="../manualpages/Vec/Vec.html">Vec</a></span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="n"><a href="../manualpages/Vec/Vec.html">Vec</a></span><span class="w"> </span><span class="n">y</span><span class="p">);</span>
<span class="n"><a href="../manualpages/PC/PCApplyTranspose.html">PCApplyTranspose</a></span><span class="p">(</span><span class="n"><a href="../manualpages/PC/PC.html">PC</a></span><span class="w"> </span><span class="n">pc</span><span class="p">,</span><span class="n"><a href="../manualpages/Vec/Vec.html">Vec</a></span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="n"><a href="../manualpages/Vec/Vec.html">Vec</a></span><span class="w"> </span><span class="n">y</span><span class="p">);</span>
</pre></div>
</div>
<p>In particular, for a preconditioner matrix, <code class="docutils notranslate"><span class="pre">B</span></code>, that has been set via
<code class="docutils notranslate"><span class="pre"><a href="../manualpages/PC/PCSetOperators.html">PCSetOperators</a>(pc,Amat,Pmat)</span></code>, the routine PCApply(pc,x,y) computes
<span class="math">\(y = B^{-1} x\)</span> by solving the linear system <span class="math">\(By = x\)</span> with
the specified preconditioner method.</p>
<p>Additional preconditioner routines are</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../manualpages/PC/PCApplyBAorAB.html">PCApplyBAorAB</a></span><span class="p">(</span><span class="n"><a href="../manualpages/PC/PC.html">PC</a></span><span class="w"> </span><span class="n">pc</span><span class="p">,</span><span class="n"><a href="../manualpages/PC/PCSide.html">PCSide</a></span><span class="w"> </span><span class="n">right</span><span class="p">,</span><span class="n"><a href="../manualpages/Vec/Vec.html">Vec</a></span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="n"><a href="../manualpages/Vec/Vec.html">Vec</a></span><span class="w"> </span><span class="n">y</span><span class="p">,</span><span class="n"><a href="../manualpages/Vec/Vec.html">Vec</a></span><span class="w"> </span><span class="n">work</span><span class="p">);</span>
<span class="n"><a href="../manualpages/PC/PCApplyBAorABTranspose.html">PCApplyBAorABTranspose</a></span><span class="p">(</span><span class="n"><a href="../manualpages/PC/PC.html">PC</a></span><span class="w"> </span><span class="n">pc</span><span class="p">,</span><span class="n"><a href="../manualpages/PC/PCSide.html">PCSide</a></span><span class="w"> </span><span class="n">right</span><span class="p">,</span><span class="n"><a href="../manualpages/Vec/Vec.html">Vec</a></span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="n"><a href="../manualpages/Vec/Vec.html">Vec</a></span><span class="w"> </span><span class="n">y</span><span class="p">,</span><span class="n"><a href="../manualpages/Vec/Vec.html">Vec</a></span><span class="w"> </span><span class="n">work</span><span class="p">);</span>
<span class="n"><a href="../manualpages/PC/PCApplyRichardson.html">PCApplyRichardson</a></span><span class="p">(</span><span class="n"><a href="../manualpages/PC/PC.html">PC</a></span><span class="w"> </span><span class="n">pc</span><span class="p">,</span><span class="n"><a href="../manualpages/Vec/Vec.html">Vec</a></span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="n"><a href="../manualpages/Vec/Vec.html">Vec</a></span><span class="w"> </span><span class="n">y</span><span class="p">,</span><span class="n"><a href="../manualpages/Vec/Vec.html">Vec</a></span><span class="w"> </span><span class="n">work</span><span class="p">,</span><span class="n"><a href="../manualpages/Sys/PetscReal.html">PetscReal</a></span><span class="w"> </span><span class="n">rtol</span><span class="p">,</span><span class="n"><a href="../manualpages/Sys/PetscReal.html">PetscReal</a></span><span class="w"> </span><span class="n">atol</span><span class="p">,</span><span class="w"> </span><span class="n"><a href="../manualpages/Sys/PetscReal.html">PetscReal</a></span><span class="w"> </span><span class="n">dtol</span><span class="p">,</span><span class="n"><a href="../manualpages/Sys/PetscInt.html">PetscInt</a></span><span class="w"> </span><span class="n">maxits</span><span class="p">,</span><span class="n"><a href="../manualpages/Sys/PetscBool.html">PetscBool</a></span><span class="w"> </span><span class="n">zeroguess</span><span class="p">,</span><span class="n"><a href="../manualpages/Sys/PetscInt.html">PetscInt</a></span><span class="w"> </span><span class="o">*</span><span class="n">outits</span><span class="p">,</span><span class="n"><a href="../manualpages/PC/PCRichardsonConvergedReason.html">PCRichardsonConvergedReason</a></span><span class="o">*</span><span class="p">);</span>
</pre></div>
</div>
<p>The first two routines apply the action of the matrix followed by the
preconditioner or the preconditioner followed by the matrix depending on
whether the <code class="docutils notranslate"><span class="pre">right</span></code> is <code class="docutils notranslate"><span class="pre"><a href="../manualpages/PC/PCSide.html">PC_LEFT</a></span></code> or <code class="docutils notranslate"><span class="pre"><a href="../manualpages/PC/PCSide.html">PC_RIGHT</a></span></code>. The final routine
applies <code class="docutils notranslate"><span class="pre">its</span></code> iterations of Richardson’s method. The last three
routines are provided to improve efficiency for certain Krylov subspace
methods.</p>
<p>A <code class="docutils notranslate"><span class="pre"><a href="../manualpages/PC/PC.html">PC</a></span></code> context that is no longer needed can be destroyed with the
command</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../manualpages/PC/PCDestroy.html">PCDestroy</a></span><span class="p">(</span><span class="n"><a href="../manualpages/PC/PC.html">PC</a></span><span class="w"> </span><span class="o">*</span><span class="n">pc</span><span class="p">);</span>
</pre></div>
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