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<section class="tex2jax_ignore mathjax_ignore" id="pctelescope">
<h1>PCTELESCOPE<a class="headerlink" href="#pctelescope" title="Link to this heading">#</a></h1>
<p>Runs a <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/KSP/KSP.html">KSP</a></span></code> solver on a sub-communicator <span id="id1">[<a class="reference internal" href="#id835" title="Dave A. May, Patrick Sanan, Karl Rupp, Matthew G. Knepley, and Barry F. Smith. Extreme-scale multigrid components within PETSc. In Proceedings of the Platform for Advanced Scientific Computing Conference, PASC '16, 5:1–5:12. New York, NY, USA, 2016. ACM. doi:10.1145/2929908.2929913.">MSR+16</a>]</span> of the communicator used by the original <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/KSP/KSP.html">KSP</a></span></code>. MPI processes not in the sub-communicator are idle during the solve. Usually used to solve the smaller coarser grid problems in multigrid (<code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PC/PCMG.html">PCMG</a></span></code>) that could not be efficiently solved on the entire communication</p>
<section id="options-database-keys">
<h2>Options Database Keys<a class="headerlink" href="#options-database-keys" title="Link to this heading">#</a></h2>
<ul class="simple">
<li><p><em><strong>-pc_telescope_reduction_factor <r> -</strong></em> factor to reduce the communicator size by. e.g. with 64 MPI ranks and r=4, the new sub-communicator will have 64/4 = 16 ranks.</p></li>
<li><p><em><strong>-pc_telescope_ignore_dm -</strong></em> flag to indicate whether an attached <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DM/DM.html">DM</a></span></code> should be ignored in constructing the new <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PC/PC.html">PC</a></span></code></p></li>
<li><p><em><strong>-pc_telescope_subcomm_type <interlaced,contiguous> -</strong></em> defines the selection of MPI processes on the sub-communicator. see <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/Sys/PetscSubcomm.html">PetscSubcomm</a></span></code> for more information.</p></li>
<li><p><em><strong>-pc_telescope_ignore_kspcomputeoperators -</strong></em> flag to indicate whether <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/KSP/KSPSetComputeOperators.html">KSPSetComputeOperators</a>()</span></code> should be used on the sub-<code class="docutils notranslate"><span class="pre"><a href="../../manualpages/KSP/KSP.html">KSP</a></span></code>.</p></li>
<li><p><em><strong>-pc_telescope_use_coarse_dm -</strong></em> flag to indicate whether the coarse <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DM/DM.html">DM</a></span></code> should be used to define the sub-communicator.</p></li>
</ul>
</section>
<section id="notes">
<h2>Notes<a class="headerlink" href="#notes" title="Link to this heading">#</a></h2>
<p>Assuming that the parent preconditioner <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PC/PC.html">PC</a></span></code> is defined on a communicator c, this implementation
creates a child sub-communicator (c’) containing fewer MPI processes than the original parent preconditioner <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PC/PC.html">PC</a></span></code>.
The preconditioner is deemed telescopic as it only calls <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/KSP/KSPSolve.html">KSPSolve</a>()</span></code> on a single
sub-communicator, in contrast with <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PC/PCREDUNDANT.html">PCREDUNDANT</a></span></code> which calls <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/KSP/KSPSolve.html">KSPSolve</a>()</span></code> on N sub-communicators.
This means there will be MPI processes which will be idle during the application of this preconditioner.
Additionally, in comparison with <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PC/PCREDUNDANT.html">PCREDUNDANT</a></span></code>, <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PC/PCTELESCOPE.html">PCTELESCOPE</a></span></code> can utilize an attached <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DM/DM.html">DM</a></span></code> to construct <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DM/DM.html">DM</a></span></code> dependent preconditioner, such as <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PC/PCMG.html">PCMG</a></span></code></p>
<p>The default type <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/KSP/KSPType.html">KSPType</a></span></code> of the sub <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/KSP/KSP.html">KSP</a></span></code> (the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/KSP/KSP.html">KSP</a></span></code> defined on c’) is <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/KSP/KSPPREONLY.html">KSPPREONLY</a></span></code>.</p>
<p>There are three setup mechanisms for <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PC/PCTELESCOPE.html">PCTELESCOPE</a></span></code>. Features support by each type are described below.
In the following, we will refer to the operators B and B’, these are the <code class="docutils notranslate"><span class="pre">Bmat</span></code> provided to the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/KSP/KSP.html">KSP</a></span></code> on the
communicators c and c’ respectively.</p>
<p>[1] Default setup
The sub-communicator c’ is created via <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/Sys/PetscSubcommCreate.html">PetscSubcommCreate</a>()</span></code>.
Any explicitly defined nullspace and near nullspace vectors attached to B with <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/Mat/MatSetNullSpace.html">MatSetNullSpace</a>()</span></code> and <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/Mat/MatSetNearNullSpace.html">MatSetNearNullSpace</a>()</span></code> are transferred to B’.
Currently there is no support for nullspaces provided with <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/Mat/MatNullSpaceSetFunction.html">MatNullSpaceSetFunction</a>()</span></code>).
No support is provided for <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/KSP/KSPSetComputeOperators.html">KSPSetComputeOperators</a>()</span></code>.
Currently there is no support for the flag <code class="docutils notranslate"><span class="pre">-pc_use_amat</span></code>.</p>
<p>[2] <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DM/DM.html">DM</a></span></code> aware setup
The sub-communicator c’ is created via <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/Sys/PetscSubcommCreate.html">PetscSubcommCreate</a>()</span></code>.
If a <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DM/DM.html">DM</a></span></code> is attached to the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PC/PC.html">PC</a></span></code>, it is re-partitioned on the sub-communicator c’.
Both the <code class="docutils notranslate"><span class="pre">Bmat</span></code> operator and the right-hand side vector are permuted into the new DOF ordering defined by the re-partitioned <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DM/DM.html">DM</a></span></code>.
Currently only support for re-partitioning a <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DMDA/DMDA.html">DMDA</a></span></code> is provided.
Any explicitly defined nullspace or near nullspace vectors attached to the original B with <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/Mat/MatSetNullSpace.html">MatSetNullSpace</a>()</span></code>
and <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/Mat/MatSetNearNullSpace.html">MatSetNearNullSpace</a>()</span></code> are extracted, re-partitioned and set on B’
(currently there is no support for nullspaces provided with <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/Mat/MatNullSpaceSetFunction.html">MatNullSpaceSetFunction</a>()</span></code>).
Support is provided for <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/KSP/KSPSetComputeOperators.html">KSPSetComputeOperators</a>()</span></code>. The user provided function and context is propagated to the sub <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/KSP/KSP.html">KSP</a></span></code>.
This is fragile since the user must ensure that their user context is valid for use on c’.
Currently there is no support for the flag <code class="docutils notranslate"><span class="pre">-pc_use_amat</span></code>.</p>
<p>[3] Coarse <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DM/DM.html">DM</a></span></code> setup
If a <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DM/DM.html">DM</a></span></code> (dmfine) is attached to the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PC/PC.html">PC</a></span></code>, dmfine is queried for a “coarse” <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DM/DM.html">DM</a></span></code> (call this dmcoarse) via <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DM/DMGetCoarseDM.html">DMGetCoarseDM</a>()</span></code>.
<code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PC/PCTELESCOPE.html">PCTELESCOPE</a></span></code> will interpret the coarse <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DM/DM.html">DM</a></span></code> as being defined on a sub-communicator of c.
The communicator associated with dmcoarse will define the c’ to be used within <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PC/PCTELESCOPE.html">PCTELESCOPE</a></span></code>.
<code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PC/PCTELESCOPE.html">PCTELESCOPE</a></span></code> will check that c’ is in fact a sub-communicator of c. If it is not, an error will be reported.
The intention of this setup type is that <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PC/PCTELESCOPE.html">PCTELESCOPE</a></span></code> will use an existing (e.g. user defined) communicator hierarchy, say as would be
available with using multi-grid on unstructured meshes.
This setup will not use the command line options <code class="docutils notranslate"><span class="pre">-pc_telescope_reduction_factor</span></code> or <code class="docutils notranslate"><span class="pre">-pc_telescope_subcomm_type</span></code>.
Any explicitly defined nullspace or near nullspace vectors attached to the B are extracted, scattered into the correct ordering consistent
with dmcoarse and set on B’
(currently there is no support for nullspaces provided with <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/Mat/MatNullSpaceSetFunction.html">MatNullSpaceSetFunction</a>()</span></code>).
There is no general method to permute field orderings, hence only <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/KSP/KSPSetComputeOperators.html">KSPSetComputeOperators</a>()</span></code> is supported.
The user must use <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/Sys/PetscObjectComposeFunction.html">PetscObjectComposeFunction</a>()</span></code> with dmfine to define the method to scatter fields from dmfine to dmcoarse.
Propagation of the user context for <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/KSP/KSPSetComputeOperators.html">KSPSetComputeOperators</a>()</span></code> on the sub <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/KSP/KSP.html">KSP</a></span></code> is attempted by querying the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DM/DM.html">DM</a></span></code> contexts associated with
dmfine and dmcoarse. Alternatively, the user may use <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/Sys/PetscObjectComposeFunction.html">PetscObjectComposeFunction</a>()</span></code> with dmcoarse to define a method which will return the appropriate user context for <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/KSP/KSPSetComputeOperators.html">KSPSetComputeOperators</a>()</span></code>.
Currently there is no support for the flag <code class="docutils notranslate"><span class="pre">-pc_use_amat</span></code>.
This setup can be invoked by the option <code class="docutils notranslate"><span class="pre">-pc_telescope_use_coarse_dm</span></code> or by calling <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PC/PCTelescopeSetUseCoarseDM.html">PCTelescopeSetUseCoarseDM</a></span></code>(pc,<code class="docutils notranslate"><span class="pre"><a href="../../manualpages/Sys/PETSC_TRUE.html">PETSC_TRUE</a></span></code>);
Further information about the user-provided methods required by this setup type are described here <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PC/PCTelescopeSetUseCoarseDM.html">PCTelescopeSetUseCoarseDM</a>()</span></code>.</p>
</section>
<section id="developer-notes">
<h2>Developer Notes<a class="headerlink" href="#developer-notes" title="Link to this heading">#</a></h2>
<p>During <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PC/PCSetUp.html">PCSetUp</a>()</span></code>, the B operator is scattered onto c’.
Within <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PC/PCApply.html">PCApply</a>()</span></code>, the RHS vector (x) is scattered into a redundant vector, xred (defined on c’).
Then, <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/KSP/KSPSolve.html">KSPSolve</a>()</span></code> is executed on the c’ communicator.</p>
<p>The communicator used within the telescoping preconditioner is defined by a <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/Sys/PetscSubcomm.html">PetscSubcomm</a></span></code> using the INTERLACED
creation routine by default (this can be changed with <code class="docutils notranslate"><span class="pre">-pc_telescope_subcomm_type</span></code>). We run the sub <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/KSP/KSP.html">KSP</a></span></code> on only
the ranks within the communicator which have a color equal to zero.</p>
<p>The telescoping preconditioner is aware of nullspaces and near nullspaces which are attached to the B operator.
In the case where B has a (near) nullspace attached, the (near) nullspace vectors are extracted from B and mapped into
a new (near) nullspace, defined on the sub-communicator, which is attached to B’ (the B operator which was scattered to c’)</p>
<p>The telescoping preconditioner can re-partition an attached <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DM/DM.html">DM</a></span></code> if it is a <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DMDA/DMDA.html">DMDA</a></span></code> (2D or 3D -
support for 1D <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DMDA/DMDA.html">DMDA</a></span></code>s is not provided). If a <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DMDA/DMDA.html">DMDA</a></span></code> is found, a topologically equivalent <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DMDA/DMDA.html">DMDA</a></span></code> is created on c’
and this new <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DM/DM.html">DM</a></span></code> is attached the sub <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/KSP/KSP.html">KSP</a></span></code>. The design of telescope is such that it should be possible to extend support
for re-partitioning other to <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DM/DM.html">DM</a></span></code>’s (e.g. <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DMPlex/DMPLEX.html">DMPLEX</a></span></code>). The user can supply a flag to ignore attached DMs.
Alternatively, user-provided re-partitioned <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DM/DM.html">DM</a></span></code>s can be used via <code class="docutils notranslate"><span class="pre">-pc_telescope_use_coarse_dm</span></code>.</p>
<p>With the default setup mode, B’ is defined by fusing rows (in order) associated with MPI processes common to c and c’.</p>
<p>When a <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DMDA/DMDA.html">DMDA</a></span></code> is attached to the parent preconditioner, B’ is defined by: (i) performing a symmetric permutation of B
into the ordering defined by the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DMDA/DMDA.html">DMDA</a></span></code> on c’, (ii) extracting the local chunks via <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/Mat/MatCreateSubMatrices.html">MatCreateSubMatrices</a>()</span></code>, (iii) fusing the
locally (sequential) matrices defined on the ranks common to c and c’ into B’ using <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/Mat/MatCreateMPIMatConcatenateSeqMat.html">MatCreateMPIMatConcatenateSeqMat</a>()</span></code></p>
<p>Limitations/improvements include the following.
<code class="docutils notranslate"><span class="pre"><a href="../../manualpages/Vec/VecPlaceArray.html">VecPlaceArray</a>()</span></code> could be used within <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PC/PCApply.html">PCApply</a>()</span></code> to improve efficiency and reduce memory usage.
A unified mechanism to query for user contexts as required by <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/KSP/KSPSetComputeOperators.html">KSPSetComputeOperators</a>()</span></code> and <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/Mat/MatNullSpaceSetFunction.html">MatNullSpaceSetFunction</a>()</span></code>.</p>
<p>The symmetric permutation used when a <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DMDA/DMDA.html">DMDA</a></span></code> is encountered is performed via explicitly assembling a permutation matrix P,
and performing P^T.A.P. Possibly it might be more efficient to use <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/Mat/MatPermute.html">MatPermute</a>()</span></code>. We opted to use P^T.A.P as it appears
<code class="docutils notranslate"><span class="pre"><a href="../../manualpages/Vec/VecPermute.html">VecPermute</a>()</span></code> does not support the use case required here. By computing P, one can permute both the operator and RHS in a
consistent manner.</p>
<p>Mapping of vectors (default setup mode) is performed in the following way.
Suppose the parent communicator size was 4, and we set a reduction factor of 2; this would give a comm size on c’ of 2.
Using the interlaced creation routine, the ranks in c with color = 0 will be rank 0 and 2.
We perform the scatter to the sub-communicator in the following way.
[1] Given a vector x defined on communicator c</p>
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span> rank(c) local values of x
------- ----------------------------------------
0 [ 0.0, 1.0, 2.0, 3.0, 4.0, 5.0 ]
1 [ 6.0, 7.0, 8.0, 9.0, 10.0, 11.0 ]
2 [ 12.0, 13.0, 14.0, 15.0, 16.0, 17.0 ]
3 [ 18.0, 19.0, 20.0, 21.0, 22.0, 23.0 ]
</pre></div>
</div>
<p>scatter into xtmp defined also on comm c, so that we have the following values</p>
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span> rank(c) local values of xtmp
------- ----------------------------------------------------------------------------
0 [ 0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0 ]
1 [ ]
2 [ 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0 ]
3 [ ]
</pre></div>
</div>
<p>The entries on rank 1 and 3 (ranks which do not have a color = 0 in c’) have no values</p>
<p>[2] Copy the values from ranks 0, 2 (indices with respect to comm c) into the vector xred which is defined on communicator c’.
Ranks 0 and 2 are the only ranks in the subcomm which have a color = 0.</p>
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span> rank(c') local values of xred
-------- ----------------------------------------------------------------------------
0 [ 0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0 ]
1 [ 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0 ]
</pre></div>
</div>
</section>
<section id="contributed-by">
<h2>Contributed by<a class="headerlink" href="#contributed-by" title="Link to this heading">#</a></h2>
<p>Dave May</p>
</section>
<section id="references">
<h2>References<a class="headerlink" href="#references" title="Link to this heading">#</a></h2>
<div class="docutils container" id="id2">
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<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id1">MSR+16</a><span class="fn-bracket">]</span></span>
<p>Dave A. May, Patrick Sanan, Karl Rupp, Matthew G. Knepley, and Barry F. Smith. Extreme-scale multigrid components within PETSc. In <em>Proceedings of the Platform for Advanced Scientific Computing Conference</em>, PASC '16, 5:1–5:12. New York, NY, USA, 2016. ACM. <a class="reference external" href="https://doi.org/10.1145/2929908.2929913">doi:10.1145/2929908.2929913</a>.</p>
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<section id="see-also">
<h2>See Also<a class="headerlink" href="#see-also" title="Link to this heading">#</a></h2>
<p><a class="reference internal" href="../../manual/ksp.html#ch-ksp"><span class="std std-ref">KSP: Linear System Solvers</span></a>, <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PC/PCTelescopeGetKSP.html">PCTelescopeGetKSP</a>()</span></code>, <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PC/PCTelescopeGetDM.html">PCTelescopeGetDM</a>()</span></code>, <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PC/PCTelescopeGetReductionFactor.html">PCTelescopeGetReductionFactor</a>()</span></code>, <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PC/PCTelescopeSetReductionFactor.html">PCTelescopeSetReductionFactor</a>()</span></code>, <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PC/PCTelescopeGetIgnoreDM.html">PCTelescopeGetIgnoreDM</a>()</span></code>, <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PC/PCTelescopeSetIgnoreDM.html">PCTelescopeSetIgnoreDM</a>()</span></code>, <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PC/PCREDUNDANT.html">PCREDUNDANT</a></span></code></p>
</section>
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<h2>Level<a class="headerlink" href="#level" title="Link to this heading">#</a></h2>
<p>advanced</p>
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<h2>Location<a class="headerlink" href="#location" title="Link to this heading">#</a></h2>
<p><A HREF="../../src/ksp/pc/impls/telescope/telescope.c.html#PCTELESCOPE">src/ksp/pc/impls/telescope/telescope.c</A></p>
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<p><a class="reference internal" href="index.html"><span class="std std-doc">Index of all PC routines</span></a><br />
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