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<section class="tex2jax_ignore mathjax_ignore" id="the-various-matrix-classes">
<h1>The Various Matrix Classes<a class="headerlink" href="#the-various-matrix-classes" title="Link to this heading">#</a></h1>
<p>PETSc provides a variety of matrix implementations, since no single
matrix format is appropriate for all problems. This section first
discusses various matrix blocking strategies and then describes the
assortment of matrix types in PETSc.</p>
<section id="matrix-blocking-strategies">
<h2>Matrix Blocking Strategies<a class="headerlink" href="#matrix-blocking-strategies" title="Link to this heading">#</a></h2>
<p>In today’s computers, the time to perform an arithmetic operation is
dominated by the time to move the data into position, not the time to
compute the arithmetic result. For example, the time to perform a
multiplication operation may be one clock cycle, while the time to move
the floating-point number from memory to the arithmetic unit may take 10
or more cycles. In order to help manage this difference in time scales,
most processors have at least three levels of memory: registers, cache,
and random access memory. (In addition, some processors have external
caches, and the complications of paging introduce another level to the
hierarchy.)</p>
<p>Thus, to achieve high performance, a code should first move data into
cache and from there move it into registers and use it repeatedly while
it remains in the cache or registers before returning it to main memory.
If a floating-point number is reused 50 times while it is in registers,
then the “hit” of 10 clock cycles to bring it into the register is not
important. But if the floating-point number is used only once, the “hit”
of 10 clock cycles becomes noticeable, resulting in disappointing flop
rates.</p>
<p>Unfortunately, the compiler controls the use of the registers, and the
hardware controls the use of the cache. Since the user has essentially
no direct control, code must be written in such a way that the compiler
and hardware cache system can perform well. Good-quality code is then
said to respect the memory hierarchy.</p>
<p>The standard approach to improving the hardware utilization is to use
blocking. That is, rather than working with individual elements in the
matrices, you employ blocks of elements. Since the use of implicit
methods in PDE-based simulations leads to matrices with a naturally
blocked structure (with a block size equal to the number of degrees of
freedom per cell), blocking is advantageous. The PETSc sparse matrix
representations use a variety of techniques for blocking, including the
following:</p>
<ul class="simple">
<li><p>Storing the matrices using a generic sparse matrix format, but
storing additional information about adjacent rows with identical
nonzero structure (so-called I-nodes); this I-node information is
used in the key computational routines to improve performance (the
default for the <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MATSEQAIJ.html">MATSEQAIJ</a></span></code> and <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MATMPIAIJ.html">MATMPIAIJ</a></span></code> formats).</p></li>
<li><p>Storing the matrices using a fixed (problem dependent) block size
(via the <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MATSEQBAIJ.html">MATSEQBAIJ</a></span></code> and <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MATMPIBAIJ.html">MATMPIBAIJ</a></span></code> formats).</p></li>
</ul>
<p>The advantage of the first approach is that it is a minimal change from
a standard sparse matrix format and brings a large percentage of the
improvement obtained via blocking. Using a fixed block size gives the
best performance, since the code can be hardwired with that particular
size (for example, in some problems the size may be 3, in others 5, and
so on), so that the compiler will then optimize for that size, removing
the overhead of small loops entirely.</p>
<p>The following table presents the floating-point performance for a basic
matrix-vector product using three approaches: a basic compressed row
storage format (using the PETSc runtime options
<code class="docutils notranslate"><span class="pre">-mat_seqaij</span> <span class="pre">-mat_nounroll)</span></code>; the same compressed row format using
I-nodes (with the option <code class="docutils notranslate"><span class="pre">-mat_seqaij</span></code>); and a fixed block size code,
with a block size of 3 for these problems (using the option
<code class="docutils notranslate"><span class="pre">-mat_seqbaij</span></code>). The rates were computed on one node of an older IBM
Power processor based system, using two test matrices. The first matrix
(ARCO1), courtesy of Rick Dean of Arco, arises in multiphase flow
simulation; it has 1,501 degrees of freedom, 26,131 matrix nonzeros, a
natural block size of 3, and a small number of well terms. The second
matrix (CFD), arises in a three-dimensional Euler flow simulation and
has 15,360 degrees of freedom, 496,000 nonzeros, and a natural block
size of 5. In addition to displaying the flop rates for matrix-vector
products, we display them for triangular solves obtained from an ILU(0)
factorization.</p>
<table class="table">
<thead>
<tr class="row-odd"><th class="head"><p>Problem</p></th>
<th class="head"><p>Block size</p></th>
<th class="head"><p>Basic</p></th>
<th class="head"><p>I-node version</p></th>
<th class="head"><p>Fixed block size</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td colspan="5"><p>Matrix-Vector Product (Mflop/sec)</p></td>
</tr>
<tr class="row-odd"><td><p>Multiphase</p></td>
<td><p>3</p></td>
<td><p>27</p></td>
<td><p>43</p></td>
<td><p>70</p></td>
</tr>
<tr class="row-even"><td><p>Euler</p></td>
<td><p>5</p></td>
<td><p>28</p></td>
<td><p>58</p></td>
<td><p>90</p></td>
</tr>
<tr class="row-odd"><td colspan="5"><p>Triangular Solves from ILU(0) (Mflop/sec)</p></td>
</tr>
<tr class="row-even"><td><p>Multiphase</p></td>
<td><p>3</p></td>
<td><p>22</p></td>
<td><p>31</p></td>
<td><p>49</p></td>
</tr>
<tr class="row-odd"><td><p>Euler</p></td>
<td><p>5</p></td>
<td><p>22</p></td>
<td><p>39</p></td>
<td><p>65</p></td>
</tr>
</tbody>
</table>
<p>These examples demonstrate that careful implementations of the basic
sequential kernels in PETSc can dramatically improve overall floating
point performance, and users can immediately benefit from such
enhancements without altering a single line of their application codes.
Note that the speeds of the I-node and fixed block operations are
several times that of the basic sparse implementations.</p>
</section>
<section id="assorted-matrix-types">
<h2>Assorted Matrix Types<a class="headerlink" href="#assorted-matrix-types" title="Link to this heading">#</a></h2>
<p>PETSc offers a variety of both sparse and dense matrix types.</p>
<section id="sequential-aij-sparse-matrices">
<h3>Sequential AIJ Sparse Matrices<a class="headerlink" href="#sequential-aij-sparse-matrices" title="Link to this heading">#</a></h3>
<p>The default matrix representation within PETSc is the general sparse AIJ
format (also called the compressed sparse row format, CSR).</p>
</section>
<section id="parallel-aij-sparse-matrices">
<h3>Parallel AIJ Sparse Matrices<a class="headerlink" href="#parallel-aij-sparse-matrices" title="Link to this heading">#</a></h3>
<p>The AIJ sparse matrix type, is the default parallel matrix format;
additional implementation details are given in <span id="id1">[<a class="reference internal" href="#id1451" title="Satish Balay, William D. Gropp, Lois Curfman McInnes, and Barry F. Smith. Efficient management of parallelism in object oriented numerical software libraries. In E. Arge, A. M. Bruaset, and H. P. Langtangen, editors, Modern Software Tools in Scientific Computing, 163–202. Birkhäuser Press, 1997.">BGMS97</a>]</span>.</p>
</section>
<section id="sequential-block-aij-sparse-matrices">
<h3>Sequential Block AIJ Sparse Matrices<a class="headerlink" href="#sequential-block-aij-sparse-matrices" title="Link to this heading">#</a></h3>
<p>The sequential and parallel block AIJ formats, which are extensions of
the AIJ formats described above, are intended especially for use with
multiclass PDEs. The block variants store matrix elements by fixed-sized
dense <code class="docutils notranslate"><span class="pre">nb</span></code> <span class="math">\(\times\)</span> <code class="docutils notranslate"><span class="pre">nb</span></code> blocks. The stored row and column
indices begin at zero.</p>
<p>The routine for creating a sequential block AIJ matrix with <code class="docutils notranslate"><span class="pre">m</span></code> rows,
<code class="docutils notranslate"><span class="pre">n</span></code> columns, and a block size of <code class="docutils notranslate"><span class="pre">nb</span></code> is</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../manualpages/Mat/MatCreateSeqBAIJ.html">MatCreateSeqBAIJ</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="kt">int</span><span class="w"> </span><span class="n">nb</span><span class="p">,</span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="p">,</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="kt">int</span><span class="w"> </span><span class="n">nz</span><span class="p">,</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nnz</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">A</span><span class="p">)</span>
</pre></div>
</div>
<p>The arguments <code class="docutils notranslate"><span class="pre">nz</span></code> and <code class="docutils notranslate"><span class="pre">nnz</span></code> can be used to preallocate matrix
memory by indicating the number of <em>block</em> nonzeros per row. For good
performance during matrix assembly, preallocation is crucial; however,
you can set <code class="docutils notranslate"><span class="pre">nz=0</span></code> and <code class="docutils notranslate"><span class="pre">nnz=NULL</span></code> for PETSc to dynamically allocate
matrix memory as needed. The PETSc users manual discusses preallocation
for the AIJ format; extension to the block AIJ format is
straightforward.</p>
<p>Note that the routine <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MatSetValuesBlocked.html">MatSetValuesBlocked</a>()</span></code> can be used for more
efficient matrix assembly when using the block AIJ format.</p>
</section>
<section id="parallel-block-aij-sparse-matrices">
<h3>Parallel Block AIJ Sparse Matrices<a class="headerlink" href="#parallel-block-aij-sparse-matrices" title="Link to this heading">#</a></h3>
<p>Parallel block AIJ matrices with block size nb can be created with the
command <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MatCreateBAIJ.html">MatCreateBAIJ</a>()</span></code></p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../manualpages/Mat/MatCreateBAIJ.html">MatCreateBAIJ</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="kt">int</span><span class="w"> </span><span class="n">nb</span><span class="p">,</span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="p">,</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="kt">int</span><span class="w"> </span><span class="n">M</span><span class="p">,</span><span class="kt">int</span><span class="w"> </span><span class="n">N</span><span class="p">,</span><span class="kt">int</span><span class="w"> </span><span class="n">d_nz</span><span class="p">,</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">d_nnz</span><span class="p">,</span><span class="kt">int</span><span class="w"> </span><span class="n">o_nz</span><span class="p">,</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">o_nnz</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">A</span><span class="p">);</span>
</pre></div>
</div>
<p><code class="docutils notranslate"><span class="pre">A</span></code> is the newly created matrix, while the arguments <code class="docutils notranslate"><span class="pre">m</span></code>, <code class="docutils notranslate"><span class="pre">n</span></code>,
<code class="docutils notranslate"><span class="pre">M</span></code>, and <code class="docutils notranslate"><span class="pre">N</span></code> indicate the number of local rows and columns and the
number of global rows and columns, respectively. Either the local or
global parameters can be replaced with <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Sys/PETSC_DECIDE.html">PETSC_DECIDE</a></span></code>, so that PETSc
will determine them. The matrix is stored with a fixed number of rows on
each processor, given by <code class="docutils notranslate"><span class="pre">m</span></code>, or determined by PETSc if <code class="docutils notranslate"><span class="pre">m</span></code> is
<code class="docutils notranslate"><span class="pre"><a href="../manualpages/Sys/PETSC_DECIDE.html">PETSC_DECIDE</a></span></code>.</p>
<p>If <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Sys/PETSC_DECIDE.html">PETSC_DECIDE</a></span></code> is not used for <code class="docutils notranslate"><span class="pre">m</span></code> and <code class="docutils notranslate"><span class="pre">n</span></code> then you must ensure
that they are chosen to be compatible with the vectors. To do so, you
first consider the product <span class="math">\(y = A x\)</span>. The <code class="docutils notranslate"><span class="pre">m</span></code> that used in
<code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MatCreateBAIJ.html">MatCreateBAIJ</a>()</span></code> must match the local size used in the
<code class="docutils notranslate"><span class="pre"><a href="../manualpages/Vec/VecCreateMPI.html">VecCreateMPI</a>()</span></code> for <code class="docutils notranslate"><span class="pre">y</span></code>. The <code class="docutils notranslate"><span class="pre">n</span></code> used must match that used as the
local size in <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Vec/VecCreateMPI.html">VecCreateMPI</a>()</span></code> for <code class="docutils notranslate"><span class="pre">x</span></code>.</p>
<p>You must set <code class="docutils notranslate"><span class="pre">d_nz=0</span></code>, <code class="docutils notranslate"><span class="pre">o_nz=0</span></code>, <code class="docutils notranslate"><span class="pre">d_nnz=NULL</span></code>, and <code class="docutils notranslate"><span class="pre">o_nnz=NULL</span></code> for
PETSc to control dynamic allocation of matrix memory space. Analogous to
<code class="docutils notranslate"><span class="pre">nz</span></code> and <code class="docutils notranslate"><span class="pre">nnz</span></code> for the routine <code class="docutils notranslate"><span class="pre"><a href="../manualpages/Mat/MatCreateSeqBAIJ.html">MatCreateSeqBAIJ</a>()</span></code>, these
arguments optionally specify block nonzero information for the diagonal
(<code class="docutils notranslate"><span class="pre">d_nz</span></code> and <code class="docutils notranslate"><span class="pre">d_nnz</span></code>) and off-diagonal (<code class="docutils notranslate"><span class="pre">o_nz</span></code> and <code class="docutils notranslate"><span class="pre">o_nnz</span></code>) parts of
the matrix. For a square global matrix, we define each processor’s
diagonal portion to be its local rows and the corresponding columns (a
square submatrix); each processor’s off-diagonal portion encompasses the
remainder of the local matrix (a rectangular submatrix). The PETSc users
manual gives an example of preallocation for the parallel AIJ matrix
format; extension to the block parallel AIJ case is straightforward.</p>
</section>
<section id="sequential-dense-matrices">
<h3>Sequential Dense Matrices<a class="headerlink" href="#sequential-dense-matrices" title="Link to this heading">#</a></h3>
<p>PETSc provides both sequential and parallel dense matrix formats, where
each processor stores its entries in a column-major array in the usual
Fortran style.</p>
</section>
<section id="parallel-dense-matrices">
<h3>Parallel Dense Matrices<a class="headerlink" href="#parallel-dense-matrices" title="Link to this heading">#</a></h3>
<p>The parallel dense matrices are partitioned by rows across the
processors, so that each local rectangular submatrix is stored in the
dense format described above.</p>
</section>
</section>
<section id="references">
<h2>References<a class="headerlink" href="#references" title="Link to this heading">#</a></h2>
<div class="docutils container" id="id2">
<div role="list" class="citation-list">
<div class="citation" id="id1451" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id1">BGMS97</a><span class="fn-bracket">]</span></span>
<p>Satish Balay, William D. Gropp, Lois Curfman McInnes, and Barry F. Smith. Efficient management of parallelism in object oriented numerical software libraries. In E. Arge, A. M. Bruaset, and H. P. Langtangen, editors, <em>Modern Software Tools in Scientific Computing</em>, 163–202. Birkhäuser Press, 1997.</p>
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