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<li class="breadcrumb-item active" aria-current="page"><span class="ellipsis">Exercise 4: Singular Value Decomposition</span></li>
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<section class="tex2jax_ignore mathjax_ignore" id="exercise-4-singular-value-decomposition">
<h1>Exercise 4: Singular Value Decomposition<a class="headerlink" href="#exercise-4-singular-value-decomposition" title="Link to this heading">#</a></h1>
<p>In this exercise we turn our attention to the Singular Value Decomposition (SVD). Remember that in positive-definite real symmetric (or complex Hermitian) matrices, singular values coincide with eigenvalues, but in general this is not the case. The SVD is defined for any matrix, even rectangular ones. Singular values are always non-negative real values.</p>
<p>This example works also by reading a matrix from a file. In particular, the matrix to be used is related to a 2D reaction-diffusion model. More details about this problem can be found at <a class="reference external" href="https://math.nist.gov/MatrixMarket/data/NEP/brussel/brussel.html" target="_blank">Matrix Market</a>.</p>
<section id="compiling">
<h2>Compiling<a class="headerlink" href="#compiling" title="Link to this heading">#</a></h2>
<p>Copy the file <a class="reference external" href="https://slepc.upv.es/release/src/svd/tutorials/ex14.c.html" target="_blank">ex14.c</a> to your directory and add these lines to the makefile</p>
<div class="highlight-make notranslate"><div class="highlight"><pre><span></span><span class="nf">ex14</span><span class="o">:</span><span class="w"> </span><span class="n">ex</span>14.<span class="n">o</span>
<span class="w"> </span>-<span class="si">${</span><span class="nv">CLINKER</span><span class="si">}</span><span class="w"> </span>-o<span class="w"> </span>ex14<span class="w"> </span>ex14.o<span class="w"> </span><span class="si">${</span><span class="nv">SLEPC_SVD_LIB</span><span class="si">}</span>
<span class="w"> </span><span class="si">${</span><span class="nv">RM</span><span class="si">}</span><span class="w"> </span>ex14.o
</pre></div>
</div>
<p>Build the executable with the command</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>make<span class="w"> </span>ex14
</pre></div>
</div>
</section>
<section id="running-the-program">
<h2>Running the Program<a class="headerlink" href="#running-the-program" title="Link to this heading">#</a></h2>
<p>In order to run the program, type the following:</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>./ex14<span class="w"> </span>-file<span class="w"> </span><span class="nv">$SLEPC_DIR</span>/share/slepc/datafiles/matrices/rdb200.petsc
</pre></div>
</div>
<p>You should get an output similar to this:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Singular</span> <span class="n">value</span> <span class="n">problem</span> <span class="n">stored</span> <span class="ow">in</span> <span class="n">file</span><span class="o">.</span>
<span class="n">Reading</span> <span class="n">REAL</span> <span class="n">matrix</span> <span class="kn">from</span><span class="w"> </span><span class="nn">a</span> <span class="n">binary</span> <span class="n">file</span><span class="o">...</span>
<span class="n">Number</span> <span class="n">of</span> <span class="n">iterations</span> <span class="n">of</span> <span class="n">the</span> <span class="n">method</span><span class="p">:</span> <span class="mi">3</span>
<span class="n">Solution</span> <span class="n">method</span><span class="p">:</span> <span class="n">cross</span>
<span class="n">Number</span> <span class="n">of</span> <span class="n">requested</span> <span class="n">singular</span> <span class="n">values</span><span class="p">:</span> <span class="mi">1</span>
<span class="n">Stopping</span> <span class="n">condition</span><span class="p">:</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-08</span><span class="p">,</span> <span class="n">maxit</span><span class="o">=</span><span class="mi">100</span>
<span class="n"><a href="../../manualpages/SVD/SVD.html">SVD</a></span> <span class="n">solve</span> <span class="n">converged</span> <span class="p">(</span><span class="mi">1</span> <span class="n">singular</span> <span class="n">triplet</span><span class="p">)</span> <span class="n">due</span> <span class="n">to</span> <span class="n">CONVERGED_TOL</span><span class="p">;</span> <span class="n">iterations</span> <span class="mi">3</span>
<span class="o">----------------------</span> <span class="o">--------------------</span>
<span class="n">sigma</span> <span class="n">relative</span> <span class="n">error</span>
<span class="o">----------------------</span> <span class="o">--------------------</span>
<span class="mf">35.007519</span> <span class="mf">1.40809e-10</span>
<span class="o">----------------------</span> <span class="o">--------------------</span>
</pre></div>
</div>
</section>
<section id="source-code-details">
<h2>Source Code Details<a class="headerlink" href="#source-code-details" title="Link to this heading">#</a></h2>
<p>The way in which the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/SVD/SVD.html">SVD</a></span></code> object works is very similar to that of <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPS.html">EPS</a></span></code>. However, some important differences exist. Examine the source code of the example program and pay attention to the differences with respect to <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPS.html">EPS</a></span></code>. After loading the matrix, the problem is solved by the following sequence of function calls:</p>
<ul class="simple">
<li><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/SVD/SVDCreate.html">SVDCreate</a>(<a href="https://petsc.org/release/manualpages/Sys/MPI_Comm/">MPI_Comm</a></span> <span class="pre">comm,<a href="../../manualpages/SVD/SVD.html">SVD</a></span> <span class="pre">*svd);</span></code></p></li>
<li><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/SVD/SVDSetOperators.html">SVDSetOperators</a></span></code>(<code class="docutils notranslate"><span class="pre"><a href="../../manualpages/SVD/SVD.html">SVD</a></span></code> <code class="docutils notranslate"><span class="pre">svd</span></code>,<a class="reference external" href="https://petsc.org/release/manualpages/Mat/Mat/" title="(in PETSc v3.24)"><span class="xref std std-doc">Mat</span></a> <code class="docutils notranslate"><span class="pre">A</span></code>,<a class="reference external" href="https://petsc.org/release/manualpages/Mat/Mat/" title="(in PETSc v3.24)"><span class="xref std std-doc">Mat</span></a> <code class="docutils notranslate"><span class="pre">B</span></code>);`</p></li>
<li><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/SVD/SVDSetFromOptions.html">SVDSetFromOptions</a>(<a href="../../manualpages/SVD/SVD.html">SVD</a></span> <span class="pre">svd);</span></code></p></li>
<li><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/SVD/SVDSolve.html">SVDSolve</a>(<a href="../../manualpages/SVD/SVD.html">SVD</a></span> <span class="pre">svd);</span></code></p></li>
<li><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/SVD/SVDGetConverged.html">SVDGetConverged</a></span></code>(<code class="docutils notranslate"><span class="pre"><a href="../../manualpages/SVD/SVD.html">SVD</a></span></code> <code class="docutils notranslate"><span class="pre">svd</span></code>, <a class="reference external" href="https://petsc.org/release/manualpages/Sys/PetscInt/" title="(in PETSc v3.24)"><span class="xref std std-doc">PetscInt</span></a> <code class="docutils notranslate"><span class="pre">*nconv</span></code>);</p></li>
<li><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/SVD/SVDGetSingularTriplet.html">SVDGetSingularTriplet</a></span></code>(<code class="docutils notranslate"><span class="pre"><a href="../../manualpages/SVD/SVD.html">SVD</a></span></code> <code class="docutils notranslate"><span class="pre">svd</span></code>,<a class="reference external" href="https://petsc.org/release/manualpages/Sys/PetscInt/" title="(in PETSc v3.24)"><span class="xref std std-doc">PetscInt</span></a> <code class="docutils notranslate"><span class="pre">i</span></code>,<a class="reference external" href="https://petsc.org/release/manualpages/Sys/PetscReal/" title="(in PETSc v3.24)"><span class="xref std std-doc">PetscReal</span></a> <code class="docutils notranslate"><span class="pre">*sigma</span></code>,<a class="reference external" href="https://petsc.org/release/manualpages/Vec/Vec/" title="(in PETSc v3.24)"><span class="xref std std-doc">Vec</span></a> <code class="docutils notranslate"><span class="pre">u</span></code>,<a class="reference external" href="https://petsc.org/release/manualpages/Vec/Vec/" title="(in PETSc v3.24)"><span class="xref std std-doc">Vec</span></a> <code class="docutils notranslate"><span class="pre">v</span></code>);</p></li>
<li><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/SVD/SVDDestroy.html">SVDDestroy</a>(<a href="../../manualpages/SVD/SVD.html">SVD</a></span> <span class="pre">svd)</span></code>;</p></li>
</ul>
<p>First, the singular value solver (<code class="docutils notranslate"><span class="pre"><a href="../../manualpages/SVD/SVD.html">SVD</a></span></code>) context is created and the matrix associated with the problem is specified. Then various options are set for customized solution. After that, the program solves the problem, retrieves the solution, and finally destroys the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/SVD/SVD.html">SVD</a></span></code> context.</p>
<p>Note that the singular value, <code class="docutils notranslate"><span class="pre">sigma</span></code>, is defined as a <a class="reference external" href="https://petsc.org/release/manualpages/Sys/PetscReal/" title="(in PETSc v3.24)"><span class="xref std std-doc">PetscReal</span></a>, and that the singular vectors are simple <a class="reference external" href="https://petsc.org/release/manualpages/Vec/Vec/" title="(in PETSc v3.24)"><span class="xref std std-doc">Vec</span></a>’s.</p>
</section>
<section id="svd-options">
<h2>SVD Options<a class="headerlink" href="#svd-options" title="Link to this heading">#</a></h2>
<p>Most of the options available in the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPS.html">EPS</a></span></code> object have their equivalent in <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/SVD/SVD.html">SVD</a></span></code>. A full list of command-line options can be obtained by running the example with the option <code class="docutils notranslate"><span class="pre">-help</span></code>.</p>
<p>To show information about the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/SVD/SVD.html">SVD</a></span></code> solver, add the <code class="docutils notranslate"><span class="pre">-svd_view</span></code> option:</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>./ex14<span class="w"> </span>-file<span class="w"> </span><span class="nv">$SLEPC_DIR</span>/share/slepc/datafiles/matrices/rdb200.petsc<span class="w"> </span>-svd_view
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>All the command-line options related to the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/SVD/SVD.html">SVD</a></span></code> object have the <code class="docutils notranslate"><span class="pre">-svd_</span></code> prefix.</p>
</div>
<p>Your output will include something like this:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../../manualpages/SVD/SVD.html">SVD</a></span> <span class="n">Object</span><span class="p">:</span> <span class="mi">1</span> <span class="n">MPI</span> <span class="n">processes</span>
<span class="nb">type</span><span class="p">:</span> <span class="n">cross</span>
<span class="n">implicit</span> <span class="n">matrix</span>
<span class="n"><a href="../../manualpages/EPS/EPS.html">EPS</a></span> <span class="n">Object</span><span class="p">:</span> <span class="p">(</span><span class="n">svd_cross_</span><span class="p">)</span> <span class="mi">1</span> <span class="n">MPI</span> <span class="n">processes</span>
<span class="nb">type</span><span class="p">:</span> <span class="n">krylovschur</span>
<span class="mi">50</span><span class="o">%</span> <span class="n">of</span> <span class="n">basis</span> <span class="n">vectors</span> <span class="n">kept</span> <span class="n">after</span> <span class="n">restart</span>
<span class="n">using</span> <span class="n">the</span> <span class="n">locking</span> <span class="n">variant</span>
<span class="n">problem</span> <span class="nb">type</span><span class="p">:</span> <span class="n">symmetric</span> <span class="n">eigenvalue</span> <span class="n">problem</span>
<span class="n">selected</span> <span class="n">portion</span> <span class="n">of</span> <span class="n">the</span> <span class="n">spectrum</span><span class="p">:</span> <span class="n">largest</span> <span class="n">real</span> <span class="n">parts</span>
<span class="n">number</span> <span class="n">of</span> <span class="n">eigenvalues</span> <span class="p">(</span><span class="n">nev</span><span class="p">):</span> <span class="mi">1</span>
<span class="n">number</span> <span class="n">of</span> <span class="n">column</span> <span class="n">vectors</span> <span class="p">(</span><span class="n">ncv</span><span class="p">):</span> <span class="mi">16</span>
<span class="n">maximum</span> <span class="n">dimension</span> <span class="n">of</span> <span class="n">projected</span> <span class="n">problem</span> <span class="p">(</span><span class="n">mpd</span><span class="p">):</span> <span class="mi">16</span>
<span class="n">maximum</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mi">100</span>
<span class="n">tolerance</span><span class="p">:</span> <span class="mf">1e-09</span>
<span class="n">convergence</span> <span class="n">test</span><span class="p">:</span> <span class="n">relative</span> <span class="n">to</span> <span class="n">the</span> <span class="n">eigenvalue</span>
<span class="n"><a href="../../manualpages/BV/BV.html">BV</a></span> <span class="n">Object</span><span class="p">:</span> <span class="p">(</span><span class="n">svd_cross_</span><span class="p">)</span> <span class="mi">1</span> <span class="n">MPI</span> <span class="n">processes</span>
<span class="nb">type</span><span class="p">:</span> <span class="n">svec</span>
<span class="mi">17</span> <span class="n">columns</span> <span class="n">of</span> <span class="k">global</span> <span class="n">length</span> <span class="mi">200</span>
<span class="n">vector</span> <span class="n">orthogonalization</span> <span class="n">method</span><span class="p">:</span> <span class="n">classical</span> <span class="n">Gram</span><span class="o">-</span><span class="n">Schmidt</span>
<span class="n">orthogonalization</span> <span class="n">refinement</span><span class="p">:</span> <span class="k">if</span> <span class="n">needed</span> <span class="p">(</span><span class="n">eta</span><span class="p">:</span> <span class="mf">0.7071</span><span class="p">)</span>
<span class="n">block</span> <span class="n">orthogonalization</span> <span class="n">method</span><span class="p">:</span> <span class="n">GS</span>
<span class="n">doing</span> <span class="n">matmult</span> <span class="k">as</span> <span class="n">a</span> <span class="n">single</span> <span class="n">matrix</span><span class="o">-</span><span class="n">matrix</span> <span class="n">product</span>
<span class="n"><a href="../../manualpages/DS/DS.html">DS</a></span> <span class="n">Object</span><span class="p">:</span> <span class="p">(</span><span class="n">svd_cross_</span><span class="p">)</span> <span class="mi">1</span> <span class="n">MPI</span> <span class="n">processes</span>
<span class="nb">type</span><span class="p">:</span> <span class="n">hep</span>
<span class="n">solving</span> <span class="n">the</span> <span class="n">problem</span> <span class="k">with</span><span class="p">:</span> <span class="n">Implicit</span> <span class="n">QR</span> <span class="n">method</span> <span class="p">(</span><span class="n">_steqr</span><span class="p">)</span>
<span class="n"><a href="../../manualpages/ST/ST.html">ST</a></span> <span class="n">Object</span><span class="p">:</span> <span class="p">(</span><span class="n">svd_cross_</span><span class="p">)</span> <span class="mi">1</span> <span class="n">MPI</span> <span class="n">processes</span>
<span class="nb">type</span><span class="p">:</span> <span class="n">shift</span>
<span class="n">shift</span><span class="p">:</span> <span class="mf">0.</span>
<span class="n">number</span> <span class="n">of</span> <span class="n">matrices</span><span class="p">:</span> <span class="mi">1</span>
<span class="n">using</span> <span class="n">a</span> <span class="n">shell</span> <span class="n">matrix</span>
<span class="n">transpose</span> <span class="n">mode</span><span class="p">:</span> <span class="n">explicit</span>
<span class="n">selected</span> <span class="n">portion</span> <span class="n">of</span> <span class="n">the</span> <span class="n">spectrum</span><span class="p">:</span> <span class="n">largest</span>
<span class="n">number</span> <span class="n">of</span> <span class="n">singular</span> <span class="n">values</span> <span class="p">(</span><span class="n">nsv</span><span class="p">):</span> <span class="mi">1</span>
<span class="n">number</span> <span class="n">of</span> <span class="n">column</span> <span class="n">vectors</span> <span class="p">(</span><span class="n">ncv</span><span class="p">):</span> <span class="mi">16</span>
<span class="n">maximum</span> <span class="n">dimension</span> <span class="n">of</span> <span class="n">projected</span> <span class="n">problem</span> <span class="p">(</span><span class="n">mpd</span><span class="p">):</span> <span class="mi">16</span>
<span class="n">maximum</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mi">100</span>
<span class="n">tolerance</span><span class="p">:</span> <span class="mf">1e-08</span>
<span class="n">convergence</span> <span class="n">test</span><span class="p">:</span> <span class="n">relative</span> <span class="n">to</span> <span class="n">the</span> <span class="n">singular</span> <span class="n">value</span>
</pre></div>
</div>
<p>The output shows all the options that are susceptible of being changed, either from the command line or from the source code of the program: the method, the portion of the spectrum (largest or smallest singular values), the number of singular values (<code class="docutils notranslate"><span class="pre">nsv</span></code>), etc.</p>
<p>Try to change some of the values, for instance:</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>./ex14<span class="w"> </span>-file<span class="w"> </span><span class="nv">$SLEPC_DIR</span>/share/slepc/datafiles/matrices/rdb200.petsc<span class="w"> </span>-svd_nsv<span class="w"> </span><span class="m">10</span><span class="w"> </span>-svd_ncv<span class="w"> </span><span class="m">40</span><span class="w"> </span>-svd_smallest
</pre></div>
</div>
<p>The “transpose mode” refers to whether the transpose of matrix A is being built explicitly or not (see <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/SVD/SVDSetImplicitTranspose.html">SVDSetImplicitTranspose</a>()</span></code> for an explanation).</p>
<p>Note that in the sample output above, the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/SVD/SVD.html">SVD</a></span></code> object contains an <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPS.html">EPS</a></span></code> object. This only happens in some <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/SVD/SVD.html">SVD</a></span></code> solver types, as detailed below.</p>
</section>
<section id="choosing-the-singular-value-solver">
<h2>Choosing the Singular Value Solver<a class="headerlink" href="#choosing-the-singular-value-solver" title="Link to this heading">#</a></h2>
<p>SLEPc provides several solvers for computing the SVD, which can be selected in the source code with the function <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/SVD/SVDSetType.html">SVDSetType</a>()</span></code>, or at run time:</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>./ex14<span class="w"> </span>-file<span class="w"> </span><span class="nv">$SLEPC_DIR</span>/share/slepc/datafiles/matrices/rdb200.petsc<span class="w"> </span>-svd_type<span class="w"> </span>trlanczos
</pre></div>
</div>
<p>Check the table <a class="reference internal" href="../manual/svd.html#tab-svdsolvers"><span class="std std-ref">List of solvers available in the SVD module</span></a>.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The <code class="docutils notranslate"><span class="pre">lapack</span></code> solver is not really a full-featured singular value solver but simply an interface to some LAPACK routines. These routines operate sequentially in dense mode and therefore are suitable only for small size problems. That method should be used only for debugging purposes.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The default solver is <code class="docutils notranslate"><span class="pre">cross</span></code>.</p>
</div>
<p>The first two solvers, <code class="docutils notranslate"><span class="pre">cross</span></code> and <code class="docutils notranslate"><span class="pre">cyclic</span></code>, are not real methods implemented in the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/SVD/SVD.html">SVD</a></span></code> module, but are two convenient ways of solving the SVD problem by making use of the eigensolvers available in the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPS.html">EPS</a></span></code> module. In those two cases, the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/SVD/SVD.html">SVD</a></span></code> object manages an <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPS.html">EPS</a></span></code> object internally, whose parameters can be set as desired (typically only the method). For example:</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>./ex14<span class="w"> </span>-file<span class="w"> </span><span class="nv">$SLEPC_DIR</span>/share/slepc/datafiles/matrices/rdb200.petsc<span class="w"> </span>-svd_type<span class="w"> </span>cyclic<span class="w"> </span>-svd_cyclic_eps_type<span class="w"> </span>lanczos<span class="w"> </span>-svd_cyclic_eps_lanczos_reorthog<span class="w"> </span><span class="nb">local</span>
</pre></div>
</div>
</section>
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