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<section class="tex2jax_ignore mathjax_ignore" id="exercise-1-standard-symmetric-eigenvalue-problem">
<h1>Exercise 1: Standard Symmetric Eigenvalue Problem<a class="headerlink" href="#exercise-1-standard-symmetric-eigenvalue-problem" title="Link to this heading">#</a></h1>
<p>This example solves a standard symmetric eigenproblem <span class="math notranslate nohighlight">\(Ax= \lambda x\)</span>, where A is the matrix resulting from the discretization of the Laplacian operator in 1 dimension by centered finite differences.</p>
<div class="math notranslate nohighlight">
\[\begin{split}A = \begin{bmatrix}
2 & -1 \\
-1 & 2 & -1 \\
& \ddots & \ddots & \ddots \\
& & -1 & 2 & -1 \\
& & & -1 & 2
\end{bmatrix}\end{split}\]</div>
<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/eps/tutorials/ex1.c.html" target="_blank">ex1.c</a> to the working directory and add these lines to the makefile</p>
<div class="highlight-make notranslate"><div class="highlight"><pre><span></span><span class="nf">ex1</span><span class="o">:</span><span class="w"> </span><span class="n">ex</span>1.<span class="n">o</span>
<span class="w"> </span>-<span class="si">${</span><span class="nv">CLINKER</span><span class="si">}</span><span class="w"> </span>-o<span class="w"> </span>ex1<span class="w"> </span>ex1.o<span class="w"> </span><span class="si">${</span><span class="nv">SLEPC_EPS_LIB</span><span class="si">}</span>
<span class="w"> </span><span class="si">${</span><span class="nv">RM</span><span class="si">}</span><span class="w"> </span>ex1.o
</pre></div>
</div>
<p>Build the executable with the command</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>make<span class="w"> </span>ex1
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Example ex1 is also available in Fortran <a class="reference external" href="https://slepc.upv.es/release/src/eps/tutorials/ex1f.F90.html" target="_blank">ex1f.F90</a>.</p>
</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 for a problem of order 50, type the following</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>./ex1<span class="w"> </span>-n<span class="w"> </span><span class="m">50</span>
</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="mi">1</span><span class="o">-</span><span class="n">D</span> <span class="n">Laplacian</span> <span class="n">Eigenproblem</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">50</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">8</span>
<span class="n">Solution</span> <span class="n">method</span><span class="p">:</span> <span class="n">krylovschur</span>
<span class="n">Number</span> <span class="n">of</span> <span class="n">requested</span> <span class="n">eigenvalues</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">Number</span> <span class="n">of</span> <span class="n">converged</span> <span class="n">eigenpairs</span><span class="p">:</span> <span class="mi">3</span>
<span class="n">k</span> <span class="o">||</span><span class="n">Ax</span><span class="o">-</span><span class="n">kx</span><span class="o">||/||</span><span class="n">kx</span><span class="o">||</span>
<span class="o">-----------------</span> <span class="o">------------------</span>
<span class="mf">3.996207</span> <span class="mf">4.30363e-10</span>
<span class="mf">3.984841</span> <span class="mf">2.08631e-09</span>
<span class="mf">3.965946</span> <span class="mf">9.98404e-09</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>Examine the source code of the sample program and locate the function calls mentioned in the following comments.</p>
<p><strong>The Options Database</strong>: All the PETSc functionality related to the options database is available in SLEPc. This allows the user to input control data at run time very easily. In this example, the function <a class="reference external" href="https://petsc.org/release/manualpages/Sys/PetscOptionsGetInt/" title="(in PETSc v3.24)"><span class="xref std std-doc">PetscOptionsGetInt</span></a>() is used to check whether the user has provided a command-line option to set the value of <code class="docutils notranslate"><span class="pre">n</span></code>, the problem dimension. If so, the variable <code class="docutils notranslate"><span class="pre">n</span></code> is set accordingly; otherwise, <code class="docutils notranslate"><span class="pre">n</span></code> remains unchanged.</p>
<p><strong>Vectors and Matrices</strong>: Usage of matrices and vectors in SLEPc is exactly the same as in PETSc. The user can create a new parallel or sequential matrix, <code class="docutils notranslate"><span class="pre">A</span></code>, with subroutine <a class="reference external" href="https://petsc.org/release/manualpages/Mat/MatCreate/" title="(in PETSc v3.24)"><span class="xref std std-doc">MatCreate</span></a>(), where the matrix format can be specified at runtime. The example creates a matrix, sets the nonzero values with <a class="reference external" href="https://petsc.org/release/manualpages/Mat/MatSetValues/" title="(in PETSc v3.24)"><span class="xref std std-doc">MatSetValues</span></a>() and then assembles it.</p>
<p><strong>Solving the Eigenvalue Problem</strong>: Usage of eigensolvers is very similar to other kinds of solvers provided by PETSc. After creating the matrix, the problem is solved by means of an <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPS.html">EPS</a></span></code> object (Eigenvalue Problem Solver) via the following sequence of function calls:</p>
<ul class="simple">
<li><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPSCreate.html">EPSCreate</a></span></code>(<code class="docutils notranslate"><span class="pre"><a href="https://petsc.org/release/manualpages/Sys/MPI_Comm/">MPI_Comm</a></span></code> <code class="docutils notranslate"><span class="pre">comm</span></code>,<code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPS.html">EPS</a></span></code> <code class="docutils notranslate"><span class="pre">*eps</span></code>);</p></li>
<li><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPSSetOperators.html">EPSSetOperators</a></span></code>(<code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPS.html">EPS</a></span></code> <code class="docutils notranslate"><span class="pre">eps</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/EPS/EPSSetProblemType.html">EPSSetProblemType</a></span></code>(<code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPS.html">EPS</a></span></code> <code class="docutils notranslate"><span class="pre">eps</span></code>,<code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPSProblemType.html">EPSProblemType</a></span></code> <code class="docutils notranslate"><span class="pre">type</span></code>);</p></li>
<li><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPSSetFromOptions.html">EPSSetFromOptions</a></span></code>(<code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPS.html">EPS</a></span></code> <code class="docutils notranslate"><span class="pre">eps</span></code>);</p></li>
<li><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPSSolve.html">EPSSolve</a></span></code>(<code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPS.html">EPS</a></span></code> <code class="docutils notranslate"><span class="pre">eps</span></code>);</p></li>
<li><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPSGetConverged.html">EPSGetConverged</a></span></code>(<code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPS.html">EPS</a></span></code> <code class="docutils notranslate"><span class="pre">eps</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/EPS/EPSGetEigenpair.html">EPSGetEigenpair</a></span></code>(<code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPS.html">EPS</a></span></code> <code class="docutils notranslate"><span class="pre">eps</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/PetscScalar/" title="(in PETSc v3.24)"><span class="xref std std-doc">PetscScalar</span></a> <code class="docutils notranslate"><span class="pre">*kr</span></code>,<a class="reference external" href="https://petsc.org/release/manualpages/Sys/PetscScalar/" title="(in PETSc v3.24)"><span class="xref std std-doc">PetscScalar</span></a> <code class="docutils notranslate"><span class="pre">*ki</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">xr</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">xi</span></code>);</p></li>
<li><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPSDestroy.html">EPSDestroy</a></span></code>(<code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPS.html">EPS</a></span></code> <code class="docutils notranslate"><span class="pre">eps</span></code>)`</p></li>
</ul>
<p>First, the eigenproblem solver (<code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPS.html">EPS</a></span></code>) context is created and the operator(s) associated with the eigensystem are set, as well as the problem type. 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/EPS/EPS.html">EPS</a></span></code> context.</p>
<p>The above function calls are very important and will be present in most SLEPc programs. In the example source code <a class="reference external" href="https://slepc.upv.es/release/src/eps/tutorials/ex1.c.html" target="_blank">ex1.c</a> you will find other functions apart from these. What do they do?</p>
</section>
<section id="playing-with-eps-options">
<h2>Playing with EPS Options<a class="headerlink" href="#playing-with-eps-options" title="Link to this heading">#</a></h2>
<p>Now we are going to experiment with different options of the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPS.html">EPS</a></span></code> object. 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 solver object:</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>./ex1<span class="w"> </span>-eps_view
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>This option internally calls the function <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPSView.html">EPSView</a>()</span></code>. Alternatively, we could include a direct call to this function in the source code. Almost all command-line options have a related function call.</p>
</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/EPS/EPS.html">EPS</a></span></code> object have the <code class="docutils notranslate"><span class="pre">-eps_</span></code> prefix.</p>
</div>
<p>This time, 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/EPS/EPS.html">EPS</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">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">eigenvalues</span> <span class="ow">in</span> <span class="n">magnitude</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-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">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="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">30</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="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="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="mi">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>
</pre></div>
</div>
<p>This option is very useful to see which solver and options the program is using.</p>
<p>Try solving a much larger problem, for instance with <code class="docutils notranslate"><span class="pre">n=400</span></code>. Note that in that case the program does not return a solution. This means that the solver has reached the maximum number of allowed iterations and the convergence criterion was not satisfied. What we can do is either increase the number of iterations</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>./ex1<span class="w"> </span>-n<span class="w"> </span><span class="m">400</span><span class="w"> </span>-eps_max_it<span class="w"> </span><span class="m">400</span>
</pre></div>
</div>
<p>or relax the convergence criterion:</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>./ex1<span class="w"> </span>-n<span class="w"> </span><span class="m">400</span><span class="w"> </span>-eps_tol<span class="w"> </span>1e-3
</pre></div>
</div>
<p>Note that in the latter case the relative error displayed by the program is significantly larger, meaning that the solution has only 3 correct decimal digits, as expected.</p>
<p>It is possible to change the number of requested eigenvalues. Try the following execution</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>./ex1<span class="w"> </span>-n<span class="w"> </span><span class="m">400</span><span class="w"> </span>-eps_nev<span class="w"> </span><span class="m">3</span><span class="w"> </span>-eps_tol<span class="w"> </span>1e-7
</pre></div>
</div>
<p>In this case, the program did not succeed to compute two of the requested eigenpairs. This is again due to the convergence criterion, which is satisfied by some eigenpairs but not for all. As in the previous case, we could further increase the number of iterations or relax the convergence criterion. Another alternative is to increase the number of column vectors (i.e., the dimension of the subspace with which the eigensolver works). This usually improves the convergence behavior at the expense of larger memory requirements.</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>./ex1<span class="w"> </span>-n<span class="w"> </span><span class="m">400</span><span class="w"> </span>-eps_nev<span class="w"> </span><span class="m">3</span><span class="w"> </span>-eps_ncv<span class="w"> </span><span class="m">24</span>
</pre></div>
</div>
<p>Note that the default value of <code class="docutils notranslate"><span class="pre">ncv</span></code> depends on the value of <code class="docutils notranslate"><span class="pre">nev</span></code>.</p>
<p>Try to set some of the above options directly in the source code by calling the related functions <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPSSetTolerances.html">EPSSetTolerances</a>()</span></code> and <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPSSetDimensions.html">EPSSetDimensions</a>()</span></code>. Modify and recompile the program. Use <code class="docutils notranslate"><span class="pre">-eps_view</span></code> to check that the values are correctly set. Is it now possible to change these options from the command-line? Does this change whether you place the calls before or after the call to <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPSSetFromOptions.html">EPSSetFromOptions</a>()</span></code>?</p>
<p>Convergence is usually bad when eigenvalues are close to each other, which is the case in this example. In order to see what is happening while the eigensolver iterates, we can use a monitor to display information associated to the convergence of eigenpairs at each iteration:</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>./ex1<span class="w"> </span>-eps_monitor
</pre></div>
</div>
<p>or</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>./ex1<span class="w"> </span>-eps_monitor_all
</pre></div>
</div>
<p>Also, in some SLEPc installations, it is possible to monitor convergence graphically with the <code class="docutils notranslate"><span class="pre">draw</span></code> viewer with <code class="docutils notranslate"><span class="pre">draw_lg</span></code> format. For example, try this:</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>./ex1<span class="w"> </span>-n<span class="w"> </span><span class="m">700</span><span class="w"> </span>-eps_nev<span class="w"> </span><span class="m">5</span><span class="w"> </span>-eps_ncv<span class="w"> </span><span class="m">35</span><span class="w"> </span>-eps_monitor_all<span class="w"> </span>draw::draw_lg<span class="w"> </span>-draw_pause<span class="w"> </span>.1
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The plot is drawn in an X11 pop-up window, so this requires that the display is set up correctly.</p>
</div>
</section>
<section id="choosing-the-eigensolver">
<h2>Choosing the Eigensolver<a class="headerlink" href="#choosing-the-eigensolver" title="Link to this heading">#</a></h2>
<p>The convergence behavior for a particular problem also depends on the properties of the eigensolver being used. SLEPc provides several eigensolvers that can be selected in the source code with the function <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPSSetType.html">EPSSetType</a>()</span></code>, or at run time:</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>./ex1<span class="w"> </span>-eps_nev<span class="w"> </span><span class="m">4</span><span class="w"> </span>-eps_type<span class="w"> </span>lobpcg
</pre></div>
</div>
<p>Check the table of <a class="reference internal" href="../manual/eps.html#tab-solvers"><span class="std std-ref">Eigenvalue solvers available in the EPS 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 eigensolver but simply an interface to some LAPACK routines. These routines operate sequentially in dense mode and therefore are suitable only for small size problems. This method should be used only for debugging purposes.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The last entries of the table (e.g., <code class="docutils notranslate"><span class="pre">arpack</span></code>) may or may not be available on your system, depending on whether they were enabled during installation of SLEPc. They are interfaces to external libraries that can be used as any other SLEPc native eigensolver, see the section <a class="reference internal" href="../manual/extra.html#sec-wrap"><span class="std std-ref">Wrappers to External Libraries</span></a>.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The default solver is <code class="docutils notranslate"><span class="pre">krylovschur</span></code> for both symmetric and non-symmetric problems.</p>
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