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<section class="tex2jax_ignore mathjax_ignore" id="mfn-matrix-function">
<span id="ch-mfn"></span><h1>MFN: Matrix Function<a class="headerlink" href="#mfn-matrix-function" title="Link to this heading">#</a></h1>
<p>The Matrix Function (<code class="docutils notranslate"><span class="pre"><a href="../../manualpages/MFN/MFN.html">MFN</a></span></code>) solver object provides algorithms that compute the action of a matrix function on a given vector, without evaluating the matrix function itself. This is not an eigenvalue problem, but there is a connection between matrix functions and some strategies for computing eigenvalues, and that is why we have this functionality in SLEPc.</p>
<section id="sec-mfn">
<h2>The Problem <span class="math notranslate nohighlight">\(f(A)v\)</span><a class="headerlink" href="#sec-mfn" title="Link to this heading">#</a></h2>
<p>The need to evaluate a function <span class="math notranslate nohighlight">\(f(A)\in\mathbb{C}^{n\times n}\)</span> of a matrix <span class="math notranslate nohighlight">\(A\in\mathbb{C}^{n\times n}\)</span> arises in many applications. There are many methods to compute matrix functions, see for instance the survey by <span id="id1">Higham and Al-Mohy [<a class="reference internal" href="#id36" title="N. J. Higham and A. H. Al-Mohy. Computing matrix functions. Acta Numerica, 19:159–208, 2010. doi:10.1017/S0962492910000036.">2010</a>]</span>. Here, we focus on the case that <span class="math notranslate nohighlight">\(A\)</span> is large and sparse, or is available only as a matrix-vector product subroutine. In such cases, it is the action of <span class="math notranslate nohighlight">\(f(A)\)</span> on a vector, <span class="math notranslate nohighlight">\(f(A)v\)</span>, that is required and not <span class="math notranslate nohighlight">\(f(A)\)</span>. For this, it is possible to adapt some of the methods used to approximate eigenvalues, such as those based on Krylov subspaces or on the concept of contour integral. The description below will be restricted to the case of Krylov methods.</p>
<p>In the sequel, we concentrate on the exponential function, which is one of the most demanded in applications, although the concepts are easily generalizable to other functions as well. Using the Taylor series expansion of <span class="math notranslate nohighlight">\(e^A\)</span>, we have</p>
<div class="math notranslate nohighlight" id="equation-eq-taylor-series">
<span class="eqno">(1)<a class="headerlink" href="#equation-eq-taylor-series" title="Link to this equation">#</a></span>\[y=e^Av=v+\frac{A}{1!}v+\frac{A^2}{2!}v+\cdots,\]</div>
<p>so, in principle, the vector <span class="math notranslate nohighlight">\(y\)</span> can be approximated by an element of the Krylov subspace <span class="math notranslate nohighlight">\(\mathcal{K}_m(A,v)\)</span> defined in equation <a class="reference internal" href="eps.html#equation-eq-krylov">(5)</a>. This is the basis of the method implemented in Expokit <span id="id2">[<a class="reference internal" href="../../manualpages/MFN/MFNEXPOKIT.html#id34" title="R. B. Sidje. Expokit: a software package for computing matrix exponentials. ACM Trans. Math. Software, 24(1):130–156, 1998. doi:10.1145/285861.285868.">Sidje, 1998</a>]</span>. Let <span class="math notranslate nohighlight">\(AV_m=V_{m+1}\underline{H}_m\)</span> be an Arnoldi decomposition, where the columns of <span class="math notranslate nohighlight">\(V_m\)</span> form an orthogonal basis of the Krylov subspace, then the approximation can be computed as</p>
<div class="math notranslate nohighlight" id="equation-eq-aprox-solution">
<span class="eqno">(2)<a class="headerlink" href="#equation-eq-aprox-solution" title="Link to this equation">#</a></span>\[\tilde y=\beta V_m\exp(H_m)e_1,\]</div>
<p>where <span class="math notranslate nohighlight">\(\beta=\|v\|_2\)</span> and <span class="math notranslate nohighlight">\(e_1\)</span> is the first coordinate vector. Hence, the problem of computing the exponential of a large matrix <span class="math notranslate nohighlight">\(A\)</span> of order <span class="math notranslate nohighlight">\(n\)</span> is reduced to computing the exponential of a small matrix <span class="math notranslate nohighlight">\(H_m\)</span> of order <span class="math notranslate nohighlight">\(m\)</span>. For the latter task, we employ algorithms implemented in the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FN.html">FN</a></span></code> auxiliary class, see section <a class="reference internal" href="aux.html#sec-fn"><span class="std std-ref">FN: Mathematical Functions</span></a>.</p>
</section>
<section id="basic-usage">
<h2>Basic Usage<a class="headerlink" href="#basic-usage" title="Link to this heading">#</a></h2>
<p>The user interface of the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/MFN/MFN.html">MFN</a></span></code> package is simpler than the interface of eigensolvers. In some ways, it is more similar to <a class="reference external" href="https://petsc.org/release/manualpages/KSP/KSP/" title="(in PETSc v3.24)"><span class="xref std std-doc">KSP</span></a>, in the sense that the solver maps a vector <span class="math notranslate nohighlight">\(v\)</span> to a vector <span class="math notranslate nohighlight">\(y\)</span>.</p>
<div class="literal-block-wrapper docutils container" id="fig-ex-mfn">
<div class="code-block-caption"><span class="caption-text">Example code for basic solution with <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/MFN/MFN.html">MFN</a></span></code></span><a class="headerlink" href="#fig-ex-mfn" title="Link to this code">#</a></div>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../../manualpages/MFN/MFN.html">MFN</a></span><span class="w"> </span><span class="n">mfn</span><span class="p">;</span><span class="w"> </span><span class="cm">/* <a href="../../manualpages/MFN/MFN.html">MFN</a> solver context */</span>
<span class="n"><a href="https://petsc.org/release/manualpages/Mat/Mat/">Mat</a></span><span class="w"> </span><span class="n">A</span><span class="p">;</span><span class="w"> </span><span class="cm">/* problem matrix */</span>
<span class="n"><a href="../../manualpages/FN/FN.html">FN</a></span><span class="w"> </span><span class="n">f</span><span class="p">;</span><span class="w"> </span><span class="cm">/* the function, exp() in this example */</span>
<span class="n"><a href="https://petsc.org/release/manualpages/Sys/PetscScalar/">PetscScalar</a></span><span class="w"> </span><span class="n">alpha</span><span class="p">;</span><span class="w"> </span><span class="cm">/* to compute exp(alpha*A) */</span>
<span class="n"><a href="https://petsc.org/release/manualpages/Vec/Vec/">Vec</a></span><span class="w"> </span><span class="n">v</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="p">;</span><span class="w"> </span><span class="cm">/* right vector and solution */</span>
<span class="n"><a href="../../manualpages/MFN/MFNCreate.html">MFNCreate</a></span><span class="p">(</span><span class="n"><a href="https://petsc.org/release/manualpages/Sys/PETSC_COMM_WORLD/">PETSC_COMM_WORLD</a></span><span class="p">,</span><span class="w"> </span><span class="o">&</span><span class="n">mfn</span><span class="p">);</span>
<span class="n"><a href="../../manualpages/MFN/MFNSetOperator.html">MFNSetOperator</a></span><span class="p">(</span><span class="n">mfn</span><span class="p">,</span><span class="w"> </span><span class="n">A</span><span class="p">);</span>
<span class="n"><a href="../../manualpages/MFN/MFNGetFN.html">MFNGetFN</a></span><span class="p">(</span><span class="n">mfn</span><span class="p">,</span><span class="w"> </span><span class="o">&</span><span class="n">f</span><span class="p">);</span>
<span class="n"><a href="../../manualpages/FN/FNSetType.html">FNSetType</a></span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="w"> </span><span class="n"><a href="../../manualpages/FN/FNEXP.html">FNEXP</a></span><span class="p">);</span>
<span class="n"><a href="../../manualpages/FN/FNSetScale.html">FNSetScale</a></span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="w"> </span><span class="n">alpha</span><span class="p">,</span><span class="w"> </span><span class="mf">1.0</span><span class="p">);</span>
<span class="n"><a href="../../manualpages/MFN/MFNSetFromOptions.html">MFNSetFromOptions</a></span><span class="p">(</span><span class="n">mfn</span><span class="p">);</span>
<span class="n"><a href="../../manualpages/MFN/MFNSolve.html">MFNSolve</a></span><span class="p">(</span><span class="n">mfn</span><span class="p">,</span><span class="w"> </span><span class="n">v</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="p">);</span>
<span class="n"><a href="../../manualpages/MFN/MFNDestroy.html">MFNDestroy</a></span><span class="p">(</span><span class="o">&</span><span class="n">mfn</span><span class="p">);</span>
</pre></div>
</div>
</div>
<p>Listing <a class="reference internal" href="#fig-ex-mfn"><span class="std std-ref">Example code for basic solution with MFN</span></a> shows a simple example with the basic steps for computing <span class="math notranslate nohighlight">\(y=\exp(\alpha A)v\)</span>. After creating the solver context with <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/MFN/MFNCreate.html">MFNCreate</a>()</span></code>, the problem matrix has to be passed with <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/MFN/MFNSetOperator.html">MFNSetOperator</a>()</span></code> and the function to compute <span class="math notranslate nohighlight">\(f(\cdot)\)</span> must be specified with the aid of the auxiliary class <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FN.html">FN</a></span></code>, see details in section <a class="reference internal" href="aux.html#sec-fn"><span class="std std-ref">FN: Mathematical Functions</span></a>. Then, a call to <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/MFN/MFNSolve.html">MFNSolve</a>()</span></code> runs the solver on a given vector <span class="math notranslate nohighlight">\(v\)</span>, returning the computed result <span class="math notranslate nohighlight">\(y\)</span>. Finally, <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/MFN/MFNDestroy.html">MFNDestroy</a>()</span></code> is used to reclaim memory. We give a few more details below.</p>
<section id="defining-the-problem">
<h3>Defining the Problem<a class="headerlink" href="#defining-the-problem" title="Link to this heading">#</a></h3>
<p>Defining the problem consists in specifying the matrix, <span class="math notranslate nohighlight">\(A\)</span>, and the function to compute, <span class="math notranslate nohighlight">\(f(\cdot)\)</span>. The problem matrix is provided with:</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../../manualpages/MFN/MFNSetOperator.html">MFNSetOperator</a></span><span class="p">(</span><span class="n"><a href="../../manualpages/MFN/MFN.html">MFN</a></span><span class="w"> </span><span class="n">mfn</span><span class="p">,</span><span class="n"><a href="https://petsc.org/release/manualpages/Mat/Mat/">Mat</a></span><span class="w"> </span><span class="n">A</span><span class="p">);</span>
</pre></div>
</div>
<p>where <code class="docutils notranslate"><span class="pre">A</span></code> should be a square matrix, stored in any allowed PETSc format including the matrix-free mechanism (see section <a class="reference internal" href="extra.html#sec-supported"><span class="std std-ref">Supported Matrix Types</span></a>). The function <span class="math notranslate nohighlight">\(f(\cdot)\)</span> is defined with an <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FN.html">FN</a></span></code> object. One possibility is to extract the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FN.html">FN</a></span></code> object handled internally by <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/MFN/MFN.html">MFN</a></span></code>:</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../../manualpages/MFN/MFNGetFN.html">MFNGetFN</a></span><span class="p">(</span><span class="n"><a href="../../manualpages/MFN/MFN.html">MFN</a></span><span class="w"> </span><span class="n">mfn</span><span class="p">,</span><span class="n"><a href="../../manualpages/FN/FN.html">FN</a></span><span class="w"> </span><span class="o">*</span><span class="n">f</span><span class="p">);</span>
</pre></div>
</div>
<p>An alternative would be to create a standalone <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FN.html">FN</a></span></code> object and pass it with <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/MFN/MFNSetFN.html">MFNSetFN</a>()</span></code>. In any case, the function is defined via its type and the relevant parameters, see section <a class="reference internal" href="aux.html#sec-fn"><span class="std std-ref">FN: Mathematical Functions</span></a> for details. The scaling parameters can be used for instance for the exponential when used in the context of ODE integration, <span class="math notranslate nohighlight">\(y=e^{tA}v\)</span>, where <span class="math notranslate nohighlight">\(t\)</span> represents the elapsed time.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Some <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/MFN/MFN.html">MFN</a></span></code> solvers may be restricted to only some types of <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FN.html">FN</a></span></code> functions.</p>
</div>
<p>In <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/MFN/MFN.html">MFN</a></span></code> it makes no sense to specify the number of eigenvalues. However, there is a related operation that allows the user to specify the size of the subspace that will be used internally by the solver (<code class="docutils notranslate"><span class="pre">ncv</span></code>, the number of column vectors of the basis):</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../../manualpages/MFN/MFNSetDimensions.html">MFNSetDimensions</a></span><span class="p">(</span><span class="n"><a href="../../manualpages/EPS/EPS.html">EPS</a></span><span class="w"> </span><span class="n">eps</span><span class="p">,</span><span class="n"><a href="https://petsc.org/release/manualpages/Sys/PetscInt/">PetscInt</a></span><span class="w"> </span><span class="n">ncv</span><span class="p">);</span>
</pre></div>
</div>
<p>This parameter can also be set at run time with the option <code class="docutils notranslate"><span class="pre">-mfn_ncv</span></code>.</p>
</section>
<section id="selecting-the-solver">
<h3>Selecting the Solver<a class="headerlink" href="#selecting-the-solver" title="Link to this heading">#</a></h3>
<div class="pst-scrollable-table-container"><table class="table" id="tab-mfnsolvers">
<caption><span class="caption-text">List of solvers available in the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/MFN/MFN.html">MFN</a></span></code> module.</span><a class="headerlink" href="#tab-mfnsolvers" title="Link to this table">#</a></caption>
<thead>
<tr class="row-odd"><th class="head"><p>Method</p></th>
<th class="head"><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/MFN/MFNType.html">MFNType</a></span></code></p></th>
<th class="head"><p>Options Database</p></th>
<th class="head"><p>Supported Functions</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p>Restarted Krylov solver</p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/MFN/MFNKRYLOV.html">MFNKRYLOV</a></span></code></p></td>
<td><p><code class="docutils notranslate"><span class="pre">krylov</span></code></p></td>
<td><p>Any</p></td>
</tr>
<tr class="row-odd"><td><p>Expokit algorithm</p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/MFN/MFNEXPOKIT.html">MFNEXPOKIT</a></span></code></p></td>
<td><p><code class="docutils notranslate"><span class="pre">expokit</span></code></p></td>
<td><p>Exponential</p></td>
</tr>
</tbody>
</table>
</div>
<p>The methods available in <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/MFN/MFN.html">MFN</a></span></code> are shown in table <a class="reference internal" href="#tab-mfnsolvers"><span class="std std-ref">List of solvers available in the MFN module.</span></a>. The solution method can be specified procedurally with:</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../../manualpages/MFN/MFNSetType.html">MFNSetType</a></span><span class="p">(</span><span class="n"><a href="../../manualpages/MFN/MFN.html">MFN</a></span><span class="w"> </span><span class="n">mfn</span><span class="p">,</span><span class="n"><a href="../../manualpages/MFN/MFNType.html">MFNType</a></span><span class="w"> </span><span class="n">method</span><span class="p">);</span>
</pre></div>
</div>
<p>or via the options database command <code class="docutils notranslate"><span class="pre">-mfn_type</span></code> followed by the method name (see table <a class="reference internal" href="#tab-mfnsolvers"><span class="std std-ref">List of solvers available in the MFN module.</span></a>).</p>
<p>Currently implemented methods are:</p>
<ul class="simple">
<li><p>A Krylov method with restarts as proposed by <span id="id3">Eiermann and Ernst [<a class="reference internal" href="../../manualpages/MFN/MFNKRYLOV.html#id39" title="M. Eiermann and O. G. Ernst. A restarted Krylov subspace method for the evaluation of matrix functions. SIAM J. Numer. Anal., 44(6):2481–2504, 2006. doi:10.1137/050633846.">2006</a>]</span>.</p></li>
<li><p>The method implemented in Expokit <span id="id4">[<a class="reference internal" href="../../manualpages/MFN/MFNEXPOKIT.html#id34" title="R. B. Sidje. Expokit: a software package for computing matrix exponentials. ACM Trans. Math. Software, 24(1):130–156, 1998. doi:10.1145/285861.285868.">Sidje, 1998</a>]</span> for the matrix exponential.</p></li>
</ul>
</section>
<section id="accuracy-and-monitors">
<h3>Accuracy and Monitors<a class="headerlink" href="#accuracy-and-monitors" title="Link to this heading">#</a></h3>
<p>In the <span class="math notranslate nohighlight">\(f(A)v\)</span> problem, there is no clear definition of residual, as opposed to the case of linear systems or eigenproblems. Still, the solvers have different ways of assessing the accuracy of the computed solution. The user can provide a tolerance and maximum number of iterations with <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/MFN/MFNSetTolerances.html">MFNSetTolerances</a>()</span></code>, but there is no guarantee that an analog of the residual is below the tolerance.</p>
<p>After the solver has finished, the number of performed (outer) iterations can be obtained with <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/MFN/MFNGetIterationNumber.html">MFNGetIterationNumber</a>()</span></code>. There are also monitors that display the error estimate, which can be activated with command-line keys <code class="docutils notranslate"><span class="pre">-mfn_monitor</span></code>, or <code class="docutils notranslate"><span class="pre">-mfn_monitor</span> <span class="pre">draw::draw_lg</span></code>. See section <a class="reference internal" href="eps.html#sec-monitor"><span class="std std-ref">Controlling and Monitoring Convergence</span></a> for additional details.</p>
<p class="rubric">References</p>
<div class="docutils container" id="id5">
<div role="list" class="citation-list">
<div class="citation" id="id42" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id3">Eie06</a><span class="fn-bracket">]</span></span>
<p>M. Eiermann and O. G. Ernst. A restarted Krylov subspace method for the evaluation of matrix functions. <em>SIAM J. Numer. Anal.</em>, 44(6):2481–2504, 2006. <a class="reference external" href="https://doi.org/10.1137/050633846">doi:10.1137/050633846</a>.</p>
</div>
<div class="citation" id="id36" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id1">Hig10</a><span class="fn-bracket">]</span></span>
<p>N. J. Higham and A. H. Al-Mohy. Computing matrix functions. <em>Acta Numerica</em>, 19:159–208, 2010. <a class="reference external" href="https://doi.org/10.1017/S0962492910000036">doi:10.1017/S0962492910000036</a>.</p>
</div>
<div class="citation" id="id37" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span>Sid98<span class="fn-bracket">]</span></span>
<span class="backrefs">(<a role="doc-backlink" href="#id2">1</a>,<a role="doc-backlink" href="#id4">2</a>)</span>
<p>R. B. Sidje. Expokit: a software package for computing matrix exponentials. <em>ACM Trans. Math. Software</em>, 24(1):130–156, 1998. <a class="reference external" href="https://doi.org/10.1145/285861.285868">doi:10.1145/285861.285868</a>.</p>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#sec-mfn">The Problem <span class="math notranslate nohighlight">\(f(A)v\)</span></a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#basic-usage">Basic Usage</a><ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#defining-the-problem">Defining the Problem</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#selecting-the-solver">Selecting the Solver</a></li>
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