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<section class="tex2jax_ignore mathjax_ignore" id="auxiliary-classes">
<span id="ch-aux"></span><h1>Auxiliary Classes<a class="headerlink" href="#auxiliary-classes" title="Link to this heading">#</a></h1>
<p>Apart from the main solver classes listed in table <a class="reference internal" href="index.html#tab-modules"><span class="std std-ref">SLEPc modules</span></a>, SLEPc contains several auxiliary classes:</p>
<ul class="simple">
<li><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/ST/ST.html">ST</a></span></code>: Spectral Transformation, fully described in chapter <a class="reference internal" href="st.html#ch-st"><span class="std std-ref">ST: Spectral Transformation</span></a>.</p></li>
<li><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FN.html">FN</a></span></code>: Mathematical Function, required in application code to represent the constituent functions of the nonlinear operator in split form (chapter <a class="reference internal" href="nep.html#ch-nep"><span class="std std-ref">NEP: Nonlinear Eigenvalue Problems</span></a>), as well as the function to be used when computing the action of a matrix function on a vector (chapter <a class="reference internal" href="mfn.html#ch-mfn"><span class="std std-ref">MFN: Matrix Function</span></a>).</p></li>
<li><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/RG/RG.html">RG</a></span></code>: Region, a way to define a region of the complex plane.</p></li>
<li><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DS/DS.html">DS</a></span></code>: Direct Solver (or Dense System), can be seen as a wrapper to LAPACK functions used within SLEPc.</p></li>
<li><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BV.html">BV</a></span></code>: Basis Vectors, provides the concept of a block of vectors that represent the basis of a subspace.</p></li>
</ul>
<p>The first three classes, <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/ST/ST.html">ST</a></span></code>, <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FN.html">FN</a></span></code> and <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/RG/RG.html">RG</a></span></code>, are relevant for end users, while <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DS/DS.html">DS</a></span></code> and <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BV.html">BV</a></span></code> are intended mainly for SLEPc developers. Below we provide a brief description of <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FN.html">FN</a></span></code>, <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/RG/RG.html">RG</a></span></code>, <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DS/DS.html">DS</a></span></code> and <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BV.html">BV</a></span></code>.</p>
<section id="sec-fn">
<h2>FN: Mathematical Functions<a class="headerlink" href="#sec-fn" title="Link to this heading">#</a></h2>
<p>The <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FN.html">FN</a></span></code> class provides a few predefined mathematical functions, including rational functions (of which polynomials are a particular case) and exponentials. Objects of this class are instantiated by providing the values of the relevant parameters. <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FN.html">FN</a></span></code> objects are created with <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FNCreate.html">FNCreate</a>()</span></code> and it is necessary to select the type of function (rational, exponential, etc.) with <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FNSetType.html">FNSetType</a>()</span></code>. Table <a class="reference internal" href="#tab-fn"><span class="std std-ref">Mathematical functions available as FN objects</span></a> lists available functions.</p>
<div class="pst-scrollable-table-container"><table class="table" id="tab-fn">
<caption><span class="caption-text">Mathematical functions available as <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FN.html">FN</a></span></code> objects</span><a class="headerlink" href="#tab-fn" title="Link to this table">#</a></caption>
<thead>
<tr class="row-odd"><th class="head"><p>Function</p></th>
<th class="head"><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FNType.html">FNType</a></span></code></p></th>
<th class="head"><p>Expression</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p>Polynomial and rational</p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FNRATIONAL.html">FNRATIONAL</a></span></code></p></td>
<td><p><span class="math notranslate nohighlight">\(p(x)/q(x)\)</span></p></td>
</tr>
<tr class="row-odd"><td><p>Exponential</p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FNEXP.html">FNEXP</a></span></code></p></td>
<td><p><span class="math notranslate nohighlight">\(e^x\)</span></p></td>
</tr>
<tr class="row-even"><td><p>Logarithm</p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FNLOG.html">FNLOG</a></span></code></p></td>
<td><p><span class="math notranslate nohighlight">\(\log x\)</span></p></td>
</tr>
<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(\varphi\)</span>-functions</p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FNPHI.html">FNPHI</a></span></code></p></td>
<td><p><span class="math notranslate nohighlight">\(\varphi_0(x)\)</span>, <span class="math notranslate nohighlight">\(\varphi_1(x)\)</span>, …</p></td>
</tr>
<tr class="row-even"><td><p>Square root</p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FNSQRT.html">FNSQRT</a></span></code></p></td>
<td><p><span class="math notranslate nohighlight">\(\sqrt{x}\)</span></p></td>
</tr>
<tr class="row-odd"><td><p>Inverse square root</p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FNINVSQRT.html">FNINVSQRT</a></span></code></p></td>
<td><p><span class="math notranslate nohighlight">\(x^{-\frac{1}{2}}\)</span></p></td>
</tr>
<tr class="row-even"><td><p>Combine two functions</p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FNCOMBINE.html">FNCOMBINE</a></span></code></p></td>
<td><p>See text</p></td>
</tr>
</tbody>
</table>
</div>
<p>Parameters common to all <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FN.html">FN</a></span></code> types are the scaling factors, which are set with:</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../../manualpages/FN/FNSetScale.html">FNSetScale</a></span><span class="p">(</span><span class="n"><a href="../../manualpages/FN/FN.html">FN</a></span><span class="w"> </span><span class="n">fn</span><span class="p">,</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="n"><a href="https://petsc.org/release/manualpages/Sys/PetscScalar/">PetscScalar</a></span><span class="w"> </span><span class="n">beta</span><span class="p">);</span>
</pre></div>
</div>
<p>where <code class="docutils notranslate"><span class="pre">alpha</span></code> multiplies the argument and <code class="docutils notranslate"><span class="pre">beta</span></code> multiplies the result. With this, the actual function is <span class="math notranslate nohighlight">\(\beta\cdot f(\alpha\cdot x)\)</span> for a given function <span class="math notranslate nohighlight">\(f(\cdot)\)</span>. For instance, an exponential function <span class="math notranslate nohighlight">\(f(x)=e^x\)</span> will turn into</p>
<div class="math notranslate nohighlight" id="equation-eq-exp-function">
<span class="eqno">(1)<a class="headerlink" href="#equation-eq-exp-function" title="Link to this equation">#</a></span>\[g(x)=\beta e^{\alpha x}.\]</div>
<p>In a rational function there are specific parameters, namely the coefficients of the numerator and denominator,</p>
<div class="math notranslate nohighlight" id="equation-eq-num-denum-coefficients">
<span class="eqno">(2)<a class="headerlink" href="#equation-eq-num-denum-coefficients" title="Link to this equation">#</a></span>\[r(x)=\frac{p(x)}{q(x)}
=\frac{\nu_{n-1}x^{n-1}+\cdots+\nu_1x+\nu_0}{\delta_{m-1}x^{m-1}+\cdots+\delta_1x+\delta_0}.\]</div>
<p>These parameters are specified with:</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../../manualpages/FN/FNRationalSetNumerator.html">FNRationalSetNumerator</a></span><span class="p">(</span><span class="n"><a href="../../manualpages/FN/FN.html">FN</a></span><span class="w"> </span><span class="n">fn</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">np</span><span class="p">,</span><span class="n"><a href="https://petsc.org/release/manualpages/Sys/PetscScalar/">PetscScalar</a></span><span class="w"> </span><span class="n">pcoeff</span><span class="p">[]);</span>
<span class="n"><a href="../../manualpages/FN/FNRationalSetDenominator.html">FNRationalSetDenominator</a></span><span class="p">(</span><span class="n"><a href="../../manualpages/FN/FN.html">FN</a></span><span class="w"> </span><span class="n">fn</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">nq</span><span class="p">,</span><span class="n"><a href="https://petsc.org/release/manualpages/Sys/PetscScalar/">PetscScalar</a></span><span class="w"> </span><span class="n">qcoeff</span><span class="p">[]);</span>
</pre></div>
</div>
<p>Here, polynomials are passed as an array with high order coefficients appearing in low indices.</p>
<p>The <span class="math notranslate nohighlight">\(\varphi\)</span>-functions are given by</p>
<div class="math notranslate nohighlight" id="equation-eq-phi-functions">
<span class="eqno">(3)<a class="headerlink" href="#equation-eq-phi-functions" title="Link to this equation">#</a></span>\[\varphi_0(x)=e^x,\qquad \varphi_1(x)=\frac{e^x-1}{x},\qquad \varphi_k(x)=\frac{\varphi_{k-1}(x)-1/(k-1)!}{x},\]</div>
<p>where the index <span class="math notranslate nohighlight">\(k\)</span> must be specified with <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FNPhiSetIndex.html">FNPhiSetIndex</a>()</span></code>.</p>
<p>Whenever the solvers need to compute <span class="math notranslate nohighlight">\(f(x)\)</span> or <span class="math notranslate nohighlight">\(f'(x)\)</span> on a given scalar <span class="math notranslate nohighlight">\(x\)</span>, the following functions are invoked:</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../../manualpages/FN/FNEvaluateFunction.html">FNEvaluateFunction</a></span><span class="p">(</span><span class="n"><a href="../../manualpages/FN/FN.html">FN</a></span><span class="w"> </span><span class="n">fn</span><span class="p">,</span><span class="n"><a href="https://petsc.org/release/manualpages/Sys/PetscScalar/">PetscScalar</a></span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="n"><a href="https://petsc.org/release/manualpages/Sys/PetscScalar/">PetscScalar</a></span><span class="w"> </span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
<span class="n"><a href="../../manualpages/FN/FNEvaluateDerivative.html">FNEvaluateDerivative</a></span><span class="p">(</span><span class="n"><a href="../../manualpages/FN/FN.html">FN</a></span><span class="w"> </span><span class="n">fn</span><span class="p">,</span><span class="n"><a href="https://petsc.org/release/manualpages/Sys/PetscScalar/">PetscScalar</a></span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="n"><a href="https://petsc.org/release/manualpages/Sys/PetscScalar/">PetscScalar</a></span><span class="w"> </span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
</pre></div>
</div>
<p>The function can also be evaluated as a matrix function, <span class="math notranslate nohighlight">\(B=f(A)\)</span>, where <span class="math notranslate nohighlight">\(A,B\)</span> are small, dense, square matrices. This is done with <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FNEvaluateFunctionMat.html">FNEvaluateFunctionMat</a>()</span></code>. Note that for a rational function, the corresponding expression would be <span class="math notranslate nohighlight">\(q(A)^{-1}p(A)\)</span>. For computing functions such as the exponential of a small matrix <span class="math notranslate nohighlight">\(A\)</span>, several methods are available. When the matrix <span class="math notranslate nohighlight">\(A\)</span> is symmetric, the default is to compute <span class="math notranslate nohighlight">\(f(A)\)</span> using the eigendecomposition <span class="math notranslate nohighlight">\(A=Q\Lambda Q^*\)</span>, for instance the exponential would be computed as <span class="math notranslate nohighlight">\(\exp(A)=Q\,\mathrm{diag}(e^{\lambda_i})Q^*\)</span>. In the general case, it is necessary to have recourse to one of the methods discussed in, e.g., <span id="id1">[<a class="reference internal" href="mfn.html#id36" title="N. J. Higham and A. H. Al-Mohy. Computing matrix functions. Acta Numerica, 19:159–208, 2010. doi:10.1017/S0962492910000036.">Higham and Al-Mohy, 2010</a>]</span>.</p>
<p>Finally, there is a mechanism to combine simple functions in order to create more complicated functions. For instance, the function</p>
<div class="math notranslate nohighlight" id="equation-eq-combined-funct">
<span class="eqno">(4)<a class="headerlink" href="#equation-eq-combined-funct" title="Link to this equation">#</a></span>\[f(x) = (1-x^2) \exp\left( \frac{-x}{1+x^2} \right)\]</div>
<p>can be represented with an expression tree with three leaves (one exponential function and two rational functions) and two interior nodes (one of them is the root, <span class="math notranslate nohighlight">\(f(x)\)</span>). Interior nodes are simply <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FN.html">FN</a></span></code> objects of type <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FNCOMBINE.html">FNCOMBINE</a></span></code> that specify how the two children must be combined (with either addition, multiplication, division or function composition):</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../../manualpages/FN/FNCombineSetChildren.html">FNCombineSetChildren</a></span><span class="p">(</span><span class="n"><a href="../../manualpages/FN/FN.html">FN</a></span><span class="w"> </span><span class="n">fn</span><span class="p">,</span><span class="n"><a href="../../manualpages/FN/FNCombineType.html">FNCombineType</a></span><span class="w"> </span><span class="n">comb</span><span class="p">,</span><span class="n"><a href="../../manualpages/FN/FN.html">FN</a></span><span class="w"> </span><span class="n">f1</span><span class="p">,</span><span class="n"><a href="../../manualpages/FN/FN.html">FN</a></span><span class="w"> </span><span class="n">f2</span><span class="p">)</span>
</pre></div>
</div>
<p>The combination of <span class="math notranslate nohighlight">\(f_1\)</span> and <span class="math notranslate nohighlight">\(f_2\)</span> with division will result in <span class="math notranslate nohighlight">\(f_1(x)/f_2(x)\)</span> and <span class="math notranslate nohighlight">\(f_2(A)^{-1}f_1(A)\)</span> in the case of matrices.</p>
</section>
<section id="sec-rg">
<h2>RG: Region<a class="headerlink" href="#sec-rg" title="Link to this heading">#</a></h2>
<p>The <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/RG/RG.html">RG</a></span></code> object defines a region of the complex plane, that can be used to specify where eigenvalues must be sought. Currently, the following types of regions are available:</p>
<ul class="simple">
<li><p>A (generalized) interval, defined as <span class="math notranslate nohighlight">\([a,b]\times[c,d]\)</span>, where the four parameters can be set with <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/RG/RGIntervalSetEndpoints.html">RGIntervalSetEndpoints</a>()</span></code>. This covers the particular cases of an interval on the real axis (setting <span class="math notranslate nohighlight">\(c=d=0\)</span>), the left halfplane <span class="math notranslate nohighlight">\([-\infty,0]\times[-\infty,+\infty]\)</span>, a quadrant, etc. See figure <a class="reference internal" href="#fig-rg-interval"><span class="std std-ref">Interval region defined via de RG class</span></a>.</p></li>
</ul>
<figure class="align-default" id="fig-rg-interval">
<img alt="Interval region defined via de RG class" src="../../_images/fig-rg-interval.svg" /><figcaption>
<p><span class="caption-text">Interval region defined via de RG class</span><a class="headerlink" href="#fig-rg-interval" title="Link to this image">#</a></p>
</figcaption>
</figure>
<ul class="simple">
<li><p>A polygon defined by its vertices, given via <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/RG/RGPolygonSetVertices.html">RGPolygonSetVertices</a>()</span></code>. See figure <a class="reference internal" href="#fig-rg-polygon"><span class="std std-ref">Polygon region defined via de RG class</span></a>.</p></li>
</ul>
<figure class="align-default" id="fig-rg-polygon">
<img alt="Polygon region defined via de RG class" src="../../_images/fig-rg-polygon.svg" /><figcaption>
<p><span class="caption-text">Polygon region defined via de RG class</span><a class="headerlink" href="#fig-rg-polygon" title="Link to this image">#</a></p>
</figcaption>
</figure>
<ul class="simple">
<li><p>An ellipse defined by its center, radius and vertical scale (1 by default), specified with <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/RG/RGEllipseSetParameters.html">RGEllipseSetParameters</a>()</span></code>. See figure <a class="reference internal" href="#fig-rg-ellipse"><span class="std std-ref">Ellipse region defined via de RG class</span></a>.</p></li>
</ul>
<figure class="align-default" id="fig-rg-ellipse">
<img alt="Ellipse region defined via de RG class" src="../../_images/fig-rg-ellipse.svg" /><figcaption>
<p><span class="caption-text">Ellipse region defined via de RG class</span><a class="headerlink" href="#fig-rg-ellipse" title="Link to this image">#</a></p>
</figcaption>
</figure>
<ul class="simple">
<li><p>A ring region similar to an ellipse but consisting of a thin stripe along the ellipse with optional start and end angles. See figure <a class="reference internal" href="#fig-rg-ring"><span class="std std-ref">Ring region defined via de RG class</span></a>. The parameters are set with <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/RG/RGRingSetParameters.html">RGRingSetParameters</a>()</span></code>.</p></li>
</ul>
<figure class="align-default" id="fig-rg-ring">
<img alt="Ring region defined via de RG class" src="../../_images/fig-rg-ring.svg" /><figcaption>
<p><span class="caption-text">Ring region defined via de RG class</span><a class="headerlink" href="#fig-rg-ring" title="Link to this image">#</a></p>
</figcaption>
</figure>
<p>Check table <a class="reference internal" href="#tab-rg"><span class="std std-ref">Regions available as RG objects</span></a> for the names that should be used in each case.</p>
<div class="pst-scrollable-table-container"><table class="table" id="tab-rg">
<caption><span class="caption-text">Regions available as <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/RG/RG.html">RG</a></span></code> objects</span><a class="headerlink" href="#tab-rg" title="Link to this table">#</a></caption>
<thead>
<tr class="row-odd"><th class="head"><p>Region Type</p></th>
<th class="head"><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/RG/RGType.html">RGType</a></span></code></p></th>
<th class="head"><p>Options Database</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p>(Generalized) Interval</p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/RG/RGINTERVAL.html">RGINTERVAL</a></span></code></p></td>
<td><p><code class="docutils notranslate"><span class="pre">interval</span></code></p></td>
</tr>
<tr class="row-odd"><td><p>Polygon</p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/RG/RGPOLYGON.html">RGPOLYGON</a></span></code></p></td>
<td><p><code class="docutils notranslate"><span class="pre">polygon</span></code></p></td>
</tr>
<tr class="row-even"><td><p>Ellipse</p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/RG/RGELLIPSE.html">RGELLIPSE</a></span></code></p></td>
<td><p><code class="docutils notranslate"><span class="pre">ellipse</span></code></p></td>
</tr>
<tr class="row-odd"><td><p>Ring</p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/RG/RGRING.html">RGRING</a></span></code></p></td>
<td><p><code class="docutils notranslate"><span class="pre">ring</span></code></p></td>
</tr>
</tbody>
</table>
</div>
<p>Sometimes it is useful to specify the complement of a certain region, e.g., the part of the complex plane outside an ellipse. This can be achieved with:</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../../manualpages/RG/RGSetComplement.html">RGSetComplement</a></span><span class="p">(</span><span class="n"><a href="../../manualpages/RG/RG.html">RG</a></span><span class="w"> </span><span class="n">rg</span><span class="p">,</span><span class="n"><a href="https://petsc.org/release/manualpages/Sys/PetscBool/">PetscBool</a></span><span class="w"> </span><span class="n">flg</span><span class="p">)</span>
</pre></div>
</div>
<p>or in the command line with <code class="docutils notranslate"><span class="pre">-rg_complement</span></code>.</p>
<p>By default, a newly created <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/RG/RG.html">RG</a></span></code> object that is not set a type nor parameters must represent the whole complex plane (the same as <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/RG/RGINTERVAL.html">RGINTERVAL</a></span></code> with values <span class="math notranslate nohighlight">\([-\infty,+\infty]\times[-\infty,+\infty]\)</span>). We call this the <em>trivial</em> region, and provide a function to test this situation:</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../../manualpages/RG/RGIsTrivial.html">RGIsTrivial</a></span><span class="p">(</span><span class="n"><a href="../../manualpages/RG/RG.html">RG</a></span><span class="w"> </span><span class="n">rg</span><span class="p">,</span><span class="n"><a href="https://petsc.org/release/manualpages/Sys/PetscBool/">PetscBool</a></span><span class="w"> </span><span class="o">*</span><span class="n">trivial</span><span class="p">)</span>
</pre></div>
</div>
<p>Another useful operation is to check whether a given point of the complex plane is inside the region or not:</p>
<div class="highlight-c notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../../manualpages/RG/RGCheckInside.html">RGCheckInside</a></span><span class="p">(</span><span class="n"><a href="../../manualpages/RG/RG.html">RG</a></span><span class="w"> </span><span class="n">rg</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">n</span><span class="p">,</span><span class="n"><a href="https://petsc.org/release/manualpages/Sys/PetscScalar/">PetscScalar</a></span><span class="w"> </span><span class="n">ar</span><span class="p">[],</span><span class="n"><a href="https://petsc.org/release/manualpages/Sys/PetscScalar/">PetscScalar</a></span><span class="w"> </span><span class="n">ai</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="o">*</span><span class="n">inside</span><span class="p">)</span>
</pre></div>
</div>
<p>Note that the point is represented as two <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>’s, similarly to eigenvalues in SLEPc.</p>
</section>
<section id="sec-ds">
<h2>DS: Direct Solver (or Dense System)<a class="headerlink" href="#sec-ds" title="Link to this heading">#</a></h2>
<p>The <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DS/DS.html">DS</a></span></code> class can be seen as a wrapper to LAPACK functions <span id="id2">[<a class="reference internal" href="../../manualpages/EPS/EPSLAPACK.html#id3" title="E. Anderson, Z. Bai, C. Bischof, L. S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen. LAPACK Users' Guide. Society for Industrial and Applied Mathematics, Philadelphia, PA, third edition, 1999.">Anderson <em>et al.</em>, 1999</a>]</span> used within SLEPc. It is mostly an internal object that need not be called by end users.</p>
<p>Many of the iterative eigensolvers implemented in SLEPc, such as Krylov-Schur or LOBPCG, perform a projection onto a low-dimensional subspace, e.g., <span class="math notranslate nohighlight">\(M=V^*AV\)</span>, where <span class="math notranslate nohighlight">\(M\)</span> is a dense matrix of small dimension (compared to the size of the original problem) and may or may not have a special structure such as Hessenberg or tridiagonal. Computing the full solution of an eigenproblem associated with matrix <span class="math notranslate nohighlight">\(M\)</span> is necessary to continue the outer iterative eigensolver, and this is typically done by calling a LAPACK function.</p>
<p>In case of parallel runs with several MPI processes, the result of the projection, <span class="math notranslate nohighlight">\(M\)</span>, is available in all processes and all of them solve the dense eigenproblem redundantly. Although this behavior can be changed slightly (see <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DS/DSSetParallel.html">DSSetParallel</a>()</span></code>), the computation done in <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DS/DS.html">DS</a></span></code> should be considered a sequential step of the overall algorithm, because we assume the size of <span class="math notranslate nohighlight">\(M\)</span> is small. That is why it is relevant for parallel efficiency to keep the size of <span class="math notranslate nohighlight">\(M\)</span> bounded, see discussion in <a class="reference internal" href="eps.html#sec-large-nev"><span class="std std-ref">Computing a Large Portion of the Spectrum</span></a>.</p>
<p>Due to the wide range of eigenproblems covered by SLEPc, the projected eigenproblem also has many variants, represented by the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DS/DSType.html">DSType</a></span></code>.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The projected problem associated with <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/MFN/MFN.html">MFN</a></span></code> solvers is a matrix function calculation, which is implemented in <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/FN/FN.html">FN</a></span></code> and not <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DS/DS.html">DS</a></span></code>.</p>
</div>
<p>Not all <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DS/DS.html">DS</a></span></code> types are covered by LAPACK subroutines, particularly <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DS/DSGHIEP.html">DSGHIEP</a></span></code>, <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DS/DSHSVD.html">DSHSVD</a></span></code>, <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DS/DSPEP.html">DSPEP</a></span></code>, and <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DS/DSNEP.html">DSNEP</a></span></code>. Hence <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DS/DS.html">DS</a></span></code> provides implementations for solving those problems. In many cases, several methods are available, and this includes the case where LAPACK provides several subroutines for the same problem. The user can select a solution method with an integer set via <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DS/DSSetMethod.html">DSSetMethod</a>()</span></code>.</p>
<p>The user interface is organized in a way that accommodates to the needs of iterative eigensolvers. The computation is split in three stages, <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DS/DSSolve.html">DSSolve</a>()</span></code>, <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DS/DSSort.html">DSSort</a>()</span></code> and <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DS/DSVectors.html">DSVectors</a>()</span></code>, which are typically called at different times. There are also convenience functions for truncating a previously computed solution, <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DS/DSTruncate.html">DSTruncate</a>()</span></code>, or to process the additional row present in <span class="math notranslate nohighlight">\(M\)</span> in case of Krylov methods, see <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DS/DSUpdateExtraRow.html">DSUpdateExtraRow</a>()</span></code>, which results in the typical arrowhead form when restarting Lanczos, for instance.</p>
<p>To manipulate matrix <span class="math notranslate nohighlight">\(M\)</span> and other matrices associated with the projected problem, the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DS/DS.html">DS</a></span></code> interface provides a list of dense matrices, see <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DS/DSMatType.html">DSMatType</a></span></code>, and operations to work with them, such as <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DS/DSGetMat.html">DSGetMat</a>()</span></code> or <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/DS/DSGetArray.html">DSGetArray</a>()</span></code>.</p>
</section>
<section id="sec-bv">
<h2>BV: Basis Vectors<a class="headerlink" href="#sec-bv" title="Link to this heading">#</a></h2>
<p>The <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BV.html">BV</a></span></code> class may be useful for advanced users, so we briefly describe it here for completeness. <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BV.html">BV</a></span></code> is a convenient way of handling a collection of vectors that often operate together, rather than working with an array of <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>. It can be seen as a generalization of <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> to a tall-skinny matrix with several columns.</p>
<div class="pst-scrollable-table-container"><table class="table" id="tab-bv">
<caption><span class="caption-text">Operations available for <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BV.html">BV</a></span></code> objects</span><a class="headerlink" href="#tab-bv" title="Link to this table">#</a></caption>
<thead>
<tr class="row-odd"><th class="head"><p>Operation</p></th>
<th class="head"><p>Block version</p></th>
<th class="head"><p>Column version</p></th>
<th class="head"><p>Vector version</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(Y=X\)</span></p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BVCopy.html">BVCopy</a>()</span></code></p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BVCopyColumn.html">BVCopyColumn</a>()</span></code></p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BVCopyVec.html">BVCopyVec</a>()</span></code></p></td>
</tr>
<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(Y=\beta Y+\alpha XQ\)</span></p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BVMult.html">BVMult</a>()</span></code></p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BVMultColumn.html">BVMultColumn</a>()</span></code></p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BVMultVec.html">BVMultVec</a>()</span></code></p></td>
</tr>
<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(M=Y^*\!AX\)</span></p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BVMatProject.html">BVMatProject</a>()</span></code></p></td>
<td><p>–</p></td>
<td><p>–</p></td>
</tr>
<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(M=Y^*X\)</span></p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BVDot.html">BVDot</a>()</span></code></p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BVDotColumn.html">BVDotColumn</a>()</span></code></p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BVDotVec.html">BVDotVec</a>()</span></code></p></td>
</tr>
<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(Y=\alpha Y\)</span></p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BVScale.html">BVScale</a>()</span></code></p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BVScaleColumn.html">BVScaleColumn</a>()</span></code></p></td>
<td><p>–</p></td>
</tr>
<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(r=\|X\|_{type}\)</span></p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BVNorm.html">BVNorm</a>()</span></code></p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BVNormColumn.html">BVNormColumn</a>()</span></code></p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BVNormVec.html">BVNormVec</a>()</span></code></p></td>
</tr>
<tr class="row-even"><td><p>Set to random values</p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BVSetRandom.html">BVSetRandom</a>()</span></code></p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BVSetRandomColumn.html">BVSetRandomColumn</a>()</span></code></p></td>
<td><p>–</p></td>
</tr>
<tr class="row-odd"><td><p>Orthogonalize</p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BVOrthogonalize.html">BVOrthogonalize</a>()</span></code></p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BVOrthogonalizeColumn.html">BVOrthogonalizeColumn</a>()</span></code></p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BVOrthogonalizeVec.html">BVOrthogonalizeVec</a>()</span></code></p></td>
</tr>
</tbody>
</table>
</div>
<p>Table <a class="reference internal" href="#tab-bv"><span class="std std-ref">Operations available for BV objects</span></a> shows a summary of the operations offered by the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BV.html">BV</a></span></code> class, with variants that operate on the whole <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BV.html">BV</a></span></code>, on a single column, or on an external <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> object. Missing variants can be achieved simply with <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> and <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> operations. Other available variants not shown in the table are <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BVMultInPlace.html">BVMultInPlace</a>()</span></code>, <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BVMultInPlaceHermitianTranspose.html">BVMultInPlaceHermitianTranspose</a>()</span></code> and <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BVOrthogonalizeSomeColumn.html">BVOrthogonalizeSomeColumn</a>()</span></code>.</p>
<p>Most SLEPc solvers use a <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BV.html">BV</a></span></code> object to represent the working subspace basis. In particular, orthogonalization operations are mostly confined within <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BV.html">BV</a></span></code>. Hence, <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BV.html">BV</a></span></code> provides options for specifying the method of orthogonalization of vectors (Gram-Schmidt) as well as the method of block orthogonalization, see <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/BV/BVSetOrthogonalization.html">BVSetOrthogonalization</a>()</span></code>.</p>
<p class="rubric">References</p>
<div class="docutils container" id="id3">
<div role="list" class="citation-list">
<div class="citation" id="id4" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id2">And99</a><span class="fn-bracket">]</span></span>
<p>E. Anderson, Z. Bai, C. Bischof, L. S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen. <em>LAPACK Users' Guide</em>. Society for Industrial and Applied Mathematics, Philadelphia, PA, third edition, 1999.</p>
</div>
<div class="citation" id="id34" 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>
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