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<section class="tex2jax_ignore mathjax_ignore" id="lme-linear-matrix-equation">
<span id="ch-lme"></span><h1>LME: Linear Matrix Equation<a class="headerlink" href="#lme-linear-matrix-equation" title="Link to this heading">#</a></h1>
<p>The Linear Matrix Equation (<code class="docutils notranslate"><span class="pre"><a href="../../manualpages/LME/LME.html">LME</a></span></code>) solver object encapsulates functionality intended for the solution of linear matrix equations such as Lyapunov, Sylvester, and their various generalizations. A matrix equation is an equation where the unknown and the coefficients are all matrices, and it is linear if there are no nonlinear terms involving the unknown. This excludes some types of equations such as the Algebraic Riccati Equation or other quadratic matrix equations, which are not considered here.</p>
<p>The general form of a linear matrix equation is</p>
<div class="math notranslate nohighlight" id="equation-eq-lme">
<span class="eqno">(1)<a class="headerlink" href="#equation-eq-lme" title="Link to this equation">#</a></span>\[AXE+DXB=C,\]</div>
<p>where <span class="math notranslate nohighlight">\(X\)</span> is the solution.</p>
<p>Since SLEPc is mainly concerned with iterative methods, so is the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/LME/LME.html">LME</a></span></code> module. This implies that <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/LME/LME.html">LME</a></span></code> can address only the case where the solution <span class="math notranslate nohighlight">\(X\)</span> has low rank. In many situations, <span class="math notranslate nohighlight">\(X\)</span> does not have low rank, which means that the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/LME/LME.html">LME</a></span></code> solver, if it converges, will truncate the solution to an artificially small rank, with a large approximation error.</p>
<div class="admonition warning">
<p class="admonition-title">Warning</p>
<p>The <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/LME/LME.html">LME</a></span></code> module should be considered experimental. As explained below, the currently implemented functionality is quite limited, and may be extended in the future.</p>
</div>
<section id="current-functionality">
<h2>Current Functionality<a class="headerlink" href="#current-functionality" title="Link to this heading">#</a></h2>
<p>The user interface of the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/LME/LME.html">LME</a></span></code> module is prepared for different types of equations, see a list in table <a class="reference internal" href="#tab-eqtype"><span class="std std-ref">Problem types considered in LME</span></a>. However, currently there is only basic support for continuous-time Lyapunov equations, and the rest are just a wish list.</p>
<div class="pst-scrollable-table-container"><table class="table" id="tab-eqtype">
<caption><span class="caption-text">Problem types considered in <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/LME/LME.html">LME</a></span></code></span><a class="headerlink" href="#tab-eqtype" title="Link to this table">#</a></caption>
<thead>
<tr class="row-odd"><th class="head"><p>Problem Type</p></th>
<th class="head"><p>Equation</p></th>
<th class="head"><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/LME/LMEProblemType.html">LMEProblemType</a></span></code></p></th>
<th class="head"><p><span class="math notranslate nohighlight">\(A\)</span></p></th>
<th class="head"><p><span class="math notranslate nohighlight">\(B\)</span></p></th>
<th class="head"><p><span class="math notranslate nohighlight">\(D\)</span></p></th>
<th class="head"><p><span class="math notranslate nohighlight">\(E\)</span></p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p>Continuous-Time Lyapunov</p></td>
<td><p><span class="math notranslate nohighlight">\(AX+XA^*=-C\)</span></p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/LME/LME_LYAPUNOV.html">LME_LYAPUNOV</a></span></code></p></td>
<td><p>yes</p></td>
<td><p><span class="math notranslate nohighlight">\(A^*\)</span></p></td>
<td><p>-</p></td>
<td><p>-</p></td>
</tr>
<tr class="row-odd"><td><p>Sylvester</p></td>
<td><p><span class="math notranslate nohighlight">\(AX+XB=C\)</span></p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/LME/LME_SYLVESTER.html">LME_SYLVESTER</a></span></code></p></td>
<td><p>yes</p></td>
<td><p>yes</p></td>
<td><p>-</p></td>
<td><p>-</p></td>
</tr>
<tr class="row-even"><td><p>Generalized Lyapunov</p></td>
<td><p><span class="math notranslate nohighlight">\(AXD^*+DXA^*=-C\)</span></p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/LME/LME_GEN_LYAPUNOV.html">LME_GEN_LYAPUNOV</a></span></code></p></td>
<td><p>yes</p></td>
<td><p><span class="math notranslate nohighlight">\(A^*\)</span></p></td>
<td><p>yes</p></td>
<td><p><span class="math notranslate nohighlight">\(D^*\)</span></p></td>
</tr>
<tr class="row-odd"><td><p>Generalized Sylvester</p></td>
<td><p><span class="math notranslate nohighlight">\(AXE+DXB=C\)</span></p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/LME/LME_GEN_SYLVESTER.html">LME_GEN_SYLVESTER</a></span></code></p></td>
<td><p>yes</p></td>
<td><p>yes</p></td>
<td><p>yes</p></td>
<td><p>yes</p></td>
</tr>
<tr class="row-even"><td><p>Discrete-Time Lyapunov</p></td>
<td><p><span class="math notranslate nohighlight">\(AXA^*-X=-C\)</span></p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/LME/LME_DT_LYAPUNOV.html">LME_DT_LYAPUNOV</a></span></code></p></td>
<td><p>yes</p></td>
<td><p>-</p></td>
<td><p>-</p></td>
<td><p><span class="math notranslate nohighlight">\(A^*\)</span></p></td>
</tr>
<tr class="row-odd"><td><p>Stein</p></td>
<td><p><span class="math notranslate nohighlight">\(AXE-X=-C\)</span></p></td>
<td><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/LME/LME_STEIN.html">LME_STEIN</a></span></code></p></td>
<td><p>yes</p></td>
<td><p>-</p></td>
<td><p>-</p></td>
<td><p>yes</p></td>
</tr>
</tbody>
</table>
</div>
<section id="continuous-time-lyapunov-equation">
<h3>Continuous-Time Lyapunov Equation<a class="headerlink" href="#continuous-time-lyapunov-equation" title="Link to this heading">#</a></h3>
<p>Given two matrices <span class="math notranslate nohighlight">\(A,C\in\mathbb{C}^{n\times n}\)</span>, with <span class="math notranslate nohighlight">\(C\)</span> Hermitian positive-definite, the continuous-time Lyapunov equation, assuming that the right-hand side matrix <span class="math notranslate nohighlight">\(C\)</span> has low rank, can be expressed as</p>
<div class="math notranslate nohighlight" id="equation-eq-clyap">
<span class="eqno">(2)<a class="headerlink" href="#equation-eq-clyap" title="Link to this equation">#</a></span>\[AX+XA^*=-C_1C_1^*,\]</div>
<p>where we have written the right-hand side in factored form, <span class="math notranslate nohighlight">\(C=C_1C_1^*\)</span>, <span class="math notranslate nohighlight">\(C_1\in\mathbb{C}^{n\times r}\)</span> with <span class="math notranslate nohighlight">\(r = \operatorname{rank}(C)\)</span>. In general, <span class="math notranslate nohighlight">\(X\)</span> will not have low rank, but our intention is to compute a low-rank approximation <span class="math notranslate nohighlight">\(\tilde X=X_1X_1^*\)</span>, <span class="math notranslate nohighlight">\(X_1\in\mathbb{C}^{n\times p}\)</span> for a given value <span class="math notranslate nohighlight">\(p\)</span>. Note that the equation has a unique solution if and only if <span class="math notranslate nohighlight">\(A\)</span> is stable, i.e., all of its eigenvalues have strictly negative real part.</p>
<p>The reason why SLEPc provides a solver for equation <a class="reference internal" href="#equation-eq-clyap">(2)</a> is that it is required in the eigensolver <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPSLYAPII.html">EPSLYAPII</a></span></code>, which implements the Lyapunov inverse iteration <span id="id1">[<a class="reference internal" href="../../manualpages/EPS/EPSLYAPII.html#id75" title="H. Elman and M. Wu. Lyapunov inverse iteration for computing a few rightmost eigenvalues of large generalized eigenvalue problems. SIAM J. Matrix Anal. Appl., 34(4):1685–1707, 2013. doi:10.1137/120897468.">Elman and Wu, 2013</a>, <a class="reference internal" href="../../manualpages/EPS/EPSLYAPII.html#id74" title="K. Meerbergen and A. Spence. Inverse iteration for purely imaginary eigenvalues with application to the detection of Hopf bifurcations in large-scale problems. SIAM J. Matrix Anal. Appl., 31(4):1982–1999, 2010. doi:10.1137/080742890.">Meerbergen and Spence, 2010</a>]</span>.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/LME/LME.html">LME</a></span></code> solvers currently implemented in SLEPc are very basic. Users interested in applying <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/LME/LME.html">LME</a></span></code> to matrix equations appearing in their applications are encouraged to contact SLEPc developers describing their use case.</p>
</div>
</section>
<section id="krylov-solver">
<h3>Krylov Solver<a class="headerlink" href="#krylov-solver" title="Link to this heading">#</a></h3>
<p>Currently, the solver for the continuous-time Lyapunov equation available in SLEPc is based on the following procedure for each column <span class="math notranslate nohighlight">\(c\)</span> of <span class="math notranslate nohighlight">\(C_1\)</span>:</p>
<ol class="arabic simple">
<li><p>Build an <em>Arnoldi factorization</em> for the Krylov subspace <span class="math notranslate nohighlight">\(\mathcal{K}_m(A,c)\)</span>,
<span class="math notranslate nohighlight">\(AV_m=V_mH_m+h_{m+1,m}v_{m+1}e_m^*\)</span>.</p></li>
<li><p>Solve the <em>compressed Lyapunov equation</em>,
<span class="math notranslate nohighlight">\(H_mY+YH_m^*=-\tilde c\,\tilde c^* \quad\mathrm{with}\quad\tilde c=V_m^*c\)</span>.</p></li>
<li><p>Set the approximate solution to <span class="math notranslate nohighlight">\(X_m=V_mYV_m^*\)</span>.</p></li>
</ol>
<p>Furthermore, our Krylov solver incorporates an Eiermann-Ernst-type restart as proposed by <span id="id2">Kressner [<a class="reference internal" href="#id77" title="D. Kressner. Memory-efficient Krylov subspace techniques for solving large-scale Lyapunov equations. In 2008 IEEE International Conference on Computer-Aided Control Systems, 613–618. IEEE, 2008. doi:10.1109/cacsd.2008.4627370.">2008</a>]</span>. The restart will discard the Arnoldi basis <span class="math notranslate nohighlight">\(V_m\)</span> but keep <span class="math notranslate nohighlight">\(H_m\)</span>, then continue the Arnoldi recurrence from <span class="math notranslate nohighlight">\(v_{m+1}\)</span>. The upper Hessenberg matrices generated at each restart are glued together. This implies that at each restart the size of the compressed Lyapunov equation grows. After convergence, once the full <span class="math notranslate nohighlight">\(Y\)</span> is available, we perform a second pass to reconstruct the successive <span class="math notranslate nohighlight">\(V_m\)</span> bases.</p>
<p>The restarted algorithm is as follows:</p>
<blockquote>
<div><p><strong>for</strong> <span class="math notranslate nohighlight">\(k=1,2,\ldots\)</span></p>
<ul class="simple">
<li><p>Run Arnoldi <span class="math notranslate nohighlight">\(AV_m^{(k)}=V_m^{(k)}H_m^{(k)}+h_{m+1,m}^{(k)}v_{m+1}^{(k)}e_m^*\)</span>.</p></li>
<li><p>Set <span class="math notranslate nohighlight">\(H_{km}=\left[\begin{array}{cc}H_{(k-1)m}&0\\h_{m+1,m}^{(k-1)}e_1e_{(k-1)m}^*&H_m^{(k)}\end{array}\right]\)</span>.</p></li>
<li><p>Solve compressed equation <span class="math notranslate nohighlight">\(H_{km}Y+YH_{km}^*=-\|c\|_2^2\,e_1e_1^*\)</span>.</p></li>
<li><p>Compute residual norm <span class="math notranslate nohighlight">\(\|R_k\|_2=h_{m+1,m}^{(k)}\|e_{km}^*Y\|_2\)</span>.</p></li>
</ul>
<p><strong>end</strong></p>
</div></blockquote>
<p>The residual norm is used for the stopping criterion.</p>
</section>
<section id="projected-problem">
<h3>Projected Problem<a class="headerlink" href="#projected-problem" title="Link to this heading">#</a></h3>
<p>As mentioned in the previous section, the projected problem is a smaller Lyapunov equation, which must be solved in each outer iteration of the method (restart). The solution of this small dense matrix equation is also implemented within the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/LME/LME.html">LME</a></span></code> module (there is no auxiliary class for this).</p>
<p>If <span class="math notranslate nohighlight">\(A\)</span> is stable, then <span class="math notranslate nohighlight">\(X\)</span> is symmetric positive semidefinite. Hence, looking at step 3 of the algorithm in the previous section, it is better to compute the Cholesky factor <span class="math notranslate nohighlight">\(Y=LL^*\)</span> to obtain the low rank factor <span class="math notranslate nohighlight">\(X_1=V_mL\)</span>. The method of Hammarling will compute <span class="math notranslate nohighlight">\(L\)</span> directly, without computing <span class="math notranslate nohighlight">\(Y\)</span> first.</p>
<p>If SLEPc has been configured with the option <code class="docutils notranslate"><span class="pre">--download-slicot</span></code> (see <a class="reference internal" href="extra.html#sec-wrap"><span class="std std-ref">Wrappers to External Libraries</span></a>), it will be possible to use SLICOT subroutines to apply Hammarling’s method, otherwise a more basic algorithm will be used.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Most of the SLICOT subroutines are implemented for real arithmetic only, so it is not possible to use them in a complex build of PETSc/SLEPc.</p>
</div>
<p>In the restarted algorithm described above, in the second pass to recalculate the successive <span class="math notranslate nohighlight">\(V_m\)</span>’s, the update is based on the truncated SVD of the Cholesky factor
<div class="math notranslate nohighlight">
\[\begin{split}L=\begin{bmatrix}Q_1 & Q_2\end{bmatrix}\begin{bmatrix}\Sigma_1 & 0 \\ 0 & \Sigma_2\end{bmatrix}\begin{bmatrix}Z_1 & Z_2\end{bmatrix}^*\quad\mathrm{with}\quad\|\Sigma_2\|_2<\varepsilon,\end{split}\]</div>
so that the final <span class="math notranslate nohighlight">\(X_1\)</span> has the appropriate rank.</p>
</section>
</section>
<section id="basic-usage">
<h2>Basic Usage<a class="headerlink" href="#basic-usage" title="Link to this heading">#</a></h2>
<p>Once the solver object has been created with <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/LME/LMECreate.html">LMECreate</a>()</span></code>, the user has to define the matrix equation to be solved. First, the type of equation must be set with <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/LME/LMESetProblemType.html">LMESetProblemType</a>()</span></code>, choosing one from table <a class="reference internal" href="#tab-eqtype"><span class="std std-ref">Problem types considered in LME</span></a>.</p>
<p>The coefficient matrices <span class="math notranslate nohighlight">\(A\)</span>, <span class="math notranslate nohighlight">\(B\)</span>, <span class="math notranslate nohighlight">\(D\)</span>, <span class="math notranslate nohighlight">\(E\)</span> must be provided via <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/LME/LMESetCoefficients.html">LMESetCoefficients</a>()</span></code>, but some of them are optional depending on the matrix equation. For Lyapunov equations, only <span class="math notranslate nohighlight">\(A\)</span> must be set, which is normally a large, sparse matrix stored in <a class="reference external" href="https://petsc.org/release/manualpages/Mat/MATAIJ/" title="(in PETSc v3.24)"><span class="xref std std-doc">MATAIJ</span></a> format. Note that in table <a class="reference internal" href="#tab-eqtype"><span class="std std-ref">Problem types considered in LME</span></a>, the notation <span class="math notranslate nohighlight">\(A^*\)</span> means that this matrix need not be passed, but the user may choose to pass an explicit transpose of matrix <span class="math notranslate nohighlight">\(A\)</span> (for improved efficiency). Also note that some of the equation types impose restrictions on the properties of the coefficient matrices (e.g., <span class="math notranslate nohighlight">\(A\)</span> stable in Lyapunov equations) and possibly on the right-hand side <span class="math notranslate nohighlight">\(C\)</span>.</p>
<p>To conclude the definition of the equation, we must pass the right-hand side <span class="math notranslate nohighlight">\(C\)</span> with <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/LME/LMESetRHS.html">LMESetRHS</a>()</span></code>. Note that <span class="math notranslate nohighlight">\(C\)</span> is expressed as the outer product of a tall-skinny matrix, <span class="math notranslate nohighlight">\(C=C_1C_1^*\)</span>. This can be represented in PETSc using a special type of matrix, <a class="reference external" href="https://petsc.org/release/manualpages/Mat/MATLRC/" title="(in PETSc v3.24)"><span class="xref std std-doc">MATLRC</span></a>, that can be created with <a class="reference external" href="https://petsc.org/release/manualpages/Mat/MatCreateLRC/" title="(in PETSc v3.24)"><span class="xref std std-doc">MatCreateLRC</span></a> passing a dense matrix <span class="math notranslate nohighlight">\(C_1\)</span>.</p>
<p>The call to <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/LME/LMESolve.html">LMESolve</a>()</span></code> will run the solver to compute the solution <span class="math notranslate nohighlight">\(X\)</span>. The rank of <span class="math notranslate nohighlight">\(X\)</span> may be prescribed by the user or selected dynamically by the solver. Next, we discuss these two options:</p>
<ul class="simple">
<li><p>To prescribe a fixed rank for <span class="math notranslate nohighlight">\(X\)</span>, we must preallocate it, creating a <a class="reference external" href="https://petsc.org/release/manualpages/Mat/MATLRC/" title="(in PETSc v3.24)"><span class="xref std std-doc">MATLRC</span></a> matrix, and then pass it via <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/LME/LMESetSolution.html">LMESetSolution</a>()</span></code> <em>before</em> calling <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/LME/LMESolve.html">LMESolve</a>()</span></code>. In this way, the solver will be restricted to a rank equal to the number of columns provided in <span class="math notranslate nohighlight">\(X_1\)</span>.</p></li>
<li><p>If we do not call <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/LME/LMESetSolution.html">LMESetSolution</a>()</span></code> beforehand, the solver will select the rank dynamically. Then after <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/LME/LMESolve.html">LMESolve</a>()</span></code> we must call <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/LME/LMEGetSolution.html">LMEGetSolution</a>()</span></code> to retrieve a <a class="reference external" href="https://petsc.org/release/manualpages/Mat/MATLRC/" title="(in PETSc v3.24)"><span class="xref std std-doc">MATLRC</span></a> matrix representing <span class="math notranslate nohighlight">\(X\)</span>.</p></li>
</ul>
<p class="rubric">References</p>
<div class="docutils container" id="id3">
<div role="list" class="citation-list">
<div class="citation" id="id76" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id1">Elm13</a><span class="fn-bracket">]</span></span>
<p>H. Elman and M. Wu. Lyapunov inverse iteration for computing a few rightmost eigenvalues of large generalized eigenvalue problems. <em>SIAM J. Matrix Anal. Appl.</em>, 34(4):1685–1707, 2013. <a class="reference external" href="https://doi.org/10.1137/120897468">doi:10.1137/120897468</a>.</p>
</div>
<div class="citation" id="id77" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id2">Kre08</a><span class="fn-bracket">]</span></span>
<p>D. Kressner. Memory-efficient Krylov subspace techniques for solving large-scale Lyapunov equations. In <em>2008 IEEE International Conference on Computer-Aided Control Systems</em>, 613–618. IEEE, 2008. <a class="reference external" href="https://doi.org/10.1109/cacsd.2008.4627370">doi:10.1109/cacsd.2008.4627370</a>.</p>
</div>
<div class="citation" id="id75" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id1">Mee10</a><span class="fn-bracket">]</span></span>
<p>K. Meerbergen and A. Spence. Inverse iteration for purely imaginary eigenvalues with application to the detection of Hopf bifurcations in large-scale problems. <em>SIAM J. Matrix Anal. Appl.</em>, 31(4):1982–1999, 2010. <a class="reference external" href="https://doi.org/10.1137/080742890">doi:10.1137/080742890</a>.</p>
</div>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#current-functionality">Current Functionality</a><ul class="nav section-nav flex-column">
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