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<section class="tex2jax_ignore mathjax_ignore" id="about">
<h1>About<a class="headerlink" href="#about" title="Link to this heading">#</a></h1>
<p>The Scalable Library for Eigenvalue Problem Computations is a software library for the solution of large sparse eigenproblems on parallel computers. It can be used for the solution of linear eigenvalue problems formulated in either standard or generalized form, as well as other related problems such as the singular value decomposition. It can also be used to solve nonlinear eigenvalue problems, either those formulated as polynomial eigenproblems or more general nonlinear problems. Finally, SLEPc provides solvers for the computation of the action of a matrix function on a vector.</p>
<p>The emphasis of the software is on methods and techniques appropriate for problems in which the associated matrices are sparse, for example, those arising after the discretization of partial differential equations. Therefore, most of the methods offered by the library are projection methods or other methods with similar properties. Examples of these methods are Krylov-Schur, Jacobi-Davidson, and Conjugate Gradient, to name a few. SLEPc provides implementations of these methods. It also provides built-in support for spectral transformations such as the shift-and-invert technique. SLEPc is a general library in the sense that it covers both Hermitian and non-Hermitian problems, with either real or complex arithmetic.</p>
<p>SLEPc is built on top of <a class="reference external" href="https://petsc.org/release/" rel="noreferer noopener" target="_blank">PETSc</a>, the Portable, Extensible Toolkit for Scientific Computation. It can be considered an extension of PETSc providing all the functionality necessary for the solution of eigenvalue problems. This means that PETSc must be previously installed in order to use SLEPc. PETSc users will find SLEPc very easy to use, since it enforces the same programming paradigm. For those users who are not familiar with PETSc yet, our recommendation is to fully understand its basic concepts before proceeding with SLEPc. Parallelism in SLEPc is obtaned by means of MPI, and in addition there is some support for GPU computing.</p>
<p>SLEPc can be used directly, and also indirectly via many <a class="reference internal" href="../material/software.html"><span class="doc std std-doc">software packages that interface to it</span></a>. In turn, SLEPc interfaces to other external software packages such as <a class="reference external" href="https://github.com/opencollab/arpack-ng" rel="noreferer noopener" target="_blank">ARPACK</a> or <a class="reference external" href="https://github.com/lobpcg/blopex" rel="noreferer noopener" target="_blank">BLOPEX</a>.</p>
<section id="linear-eigensolvers">
<h2>Linear Eigensolvers<a class="headerlink" href="#linear-eigensolvers" title="Link to this heading">#</a></h2>
<blockquote>
<div><p>Parallel Solution of Large-scale Sparse Linear Eigenvalue Problems</p>
<div class="math notranslate nohighlight">
\[\begin{split}Ax=\lambda x \\
Ax=\lambda Bx\end{split}\]</div>
</div></blockquote>
<p>The solvers for linear eigenvalue problems currently provided by SLEPc are the following:</p>
<ul class="simple">
<li><p><strong>Krylov-Schur</strong> , which is a variation of Arnoldi/Lanczos with a very effective restarting technique.</p></li>
<li><p>Generalized Davidson, a simple iteration based on the subspace expansion by the preconditioned residual.</p></li>
<li><p>Jacobi-Davidson, a preconditioned eigensolver with an effective correction equation.</p></li>
<li><p>Locally optimal block preconditioned conjugate gradient (LOBPCG).</p></li>
<li><p>Rayleigh quotient conjugate gradient minimization with Rayleigh-Ritz acceleration.</p></li>
<li><p>Contour integral spectrum slicing, based on Sakurai-Sugiura method.</p></li>
</ul>
<p>For a full list of implemented solvers, check the Users Manual chapter <a class="reference internal" href="../documentation/manual/eps.html#ch-eps"><span class="std std-ref">EPS: Eigenvalue Problem Solver</span></a>.</p>
</section>
<section id="nonlinear-eigensolvers">
<h2>Nonlinear Eigensolvers<a class="headerlink" href="#nonlinear-eigensolvers" title="Link to this heading">#</a></h2>
<blockquote>
<div><p>Polynomial and General Nonlinear Eigenproblems</p>
<div class="math notranslate nohighlight">
\[\begin{split}P(\lambda)x=0 \\
T(\lambda)x=0\end{split}\]</div>
</div></blockquote>
<p>Apart from the linear eigenvalue solvers, SLEPc also provides specific solvers for nonlinear eigenvalue problems, either polynomial (PEP) or general nonlinear (NEP).</p>
<p>We provide a robust and efficient implementation of TOAR, a Krylov subspace method specific for the polynomial eigenvalue problem, as well as a polynomial Jacobi-Davidson. It is also possible to linearize the polynomial problem and use any of the above mentioned linear eigensolvers. For additional details, check the Users Manual chapter <a class="reference internal" href="../documentation/manual/pep.html#ch-pep"><span class="std std-ref">PEP: Polynomial Eigenvalue Problems</span></a>.</p>
<p>For general nonlinear eigenproblems, the user provides the nonlinear function <span class="math notranslate nohighlight">\(T(\cdot)\)</span> either by means of callback routines or in split form (matrices combined with scalar nonlinear functions). So far, the following solvers are available:</p>
<ul class="simple">
<li><p>NLEIGS.</p></li>
<li><p>Contour integral.</p></li>
<li><p>Polynomial interpolation that uses a PEP solver.</p></li>
<li><p>Basic solvers:</p>
<ul>
<li><p>Residual inverse iteration with varying shift.</p></li>
<li><p>Successive linear problems, a Newton-type iteration based on first order Taylor approximation.</p></li>
<li><p>Nonlinear Arnoldi.</p></li>
</ul>
</li>
</ul>
<p>For a full list of nonlinear eigensolvers, check the Users Manual chapter <a class="reference internal" href="../documentation/manual/nep.html#ch-nep"><span class="std std-ref">NEP: Nonlinear Eigenvalue Problems</span></a>.</p>
</section>
<section id="solvers-for-svd-and-related-problems">
<h2>Solvers for SVD and Related Problems<a class="headerlink" href="#solvers-for-svd-and-related-problems" title="Link to this heading">#</a></h2>
<blockquote>
<div><p>Partial SVD</p>
<div class="math notranslate nohighlight">
\[Av=\sigma u\]</div>
</div></blockquote>
<p>In addition to eigensolvers, SLEPc includes functionality for computing the partial singular value decomposition (SVD) of a large sparse rectangular matrix. For the computation of singular values and vectors, any of the above mentioned linear eigensolvers can be used, together with some specific SVD solvers such as Lanczos and thick-restart Lanczos bidiagonalization.</p>
<p>SLEPc also provides functionality for other related problems such as the GSVD and the HSVD. For additional details, check the Users Manual chapter <a class="reference internal" href="../documentation/manual/svd.html#ch-svd"><span class="std std-ref">SVD: Singular Value Decomposition</span></a>.</p>
</section>
<section id="matrix-functions">
<h2>Matrix Functions<a class="headerlink" href="#matrix-functions" title="Link to this heading">#</a></h2>
<blockquote>
<div><p>Matrix Function</p>
<div class="math notranslate nohighlight">
\[w=f(\alpha A)v\]</div>
</div></blockquote>
<p>There are also solvers for computing the action of the function of a large sparse matrix on a vector (MFN). For this, SLEPc currently provides a restarted Krylov method, where the function can be the exponential, the square root, a rational function, or combinations thereof. More details are provided in the corrsponding chapter of the Users Manual <a class="reference internal" href="../documentation/manual/mfn.html#ch-mfn"><span class="std std-ref">MFN: Matrix Function</span></a>.</p>
</section>
<section id="highlights">
<h2>Highlights<a class="headerlink" href="#highlights" title="Link to this heading">#</a></h2>
<ul class="simple">
<li><p>Easy programming with PETSc’s object-oriented style. See an example in <a class="reference external" href="https://slepc.upv.es/release/src/eps/tutorials/ex1.c.html" rel="noreferer noopener" target="_blank">C</a> and <a class="reference external" href="https://slepc.upv.es/release/src/eps/tutorials/ex1f.F90.html" rel="noreferer noopener" target="_blank">Fortran</a>. More examples are available in the <a class="reference internal" href="../documentation/index.html"><span class="doc std std-doc">documentation section</span></a>.</p></li>
<li><p>Data-structure neutral implementation. Problems can be solved with matrices stored in parallel and serial, sparse and dense formats, and even without explicit storage.</p></li>
<li><p>Run-time flexibility, giving full control over the solution process. See some command-line examples below.</p></li>
<li><p>Portability to a wide range of parallel platforms, including Linux clusters, MacOSX, and many others.</p></li>
<li><p>Usable from code written in C, C++, and Fortran0, as well as Python (note that slepc4py is now part of the SLEPc distribution).</p></li>
<li><p>Extensive documentation, including a users manual, on-line tutorial exercises, example programs and on-line manual pages for every subroutine.</p></li>
<li><p>Seamless integration of other eigensolver packages such as <a class="reference external" href="https://github.com/opencollab/arpack-ng" rel="noreferer noopener" target="_blank">ARPACK</a> or <a class="reference external" href="https://github.com/lobpcg/blopex" rel="noreferer noopener" target="_blank">BLOPEX</a>.</p></li>
</ul>
</section>
<section id="command-line-examples">
<h2>Command-line Examples<a class="headerlink" href="#command-line-examples" title="Link to this heading">#</a></h2>
<p>The following examples illustrate the use of a SLEPc program via command-line options. All options are also available with an equivalent programmatic interface.</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>./exeps<span class="w"> </span>-eps_view<span class="w"> </span>-eps_monitor
<span class="gp">$ </span>./exeps<span class="w"> </span>-eps_type<span class="w"> </span>krylovschur<span class="w"> </span>-eps_smallest_real
<span class="gp">$ </span>./exeps<span class="w"> </span>-eps_type<span class="w"> </span>lanczos<span class="w"> </span>-eps_nev<span class="w"> </span><span class="m">6</span><span class="w"> </span>-eps_ncv<span class="w"> </span><span class="m">24</span>
<span class="gp">$ </span>./exeps<span class="w"> </span>-eps_type<span class="w"> </span>krylovschur<span class="w"> </span>-eps_nev<span class="w"> </span><span class="m">100</span><span class="w"> </span>-eps_mpd<span class="w"> </span><span class="m">40</span>
<span class="gp">$ </span>./exeps<span class="w"> </span>-eps_type<span class="w"> </span>arnoldi<span class="w"> </span>-eps_tol<span class="w"> </span>1e-8<span class="w"> </span>-eps_max_it<span class="w"> </span><span class="m">2000</span>
<span class="gp">$ </span>./exeps<span class="w"> </span>-eps_type<span class="w"> </span>gd<span class="w"> </span>-eps_gd_plusk<span class="w"> </span><span class="m">1</span>
<span class="gp">$ </span>./exeps<span class="w"> </span>-eps_type<span class="w"> </span>jd<span class="w"> </span>-eps_jd_blocksize<span class="w"> </span><span class="m">2</span><span class="w"> </span>-eps_jd_krylov_start
<span class="gp">$ </span>./exeps<span class="w"> </span>-eps_type<span class="w"> </span>subspace<span class="w"> </span>-eps_hermitian<span class="w"> </span>-log_view
<span class="gp">$ </span>./exeps<span class="w"> </span>-eps_type<span class="w"> </span>arpack<span class="w"> </span>-eps_view_values<span class="w"> </span>draw<span class="w"> </span>-draw_pause<span class="w"> </span>-1
<span class="gp">$ </span>./exeps<span class="w"> </span>-eps_target<span class="w"> </span><span class="m">2</span>.1
<span class="gp">$ </span>./exeps<span class="w"> </span>-eps_target<span class="w"> </span><span class="m">2</span>.1<span class="w"> </span>-eps_harmonic
<span class="gp">$ </span>./exeps<span class="w"> </span>-eps_target<span class="w"> </span><span class="m">2</span>.1<span class="w"> </span>-st_type<span class="w"> </span>sinvert
<span class="gp">$ </span>./exsvd<span class="w"> </span>-svd_type<span class="w"> </span>trlanczos<span class="w"> </span>-svd_nsv<span class="w"> </span><span class="m">4</span>
<span class="gp">$ </span>./exsvd<span class="w"> </span>-svd_type<span class="w"> </span>cross<span class="w"> </span>-svd_cross_eps_type<span class="w"> </span>krylovschur<span class="w"> </span>-svd_smallest<span class="w"> </span>-svd_monitor_lg<span class="w"> </span>-svd_ncv<span class="w"> </span><span class="m">30</span>
<span class="gp">$ </span>./expep<span class="w"> </span>-pep_nev<span class="w"> </span><span class="m">6</span><span class="w"> </span>-pep_tol<span class="w"> </span>1e-5
<span class="gp">$ </span>./expep<span class="w"> </span>-pep_type<span class="w"> </span>linear<span class="w"> </span>-pep_linear_linearization<span class="w"> </span><span class="m">1</span>,0<span class="w"> </span>-pep_linear_explicitmatrix
<span class="gp">$ </span>./expep<span class="w"> </span>-pep_type<span class="w"> </span>toar
</pre></div>
</div>
</section>
<section id="scheme-of-slepc-classes">
<h2>Scheme of SLEPc Classes<a class="headerlink" href="#scheme-of-slepc-classes" title="Link to this heading">#</a></h2>
<p>The following scheme represents the functionality provided by SLEPc and how it relates to PETSc.</p>
<ul class="simple">
<li><p>SLEPc provides five main objects: <code class="docutils notranslate"><span class="pre"><a href="../manualpages/EPS/EPS.html">EPS</a></span></code> (Eigenvalue Problem Solver), <code class="docutils notranslate"><span class="pre"><a href="../manualpages/SVD/SVD.html">SVD</a></span></code> (Singular Value Decomposition), <code class="docutils notranslate"><span class="pre"><a href="../manualpages/PEP/PEP.html">PEP</a></span></code> (Polynomial Eigenvalue Problem), <code class="docutils notranslate"><span class="pre"><a href="../manualpages/NEP/NEP.html">NEP</a></span></code> (Nonlinear Eigenvalue Problem), and <code class="docutils notranslate"><span class="pre"><a href="../manualpages/MFN/MFN.html">MFN</a></span></code> (Matrix Function).</p></li>
<li><p>These objects occupy a level of abstraction similar to other PETSc solvers such as <code class="docutils notranslate"><span class="pre"><a href="https://petsc.org/release/manualpages/KSP/KSP/">KSP</a></span></code> or <code class="docutils notranslate"><span class="pre"><a href="https://petsc.org/release/manualpages/SNES/SNES/">SNES</a></span></code> and use low-level infrastructure such as <code class="docutils notranslate"><span class="pre"><a href="https://petsc.org/release/manualpages/Mat/Mat/">Mat</a></span></code> and <code class="docutils notranslate"><span class="pre"><a href="https://petsc.org/release/manualpages/Vec/Vec/">Vec</a></span></code>.</p></li>
<li><p>The shaded blocks represent the generic interface of the object while the white boxes represent different implementations. The programmer usually interacts with the object via its interface and the particular implementation is typically picked at run time.</p></li>
<li><p><code class="docutils notranslate"><span class="pre"><a href="../manualpages/ST/ST.html">ST</a></span></code> (Spectral Transformation) is used in combination with most solvers to compute interior eigenvalues.</p></li>
<li><p>Additionally, several auxiliary classes are provided: <code class="docutils notranslate"><span class="pre"><a href="../manualpages/BV/BV.html">BV</a></span></code> (Basis Vectors), <code class="docutils notranslate"><span class="pre"><a href="../manualpages/DS/DS.html">DS</a></span></code> (Dense System), <code class="docutils notranslate"><span class="pre"><a href="../manualpages/RG/RG.html">RG</a></span></code> (Region), <code class="docutils notranslate"><span class="pre"><a href="../manualpages/FN/FN.html">FN</a></span></code> (Mathematical Function).</p></li>
</ul>
<img src="../_static/images/manual/svg/fig-slepc.svg" alt="petsc-slepc" width="120%" align="center">
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