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<h1>Exercise 5: Problem without Explicit Matrix Storage<a class="headerlink" href="#exercise-5-problem-without-explicit-matrix-storage" title="Link to this heading">#</a></h1>
<p>In many applications, it may be better to keep the matrix (or matrices) that define the eigenvalue problem implicit, that is, without storing its nonzero entries explicitly. An example is when we have a matrix-vector routine available. SLEPc allows easy management of this case. This exercise tries to illustrate it by solving a standard symmetric eigenproblem corresponding to the Laplacian operator in 2 dimensions in which the matrix is not built explicitly.</p>
<section id="compiling">
<h2>Compiling<a class="headerlink" href="#compiling" title="Link to this heading">#</a></h2>
<p>Copy the file <a class="reference external" href="https://slepc.upv.es/release/src/eps/tutorials/ex3.c.html" target="_blank">ex3.c</a> to your directory and add these lines to the makefile</p>
<div class="highlight-make notranslate"><div class="highlight"><pre><span></span><span class="nf">ex3</span><span class="o">:</span><span class="w"> </span><span class="n">ex</span>3.<span class="n">o</span>
<span class="w"> </span>-<span class="si">${</span><span class="nv">CLINKER</span><span class="si">}</span><span class="w"> </span>-o<span class="w"> </span>ex3<span class="w"> </span>ex3.o<span class="w"> </span><span class="si">${</span><span class="nv">SLEPC_EPS_LIB</span><span class="si">}</span>
<span class="w"> </span><span class="si">${</span><span class="nv">RM</span><span class="si">}</span><span class="w"> </span>ex3.o
</pre></div>
</div>
<p>Build the executable with the command</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>make<span class="w"> </span>ex3
</pre></div>
</div>
</section>
<section id="source-code-details">
<h2>Source Code Details<a class="headerlink" href="#source-code-details" title="Link to this heading">#</a></h2>
<p>PETSc provides support for matrix-free problems via the <em>shell</em> matrix type.
This kind of matrices is created with a call to <a class="reference external" href="https://petsc.org/release/manualpages/Mat/MatCreateShell/" title="(in PETSc v3.24)"><span class="xref std std-doc">MatCreateShell</span></a>(), and their operations are specified with <a class="reference external" href="https://petsc.org/release/manualpages/Mat/MatShellSetOperation/" title="(in PETSc v3.24)"><span class="xref std std-doc">MatShellSetOperation</span></a>().
For basic use of these matrices with <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPS.html">EPS</a></span></code> solvers only the matrix-vector product operation is required. In the example, this operation is performed by a separate function <code class="docutils notranslate"><span class="pre">MatMult_Laplacian2D</span></code>().</p>
</section>
<section id="running-the-program">
<h2>Running the Program<a class="headerlink" href="#running-the-program" title="Link to this heading">#</a></h2>
<p>Run the program without any spectral transformation options. For instance:</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>./ex3<span class="w"> </span>-eps_type<span class="w"> </span>subspace<span class="w"> </span>-eps_tol<span class="w"> </span>1e-9<span class="w"> </span>-eps_nev<span class="w"> </span><span class="m">8</span>
</pre></div>
</div>
<p>Now try running the program with shift-and-invert to get the eigenvalues closest to the origin</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>./ex3<span class="w"> </span>-eps_target<span class="w"> </span><span class="m">0</span>.0<span class="w"> </span>-st_type<span class="w"> </span>sinvert
</pre></div>
</div>
<p>Note that the above command yields a run-time error. Observe the information printed on the screen and try to deduce the reason of the error. In this case, the error is due to the fact that SLEPc tries to use a direct linear solver within the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/ST/ST.html">ST</a></span></code> object, and this is not possible unless the matrix has been created explicitly as in previous examples.</p>
<p>There are more chances to have success if an inexact shift-and-invert scheme is used. Try using an interative linear solver without preconditioning:</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>./ex3<span class="w"> </span>-eps_target<span class="w"> </span><span class="m">0</span>.0<span class="w"> </span>-st_type<span class="w"> </span>sinvert<span class="w"> </span>-st_ksp_rtol<span class="w"> </span>1e-10<span class="w"> </span>-st_ksp_type<span class="w"> </span>gmres<span class="w"> </span>-st_pc_type<span class="w"> </span>none
</pre></div>
</div>
<p>You can also try with a nonzero target.</p>
<p>This example is slow, because the iterative linear solver takes a lot of iterations to reach the required precision (add <code class="docutils notranslate"><span class="pre">-st_ksp_converged_reason</span></code> to monitor the convergence of the linear solver). In order to alleviate this problem, a preconditioner should be used. However, a powerful preconditioner such as ILU cannot be used in this case, for the same reason a direct solver is not available. The only possibility is to use a simple preconditioner such as Jacobi. Try running the last example again with <code class="docutils notranslate"><span class="pre">-st_pc_type</span> <span class="pre">jacobi</span></code>. Note that this works because the <code class="docutils notranslate"><span class="pre">MATOP_GET_DIAGONAL</span></code> has been defined in our program. In this particular example the Jacobi preconditioner does not help reduce the linear iterations since the matrix is not strongly diagonally dominant.</p>
<p>Some of the above difficulties can be avoided by using a preconditioned eigensolver, as in the examples shown in previous exercises.</p>
</section>
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