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<h2 id="Exercises">Exercises<a class="anchor-link" href="#Exercises"></a></h2>
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<h3 id="Exercise-1">Exercise 1<a class="anchor-link" href="#Exercise-1"></a></h3><p>In the parallel implementation of the Jacobi method in previous notebook, we assumed that the method runs for a given number of iterations. However, other stopping criteria are used in practice. The following sequential code implements a version of Jacobi in which the method iterates until the norm of the difference between u and u_new is below a tolerance.</p>
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<div class="highlight hl-julia"><pre><span></span><span class="k">function</span><span class="w"> </span><span class="n">jacobi_with_tol</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">tol</span><span class="p">)</span>
<span class="w"> </span><span class="n">u</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">2</span><span class="p">)</span>
<span class="w"> </span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
<span class="w"> </span><span class="n">u</span><span class="p">[</span><span class="k">end</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span>
<span class="w"> </span><span class="n">u_new</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">copy</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
<span class="w"> </span><span class="n">increment</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">similar</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
<span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="nb">true</span>
<span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">2</span><span class="o">:</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
<span class="w"> </span><span class="n">u_new</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0.5</span><span class="o">*</span><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">+</span><span class="n">u</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">])</span>
<span class="w"> </span><span class="k">end</span>
<span class="w"> </span><span class="n">increment</span><span class="w"> </span><span class="o">.=</span><span class="w"> </span><span class="n">u_new</span><span class="w"> </span><span class="o">.-</span><span class="w"> </span><span class="n">u</span>
<span class="w"> </span><span class="n">norm_increment</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0.0</span>
<span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">:</span><span class="n">n</span>
<span class="w"> </span><span class="n">increment_i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">increment</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
<span class="w"> </span><span class="n">norm_increment</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">increment_i</span><span class="o">*</span><span class="n">increment_i</span>
<span class="w"> </span><span class="k">end</span>
<span class="w"> </span><span class="n">norm_increment</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">sqrt</span><span class="p">(</span><span class="n">norm_increment</span><span class="p">)</span>
<span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">norm_increment</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">tol</span><span class="o">*</span><span class="n">n</span>
<span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">u_new</span>
<span class="w"> </span><span class="k">end</span>
<span class="w"> </span><span class="n">u</span><span class="p">,</span><span class="w"> </span><span class="n">u_new</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">u_new</span><span class="p">,</span><span class="w"> </span><span class="n">u</span>
<span class="w"> </span><span class="k">end</span>
<span class="w"> </span><span class="n">u</span>
<span class="k">end</span>
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<div class="highlight hl-julia"><pre><span></span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">10</span>
<span class="n">tol</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1e-12</span>
<span class="n">jacobi_with_tol</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">tol</span><span class="p">)</span>
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<p>Implement a parallel version of this algorithm. Recommended: start with the parallel implementation given in the previous notebook (see function <code>jacobi_mpi</code>) and introduce the new stopping criteria. Think carefully about which MPI operations you need to use in this case.</p>
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