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<div class="section" id="python-interface">
<h1>Python Interface<a class="headerlink" href="#python-interface" title="Permalink to this headline">¶</a></h1>
<dl class="py function">
<dt id="primme.svds">
<code class="sig-prename descclassname">primme.</code><code class="sig-name descname">svds</code><span class="sig-paren">(</span><em class="sig-param"><span class="n">A</span></em>, <em class="sig-param"><span class="n">k</span><span class="o">=</span><span class="default_value">6</span></em>, <em class="sig-param"><span class="n">ncv</span><span class="o">=</span><span class="default_value">None</span></em>, <em class="sig-param"><span class="n">tol</span><span class="o">=</span><span class="default_value">0</span></em>, <em class="sig-param"><span class="n">which</span><span class="o">=</span><span class="default_value">'LM'</span></em>, <em class="sig-param"><span class="n">v0</span><span class="o">=</span><span class="default_value">None</span></em>, <em class="sig-param"><span class="n">maxiter</span><span class="o">=</span><span class="default_value">None</span></em>, <em class="sig-param"><span class="n">return_singular_vectors</span><span class="o">=</span><span class="default_value">True</span></em>, <em class="sig-param"><span class="n">precAHA</span><span class="o">=</span><span class="default_value">None</span></em>, <em class="sig-param"><span class="n">precAAH</span><span class="o">=</span><span class="default_value">None</span></em>, <em class="sig-param"><span class="n">precAug</span><span class="o">=</span><span class="default_value">None</span></em>, <em class="sig-param"><span class="n">u0</span><span class="o">=</span><span class="default_value">None</span></em>, <em class="sig-param"><span class="n">orthou0</span><span class="o">=</span><span class="default_value">None</span></em>, <em class="sig-param"><span class="n">orthov0</span><span class="o">=</span><span class="default_value">None</span></em>, <em class="sig-param"><span class="n">return_stats</span><span class="o">=</span><span class="default_value">False</span></em>, <em class="sig-param"><span class="n">maxBlockSize</span><span class="o">=</span><span class="default_value">0</span></em>, <em class="sig-param"><span class="n">method</span><span class="o">=</span><span class="default_value">None</span></em>, <em class="sig-param"><span class="n">methodStage1</span><span class="o">=</span><span class="default_value">None</span></em>, <em class="sig-param"><span class="n">methodStage2</span><span class="o">=</span><span class="default_value">None</span></em>, <em class="sig-param"><span class="n">return_history</span><span class="o">=</span><span class="default_value">False</span></em>, <em class="sig-param"><span class="n">convtest</span><span class="o">=</span><span class="default_value">None</span></em>, <em class="sig-param"><span class="o">**</span><span class="n">kargs</span></em><span class="sig-paren">)</span><a class="headerlink" href="#primme.svds" title="Permalink to this definition">¶</a></dt>
<dd><p>Compute k singular values and vectors of the matrix A.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>A</strong> (<em>{sparse matrix</em><em>, </em><em>LinearOperator}</em>) – Array to compute the SVD on, of shape (M, N)</p></li>
<li><p><strong>k</strong> (<em>int</em><em>, </em><em>optional</em>) – Number of singular values and vectors to compute.
Must be 1 <= k < min(A.shape).</p></li>
<li><p><strong>ncv</strong> (<em>int</em><em>, </em><em>optional</em>) – The maximum size of the basis</p></li>
<li><p><strong>tol</strong> (<em>float</em><em>, </em><em>optional</em>) – <p>Tolerance for singular values. Zero (default) means 10**4 times the machine precision.</p>
<p>A triplet <code class="docutils literal notranslate"><span class="pre">(u,sigma,v)</span></code> is marked as converged when
(||A*v - sigma*u||**2 + ||A.H*u - sigma*v||**2)**.5
is less than “tol” * ||A||, or close to the minimum tolerance that
the method can achieve. See the note.</p>
<p>The value is ignored if convtest is provided.</p>
</p></li>
<li><p><strong>which</strong> (<em>str</em><em> [</em><em>'LM' | 'SM'</em><em>] or </em><em>number</em><em>, </em><em>optional</em>) – <p>Which <cite>k</cite> singular values to find:</p>
<blockquote>
<div><ul>
<li><p>’LM’ : largest singular values</p></li>
<li><p>’SM’ : smallest singular values</p></li>
<li><p>number : closest singular values to (referred as sigma later)</p></li>
</ul>
</div></blockquote>
</p></li>
<li><p><strong>u0</strong> (<em>ndarray</em><em>, </em><em>optional</em>) – <p>Initial guesses for the left singular vectors.</p>
<p>If only u0 or v0 is provided, the other is computed. If both are
provided, u0 and v0 should have the same number of columns.</p>
</p></li>
<li><p><strong>v0</strong> (<em>ndarray</em><em>, </em><em>optional</em>) – Initial guesses for the right singular vectors.</p></li>
<li><p><strong>maxiter</strong> (<em>int</em><em>, </em><em>optional</em>) – Maximum number of matvecs with A and A.H.</p></li>
<li><p><strong>precAHA</strong> (<em>{N x N matrix</em><em>, </em><em>array</em><em>, </em><em>sparse matrix</em><em>, </em><em>LinearOperator}</em><em>, </em><em>optional</em>) – Approximate inverse of (A.H*A - sigma**2*I). If provided and M>=N, it
usually accelerates the convergence.</p></li>
<li><p><strong>precAAH</strong> (<em>{M x M matrix</em><em>, </em><em>array</em><em>, </em><em>sparse matrix</em><em>, </em><em>LinearOperator}</em><em>, </em><em>optional</em>) – Approximate inverse of (A*A.H - sigma**2*I). If provided and M<N, it
usually accelerates the convergence.</p></li>
<li><p><strong>precAug</strong> (<em>{</em><em>(</em><em>M+N</em><em>) </em><em>x</em><em> (</em><em>M+N</em><em>) </em><em>matrix</em><em>, </em><em>array</em><em>, </em><em>sparse matrix</em><em>, </em><em>LinearOperator}</em><em>, </em><em>optional</em>) – Approximate inverse of ([zeros() A.H; zeros() A] - sigma*I).</p></li>
<li><p><strong>orthou0</strong> (<em>ndarray</em><em>, </em><em>optional</em>) – <p>Left orthogonal vector constrain.</p>
<p>Seek singular triplets orthogonal to orthou0 and orthov0. The provided vectors
<em>should</em> be orthonormal. If only orthou0 or orthov0 is provided, the other
is computed. Useful to avoid converging to previously computed solutions.</p>
</p></li>
<li><p><strong>orthov0</strong> (<em>ndarray</em><em>, </em><em>optional</em>) – Right orthogonal vector constrain. See orthou0.</p></li>
<li><p><strong>maxBlockSize</strong> (<em>int</em><em>, </em><em>optional</em>) – Maximum number of vectors added at every iteration.</p></li>
<li><p><strong>convtest</strong> (<em>callable</em>) – <p>User-defined function to mark an approximate singular triplet as converged.</p>
<p>The function is called as convtest(sval, svecleft, svecright, resNorm)
and returns True if the triplet with value <cite>sval</cite>, left vector <cite>svecleft</cite>,
right vector <cite>svecright</cite>, and residual norm <cite>resNorm</cite> is considered converged.</p>
</p></li>
<li><p><strong>return_stats</strong> (<em>bool</em><em>, </em><em>optional</em>) – If True, the function returns extra information (see stats in Returns).</p></li>
<li><p><strong>return_history</strong> (<em>bool</em><em>, </em><em>optional</em>) – If True, the function returns performance information at every iteration</p></li>
</ul>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><p><ul class="simple">
<li><p><strong>u</strong> (<em>ndarray, shape=(M, k), optional</em>) – Unitary matrix having left singular vectors as columns.
Returned if <cite>return_singular_vectors</cite> is True.</p></li>
<li><p><strong>s</strong> (<em>ndarray, shape=(k,)</em>) – The singular values.</p></li>
<li><p><strong>vt</strong> (<em>ndarray, shape=(k, N), optional</em>) – Unitary matrix having right singular vectors as rows.
Returned if <cite>return_singular_vectors</cite> is True.</p></li>
<li><p><strong>stats</strong> (<em>dict, optional (if return_stats)</em>) – Extra information reported by PRIMME:</p>
<ul>
<li><p>”numOuterIterations”: number of outer iterations</p></li>
<li><p>”numRestarts”: number of restarts</p></li>
<li><p>”numMatvecs”: number of matvecs with A and A.H</p></li>
<li><p>”numPreconds”: cumulative number of applications of precAHA, precAAH
and precAug</p></li>
<li><p>”elapsedTime”: time that took</p></li>
<li><p>”rnorms” : (||A*v[:,i] - sigma[i]*u[:,i]||**2 + ||A.H*u[:,i] - sigma[i]*v[:,i]||**2)**.5</p></li>
<li><p>”hist” : (if return_history) report at every outer iteration of:</p>
<ul>
<li><p>”elapsedTime”: time spent up to now</p></li>
<li><p>”numMatvecs”: number of A*v and A.H*v spent up to now</p></li>
<li><p>”nconv”: number of converged triplets</p></li>
<li><p>”sval”: singular value of the first unconverged triplet</p></li>
<li><p>”resNorm”: residual norm of the first unconverged triplet</p></li>
</ul>
</li>
</ul>
</li>
</ul>
</p>
</dd>
</dl>
<p class="rubric">Notes</p>
<p>The default method used is the hybrid method, which first solves the
equivalent eigenvalue problem A.H*A or A*A.H (normal equations) and then
refines the solution solving the augmented problem. The minimum tolerance
that this method can achieve is ||A||*epsilon, where epsilon is the
machine precision. However it may not return triplets with singular values
smaller than ||A||*epsilon if “tol” is smaller than ||A||*epsilon/sigma.</p>
<p>This function is a wrapper to PRIMME functions to find singular values and
vectors <a class="footnote-reference brackets" href="#id2" id="id1">1</a>.</p>
<p class="rubric">References</p>
<dl class="footnote brackets">
<dt class="label" id="id2"><span class="brackets"><a class="fn-backref" href="#id1">1</a></span></dt>
<dd><p>PRIMME Software, <a class="reference external" href="https://github.com/primme/primme">https://github.com/primme/primme</a></p>
</dd>
<dt class="label" id="id3"><span class="brackets">2</span></dt>
<dd><p>L. Wu, E. Romero and A. Stathopoulos, PRIMME_SVDS: A High-
Performance Preconditioned SVD Solver for Accurate Large-Scale
Computations. <a class="reference external" href="https://arxiv.org/abs/1607.01404">https://arxiv.org/abs/1607.01404</a></p>
</dd>
</dl>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="pyeigsh.html#primme.eigsh" title="primme.eigsh"><code class="xref py py-func docutils literal notranslate"><span class="pre">primme.eigsh()</span></code></a></dt><dd><p>eigenvalue decomposition for a sparse symmetrix/complex Hermitian matrix A</p>
</dd>
<dt><code class="xref py py-func docutils literal notranslate"><span class="pre">scipy.sparse.linalg.eigs()</span></code></dt><dd><p>eigenvalues and eigenvectors for a general (nonsymmetric) matrix A</p>
</dd>
</dl>
</div>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">primme</span><span class="o">,</span> <span class="nn">scipy.sparse</span>
<span class="gp">>>> </span><span class="n">A</span> <span class="o">=</span> <span class="n">scipy</span><span class="o">.</span><span class="n">sparse</span><span class="o">.</span><span class="n">spdiags</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">11</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">100</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span> <span class="c1"># sparse diag. rect. matrix</span>
<span class="gp">>>> </span><span class="n">svecs_left</span><span class="p">,</span> <span class="n">svals</span><span class="p">,</span> <span class="n">svecs_right</span> <span class="o">=</span> <span class="n">primme</span><span class="o">.</span><span class="n">svds</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-6</span><span class="p">,</span> <span class="n">which</span><span class="o">=</span><span class="s1">'LM'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">svals</span> <span class="c1"># the three largest singular values of A</span>
<span class="go">array([10., 9., 8.])</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">primme</span><span class="o">,</span> <span class="nn">scipy.sparse</span><span class="o">,</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="gp">>>> </span><span class="n">A</span> <span class="o">=</span> <span class="n">scipy</span><span class="o">.</span><span class="n">sparse</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">10000</span><span class="p">,</span> <span class="mi">100</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">prec</span> <span class="o">=</span> <span class="n">scipy</span><span class="o">.</span><span class="n">sparse</span><span class="o">.</span><span class="n">spdiags</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">reciprocal</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">multiply</span><span class="p">(</span><span class="n">A</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)),</span>
<span class="gp">... </span> <span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">100</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span> <span class="c1"># square diag. preconditioner</span>
<span class="gp">>>> </span><span class="c1"># the three smallest singular values of A, using preconditioning</span>
<span class="gp">>>> </span><span class="n">svecs_left</span><span class="p">,</span> <span class="n">svals</span><span class="p">,</span> <span class="n">svecs_right</span> <span class="o">=</span> <span class="n">primme</span><span class="o">.</span><span class="n">svds</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">which</span><span class="o">=</span><span class="s1">'SM'</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-6</span><span class="p">,</span> <span class="n">precAHA</span><span class="o">=</span><span class="n">prec</span><span class="p">)</span>
<span class="gp">>>> </span><span class="p">[</span><span class="s2">"</span><span class="si">%.5f</span><span class="s2">"</span> <span class="o">%</span> <span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">svals</span><span class="o">.</span><span class="n">flat</span><span class="p">]</span>
<span class="go">['4.57263', '4.78752', '4.82229']</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="c1"># Giving the matvecs as functions</span>
<span class="gp">>>> </span><span class="kn">import</span> <span class="nn">primme</span><span class="o">,</span> <span class="nn">scipy.sparse</span><span class="o">,</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="gp">>>> </span><span class="n">Bdiag</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="mi">100</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">Bdiagr</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="mi">100</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">100</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">)),</span> <span class="n">axis</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="mi">200</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span>
<span class="gp">>>> </span><span class="k">def</span> <span class="nf">Bmatmat</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="gp">... </span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span> <span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="mi">100</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span>
<span class="gp">... </span> <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">((</span><span class="n">Bdiag</span> <span class="o">*</span> <span class="n">x</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">100</span><span class="p">,</span> <span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">)))</span>
<span class="gp">...</span>
<span class="gp">>>> </span><span class="k">def</span> <span class="nf">Brmatmat</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="gp">... </span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span> <span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="mi">200</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span>
<span class="gp">... </span> <span class="k">return</span> <span class="p">(</span><span class="n">Bdiagr</span> <span class="o">*</span> <span class="n">x</span><span class="p">)[</span><span class="mi">0</span><span class="p">:</span><span class="mi">100</span><span class="p">,:]</span>
<span class="gp">...</span>
<span class="gp">>>> </span><span class="n">B</span> <span class="o">=</span> <span class="n">scipy</span><span class="o">.</span><span class="n">sparse</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">LinearOperator</span><span class="p">((</span><span class="mi">200</span><span class="p">,</span><span class="mi">100</span><span class="p">),</span> <span class="n">matvec</span><span class="o">=</span><span class="n">Bmatmat</span><span class="p">,</span> <span class="n">matmat</span><span class="o">=</span><span class="n">Bmatmat</span><span class="p">,</span> <span class="n">rmatvec</span><span class="o">=</span><span class="n">Brmatmat</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">svecs_left</span><span class="p">,</span> <span class="n">svals</span><span class="p">,</span> <span class="n">svecs_right</span> <span class="o">=</span> <span class="n">primme</span><span class="o">.</span><span class="n">svds</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="n">which</span><span class="o">=</span><span class="s1">'LM'</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-6</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">svals</span>
<span class="go">array([99., 98., 97., 96., 95.])</span>
</pre></div>
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