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fdmath
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Table of contents
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="#meanas.eigensolvers" class="md-nav__link">
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<span class="md-ellipsis">
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<code class="doc-symbol doc-symbol-toc doc-symbol-module"></code> eigensolvers
|
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</span>
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</a>
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<nav class="md-nav" aria-label=" eigensolvers">
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<a href="#meanas.eigensolvers.power_iteration" class="md-nav__link">
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<span class="md-ellipsis">
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<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> power_iteration
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</a>
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<a href="#meanas.eigensolvers.rayleigh_quotient_iteration" class="md-nav__link">
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|
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<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> rayleigh_quotient_iteration
|
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|
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</span>
|
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</a>
|
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|
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<a href="#meanas.eigensolvers.signed_eigensolve" class="md-nav__link">
|
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<span class="md-ellipsis">
|
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|
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<code class="doc-symbol doc-symbol-toc doc-symbol-function"></code> signed_eigensolve
|
||
|
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|
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</a>
|
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|
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|
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|
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<article class="md-content__inner md-typeset">
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|
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<h1 id="eigensolvers">eigensolvers<a class="headerlink" href="#eigensolvers" title="Permanent link">¶</a></h1>
|
||
|
||
|
||
<div class="doc doc-object doc-module">
|
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|
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|
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|
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<h2 id="meanas.eigensolvers" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-module"></code> <span class="doc doc-object-name doc-module-name">meanas.eigensolvers</span>
|
||
|
||
|
||
<a href="#meanas.eigensolvers" class="headerlink" title="Permanent link">¶</a></h2>
|
||
|
||
<div class="doc doc-contents first">
|
||
|
||
<p>Solvers for eigenvalue / eigenvector problems</p>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
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<div class="doc doc-children">
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|
||
|
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|
||
<div class="doc doc-object doc-function">
|
||
|
||
|
||
<h3 id="meanas.eigensolvers.power_iteration" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-function"></code> <span class="doc doc-object-name doc-function-name">power_iteration</span>
|
||
|
||
|
||
<a href="#meanas.eigensolvers.power_iteration" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">power_iteration</span><span class="p">(</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">operator</span><span class="p">:</span> <span class="n">spmatrix</span><span class="p">,</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a> <span class="n">guess_vector</span><span class="p">:</span> <span class="n">NDArray</span><span class="p">[</span><span class="n">complex128</span><span class="p">]</span> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span><span class="p">,</span>
|
||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a> <span class="n">iterations</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">20</span><span class="p">,</span>
|
||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="p">)</span> <span class="o">-></span> <span class="nb">tuple</span><span class="p">[</span><span class="nb">complex</span><span class="p">,</span> <span class="n">NDArray</span><span class="p">[</span><span class="n">numpy</span><span class="o">.</span><span class="n">complex128</span><span class="p">]]</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Use power iteration to estimate the dominant eigenvector of a matrix.</p>
|
||
|
||
|
||
<p><span class="doc-section-title">Parameters:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Name</th>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
<th>Default</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>operator</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="scipy.sparse.spmatrix">spmatrix</span></code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Matrix to analyze.</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<em>required</em>
|
||
</td>
|
||
</tr>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>guess_vector</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="numpy.typing.NDArray">NDArray</span>[<span title="numpy.complex128">complex128</span>] | None</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Starting point for the eigenvector. Default is a randomly chosen vector.</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<code>None</code>
|
||
</td>
|
||
</tr>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>iterations</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="int">int</span></code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Number of iterations to perform. Default 20.</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<code>20</code>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
<p><span class="doc-section-title">Returns:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code><span title="tuple">tuple</span>[<span title="complex">complex</span>, <span title="numpy.typing.NDArray">NDArray</span>[<span title="numpy.complex128">complex128</span>]]</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>(Largest-magnitude eigenvalue, Corresponding eigenvector estimate)</p>
|
||
</div>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-function">
|
||
|
||
|
||
<h3 id="meanas.eigensolvers.rayleigh_quotient_iteration" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-function"></code> <span class="doc doc-object-name doc-function-name">rayleigh_quotient_iteration</span>
|
||
|
||
|
||
<a href="#meanas.eigensolvers.rayleigh_quotient_iteration" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">rayleigh_quotient_iteration</span><span class="p">(</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">operator</span><span class="p">:</span> <span class="n">spmatrix</span> <span class="o">|</span> <span class="n">LinearOperator</span><span class="p">,</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a> <span class="n">guess_vector</span><span class="p">:</span> <span class="n">NDArray</span><span class="p">[</span><span class="n">complex128</span><span class="p">],</span>
|
||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a> <span class="n">iterations</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">40</span><span class="p">,</span>
|
||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a> <span class="n">tolerance</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">1e-13</span><span class="p">,</span>
|
||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a> <span class="n">solver</span><span class="p">:</span> <span class="n">Callable</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="n">NDArray</span><span class="p">[</span><span class="n">complex128</span><span class="p">]]</span>
|
||
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span><span class="p">,</span>
|
||
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="p">)</span> <span class="o">-></span> <span class="nb">tuple</span><span class="p">[</span><span class="nb">complex</span><span class="p">,</span> <span class="n">NDArray</span><span class="p">[</span><span class="n">numpy</span><span class="o">.</span><span class="n">complex128</span><span class="p">]]</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Use Rayleigh quotient iteration to refine an eigenvector guess.</p>
|
||
|
||
|
||
<p><span class="doc-section-title">Parameters:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Name</th>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
<th>Default</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>operator</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="scipy.sparse.spmatrix">spmatrix</span> | <span title="scipy.sparse.linalg.LinearOperator">LinearOperator</span></code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Matrix to analyze.</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<em>required</em>
|
||
</td>
|
||
</tr>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>guess_vector</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="numpy.typing.NDArray">NDArray</span>[<span title="numpy.complex128">complex128</span>]</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Eigenvector to refine.</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<em>required</em>
|
||
</td>
|
||
</tr>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>iterations</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="int">int</span></code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Maximum number of iterations to perform. Default 40.</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<code>40</code>
|
||
</td>
|
||
</tr>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>tolerance</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="float">float</span></code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Stop iteration if <code>(A - I*eigenvalue) @ v < num_vectors * tolerance</code>,
|
||
Default 1e-13.</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<code>1e-13</code>
|
||
</td>
|
||
</tr>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>solver</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="collections.abc.Callable">Callable</span>[..., <span title="numpy.typing.NDArray">NDArray</span>[<span title="numpy.complex128">complex128</span>]] | None</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Solver function of the form <code>x = solver(A, b)</code>.
|
||
By default, use scipy.sparse.spsolve for sparse matrices and
|
||
scipy.sparse.bicgstab for general LinearOperator instances.</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<code>None</code>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
<p><span class="doc-section-title">Returns:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code><span title="tuple">tuple</span>[<span title="complex">complex</span>, <span title="numpy.typing.NDArray">NDArray</span>[<span title="numpy.complex128">complex128</span>]]</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>(eigenvalues, eigenvectors)</p>
|
||
</div>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="doc doc-object doc-function">
|
||
|
||
|
||
<h3 id="meanas.eigensolvers.signed_eigensolve" class="doc doc-heading">
|
||
<code class="doc-symbol doc-symbol-heading doc-symbol-function"></code> <span class="doc doc-object-name doc-function-name">signed_eigensolve</span>
|
||
|
||
|
||
<a href="#meanas.eigensolvers.signed_eigensolve" class="headerlink" title="Permanent link">¶</a></h3>
|
||
<div class="doc-signature highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">signed_eigensolve</span><span class="p">(</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">operator</span><span class="p">:</span> <span class="n">spmatrix</span> <span class="o">|</span> <span class="n">LinearOperator</span><span class="p">,</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a> <span class="n">how_many</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span>
|
||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a> <span class="n">negative</span><span class="p">:</span> <span class="nb">bool</span> <span class="o">=</span> <span class="kc">False</span><span class="p">,</span>
|
||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="p">)</span> <span class="o">-></span> <span class="nb">tuple</span><span class="p">[</span>
|
||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a> <span class="n">NDArray</span><span class="p">[</span><span class="n">numpy</span><span class="o">.</span><span class="n">complex128</span><span class="p">],</span> <span class="n">NDArray</span><span class="p">[</span><span class="n">numpy</span><span class="o">.</span><span class="n">complex128</span><span class="p">]</span>
|
||
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="p">]</span>
|
||
</code></pre></div>
|
||
|
||
<div class="doc doc-contents ">
|
||
|
||
<p>Find the largest-magnitude positive-only (or negative-only) eigenvalues and
|
||
eigenvectors of the provided matrix.</p>
|
||
|
||
|
||
<p><span class="doc-section-title">Parameters:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Name</th>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
<th>Default</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>operator</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="scipy.sparse.spmatrix">spmatrix</span> | <span title="scipy.sparse.linalg.LinearOperator">LinearOperator</span></code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Matrix to analyze.</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<em>required</em>
|
||
</td>
|
||
</tr>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>how_many</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="int">int</span></code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>How many eigenvalues to find.</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<em>required</em>
|
||
</td>
|
||
</tr>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code>negative</code>
|
||
</td>
|
||
<td>
|
||
<code><span title="bool">bool</span></code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>Whether to find negative-only eigenvalues.
|
||
Default False (positive only).</p>
|
||
</div>
|
||
</td>
|
||
<td>
|
||
<code>False</code>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
<p><span class="doc-section-title">Returns:</span></p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Type</th>
|
||
<th>Description</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code><span title="numpy.typing.NDArray">NDArray</span>[<span title="numpy.complex128">complex128</span>]</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p>(sorted list of eigenvalues, 2D ndarray of corresponding eigenvectors)</p>
|
||
</div>
|
||
</td>
|
||
</tr>
|
||
<tr class="doc-section-item">
|
||
<td>
|
||
<code><span title="numpy.typing.NDArray">NDArray</span>[<span title="numpy.complex128">complex128</span>]</code>
|
||
</td>
|
||
<td>
|
||
<div class="doc-md-description">
|
||
<p><code>eigenvectors[:, k]</code> corresponds to the k-th eigenvalue</p>
|
||
</div>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
|
||
|
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