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<body>
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<div id="org-div-home-and-up">
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<a accesskey="h" href=""> UP </a>
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<a accesskey="H" href="index.html"> HOME </a>
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</div><div id="content">
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<h1 class="title">Sherman-Morrison-Woodbury</h1>
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<div id="table-of-contents">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#orge919fab">1. Headers</a></li>
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<li><a href="#org4a47370">2. Naïve Sherman-Morrison</a>
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<ul>
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<li><a href="#orgb4a168d">2.1. <code>qmckl_sherman_morrison</code></a>
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<ul>
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<li><a href="#orgaa63394">2.1.1. Requirements</a></li>
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<li><a href="#org2271b61">2.1.2. C header</a></li>
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<li><a href="#orga222a45">2.1.3. C source</a></li>
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<li><a href="#orgd6f8084">2.1.4. Performance</a></li>
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</ul>
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</li>
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</ul>
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</li>
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<li><a href="#org1b563f4">3. Woodbury 2x2</a>
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<ul>
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<li><a href="#org2d96bed">3.1. <code>qmckl_woodbury_2</code></a>
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<ul>
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<li><a href="#org65a0cbb">3.1.1. Requirements</a></li>
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<li><a href="#org65b30e6">3.1.2. C header</a></li>
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<li><a href="#org1ce7db0">3.1.3. C source</a></li>
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<li><a href="#org97b246f">3.1.4. Performance</a></li>
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</ul>
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</li>
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</ul>
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</li>
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<li><a href="#org1f42d98">4. Woodbury 3x3</a>
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<ul>
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<li><a href="#orge77cc2f">4.1. <code>qmckl_woodbury_3</code></a>
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<ul>
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<li><a href="#org239d39b">4.1.1. Requirements</a></li>
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<li><a href="#org37cfe45">4.1.2. C header</a></li>
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<li><a href="#org8d81023">4.1.3. C source</a></li>
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<li><a href="#orge0c6c9c">4.1.4. Performance…</a></li>
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</ul>
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</li>
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</ul>
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</li>
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<li><a href="#orge6c825a">5. Sherman-Morrison with update splitting</a>
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<ul>
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<li><a href="#orgfda3705">5.1. <code>qmckl_sherman_morrison_splitting</code></a>
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<ul>
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<li><a href="#orge595ed9">5.1.1. Requirements</a></li>
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<li><a href="#orgd807e82">5.1.2. C header</a></li>
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<li><a href="#org1944926">5.1.3. C source</a></li>
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<li><a href="#org18648d1">5.1.4. Performance…</a></li>
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</ul>
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</li>
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</ul>
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</li>
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<li><a href="#org706b4f0">6. Woodbury 3x3 and 2x2 with Sherman-Morrison and update splitting</a>
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<ul>
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<li><a href="#orgbd2e70a">6.1. <code>qmckl_sherman_morrison_smw32s</code></a>
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<ul>
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<li><a href="#org2472e42">6.1.1. Requirements</a></li>
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<li><a href="#orgacdda34">6.1.2. C header</a></li>
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<li><a href="#orgaf1d6d9">6.1.3. C source</a></li>
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<li><a href="#org86d6360">6.1.4. Performance…</a></li>
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</ul>
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</li>
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</ul>
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</li>
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<li><a href="#org9577fed">7. Helper Functions</a>
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<ul>
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<li><a href="#orgb3d1ef9">7.1. <code>qmckl_slagel_splitting</code></a>
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<ul>
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<li><a href="#org37877c7">7.1.1. Requirements</a></li>
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<li><a href="#orgcc799b1">7.1.2. C header</a></li>
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<li><a href="#orge0463aa">7.1.3. C source</a></li>
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<li><a href="#org3048b08">7.1.4. Performance</a></li>
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</ul>
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</li>
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</ul>
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</li>
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<li><a href="#org97008df">8. End of files</a></li>
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|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orge919fab" class="outline-2">
|
|
<h2 id="orge919fab"><span class="section-number-2">1</span> Headers</h2>
|
|
<div class="outline-text-2" id="text-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-c"><span style="color: #483d8b;">#include</span> <span style="color: #8b2252;">"qmckl.h"</span>
|
|
<span style="color: #483d8b;">#include</span> <span style="color: #8b2252;">"assert.h"</span>
|
|
<span style="color: #483d8b;">#ifdef</span> HAVE_CONFIG_H
|
|
<span style="color: #483d8b;">#include</span> <span style="color: #8b2252;">"config.h"</span>
|
|
<span style="color: #483d8b;">#endif</span>
|
|
<span style="color: #483d8b;">#include</span> <span style="color: #8b2252;"><math.h></span>
|
|
|
|
<span style="color: #228b22;">int</span> <span style="color: #0000ff;">main</span>() {
|
|
<span style="color: #228b22;">qmckl_context</span> <span style="color: #a0522d;">context</span>;
|
|
context = qmckl_context_create();
|
|
<span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #a0522d;">rc</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org4a47370" class="outline-2">
|
|
<h2 id="org4a47370"><span class="section-number-2">2</span> Naïve Sherman-Morrison</h2>
|
|
<div class="outline-text-2" id="text-2">
|
|
</div>
|
|
<div id="outline-container-orgb4a168d" class="outline-3">
|
|
<h3 id="orgb4a168d"><span class="section-number-3">2.1</span> <code>qmckl_sherman_morrison</code></h3>
|
|
<div class="outline-text-3" id="text-2-1">
|
|
<p>
|
|
This is the simplest of the available Sherman-Morrison-Woodbury kernels. It applies rank-1 updates one by one in
|
|
the order that is given. It only checks if the denominator in the Sherman-Morrison formula is not too close to
|
|
zero when an update is evaluated. It will exit with an error code of the denominator is too close to zero.
|
|
</p>
|
|
|
|
<p>
|
|
The formula for any update \(u_j\) (index \(j\) is suppresed for clarity) that is applied is
|
|
\[
|
|
(S + uv^T)^{-1} = S^{-1} - \frac{S^{-1} uv^T S^{-1}}{1 + v^T S^{-1} u}
|
|
\]
|
|
</p>
|
|
|
|
<p>
|
|
where
|
|
\(S\) is the Slater-matrix,
|
|
\(u\) and \(v^T\) are the column and row vectors containing the updates,
|
|
\(S^{-1}\) is the inverse of the Slater-matrix.
|
|
</p>
|
|
|
|
<p>
|
|
Even though the Slater-matrix \(S\) with all updates applied at once is invertable, during the course of applying
|
|
updates to the inverse Slater-matrix \(S^{-1}\) one-by-one it can happen that one of the intermediate inverse
|
|
matrices \(S^{-1}\) becomes singular. Therefore a global threshold value \(\epsilon\) is defined that is used to
|
|
evaluate each individual update \(u_j\) when it is applied.
|
|
</p>
|
|
|
|
<p>
|
|
This value sets the lower bound for which the
|
|
denominator \(1+v_j^TS^{-1}u_j\) is considered to be too small and will most probably result in a singular matrix
|
|
\(S\), or at least in an inverse of \(S\) of very poor numerical quality. Therefore, when \(1+v_j^TS^{-1}u_j \geq \epsilon\),
|
|
the update is applied as usual and the kernel exits with return code \texttt{QMCKL_SUCCESS}.
|
|
If \(1+v_j^TS^{-1}u_j \leq \epsilon\) the update is rejected and the kernel exits with return code \texttt{QMCKL_FAILURE}.
|
|
</p>
|
|
|
|
<p>
|
|
If the determinant of the Slater-matrix is passed, it will be updated to the determinant resulting
|
|
from applying the updates to the original matrix.
|
|
</p>
|
|
|
|
<table id="org4f684ee" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<colgroup>
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
</colgroup>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left">qmckl<sub>context</sub></td>
|
|
<td class="org-left">context</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Global state</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">uint64<sub>t</sub></td>
|
|
<td class="org-left">LDS</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Leading dimension of Slater<sub>inv</sub></td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">uint64<sub>t</sub></td>
|
|
<td class="org-left">Dim</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Dimension of Slater<sub>inv</sub></td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">uint64<sub>t</sub></td>
|
|
<td class="org-left">N<sub>updates</sub></td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Number of rank-1 updates to be applied to Slater<sub>inv</sub></td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">double</td>
|
|
<td class="org-left">Updates[N<sub>updates</sub>*Dim]</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Array containing the updates</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">uint64<sub>t</sub></td>
|
|
<td class="org-left">Updates<sub>index</sub>[N<sub>updates</sub>]</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Array containing the rank-1 updates</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">double</td>
|
|
<td class="org-left">breakdown</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Break-down parameter on which to fail or not</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">double</td>
|
|
<td class="org-left">Slater<sub>inv</sub>[LDS*Dim]</td>
|
|
<td class="org-left">inout</td>
|
|
<td class="org-left">Array containing the inverse of a Slater-matrix</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">double*</td>
|
|
<td class="org-left">determinant</td>
|
|
<td class="org-left">inout</td>
|
|
<td class="org-left">Determinant of the Slater-matrix</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
|
|
<div id="outline-container-orgaa63394" class="outline-4">
|
|
<h4 id="orgaa63394"><span class="section-number-4">2.1.1</span> Requirements</h4>
|
|
<div class="outline-text-4" id="text-2-1-1">
|
|
<ul class="org-ul">
|
|
<li><code>context</code> is not <code>QMCKL_NULL_CONTEXT</code></li>
|
|
<li><code>LDS >= 2</code></li>
|
|
<li><code>Dim >= 2</code></li>
|
|
<li><code>N_updates >= 1</code></li>
|
|
<li><code>Updates</code> is allocated with \(N_updates \times Dim\) elements</li>
|
|
<li><code>Updates_index</code> is allocated with \(N_updates\) elements</li>
|
|
<li><code>breakdown</code> is a small number such that \(0 < breakdown << 1\)</li>
|
|
<li><code>Slater_inv</code> is allocated with \(Dim \times Dim\) elements</li>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org2271b61" class="outline-4">
|
|
<h4 id="org2271b61"><span class="section-number-4">2.1.2</span> C header</h4>
|
|
<div class="outline-text-4" id="text-2-1-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-c"><span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_sherman_morrison</span> (
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">context</span> <span style="color: #a0522d;">qmckl_context</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">LDS</span> <span style="color: #a0522d;">uint64_t</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">Dim</span> <span style="color: #a0522d;">uint64_t</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">N_updates</span> <span style="color: #a0522d;">uint64_t</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">Updates</span>* <span style="color: #228b22;">double</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">Updates_index</span>* <span style="color: #a0522d;">uint64_t</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">breakdown</span> <span style="color: #228b22;">double</span>,
|
|
<span style="color: #228b22;">Slater_inv</span>* <span style="color: #228b22;">double</span>,
|
|
<span style="color: #228b22;">determinant</span>* <span style="color: #228b22;">double</span>* );
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orga222a45" class="outline-4">
|
|
<h4 id="orga222a45"><span class="section-number-4">2.1.3</span> C source</h4>
|
|
<div class="outline-text-4" id="text-2-1-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-c"><span style="color: #483d8b;">#include</span> <span style="color: #8b2252;"><stdbool.h></span>
|
|
<span style="color: #483d8b;">#include</span> <span style="color: #8b2252;"><math.h></span>
|
|
<span style="color: #483d8b;">#include</span> <span style="color: #8b2252;">"qmckl.h"</span>
|
|
|
|
<span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_sherman_morrison</span>(<span style="color: #a020f0;">const</span> <span style="color: #228b22;">qmckl_context</span> <span style="color: #a0522d;">context</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">LDS</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">Dim</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">N_updates</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span>* <span style="color: #a0522d;">Updates</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">uint64_t</span>* <span style="color: #a0522d;">Updates_index</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> <span style="color: #a0522d;">breakdown</span>,
|
|
<span style="color: #228b22;">double</span>* <span style="color: #a0522d;">Slater_inv</span>,
|
|
<span style="color: #228b22;">double</span>* <span style="color: #a0522d;">determinant</span>) {
|
|
|
|
<span style="color: #a020f0;">if</span> (qmckl_context_check(context) == QMCKL_NULL_CONTEXT) {
|
|
<span style="color: #a020f0;">return</span> QMCKL_NULL_CONTEXT;
|
|
}
|
|
|
|
<span style="color: #228b22;">double</span> <span style="color: #a0522d;">C</span>[Dim];
|
|
<span style="color: #228b22;">double</span> <span style="color: #a0522d;">D</span>[Dim];
|
|
|
|
<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">l</span> = 0;
|
|
// <span style="color: #b22222;">For each update</span>
|
|
<span style="color: #a020f0;">while</span> (l < N_updates) {
|
|
// <span style="color: #b22222;">C = A^{-1} x U_l</span>
|
|
<span style="color: #a020f0;">for</span> (<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">i</span> = 0; i < Dim; i++) {
|
|
C[i] = 0;
|
|
<span style="color: #a020f0;">for</span> (<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">j</span> = 0; j < Dim; j++) {
|
|
C[i] += Slater_inv[i * LDS + j] * Updates[l * Dim + j];
|
|
}
|
|
}
|
|
|
|
// <span style="color: #b22222;">Denominator</span>
|
|
<span style="color: #228b22;">double</span> <span style="color: #a0522d;">den</span> = 1 + C[Updates_index[l] - 1];
|
|
|
|
<span style="color: #a020f0;">if</span> (fabs(den) < breakdown) {
|
|
<span style="color: #a020f0;">return</span> QMCKL_FAILURE;
|
|
}
|
|
<span style="color: #228b22;">double</span> <span style="color: #a0522d;">iden</span> = 1 / den;
|
|
|
|
// <span style="color: #b22222;">Update det(A)</span>
|
|
<span style="color: #a020f0;">if</span> (determinant != <span style="color: #008b8b;">NULL</span>)
|
|
*determinant *= den;
|
|
|
|
// <span style="color: #b22222;">D = v^T x A^{-1}</span>
|
|
<span style="color: #a020f0;">for</span> (<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">j</span> = 0; j < Dim; j++) {
|
|
D[j] = Slater_inv[(Updates_index[l] - 1) * LDS + j];
|
|
}
|
|
|
|
// <span style="color: #b22222;">A^{-1} = A^{-1} - C x D / den</span>
|
|
<span style="color: #a020f0;">for</span> (<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">i</span> = 0; i < Dim; i++) {
|
|
<span style="color: #a020f0;">for</span> (<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">j</span> = 0; j < Dim; j++) {
|
|
<span style="color: #228b22;">double</span> <span style="color: #a0522d;">update</span> = C[i] * D[j] * iden;
|
|
Slater_inv[i * LDS + j] -= update;
|
|
}
|
|
}
|
|
|
|
l += 1;
|
|
}
|
|
|
|
<span style="color: #a020f0;">return</span> QMCKL_SUCCESS;
|
|
}
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgd6f8084" class="outline-4">
|
|
<h4 id="orgd6f8084"><span class="section-number-4">2.1.4</span> Performance</h4>
|
|
<div class="outline-text-4" id="text-2-1-4">
|
|
<p>
|
|
This function performs best when there is only 1 rank-1 update in the update cycle. It is not useful to
|
|
use Sherman-Morrison with update splitting for these cycles since splitting can never resolve a situation
|
|
where applying the update causes singular behaviour.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org1b563f4" class="outline-2">
|
|
<h2 id="org1b563f4"><span class="section-number-2">3</span> Woodbury 2x2</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
</div>
|
|
<div id="outline-container-org2d96bed" class="outline-3">
|
|
<h3 id="org2d96bed"><span class="section-number-3">3.1</span> <code>qmckl_woodbury_2</code></h3>
|
|
<div class="outline-text-3" id="text-3-1">
|
|
<p>
|
|
The Woodbury 2x2 kernel. It is used to apply two rank-1 updates at once. The formula used in
|
|
this algorithm is called the Woodbury Matrix Identity
|
|
\[
|
|
(S + U V)^{-1} = S^{-1} - C B^{-1} D
|
|
\]
|
|
where
|
|
\(S\) is the Slater-matrix
|
|
\(U\) and \(V\) are the matrices containing the updates and the canonical basis matrix
|
|
\(S^{-1}\) is the inverse of the Slater-matrix
|
|
\(C:= S^{-1}U\), a Dim \(\times 2\) matrix
|
|
\(B := 1 + VC\), the \(2 \times 2\) matrix that is going to be inverted
|
|
\(D := VS^{-1}\), a \(2 \times Dim\) matrix
|
|
</p>
|
|
|
|
<p>
|
|
If the determinant of the Slater-matrix is passed, it will be updated to the determinant resulting
|
|
from applying the updates to the original matrix.
|
|
</p>
|
|
|
|
|
|
|
|
<table id="orgdc528ea" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<colgroup>
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
</colgroup>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left">qmckl<sub>context</sub></td>
|
|
<td class="org-left">context</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Global state</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">uint64<sub>t</sub></td>
|
|
<td class="org-left">LDS</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Leading dimension of Slater<sub>inv</sub></td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">uint64<sub>t</sub></td>
|
|
<td class="org-left">Dim</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Dimension of Slater<sub>inv</sub></td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">double</td>
|
|
<td class="org-left">Updates[2*Dim]</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Array containing the updates</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">uint64<sub>t</sub></td>
|
|
<td class="org-left">Updates<sub>index</sub>[2]</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Array containing the rank-1 updates</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">double</td>
|
|
<td class="org-left">breakdown</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Break-down parameter on which to fail or not</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">double</td>
|
|
<td class="org-left">Slater<sub>inv</sub>[LDS*Dim]</td>
|
|
<td class="org-left">inout</td>
|
|
<td class="org-left">Array containing the inverse of a Slater-matrix</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">double*</td>
|
|
<td class="org-left">determinant</td>
|
|
<td class="org-left">inout</td>
|
|
<td class="org-left">Determinant of Slater-matrix</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
|
|
<div id="outline-container-org65a0cbb" class="outline-4">
|
|
<h4 id="org65a0cbb"><span class="section-number-4">3.1.1</span> Requirements</h4>
|
|
<div class="outline-text-4" id="text-3-1-1">
|
|
<ul class="org-ul">
|
|
<li><code>context</code> is not <code>qmckl_null_context</code></li>
|
|
<li><code>LDS >= 2</code></li>
|
|
<li><code>Dim >= 2</code></li>
|
|
<li><code>Updates</code> is allocated with \(2 \times Dim\) elements</li>
|
|
<li><code>Updates_index</code> is allocated with \(2\) elements</li>
|
|
<li><code>breakdown</code> is a small number such that \(0 < breakdown << 1\)</li>
|
|
<li><code>Slater_inv</code> is allocated with \(Dim \times Dim\) elements</li>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org65b30e6" class="outline-4">
|
|
<h4 id="org65b30e6"><span class="section-number-4">3.1.2</span> C header</h4>
|
|
<div class="outline-text-4" id="text-3-1-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-c"><span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_woodbury_2</span> (
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">context</span> <span style="color: #a0522d;">qmckl_context</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">LDS</span> <span style="color: #a0522d;">uint64_t</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">Dim</span> <span style="color: #a0522d;">uint64_t</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">Updates</span>* <span style="color: #228b22;">double</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">Updates_index</span>* <span style="color: #a0522d;">uint64_t</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">breakdown</span> <span style="color: #228b22;">double</span>,
|
|
<span style="color: #228b22;">Slater_inv</span>* <span style="color: #228b22;">double</span>,
|
|
<span style="color: #228b22;">determinant</span>* <span style="color: #228b22;">double</span>* );
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org1ce7db0" class="outline-4">
|
|
<h4 id="org1ce7db0"><span class="section-number-4">3.1.3</span> C source</h4>
|
|
<div class="outline-text-4" id="text-3-1-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-c"><span style="color: #483d8b;">#include</span> <span style="color: #8b2252;"><stdbool.h></span>
|
|
<span style="color: #483d8b;">#include</span> <span style="color: #8b2252;"><math.h></span>
|
|
<span style="color: #483d8b;">#include</span> <span style="color: #8b2252;">"qmckl.h"</span>
|
|
|
|
<span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_woodbury_2</span>(<span style="color: #a020f0;">const</span> <span style="color: #228b22;">qmckl_context</span> <span style="color: #a0522d;">context</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">LDS</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">Dim</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span>* <span style="color: #a0522d;">Updates</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">uint64_t</span>* <span style="color: #a0522d;">Updates_index</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> <span style="color: #a0522d;">breakdown</span>,
|
|
<span style="color: #228b22;">double</span>* <span style="color: #a0522d;">Slater_inv</span>,
|
|
<span style="color: #228b22;">double</span>* <span style="color: #a0522d;">determinant</span>) {
|
|
/*
|
|
<span style="color: #b22222;"> C := S^{-1} * U, dim x 2</span>
|
|
<span style="color: #b22222;"> B := 1 + V * C, 2 x 2</span>
|
|
<span style="color: #b22222;"> D := V * S^{-1}, 2 x dim</span>
|
|
*/
|
|
|
|
<span style="color: #a020f0;">if</span> (qmckl_context_check(context) == QMCKL_NULL_CONTEXT) {
|
|
<span style="color: #a020f0;">return</span> QMCKL_NULL_CONTEXT;
|
|
}
|
|
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">row1</span> = (Updates_index[0] - 1);
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">row2</span> = (Updates_index[1] - 1);
|
|
|
|
// <span style="color: #b22222;">Compute C = S_inv * U !! NON-STANDARD MATRIX MULTIPLICATION BECAUSE</span>
|
|
// <span style="color: #b22222;">OF LAYOUT OF 'Updates' !!</span>
|
|
<span style="color: #228b22;">double</span> <span style="color: #a0522d;">C</span>[2 * Dim];
|
|
<span style="color: #a020f0;">for</span> (<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">i</span> = 0; i < Dim; i++) {
|
|
<span style="color: #a020f0;">for</span> (<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">j</span> = 0; j < 2; j++) {
|
|
C[i * 2 + j] = 0;
|
|
<span style="color: #a020f0;">for</span> (<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">k</span> = 0; k < Dim; k++) {
|
|
C[i * 2 + j] += Slater_inv[i * LDS + k] * Updates[Dim * j + k];
|
|
}
|
|
}
|
|
}
|
|
|
|
// <span style="color: #b22222;">Compute B = 1 + V * C</span>
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> <span style="color: #a0522d;">B0</span> = C[row1 * 2] + 1;
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> <span style="color: #a0522d;">B1</span> = C[row1 * 2 + 1];
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> <span style="color: #a0522d;">B2</span> = C[row2 * 2];
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> <span style="color: #a0522d;">B3</span> = C[row2 * 2 + 1] + 1;
|
|
|
|
// <span style="color: #b22222;">Check if determinant of inverted matrix is not zero</span>
|
|
<span style="color: #228b22;">double</span> <span style="color: #a0522d;">det</span> = B0 * B3 - B1 * B2;
|
|
<span style="color: #a020f0;">if</span> (fabs(det) < breakdown) {
|
|
<span style="color: #a020f0;">return</span> QMCKL_FAILURE;
|
|
}
|
|
|
|
// <span style="color: #b22222;">Update det(S) when passed</span>
|
|
<span style="color: #a020f0;">if</span> (determinant != <span style="color: #008b8b;">NULL</span>)
|
|
*determinant *= det;
|
|
|
|
// <span style="color: #b22222;">Compute B^{-1} with explicit formula for 2x2 inversion</span>
|
|
<span style="color: #228b22;">double</span> <span style="color: #a0522d;">Binv</span>[4], <span style="color: #a0522d;">idet</span> = 1.0 / det;
|
|
Binv[0] = idet * B3;
|
|
Binv[1] = -1.0 * idet * B1;
|
|
Binv[2] = -1.0 * idet * B2;
|
|
Binv[3] = idet * B0;
|
|
|
|
// <span style="color: #b22222;">Compute tmp = B^{-1} x (V.S^{-1})</span>
|
|
<span style="color: #228b22;">double</span> <span style="color: #a0522d;">tmp</span>[2 * Dim];
|
|
<span style="color: #a020f0;">for</span> (<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">i</span> = 0; i < 2; i++) {
|
|
<span style="color: #a020f0;">for</span> (<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">j</span> = 0; j < Dim; j++) {
|
|
tmp[i * Dim + j] = Binv[i * 2] * Slater_inv[row1 * LDS + j];
|
|
tmp[i * Dim + j] += Binv[i * 2 + 1] * Slater_inv[row2 * LDS + j];
|
|
}
|
|
}
|
|
|
|
// <span style="color: #b22222;">Compute (S + U V)^{-1} = S^{-1} - C x tmp</span>
|
|
<span style="color: #a020f0;">for</span> (<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">i</span> = 0; i < Dim; i++) {
|
|
<span style="color: #a020f0;">for</span> (<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">j</span> = 0; j < Dim; j++) {
|
|
Slater_inv[i * LDS + j] -= C[i * 2] * tmp[j];
|
|
Slater_inv[i * LDS + j] -= C[i * 2 + 1] * tmp[Dim + j];
|
|
}
|
|
}
|
|
|
|
<span style="color: #a020f0;">return</span> QMCKL_SUCCESS;
|
|
}
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org97b246f" class="outline-4">
|
|
<h4 id="org97b246f"><span class="section-number-4">3.1.4</span> Performance</h4>
|
|
<div class="outline-text-4" id="text-3-1-4">
|
|
<p>
|
|
This function is most efficient when used in cases where there are only 2 rank-1 updates and
|
|
it is sure they will not result in a singular matrix.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org1f42d98" class="outline-2">
|
|
<h2 id="org1f42d98"><span class="section-number-2">4</span> Woodbury 3x3</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
</div>
|
|
<div id="outline-container-orge77cc2f" class="outline-3">
|
|
<h3 id="orge77cc2f"><span class="section-number-3">4.1</span> <code>qmckl_woodbury_3</code></h3>
|
|
<div class="outline-text-3" id="text-4-1">
|
|
<p>
|
|
The 3x3 version of the Woodbury 2x2 kernel. It is used to apply three
|
|
rank-1 updates at once. The formula used in this kernel is the same as for Woodbury 2x2,
|
|
except for the sizes of the following matrices:
|
|
</p>
|
|
|
|
<p>
|
|
\(C:= S^{-1}U\), a Dim \(\times 3\) matrix
|
|
\(B := 1 + VC\), the \(3 \times 3\) matrix that is going to be inverted
|
|
\(D := VS^{-1}\), a \(3 \times Dim\) matrix
|
|
</p>
|
|
|
|
<p>
|
|
If the determinant of the Slater-matrix is passed, it will be updated to the determinant resulting
|
|
from applying the updates to the original matrix.
|
|
</p>
|
|
|
|
|
|
|
|
<table id="org1ec8f46" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<colgroup>
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
</colgroup>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left">qmckl<sub>context</sub></td>
|
|
<td class="org-left">context</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Global state</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">uint64<sub>t</sub></td>
|
|
<td class="org-left">LDS</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Leading dimension of Slater<sub>inv</sub></td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">uint64<sub>t</sub></td>
|
|
<td class="org-left">Dim</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Dimension of Slater<sub>inv</sub></td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">double</td>
|
|
<td class="org-left">Updates[3*Dim]</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Array containing the updates</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">uint64<sub>t</sub></td>
|
|
<td class="org-left">Updates<sub>index</sub>[3]</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Array containing the rank-1 updates</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">double</td>
|
|
<td class="org-left">breakdown</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Break-down parameter on which to fail or not</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">double</td>
|
|
<td class="org-left">Slater<sub>inv</sub>[LDS*Dim]</td>
|
|
<td class="org-left">inout</td>
|
|
<td class="org-left">Array containing the inverse of a Slater-matrix</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">double*</td>
|
|
<td class="org-left">determinant</td>
|
|
<td class="org-left">inout</td>
|
|
<td class="org-left">Determinant of Slater-matrix</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
|
|
<div id="outline-container-org239d39b" class="outline-4">
|
|
<h4 id="org239d39b"><span class="section-number-4">4.1.1</span> Requirements</h4>
|
|
<div class="outline-text-4" id="text-4-1-1">
|
|
<ul class="org-ul">
|
|
<li><code>context</code> is not <code>qmckl_null_context</code></li>
|
|
<li><code>LDS >= 2</code></li>
|
|
<li><code>Dim >= 2</code></li>
|
|
<li><code>Updates</code> is allocated with \(3 \times Dim\) elements</li>
|
|
<li><code>Updates_index</code> is allocated with \(3\) elements</li>
|
|
<li><code>breakdown</code> is a small number such that \(0 < breakdown << 1\)</li>
|
|
<li><code>Slater_inv</code> is allocated with \(Dim \times Dim\) elements</li>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org37cfe45" class="outline-4">
|
|
<h4 id="org37cfe45"><span class="section-number-4">4.1.2</span> C header</h4>
|
|
<div class="outline-text-4" id="text-4-1-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-c"><span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_woodbury_3</span> (
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">context</span> <span style="color: #a0522d;">qmckl_context</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">LDS</span> <span style="color: #a0522d;">uint64_t</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">Dim</span> <span style="color: #a0522d;">uint64_t</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">Updates</span>* <span style="color: #228b22;">double</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">Updates_index</span>* <span style="color: #a0522d;">uint64_t</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">breakdown</span> <span style="color: #228b22;">double</span>,
|
|
<span style="color: #228b22;">Slater_inv</span>* <span style="color: #228b22;">double</span>,
|
|
<span style="color: #228b22;">determinant</span>* <span style="color: #228b22;">double</span>* );
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org8d81023" class="outline-4">
|
|
<h4 id="org8d81023"><span class="section-number-4">4.1.3</span> C source</h4>
|
|
<div class="outline-text-4" id="text-4-1-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-c"><span style="color: #483d8b;">#include</span> <span style="color: #8b2252;"><stdbool.h></span>
|
|
<span style="color: #483d8b;">#include</span> <span style="color: #8b2252;"><math.h></span>
|
|
<span style="color: #483d8b;">#include</span> <span style="color: #8b2252;">"qmckl.h"</span>
|
|
|
|
<span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_woodbury_3</span>(<span style="color: #a020f0;">const</span> <span style="color: #228b22;">qmckl_context</span> <span style="color: #a0522d;">context</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">LDS</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">Dim</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span>* <span style="color: #a0522d;">Updates</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">uint64_t</span>* <span style="color: #a0522d;">Updates_index</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> <span style="color: #a0522d;">breakdown</span>,
|
|
<span style="color: #228b22;">double</span>* <span style="color: #a0522d;">Slater_inv</span>,
|
|
<span style="color: #228b22;">double</span>* <span style="color: #a0522d;">determinant</span>) {
|
|
/*
|
|
<span style="color: #b22222;"> C := S^{-1} * U, dim x 3</span>
|
|
<span style="color: #b22222;"> B := 1 + V * C, 3 x 3</span>
|
|
<span style="color: #b22222;"> D := V * S^{-1}, 3 x dim</span>
|
|
*/
|
|
|
|
<span style="color: #a020f0;">if</span> (qmckl_context_check(context) == QMCKL_NULL_CONTEXT) {
|
|
<span style="color: #a020f0;">return</span> QMCKL_NULL_CONTEXT;
|
|
}
|
|
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">row1</span> = (Updates_index[0] - 1);
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">row2</span> = (Updates_index[1] - 1);
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">row3</span> = (Updates_index[2] - 1);
|
|
|
|
// <span style="color: #b22222;">Compute C = S_inv * U !! NON-STANDARD MATRIX MULTIPLICATION BECAUSE</span>
|
|
// <span style="color: #b22222;">OF LAYOUT OF 'Updates' !!</span>
|
|
<span style="color: #228b22;">double</span> <span style="color: #a0522d;">C</span>[3 * Dim];
|
|
<span style="color: #a020f0;">for</span> (<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">i</span> = 0; i < Dim; i++) {
|
|
<span style="color: #a020f0;">for</span> (<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">j</span> = 0; j < 3; j++) {
|
|
C[i * 3 + j] = 0;
|
|
<span style="color: #a020f0;">for</span> (<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">k</span> = 0; k < Dim; k++) {
|
|
C[i * 3 + j] += Slater_inv[i * LDS + k] * Updates[Dim * j + k];
|
|
}
|
|
}
|
|
}
|
|
|
|
// <span style="color: #b22222;">Compute B = 1 + V.C</span>
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> <span style="color: #a0522d;">B0</span> = C[row1 * 3] + 1;
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> <span style="color: #a0522d;">B1</span> = C[row1 * 3 + 1];
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> <span style="color: #a0522d;">B2</span> = C[row1 * 3 + 2];
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> <span style="color: #a0522d;">B3</span> = C[row2 * 3];
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> <span style="color: #a0522d;">B4</span> = C[row2 * 3 + 1] + 1;
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> <span style="color: #a0522d;">B5</span> = C[row2 * 3 + 2];
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> <span style="color: #a0522d;">B6</span> = C[row3 * 3];
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> <span style="color: #a0522d;">B7</span> = C[row3 * 3 + 1];
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> <span style="color: #a0522d;">B8</span> = C[row3 * 3 + 2] + 1;
|
|
|
|
// <span style="color: #b22222;">Check if determinant of B is not too close to zero</span>
|
|
<span style="color: #228b22;">double</span> <span style="color: #a0522d;">det</span>;
|
|
det = B0 * (B4 * B8 - B5 * B7) - B1 * (B3 * B8 - B5 * B6) +
|
|
B2 * (B3 * B7 - B4 * B6);
|
|
<span style="color: #a020f0;">if</span> (fabs(det) < breakdown) {
|
|
<span style="color: #a020f0;">return</span> QMCKL_FAILURE;
|
|
}
|
|
|
|
// <span style="color: #b22222;">Update det(Slater) if passed</span>
|
|
<span style="color: #a020f0;">if</span> (determinant != <span style="color: #008b8b;">NULL</span>)
|
|
*determinant *= det;
|
|
|
|
// <span style="color: #b22222;">Compute B^{-1} with explicit formula for 3x3 inversion</span>
|
|
<span style="color: #228b22;">double</span> <span style="color: #a0522d;">Binv</span>[9], <span style="color: #a0522d;">idet</span> = 1.0 / det;
|
|
Binv[0] = (B4 * B8 - B7 * B5) * idet;
|
|
Binv[1] = -(B1 * B8 - B7 * B2) * idet;
|
|
Binv[2] = (B1 * B5 - B4 * B2) * idet;
|
|
Binv[3] = -(B3 * B8 - B6 * B5) * idet;
|
|
Binv[4] = (B0 * B8 - B6 * B2) * idet;
|
|
Binv[5] = -(B0 * B5 - B3 * B2) * idet;
|
|
Binv[6] = (B3 * B7 - B6 * B4) * idet;
|
|
Binv[7] = -(B0 * B7 - B6 * B1) * idet;
|
|
Binv[8] = (B0 * B4 - B3 * B1) * idet;
|
|
|
|
// <span style="color: #b22222;">Compute tmp = B^{-1} x (V.S^{-1})</span>
|
|
<span style="color: #228b22;">double</span> <span style="color: #a0522d;">tmp</span>[3 * Dim];
|
|
<span style="color: #a020f0;">for</span> (<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">i</span> = 0; i < 3; i++) {
|
|
<span style="color: #a020f0;">for</span> (<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">j</span> = 0; j < Dim; j++) {
|
|
tmp[i * Dim + j] = Binv[i * 3] * Slater_inv[row1 * LDS + j];
|
|
tmp[i * Dim + j] += Binv[i * 3 + 1] * Slater_inv[row2 * LDS + j];
|
|
tmp[i * Dim + j] += Binv[i * 3 + 2] * Slater_inv[row3 * LDS + j];
|
|
}
|
|
}
|
|
|
|
// <span style="color: #b22222;">Compute (S + U V)^{-1} = S^{-1} - C x tmp</span>
|
|
<span style="color: #a020f0;">for</span> (<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">i</span> = 0; i < Dim; i++) {
|
|
<span style="color: #a020f0;">for</span> (<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">j</span> = 0; j < Dim; j++) {
|
|
Slater_inv[i * LDS + j] -= C[i * 3] * tmp[j];
|
|
Slater_inv[i * LDS + j] -= C[i * 3 + 1] * tmp[Dim + j];
|
|
Slater_inv[i * LDS + j] -= C[i * 3 + 2] * tmp[2 * Dim + j];
|
|
}
|
|
}
|
|
|
|
<span style="color: #a020f0;">return</span> QMCKL_SUCCESS;
|
|
}
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orge0c6c9c" class="outline-4">
|
|
<h4 id="orge0c6c9c"><span class="section-number-4">4.1.4</span> Performance…</h4>
|
|
<div class="outline-text-4" id="text-4-1-4">
|
|
<p>
|
|
This function is most efficient when used in cases where there are only 3 rank-1 updates and
|
|
it is sure they will not result in a singular matrix.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orge6c825a" class="outline-2">
|
|
<h2 id="orge6c825a"><span class="section-number-2">5</span> Sherman-Morrison with update splitting</h2>
|
|
<div class="outline-text-2" id="text-5">
|
|
</div>
|
|
<div id="outline-container-orgfda3705" class="outline-3">
|
|
<h3 id="orgfda3705"><span class="section-number-3">5.1</span> <code>qmckl_sherman_morrison_splitting</code></h3>
|
|
<div class="outline-text-3" id="text-5-1">
|
|
<p>
|
|
This is a variation on the 'Naive' Sherman-Morrison kernel. Whenever the denominator \(1+v_j^T S^{-1} u_j\) in
|
|
the Sherman-Morrison formula is deemed to be too close to zero, the update \(u_j\) is split in half:
|
|
\(u_j \rightarrow \frac{1}{2} u_j\). One half is applied immediately –necessarily increasing the value of the
|
|
denominator because of the split– while the other halve is put in a queue that will be applied when all the
|
|
remaining updates have been treated.
|
|
</p>
|
|
|
|
<p>
|
|
The kernel is executed recursively until the queue is eiter empty and all
|
|
updates are applied successfully, or the size of the queue equals the number of initial updates. In the last
|
|
case the Slater-matrix that would have resulted from applying the updates is singular and therefore the
|
|
kernel exits with an exit code.
|
|
</p>
|
|
|
|
<p>
|
|
If the determinant of the Slater-matrix is passed, it will be updated to the determinant resulting
|
|
from applying the updates to the original matrix.
|
|
</p>
|
|
|
|
<table id="org1352a37" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<colgroup>
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
</colgroup>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left">qmckl<sub>context</sub></td>
|
|
<td class="org-left">context</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Global state</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">uint64<sub>t</sub></td>
|
|
<td class="org-left">LDS</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Leading dimension of Slater<sub>inv</sub></td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">uint64<sub>t</sub></td>
|
|
<td class="org-left">Dim</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Dimension of Slater<sub>inv</sub></td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">uint64<sub>t</sub></td>
|
|
<td class="org-left">N<sub>updates</sub></td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Number of rank-1 updates to be applied to Slater<sub>inv</sub></td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">double</td>
|
|
<td class="org-left">Updates[N<sub>updates</sub>*Dim]</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Array containing the updates</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">uint64<sub>t</sub></td>
|
|
<td class="org-left">Updates<sub>index</sub>[N<sub>updates</sub>]</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Array containing the rank-1 updates</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">double</td>
|
|
<td class="org-left">breakdown</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Break-down parameter on which to fail or not</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">double</td>
|
|
<td class="org-left">Slater<sub>inv</sub>[LDS*Dim]</td>
|
|
<td class="org-left">inout</td>
|
|
<td class="org-left">Array containing the inverse of a Slater-matrix</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">double*</td>
|
|
<td class="org-left">determinant</td>
|
|
<td class="org-left">inout</td>
|
|
<td class="org-left">Determinant of the Slater-matrix</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
<p>
|
|
If the determinant of the Slater-matrix is passed, it will be updated to the determinant resulting
|
|
from applying the updates to the original matrix.
|
|
</p>
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-orge595ed9" class="outline-4">
|
|
<h4 id="orge595ed9"><span class="section-number-4">5.1.1</span> Requirements</h4>
|
|
<div class="outline-text-4" id="text-5-1-1">
|
|
<ul class="org-ul">
|
|
<li><code>context</code> is not <code>QMCKL_NULL_CONTEXT</code></li>
|
|
<li><code>LDS >= 2</code></li>
|
|
<li><code>Dim >= 2</code></li>
|
|
<li><code>N_updates >= 1</code></li>
|
|
<li><code>Updates</code> is allocated with \(N_updates \times Dim\) elements</li>
|
|
<li><code>Updates_index</code> is allocated with \(N_updates\) elements</li>
|
|
<li><code>breakdown</code> is a small number such that \(0 < breakdown << 1\)</li>
|
|
<li><code>Slater_inv</code> is allocated with \(Dim \times Dim\) elements</li>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgd807e82" class="outline-4">
|
|
<h4 id="orgd807e82"><span class="section-number-4">5.1.2</span> C header</h4>
|
|
<div class="outline-text-4" id="text-5-1-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-c"><span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_sherman_morrison_splitting</span> (
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">context</span> <span style="color: #a0522d;">qmckl_context</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">LDS</span> <span style="color: #a0522d;">uint64_t</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">Dim</span> <span style="color: #a0522d;">uint64_t</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">N_updates</span> <span style="color: #a0522d;">uint64_t</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">Updates</span>* <span style="color: #228b22;">double</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">Updates_index</span>* <span style="color: #a0522d;">uint64_t</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">breakdown</span> <span style="color: #228b22;">double</span>,
|
|
<span style="color: #228b22;">Slater_inv</span>* <span style="color: #228b22;">double</span>,
|
|
<span style="color: #228b22;">determinant</span>* <span style="color: #228b22;">double</span>* );
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org1944926" class="outline-4">
|
|
<h4 id="org1944926"><span class="section-number-4">5.1.3</span> C source</h4>
|
|
<div class="outline-text-4" id="text-5-1-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-c"><span style="color: #483d8b;">#include</span> <span style="color: #8b2252;"><stdbool.h></span>
|
|
<span style="color: #483d8b;">#include</span> <span style="color: #8b2252;">"qmckl.h"</span>
|
|
|
|
<span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_sherman_morrison_splitting</span>(<span style="color: #a020f0;">const</span> <span style="color: #228b22;">qmckl_context</span> <span style="color: #a0522d;">context</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">LDS</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">Dim</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">N_updates</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span>* <span style="color: #a0522d;">Updates</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">uint64_t</span>* <span style="color: #a0522d;">Updates_index</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> <span style="color: #a0522d;">breakdown</span>,
|
|
<span style="color: #228b22;">double</span>* <span style="color: #a0522d;">Slater_inv</span>,
|
|
<span style="color: #228b22;">double</span>* <span style="color: #a0522d;">determinant</span>) {
|
|
|
|
<span style="color: #a020f0;">if</span> (qmckl_context_check(context) == QMCKL_NULL_CONTEXT) {
|
|
<span style="color: #a020f0;">return</span> QMCKL_NULL_CONTEXT;
|
|
}
|
|
|
|
<span style="color: #228b22;">double</span> <span style="color: #a0522d;">later_updates</span>[Dim * N_updates];
|
|
<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">later_index</span>[N_updates];
|
|
<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">later</span> = 0;
|
|
|
|
(<span style="color: #228b22;">void</span>) qmckl_slagel_splitting(LDS, Dim, N_updates, Updates, Updates_index,
|
|
breakdown, Slater_inv, later_updates, later_index, &later, determinant);
|
|
|
|
<span style="color: #a020f0;">if</span> (later > 0) {
|
|
(<span style="color: #228b22;">void</span>) qmckl_sherman_morrison_splitting(context, LDS, Dim, later,
|
|
later_updates, later_index, breakdown, Slater_inv, determinant);
|
|
}
|
|
|
|
<span style="color: #a020f0;">return</span> QMCKL_SUCCESS;
|
|
}
|
|
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org18648d1" class="outline-4">
|
|
<h4 id="org18648d1"><span class="section-number-4">5.1.4</span> Performance…</h4>
|
|
<div class="outline-text-4" id="text-5-1-4">
|
|
<p>
|
|
This kernel performs best when there are 2 or more rank-1 update cycles and fail-rate is high.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org706b4f0" class="outline-2">
|
|
<h2 id="org706b4f0"><span class="section-number-2">6</span> Woodbury 3x3 and 2x2 with Sherman-Morrison and update splitting</h2>
|
|
<div class="outline-text-2" id="text-6">
|
|
</div>
|
|
<div id="outline-container-orgbd2e70a" class="outline-3">
|
|
<h3 id="orgbd2e70a"><span class="section-number-3">6.1</span> <code>qmckl_sherman_morrison_smw32s</code></h3>
|
|
<div class="outline-text-3" id="text-6-1">
|
|
<p>
|
|
The Woodbury 3x3 and 2x2 kernel with Sherman-Morrison and update splitting combines the low-level Woodbury 3x3 kernel,
|
|
the Woobury 2x2 kernel and Sherman-Morrison with update splitting. It works the almost the same as Woodbury 3x3 with
|
|
Sherman-Morrison and update splitting, except that when there is a remainder of two rank-1 updates, it is first tried
|
|
with Woodbury 2x2 instead of sending them all to Sherman-Morrison with update splitting. For example, in the case of
|
|
5 updates the updates are applied in 1 block of 3 updates end 1 block of 2 updates.
|
|
</p>
|
|
|
|
<p>
|
|
If the determinant of the Slater-matrix is passed, it will be updated to the determinant resulting
|
|
from applying the updates to the original matrix.
|
|
</p>
|
|
|
|
<table id="org5dd7c62" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<colgroup>
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
</colgroup>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left">qmckl<sub>context</sub></td>
|
|
<td class="org-left">context</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Global state</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">uint64<sub>t</sub></td>
|
|
<td class="org-left">LDS</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Leading dimension of Slater<sub>inv</sub></td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">uint64<sub>t</sub></td>
|
|
<td class="org-left">Dim</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Dimension of Slater<sub>inv</sub></td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">uint64<sub>t</sub></td>
|
|
<td class="org-left">N<sub>updates</sub></td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Number of rank-1 updates to be applied to Slater<sub>inv</sub></td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">double</td>
|
|
<td class="org-left">Updates[N<sub>updates</sub>*Dim]</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Array containing the updates</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">uint64<sub>t</sub></td>
|
|
<td class="org-left">Updates<sub>index</sub>[N<sub>updates</sub>]</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Array containing the rank-1 updates</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">double</td>
|
|
<td class="org-left">breakdown</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Break-down parameter on which to fail or not</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">double</td>
|
|
<td class="org-left">Slater<sub>inv</sub>[LDS*Dim]</td>
|
|
<td class="org-left">inout</td>
|
|
<td class="org-left">Array containing the inverse of a Slater-matrix</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">double*</td>
|
|
<td class="org-left">determinant</td>
|
|
<td class="org-left">inout</td>
|
|
<td class="org-left">Determinant of the Slater-matrix</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-org2472e42" class="outline-4">
|
|
<h4 id="org2472e42"><span class="section-number-4">6.1.1</span> Requirements</h4>
|
|
<div class="outline-text-4" id="text-6-1-1">
|
|
<ul class="org-ul">
|
|
<li><code>context</code> is not <code>QMCKL_NULL_CONTEXT</code></li>
|
|
<li><code>LDS >= 2</code></li>
|
|
<li><code>Dim >= 2</code></li>
|
|
<li><code>N_updates >= 1</code></li>
|
|
<li><code>Updates</code> is allocated with \(N_updates \times Dim\) elements</li>
|
|
<li><code>Updates_index</code> is allocated with \(N_updates\) elements</li>
|
|
<li><code>breakdown</code> is a small number such that \(0 < breakdown << 1\)</li>
|
|
<li><code>Slater_inv</code> is allocated with \(Dim \times Dim\) elements</li>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgacdda34" class="outline-4">
|
|
<h4 id="orgacdda34"><span class="section-number-4">6.1.2</span> C header</h4>
|
|
<div class="outline-text-4" id="text-6-1-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-c"><span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_sherman_morrison_smw32s</span> (
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">context</span> <span style="color: #a0522d;">qmckl_context</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">LDS</span> <span style="color: #a0522d;">uint64_t</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">Dim</span> <span style="color: #a0522d;">uint64_t</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">N_updates</span> <span style="color: #a0522d;">uint64_t</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">Updates</span>* <span style="color: #228b22;">double</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">Updates_index</span>* <span style="color: #a0522d;">uint64_t</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">breakdown</span> <span style="color: #228b22;">double</span>,
|
|
<span style="color: #228b22;">Slater_inv</span>* <span style="color: #228b22;">double</span>,
|
|
<span style="color: #228b22;">determinant</span>* <span style="color: #228b22;">double</span>* );
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgaf1d6d9" class="outline-4">
|
|
<h4 id="orgaf1d6d9"><span class="section-number-4">6.1.3</span> C source</h4>
|
|
<div class="outline-text-4" id="text-6-1-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-c"><span style="color: #483d8b;">#include</span> <span style="color: #8b2252;"><stdbool.h></span>
|
|
<span style="color: #483d8b;">#include</span> <span style="color: #8b2252;">"qmckl.h"</span>
|
|
|
|
<span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_sherman_morrison_smw32s</span>(<span style="color: #a020f0;">const</span> <span style="color: #228b22;">qmckl_context</span> <span style="color: #a0522d;">context</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">LDS</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">Dim</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">N_updates</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span>* <span style="color: #a0522d;">Updates</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">uint64_t</span>* <span style="color: #a0522d;">Updates_index</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> <span style="color: #a0522d;">breakdown</span>,
|
|
<span style="color: #228b22;">double</span>* <span style="color: #a0522d;">Slater_inv</span>,
|
|
<span style="color: #228b22;">double</span>* <span style="color: #a0522d;">determinant</span>) {
|
|
|
|
<span style="color: #a020f0;">if</span> (qmckl_context_check(context) == QMCKL_NULL_CONTEXT) {
|
|
<span style="color: #a020f0;">return</span> QMCKL_NULL_CONTEXT;
|
|
}
|
|
|
|
<span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #a0522d;">rc</span>;
|
|
|
|
<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">n_of_3blocks</span> = N_updates / 3;
|
|
<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">remainder</span> = N_updates % 3;
|
|
<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">length_3block</span> = 3 * Dim;
|
|
|
|
// <span style="color: #b22222;">Apply first 3*n_of_3blocks updates in n_of_3blocks blocks of 3 updates with</span>
|
|
// <span style="color: #b22222;">Woodbury 3x3 kernel</span>
|
|
<span style="color: #228b22;">double</span> <span style="color: #a0522d;">later_updates</span>[Dim * N_updates];
|
|
<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">later_index</span>[N_updates];
|
|
<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">later</span> = 0;
|
|
<span style="color: #a020f0;">if</span> (n_of_3blocks > 0) {
|
|
<span style="color: #a020f0;">for</span> (<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">i</span> = 0; i < n_of_3blocks; i++) {
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> *<span style="color: #a0522d;">Updates_3block</span> = &Updates[i * length_3block];
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">uint64_t</span> *<span style="color: #a0522d;">Updates_index_3block</span> = &Updates_index[i * 3];
|
|
rc = qmckl_woodbury_3(context, LDS, Dim, Updates_3block, Updates_index_3block, breakdown, Slater_inv, determinant);
|
|
<span style="color: #a020f0;">if</span> (rc != 0) { // <span style="color: #b22222;">Send the entire block to slagel_splitting</span>
|
|
<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">l</span> = 0;
|
|
(<span style="color: #228b22;">void</span>) qmckl_slagel_splitting(LDS, Dim, 3, Updates_3block, Updates_index_3block,
|
|
breakdown, Slater_inv, later_updates + (Dim * later), later_index + later, &l, determinant);
|
|
later = later + l;
|
|
}
|
|
}
|
|
}
|
|
|
|
// <span style="color: #b22222;">Apply last remaining block of 2 updates with Woodbury 2x2 kernel</span>
|
|
<span style="color: #a020f0;">if</span> (remainder == 2) {
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> *<span style="color: #a0522d;">Updates_2block</span> = &Updates[n_of_3blocks * length_3block];
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">uint64_t</span> *<span style="color: #a0522d;">Updates_index_2block</span> = &Updates_index[3 * n_of_3blocks];
|
|
rc = qmckl_woodbury_2(context, LDS, Dim, Updates_2block, Updates_index_2block, breakdown, Slater_inv, determinant);
|
|
<span style="color: #a020f0;">if</span> (rc != 0) { // <span style="color: #b22222;">Send the entire block to slagel_splitting</span>
|
|
<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">l</span> = 0;
|
|
(<span style="color: #228b22;">void</span>) qmckl_slagel_splitting(LDS, Dim, 2, Updates_2block, Updates_index_2block,
|
|
breakdown, Slater_inv, later_updates + (Dim * later), later_index + later, &l, determinant);
|
|
later = later + l;
|
|
}
|
|
}
|
|
// <span style="color: #b22222;">Apply last remaining update with slagel_splitting</span>
|
|
<span style="color: #a020f0;">else</span> <span style="color: #a020f0;">if</span> (remainder == 1) {
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> *<span style="color: #a0522d;">Updates_1block</span> = &Updates[n_of_3blocks * length_3block];
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">uint64_t</span> *<span style="color: #a0522d;">Updates_index_1block</span> = &Updates_index[3 * n_of_3blocks];
|
|
<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">l</span> = 0;
|
|
(<span style="color: #228b22;">void</span>) qmckl_slagel_splitting(LDS, Dim, 1, Updates_1block, Updates_index_1block,
|
|
breakdown, Slater_inv, later_updates + (Dim * later), later_index + later, &l, determinant);
|
|
later = later + l;
|
|
}
|
|
|
|
<span style="color: #a020f0;">if</span> (later > 0) {
|
|
(<span style="color: #228b22;">void</span>) qmckl_sherman_morrison_splitting(context, LDS, Dim, later, later_updates, later_index, breakdown, Slater_inv, determinant);
|
|
}
|
|
<span style="color: #a020f0;">return</span> QMCKL_SUCCESS;
|
|
}
|
|
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org86d6360" class="outline-4">
|
|
<h4 id="org86d6360"><span class="section-number-4">6.1.4</span> Performance…</h4>
|
|
<div class="outline-text-4" id="text-6-1-4">
|
|
<p>
|
|
This kernel performs best for update cycles with 2 or more rank-1 updates and the fail-rate is low.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org9577fed" class="outline-2">
|
|
<h2 id="org9577fed"><span class="section-number-2">7</span> Helper Functions</h2>
|
|
<div class="outline-text-2" id="text-7">
|
|
<p>
|
|
Private helper-functions that are used by the Sherman-Morrison-Woodbury kernels.
|
|
These functions can only be used internally by the kernels in this module.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-orgb3d1ef9" class="outline-3">
|
|
<h3 id="orgb3d1ef9"><span class="section-number-3">7.1</span> <code>qmckl_slagel_splitting</code></h3>
|
|
<div class="outline-text-3" id="text-7-1">
|
|
<p>
|
|
<code>qmckl_slagel_splitting</code> is the non-recursive, inner part of the 'Sherman-Morrison with update splitting'-kernel.
|
|
It is used internally to apply a collection of \(N\) rank-1 updates to the inverse Slater-matrix \(S^{-1}\) and
|
|
splitting an update in two equal pieces if necessary. In case of a split, it applies the first half of the update,
|
|
while putting the second half in a waiting queue to be applied at the end.
|
|
</p>
|
|
|
|
<p>
|
|
Therefore, when \(1+v_j^TS^{-1}u_j \geq \epsilon\), the update is applied as usual. Otherwise, \(u_j\) will be redefined
|
|
as \(\frac{1}{2}u_j\). One half is applied immediately, the other half will be applied at the end of the algorithm, using vectors
|
|
\(u_{j'}=\frac{1}{2}u_j\) and \(v_{j'}^T=v_{j}^T\), which are stored in the array \texttt{later_updates}.
|
|
</p>
|
|
|
|
<p>
|
|
If the determinant of the Slater-matrix is passed, it will be updated to the determinant resulting
|
|
from applying the updates to the original matrix.
|
|
</p>
|
|
|
|
<table id="org3269f2f" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<colgroup>
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
</colgroup>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left">uint64<sub>t</sub></td>
|
|
<td class="org-left">LDS</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Leading dimension of Slater<sub>inv</sub></td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">uint64<sub>t</sub></td>
|
|
<td class="org-left">Dim</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Dimension of Slater<sub>inv</sub></td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">uint64<sub>t</sub></td>
|
|
<td class="org-left">N<sub>updates</sub></td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Number of rank-1 updates to be applied to Slater<sub>inv</sub></td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">double</td>
|
|
<td class="org-left">Updates[N<sub>updates</sub>*Dim]</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Array containing the rank-1 updates</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">uint64<sub>t</sub></td>
|
|
<td class="org-left">Updates<sub>index</sub>[N<sub>updates</sub>]</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Array containing positions of the rank-1 updates</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">double</td>
|
|
<td class="org-left">breakdown</td>
|
|
<td class="org-left">in</td>
|
|
<td class="org-left">Break-down parameter on which to fail or not</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">double</td>
|
|
<td class="org-left">Slater<sub>inv</sub>[LDS*Dim]</td>
|
|
<td class="org-left">inout</td>
|
|
<td class="org-left">Array containing the inverse Slater-matrix</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">double</td>
|
|
<td class="org-left">later<sub>updates</sub>[Dim * N<sub>updates</sub>]</td>
|
|
<td class="org-left">inout</td>
|
|
<td class="org-left">Array containing the split updates for later</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">uint64<sub>t</sub></td>
|
|
<td class="org-left">later<sub>index</sub>[N<sub>updates</sub>]</td>
|
|
<td class="org-left">inout</td>
|
|
<td class="org-left">Array containing the positions of the split updates for later</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">uint64<sub>t</sub></td>
|
|
<td class="org-left">later</td>
|
|
<td class="org-left">inout</td>
|
|
<td class="org-left">Number of split updates for later</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">double*</td>
|
|
<td class="org-left">determinant</td>
|
|
<td class="org-left">inout</td>
|
|
<td class="org-left">Determinant of the Slater-matrix</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-org37877c7" class="outline-4">
|
|
<h4 id="org37877c7"><span class="section-number-4">7.1.1</span> Requirements</h4>
|
|
<div class="outline-text-4" id="text-7-1-1">
|
|
<ul class="org-ul">
|
|
<li><code>LDS >= 2</code></li>
|
|
<li><code>Dim >= 2</code></li>
|
|
<li><code>N_updates >= 1</code></li>
|
|
<li><code>Updates</code> is allocated with \(N_updates \times Dim\) elements</li>
|
|
<li><code>Updates_index</code> is allocated with \(N_updates\) elements</li>
|
|
<li><code>breakdown</code> is a small number such that \(0 < breakdown << 1\)</li>
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<li><code>Slater_inv</code> is allocated with \(Dim \times Dim\) elements</li>
|
|
<li><code>later_updates</code> is allocated with \(later \times Dim\) elements</li>
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|
<li><code>later_index</code> is allocated with \(N_updates\) elements</li>
|
|
<li><code>later >= 0</code></li>
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|
</ul>
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|
</div>
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|
</div>
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|
|
|
<div id="outline-container-orgcc799b1" class="outline-4">
|
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<h4 id="orgcc799b1"><span class="section-number-4">7.1.2</span> C header</h4>
|
|
<div class="outline-text-4" id="text-7-1-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-c"><span style="color: #228b22;">double</span> <span style="color: #0000ff;">qmckl_slagel_splitting</span> (
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">LDS</span> <span style="color: #a0522d;">uint64_t</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">Dim</span> <span style="color: #a0522d;">uint64_t</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">N_updates</span> <span style="color: #a0522d;">uint64_t</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">Updates</span>* <span style="color: #228b22;">double</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">Updates_index</span>* <span style="color: #a0522d;">uint64_t</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">breakdown</span> <span style="color: #228b22;">double</span>,
|
|
<span style="color: #228b22;">Slater_inv</span>* <span style="color: #228b22;">double</span>,
|
|
<span style="color: #228b22;">later_updates</span>* <span style="color: #228b22;">double</span>,
|
|
<span style="color: #228b22;">later_index</span>* <span style="color: #a0522d;">uint64_t</span>,
|
|
<span style="color: #228b22;">later</span>* <span style="color: #a0522d;">uint64_t</span>,
|
|
<span style="color: #228b22;">determinant</span>* <span style="color: #228b22;">double</span>* );
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orge0463aa" class="outline-4">
|
|
<h4 id="orge0463aa"><span class="section-number-4">7.1.3</span> C source</h4>
|
|
<div class="outline-text-4" id="text-7-1-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-c"><span style="color: #483d8b;">#include</span> <span style="color: #8b2252;"><stdbool.h></span>
|
|
<span style="color: #483d8b;">#include</span> <span style="color: #8b2252;"><math.h></span>
|
|
<span style="color: #483d8b;">#include</span> <span style="color: #8b2252;">"qmckl.h"</span>
|
|
|
|
<span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_slagel_splitting</span>(<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">LDS</span>,
|
|
<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">Dim</span>,
|
|
<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">N_updates</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> *<span style="color: #a0522d;">Updates</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">uint64_t</span> *<span style="color: #a0522d;">Updates_index</span>,
|
|
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> <span style="color: #a0522d;">breakdown</span>,
|
|
<span style="color: #228b22;">double</span> *<span style="color: #a0522d;">Slater_inv</span>,
|
|
<span style="color: #228b22;">double</span> *<span style="color: #a0522d;">later_updates</span>,
|
|
<span style="color: #228b22;">uint64_t</span> *<span style="color: #a0522d;">later_index</span>,
|
|
<span style="color: #228b22;">uint64_t</span> *<span style="color: #a0522d;">later</span>,
|
|
<span style="color: #228b22;">double</span> *<span style="color: #a0522d;">determinant</span>) {
|
|
// <span style="color: #b22222;">#ifdef DEBUG // Leave commented out since debugging information is not yet implemented in QMCkl.</span>
|
|
// <span style="color: #b22222;">std::cerr << "Called slagel_splitting with " << N_updates << " updates" << std::endl;</span>
|
|
// <span style="color: #b22222;">#endif</span>
|
|
|
|
<span style="color: #228b22;">double</span> <span style="color: #a0522d;">C</span>[Dim];
|
|
<span style="color: #228b22;">double</span> <span style="color: #a0522d;">D</span>[Dim];
|
|
|
|
<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">l</span> = 0;
|
|
// <span style="color: #b22222;">For each update</span>
|
|
<span style="color: #a020f0;">while</span> (l < N_updates) {
|
|
// <span style="color: #b22222;">C = S^{-1} x U_l</span>
|
|
<span style="color: #a020f0;">for</span> (<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">i</span> = 0; i < Dim; i++) {
|
|
C[i] = 0;
|
|
<span style="color: #a020f0;">for</span> (<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">j</span> = 0; j < Dim; j++) {
|
|
C[i] += Slater_inv[i * LDS + j] * Updates[l * Dim + j];
|
|
}
|
|
}
|
|
|
|
// <span style="color: #b22222;">Denominator</span>
|
|
<span style="color: #228b22;">double</span> <span style="color: #a0522d;">den</span> = 1 + C[Updates_index[l] - 1];
|
|
<span style="color: #a020f0;">if</span> (fabs(den) < breakdown) { // <span style="color: #b22222;">Here is decided to split the update, or not.</span>
|
|
|
|
// <span style="color: #b22222;">U_l = U_l / 2: split the update in 2 equal halves and save the second halve in later_updates</span>
|
|
<span style="color: #a020f0;">for</span> (<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">i</span> = 0; i < Dim; i++) {
|
|
later_updates[*later * Dim + i] = Updates[l * Dim + i] / 2.0;
|
|
C[i] /= 2.0;
|
|
}
|
|
later_index[*later] = Updates_index[l];
|
|
(*later)++;
|
|
|
|
den = 1 + C[Updates_index[l] - 1];
|
|
} // <span style="color: #b22222;">From here onwards we continue with applying the first havel of the update to Slater_inv</span>
|
|
<span style="color: #228b22;">double</span> <span style="color: #a0522d;">iden</span> = 1 / den;
|
|
|
|
<span style="color: #a020f0;">if</span> (determinant != <span style="color: #008b8b;">NULL</span>)
|
|
*determinant *= den;
|
|
|
|
// <span style="color: #b22222;">D = v^T x S^{-1}</span>
|
|
<span style="color: #a020f0;">for</span> (<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">j</span> = 0; j < Dim; j++) {
|
|
D[j] = Slater_inv[(Updates_index[l] - 1) * LDS + j];
|
|
}
|
|
|
|
// <span style="color: #b22222;">S^{-1} = S^{-1} - C x D / den</span>
|
|
<span style="color: #a020f0;">for</span> (<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">i</span> = 0; i < Dim; i++) {
|
|
<span style="color: #a020f0;">for</span> (<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">j</span> = 0; j < Dim; j++) {
|
|
<span style="color: #228b22;">double</span> <span style="color: #a0522d;">update</span> = C[i] * D[j] * iden;
|
|
Slater_inv[i * LDS + j] -= update;
|
|
}
|
|
}
|
|
l += 1;
|
|
}
|
|
|
|
<span style="color: #a020f0;">return</span> QMCKL_SUCCESS;
|
|
}
|
|
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org3048b08" class="outline-4">
|
|
<h4 id="org3048b08"><span class="section-number-4">7.1.4</span> Performance</h4>
|
|
<div class="outline-text-4" id="text-7-1-4">
|
|
<p>
|
|
This function cannot be used by itself and is used in Sherman-Morrison with update splitting and Woodbury 3x3 and 2x2
|
|
with Sherman-Morrison and update splitting. Please look at the performance reccomendations for those two kernels.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org97008df" class="outline-2">
|
|
<h2 id="org97008df"><span class="section-number-2">8</span> End of files</h2>
|
|
<div class="outline-text-2" id="text-8">
|
|
<div class="org-src-container">
|
|
<pre class="src src-c"> <span style="color: #0000ff;">assert</span> (<span style="color: #228b22;">qmckl_context_destroy</span>(<span style="color: #a0522d;">context</span>) == QMCKL_SUCCESS);
|
|
<span style="color: #a020f0;">return</span> 0;
|
|
|
|
}
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: TREX CoE</p>
|
|
<p class="date">Created: 2022-10-13 Thu 09:55</p>
|
|
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
|
</div>
|
|
</body>
|
|
</html>
|