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1687 lines
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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="#org07e024f">1. Headers</a></li>
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<li><a href="#orge0fb915">2. Naïve Sherman-Morrison</a>
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<ul>
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<li><a href="#org49ac43b">2.1. <code>qmckl_sherman_morrison</code></a>
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<ul>
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<li><a href="#org5388d8e">2.1.1. Requirements</a></li>
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<li><a href="#orgc73fcc8">2.1.2. C header</a></li>
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<li><a href="#org9320046">2.1.3. C source</a></li>
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<li><a href="#orgce4160e">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="#org19f3223">3. Woodbury 2x2</a>
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<ul>
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<li><a href="#org611a883">3.1. <code>qmckl_woodbury_2</code></a>
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<ul>
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<li><a href="#org2e80e84">3.1.1. Requirements</a></li>
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<li><a href="#orgdfcc161">3.1.2. C header</a></li>
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<li><a href="#orgd7bbc1e">3.1.3. C source</a></li>
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<li><a href="#org7c85d41">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="#orga8ec6eb">4. Woodbury 3x3</a>
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<ul>
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<li><a href="#orge943028">4.1. <code>qmckl_woodbury_3</code></a>
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<ul>
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<li><a href="#orge4ab892">4.1.1. Requirements</a></li>
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<li><a href="#org0223aa7">4.1.2. C header</a></li>
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<li><a href="#org79505b6">4.1.3. C source</a></li>
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<li><a href="#org259a5d0">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="#orgef7c5d7">5. Sherman-Morrison with update splitting</a>
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<ul>
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<li><a href="#orgb7f6c99">5.1. <code>qmckl_sherman_morrison_splitting</code></a>
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<ul>
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<li><a href="#org261772d">5.1.1. Requirements</a></li>
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<li><a href="#org1909147">5.1.2. C header</a></li>
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<li><a href="#org682aaaa">5.1.3. C source</a></li>
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<li><a href="#orgf97483d">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="#org018790f">6. Woodbury 3x3 and 2x2 with Sherman-Morrison and update splitting</a>
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<ul>
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<li><a href="#org16acd3b">6.1. <code>qmckl_sherman_morrison_smw32s</code></a>
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<ul>
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<li><a href="#org48302eb">6.1.1. Requirements</a></li>
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<li><a href="#org54b7c4d">6.1.2. C header</a></li>
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<li><a href="#orga0c421a">6.1.3. C source</a></li>
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<li><a href="#orgf0f8d41">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="#org8c53f21">7. Helper Functions</a>
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<ul>
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<li><a href="#org1424031">7.1. <code>qmckl_slagel_splitting</code></a>
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<ul>
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<li><a href="#org1a923af">7.1.1. Requirements</a></li>
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<li><a href="#orgb5966b9">7.1.2. C header</a></li>
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<li><a href="#org9bbcc2b">7.1.3. C source</a></li>
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<li><a href="#orgf83c92e">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="#org1383df4">8. End of files</a></li>
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|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org07e024f" class="outline-2">
|
|
<h2 id="org07e024f"><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-orge0fb915" class="outline-2">
|
|
<h2 id="orge0fb915"><span class="section-number-2">2</span> Naïve Sherman-Morrison</h2>
|
|
<div class="outline-text-2" id="text-2">
|
|
</div>
|
|
<div id="outline-container-org49ac43b" class="outline-3">
|
|
<h3 id="org49ac43b"><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>
|
|
|
|
<table id="org6520a06" 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">Dim</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">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>[Dim*Dim]</td>
|
|
<td class="org-left">inout</td>
|
|
<td class="org-left">Array containing the inverse of a Slater-matrix</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
|
|
<div id="outline-container-org5388d8e" class="outline-4">
|
|
<h4 id="org5388d8e"><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>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-orgc73fcc8" class="outline-4">
|
|
<h4 id="orgc73fcc8"><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;">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;">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> );
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org9320046" class="outline-4">
|
|
<h4 id="org9320046"><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;">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: #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 * Dim + 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;">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) * Dim + 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 * Dim + j] -= update;
|
|
}
|
|
}
|
|
|
|
l += 1;
|
|
}
|
|
|
|
<span style="color: #a020f0;">return</span> QMCKL_SUCCESS;
|
|
}
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgce4160e" class="outline-4">
|
|
<h4 id="orgce4160e"><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-org19f3223" class="outline-2">
|
|
<h2 id="org19f3223"><span class="section-number-2">3</span> Woodbury 2x2</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
</div>
|
|
<div id="outline-container-org611a883" class="outline-3">
|
|
<h3 id="org611a883"><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>
|
|
|
|
|
|
<table id="org5890d2f" 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">Dim</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">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>[Dim*Dim]</td>
|
|
<td class="org-left">inout</td>
|
|
<td class="org-left">Array containing the inverse of a Slater-matrix</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
|
|
<div id="outline-container-org2e80e84" class="outline-4">
|
|
<h4 id="org2e80e84"><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>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-orgdfcc161" class="outline-4">
|
|
<h4 id="orgdfcc161"><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;">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;">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> );
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgd7bbc1e" class="outline-4">
|
|
<h4 id="orgd7bbc1e"><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;">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: #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 * Dim + 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;">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 * Dim + j];
|
|
tmp[i * Dim + j] += Binv[i * 2 + 1] * Slater_inv[row2 * Dim + 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 * Dim + j] -= C[i * 2] * tmp[j];
|
|
Slater_inv[i * Dim + j] -= C[i * 2 + 1] * tmp[Dim + j];
|
|
}
|
|
}
|
|
|
|
<span style="color: #a020f0;">return</span> QMCKL_SUCCESS;
|
|
}
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org7c85d41" class="outline-4">
|
|
<h4 id="org7c85d41"><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-orga8ec6eb" class="outline-2">
|
|
<h2 id="orga8ec6eb"><span class="section-number-2">4</span> Woodbury 3x3</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
</div>
|
|
<div id="outline-container-orge943028" class="outline-3">
|
|
<h3 id="orge943028"><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>
|
|
|
|
<table id="org798f613" 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">Dim</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">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>[Dim*Dim]</td>
|
|
<td class="org-left">inout</td>
|
|
<td class="org-left">Array containing the inverse of a Slater-matrix</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
|
|
<div id="outline-container-orge4ab892" class="outline-4">
|
|
<h4 id="orge4ab892"><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>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-org0223aa7" class="outline-4">
|
|
<h4 id="org0223aa7"><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;">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;">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> );
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org79505b6" class="outline-4">
|
|
<h4 id="org79505b6"><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;">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: #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 * Dim + 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;">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 * Dim + j];
|
|
tmp[i * Dim + j] += Binv[i * 3 + 1] * Slater_inv[row2 * Dim + j];
|
|
tmp[i * Dim + j] += Binv[i * 3 + 2] * Slater_inv[row3 * Dim + 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 * Dim + j] -= C[i * 3] * tmp[j];
|
|
Slater_inv[i * Dim + j] -= C[i * 3 + 1] * tmp[Dim + j];
|
|
Slater_inv[i * Dim + 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-org259a5d0" class="outline-4">
|
|
<h4 id="org259a5d0"><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-orgef7c5d7" class="outline-2">
|
|
<h2 id="orgef7c5d7"><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-orgb7f6c99" class="outline-3">
|
|
<h3 id="orgb7f6c99"><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>
|
|
|
|
<table id="orgcf10357" 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">Dim</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">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>[Dim*Dim]</td>
|
|
<td class="org-left">inout</td>
|
|
<td class="org-left">Array containing the inverse of a Slater-matrix</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
|
|
<div id="outline-container-org261772d" class="outline-4">
|
|
<h4 id="org261772d"><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>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-org1909147" class="outline-4">
|
|
<h4 id="org1909147"><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;">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;">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> );
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org682aaaa" class="outline-4">
|
|
<h4 id="org682aaaa"><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;">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: #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(Dim, N_updates, Updates, Updates_index,
|
|
breakdown, Slater_inv, later_updates, later_index, &later);
|
|
|
|
<span style="color: #a020f0;">if</span> (later > 0) {
|
|
(<span style="color: #228b22;">void</span>) qmckl_sherman_morrison_splitting(context, Dim, later,
|
|
later_updates, later_index, breakdown, Slater_inv);
|
|
}
|
|
|
|
<span style="color: #a020f0;">return</span> QMCKL_SUCCESS;
|
|
}
|
|
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgf97483d" class="outline-4">
|
|
<h4 id="orgf97483d"><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-org018790f" class="outline-2">
|
|
<h2 id="org018790f"><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-org16acd3b" class="outline-3">
|
|
<h3 id="org16acd3b"><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>
|
|
|
|
<table id="org36d3011" 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">Dim</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">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>[Dim*Dim]</td>
|
|
<td class="org-left">inout</td>
|
|
<td class="org-left">Array containing the inverse of a Slater-matrix</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
|
|
<div id="outline-container-org48302eb" class="outline-4">
|
|
<h4 id="org48302eb"><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>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-org54b7c4d" class="outline-4">
|
|
<h4 id="org54b7c4d"><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;">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;">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> );
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orga0c421a" class="outline-4">
|
|
<h4 id="orga0c421a"><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;">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: #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, Dim, Updates_3block, Updates_index_3block, breakdown, Slater_inv);
|
|
<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;
|
|
rc = qmckl_slagel_splitting(Dim, 3, Updates_3block, Updates_index_3block,
|
|
breakdown, Slater_inv, later_updates + (Dim * later), later_index + later, &l);
|
|
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, Dim, Updates_2block, Updates_index_2block, breakdown, Slater_inv);
|
|
<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(Dim, 2, Updates_2block, Updates_index_2block,
|
|
breakdown, Slater_inv, later_updates + (Dim * later), later_index + later, &l);
|
|
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(Dim, 1, Updates_1block, Updates_index_1block,
|
|
breakdown, Slater_inv, later_updates + (Dim * later), later_index + later, &l);
|
|
later = later + l;
|
|
}
|
|
|
|
<span style="color: #a020f0;">if</span> (later > 0) {
|
|
(<span style="color: #228b22;">void</span>) qmckl_sherman_morrison_splitting(context, Dim, later, later_updates, later_index, breakdown, Slater_inv);
|
|
}
|
|
<span style="color: #a020f0;">return</span> QMCKL_SUCCESS;
|
|
}
|
|
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgf0f8d41" class="outline-4">
|
|
<h4 id="orgf0f8d41"><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-org8c53f21" class="outline-2">
|
|
<h2 id="org8c53f21"><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-org1424031" class="outline-3">
|
|
<h3 id="org1424031"><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>
|
|
|
|
<table id="orgc70c7a0" 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">Dim</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">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>[Dim*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>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-org1a923af" class="outline-4">
|
|
<h4 id="org1a923af"><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>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>
|
|
<li><code>later_updates</code> is allocated with \(later \times Dim\) elements</li>
|
|
<li><code>later_index</code> is allocated with \(N_updates\) elements</li>
|
|
<li><code>later >= 0</code></li>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgb5966b9" class="outline-4">
|
|
<h4 id="orgb5966b9"><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;">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;">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> );
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org9bbcc2b" class="outline-4">
|
|
<h4 id="org9bbcc2b"><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;">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: #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 * Dim + 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: #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) * Dim + 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 * Dim + j] -= update;
|
|
}
|
|
}
|
|
l += 1;
|
|
}
|
|
|
|
<span style="color: #a020f0;">return</span> QMCKL_SUCCESS;
|
|
}
|
|
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgf83c92e" class="outline-4">
|
|
<h4 id="orgf83c92e"><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-org1383df4" class="outline-2">
|
|
<h2 id="org1383df4"><span class="section-number-2">8</span> End of files</h2>
|
|
<div class="outline-text-2" id="text-8">
|
|
<div class="org-src-container">
|
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<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);
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<span style="color: #a020f0;">return</span> 0;
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}
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</pre>
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</div>
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</div>
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</div>
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: TREX CoE</p>
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<p class="date">Created: 2021-10-11 Mon 22:18</p>
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<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
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</div>
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</body>
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</html>
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