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{"documenter":{"julia_version":"1.11.2","generation_timestamp":"2025-01-03T14:38:17","documenter_version":"1.8.0"}}
{"documenter":{"julia_version":"1.11.2","generation_timestamp":"2025-01-04T03:00:16","documenter_version":"1.8.0"}}
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<!DOCTYPE html>
<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Home · CGcoefficient.jl</title><meta name="title" content="Home · CGcoefficient.jl"/><meta property="og:title" content="Home · CGcoefficient.jl"/><meta property="twitter:title" content="Home · CGcoefficient.jl"/><meta name="description" content="Documentation for CGcoefficient.jl."/><meta property="og:description" content="Documentation for CGcoefficient.jl."/><meta property="twitter:description" content="Documentation for CGcoefficient.jl."/><script data-outdated-warner src="assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" 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type="text/css" href="assets/themes/catppuccin-frappe.css" data-theme-name="catppuccin-frappe"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/catppuccin-latte.css" data-theme-name="catppuccin-latte"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href><img src="assets/logo.svg" alt="CGcoefficient.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit"><a href>CGcoefficient.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li class="is-active"><a class="tocitem" href>Home</a><ul class="internal"><li><a class="tocitem" href="#Install"><span>Install</span></a></li><li><a class="tocitem" href="#Usage"><span>Usage</span></a></li><li><a class="tocitem" href="#About"><span>About</span></a></li></ul></li><li><a class="tocitem" href="api/">API</a></li><li><span class="tocitem">Formula</span><ul><li><a class="tocitem" href="wigner/">Wigner Symbols</a></li><li><a class="tocitem" href="lsjj/">LS-jj recoupling</a></li><li><a class="tocitem" href="moshinsky/">Moshinsky</a></li><li><a class="tocitem" href="pfrational/">Prime Factorization</a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Home</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Home</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/0382/CGcoefficient.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/0382/CGcoefficient.jl/blob/master/docs/src/index.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Home"><a class="docs-heading-anchor" href="#Home">Home</a><a id="Home-1"></a><a class="docs-heading-anchor-permalink" href="#Home" title="Permalink"></a></h1><p>A package to calculate CG-coefficient, Racha coefficient, and Wigner 3j, 6j, 9j symbols. We offer two versions of API for all these coefficients.</p><ol><li>The exact functions return <code>SqrtRational</code>, which are designed for demonstration. They use <code>BigInt</code> in the internal calculation, and do not cache the binomial table, so they are not efficient.</li><li>The floating-point functions return <code>Float64</code>, which are designed for numeric calculation. They use <code>Int, Float64</code> in the internal calculation, and you should pre-call <a href="api/#CGcoefficient.wigner_init_float"><code>wigner_init_float</code></a> to calculate and cache the binomial table for later calculation. They may give inaccurate result for vary large angular momentum, due to floating-point arithmetic. You can find the max error at here: <a href="https://github.com/0382/wigner-benchmark">wigner-benchmark</a>.</li></ol><h2 id="Install"><a class="docs-heading-anchor" href="#Install">Install</a><a id="Install-1"></a><a class="docs-heading-anchor-permalink" href="#Install" title="Permalink"></a></h2><p>Just install with Julia REPL and enjoy it.</p><pre><code class="language-julia-repl hljs">pkg&gt; add CGcoefficient</code></pre><h2 id="Usage"><a class="docs-heading-anchor" href="#Usage">Usage</a><a id="Usage-1"></a><a class="docs-heading-anchor-permalink" href="#Usage" title="Permalink"></a></h2><pre><code class="language-julia hljs">using CGcoefficient
sixJ(1,2,3,4,5,6)</code></pre><p class="math-container">\[\frac{1}{3}\sqrt{\frac{2}{715}}\]</p><p>In a markdown enviroment, such as jupyter notebook, it will give you a latex output. You can also do some arithmetics with the result, thus do arithmetics using the <code>SqrtRational</code> type.</p><pre><code class="language-julia hljs">x = sixJ(1,2,3,4,5,6) * exact_sqrt(1//7) * exact_sqrt(1//13) * iphase(2+3+5+6)</code></pre><p class="math-container">\[\frac{1}{3}\sqrt{\frac{2}{65065}}\]</p><p>The result is not simplified by default, you can use <code>simplify</code> function to simplify it.</p><pre><code class="language-julia hljs">simplify(x)</code></pre><p class="math-container">\[\frac{1}{39}\sqrt{\frac{2}{385}}\]</p><p>In a console enviroment it will give out a text output.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; nineJ(1,2,3,5,4,3,6,6,0)</code><code class="nohighlight hljs ansi" style="display:block;">1//39√(2//385)</code></pre><p>You can convert a <code>SqrtRational</code> in to <code>BigFloat</code>,</p><pre><code class="language-julia hljs">t = sixJ(17/2,9,19/2,15/2,10,15/2)
float(t)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">0.006625060239168051702701949513657948806045797449799216745233646948167908577160755</code></pre><p>Calling <code>wigner_init_float</code> first to pre-calculate and cache binomial table.</p><pre><code class="language-julia hljs">wigner_init_float(10, &quot;Jmax&quot;, 6)</code></pre><p>Then call a float version function</p><pre><code class="language-julia hljs">f6j(17,18,19,15,20,15)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">0.006625060239168052</code></pre><h2 id="About"><a class="docs-heading-anchor" href="#About">About</a><a id="About-1"></a><a class="docs-heading-anchor-permalink" href="#About" title="Permalink"></a></h2><p>This package is inspired by Ref <sup class="footnote-reference"><a id="citeref-1" href="#footnote-1">[1]</a></sup>. See <a href="https://github.com/ManyBodyPhysics/CENS/blob/master/MBPT/VEffective/bhf-modules.f90">CENS-MBPT</a> for details.</p><p>The idea is to simplify 3nj Symbols to sum combinations of binomial coefficients. We can calculate binomial coefficients by Pascal&#39;s Triangle, and store them first. Then we calculate 3nj Symbols using the stored binomial coefficients.</p><p>In this package, we just use the builtin <code>binomial</code> function for exact calculation. Only the float version uses stored <code>binomial</code>s.</p><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a>T. Engeland and M. Hjorth-Jensen, the Oslo-FCI code. <a href="https://github.com/ManyBodyPhysics/CENS">https://github.com/ManyBodyPhysics/CENS</a>.</li></ul></section></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="api/">API »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Friday 3 January 2025 14:38">Friday 3 January 2025</span>. Using Julia version 1.11.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
float(t)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">0.006625060239168051702701949513657948806045797449799216745233646948167908577160755</code></pre><p>Calling <code>wigner_init_float</code> first to pre-calculate and cache binomial table.</p><pre><code class="language-julia hljs">wigner_init_float(10, &quot;Jmax&quot;, 6)</code></pre><p>Then call a float version function</p><pre><code class="language-julia hljs">f6j(17,18,19,15,20,15)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">0.006625060239168052</code></pre><h2 id="About"><a class="docs-heading-anchor" href="#About">About</a><a id="About-1"></a><a class="docs-heading-anchor-permalink" href="#About" title="Permalink"></a></h2><p>This package is inspired by Ref <sup class="footnote-reference"><a id="citeref-1" href="#footnote-1">[1]</a></sup>. See <a href="https://github.com/ManyBodyPhysics/CENS/blob/master/MBPT/VEffective/bhf-modules.f90">CENS-MBPT</a> for details.</p><p>The idea is to simplify 3nj Symbols to sum combinations of binomial coefficients. We can calculate binomial coefficients by Pascal&#39;s Triangle, and store them first. Then we calculate 3nj Symbols using the stored binomial coefficients.</p><p>In this package, we just use the builtin <code>binomial</code> function for exact calculation. Only the float version uses stored <code>binomial</code>s.</p><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a>T. Engeland and M. Hjorth-Jensen, the Oslo-FCI code. <a href="https://github.com/ManyBodyPhysics/CENS">https://github.com/ManyBodyPhysics/CENS</a>.</li></ul></section></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="api/">API »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Saturday 4 January 2025 03:00">Saturday 4 January 2025</span>. Using Julia version 1.11.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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