File: C:/Ruby27-x64/share/doc/ruby/html/Jacobian.html
<!DOCTYPE html>
<html>
<head>
<meta charset="UTF-8">
<title>module Jacobian - RDoc Documentation</title>
<script type="text/javascript">
var rdoc_rel_prefix = "./";
var index_rel_prefix = "./";
</script>
<script src="./js/navigation.js" defer></script>
<script src="./js/search.js" defer></script>
<script src="./js/search_index.js" defer></script>
<script src="./js/searcher.js" defer></script>
<script src="./js/darkfish.js" defer></script>
<link href="./css/fonts.css" rel="stylesheet">
<link href="./css/rdoc.css" rel="stylesheet">
<body id="top" role="document" class="module">
<nav role="navigation">
<div id="project-navigation">
<div id="home-section" role="region" title="Quick navigation" class="nav-section">
<h2>
<a href="./index.html" rel="home">Home</a>
</h2>
<div id="table-of-contents-navigation">
<a href="./table_of_contents.html#pages">Pages</a>
<a href="./table_of_contents.html#classes">Classes</a>
<a href="./table_of_contents.html#methods">Methods</a>
</div>
</div>
<div id="search-section" role="search" class="project-section initially-hidden">
<form action="#" method="get" accept-charset="utf-8">
<div id="search-field-wrapper">
<input id="search-field" role="combobox" aria-label="Search"
aria-autocomplete="list" aria-controls="search-results"
type="text" name="search" placeholder="Search" spellcheck="false"
title="Type to search, Up and Down to navigate, Enter to load">
</div>
<ul id="search-results" aria-label="Search Results"
aria-busy="false" aria-expanded="false"
aria-atomic="false" class="initially-hidden"></ul>
</form>
</div>
</div>
<div id="class-metadata">
<!-- Method Quickref -->
<div id="method-list-section" class="nav-section">
<h3>Methods</h3>
<ul class="link-list" role="directory">
<li ><a href="#method-i-dfdxi">#dfdxi</a>
<li ><a href="#method-i-isEqual">#isEqual</a>
<li ><a href="#method-i-jacobian">#jacobian</a>
</ul>
</div>
</div>
</nav>
<main role="main" aria-labelledby="module-Jacobian">
<h1 id="module-Jacobian" class="module">
module Jacobian
</h1>
<section class="description">
<p>require 'bigdecimal/jacobian'</p>
<p>Provides methods to compute the <a href="Jacobian.html"><code>Jacobian</code></a> matrix of a set of equations at a point x. In the methods below:</p>
<p>f is an <a href="Object.html"><code>Object</code></a> which is used to compute the <a href="Jacobian.html"><code>Jacobian</code></a> matrix of the equations. It must provide the following methods:</p>
<dl class="rdoc-list note-list"><dt>f.values(x)
<dd>
<p>returns the values of all functions at x</p>
</dd><dt>f.zero
<dd>
<p>returns 0.0</p>
</dd><dt>f.one
<dd>
<p>returns 1.0</p>
</dd><dt>f.two
<dd>
<p>returns 2.0</p>
</dd><dt>f.ten
<dd>
<p>returns 10.0</p>
</dd><dt>f.eps
<dd>
<p>returns the convergence criterion (epsilon value) used to determine whether two values are considered equal. If |a-b| < epsilon, the two values are considered equal.</p>
</dd></dl>
<p>x is the point at which to compute the <a href="Jacobian.html"><code>Jacobian</code></a>.</p>
<p>fx is f.values(x).</p>
</section>
<section id="5Buntitled-5D" class="documentation-section">
<section id="public-instance-5Buntitled-5D-method-details" class="method-section">
<header>
<h3>Public Instance Methods</h3>
</header>
<div id="method-i-dfdxi" class="method-detail ">
<div class="method-heading">
<span class="method-name">dfdxi</span><span
class="method-args">(f,fx,x,i)</span>
<span class="method-click-advice">click to toggle source</span>
</div>
<div class="method-description">
<p>Computes the derivative of <a href="i">f</a> at <a href="i">x</a>. fx is the value of f at x.</p>
<div class="method-source-code" id="dfdxi-source">
<pre><span class="ruby-comment"># File ext/bigdecimal/lib/bigdecimal/jacobian.rb, line 47</span>
<span class="ruby-keyword">def</span> <span class="ruby-identifier ruby-title">dfdxi</span>(<span class="ruby-identifier">f</span>,<span class="ruby-identifier">fx</span>,<span class="ruby-identifier">x</span>,<span class="ruby-identifier">i</span>)
<span class="ruby-identifier">nRetry</span> = <span class="ruby-value">0</span>
<span class="ruby-identifier">n</span> = <span class="ruby-identifier">x</span>.<span class="ruby-identifier">size</span>
<span class="ruby-identifier">xSave</span> = <span class="ruby-identifier">x</span>[<span class="ruby-identifier">i</span>]
<span class="ruby-identifier">ok</span> = <span class="ruby-value">0</span>
<span class="ruby-identifier">ratio</span> = <span class="ruby-identifier">f</span>.<span class="ruby-identifier">ten</span><span class="ruby-operator">*</span><span class="ruby-identifier">f</span>.<span class="ruby-identifier">ten</span><span class="ruby-operator">*</span><span class="ruby-identifier">f</span>.<span class="ruby-identifier">ten</span>
<span class="ruby-identifier">dx</span> = <span class="ruby-identifier">x</span>[<span class="ruby-identifier">i</span>].<span class="ruby-identifier">abs</span><span class="ruby-operator">/</span><span class="ruby-identifier">ratio</span>
<span class="ruby-identifier">dx</span> = <span class="ruby-identifier">fx</span>[<span class="ruby-identifier">i</span>].<span class="ruby-identifier">abs</span><span class="ruby-operator">/</span><span class="ruby-identifier">ratio</span> <span class="ruby-keyword">if</span> <span class="ruby-identifier">isEqual</span>(<span class="ruby-identifier">dx</span>,<span class="ruby-identifier">f</span>.<span class="ruby-identifier">zero</span>,<span class="ruby-identifier">f</span>.<span class="ruby-identifier">zero</span>,<span class="ruby-identifier">f</span>.<span class="ruby-identifier">eps</span>)
<span class="ruby-identifier">dx</span> = <span class="ruby-identifier">f</span>.<span class="ruby-identifier">one</span><span class="ruby-operator">/</span><span class="ruby-identifier">f</span>.<span class="ruby-identifier">ten</span> <span class="ruby-keyword">if</span> <span class="ruby-identifier">isEqual</span>(<span class="ruby-identifier">dx</span>,<span class="ruby-identifier">f</span>.<span class="ruby-identifier">zero</span>,<span class="ruby-identifier">f</span>.<span class="ruby-identifier">zero</span>,<span class="ruby-identifier">f</span>.<span class="ruby-identifier">eps</span>)
<span class="ruby-keyword">until</span> <span class="ruby-identifier">ok</span><span class="ruby-operator">></span><span class="ruby-value">0</span> <span class="ruby-keyword">do</span>
<span class="ruby-identifier">deriv</span> = []
<span class="ruby-identifier">nRetry</span> <span class="ruby-operator">+=</span> <span class="ruby-value">1</span>
<span class="ruby-keyword">if</span> <span class="ruby-identifier">nRetry</span> <span class="ruby-operator">></span> <span class="ruby-value">100</span>
<span class="ruby-identifier">raise</span> <span class="ruby-string">"Singular Jacobian matrix. No change at x["</span> <span class="ruby-operator">+</span> <span class="ruby-identifier">i</span>.<span class="ruby-identifier">to_s</span> <span class="ruby-operator">+</span> <span class="ruby-string">"]"</span>
<span class="ruby-keyword">end</span>
<span class="ruby-identifier">dx</span> = <span class="ruby-identifier">dx</span><span class="ruby-operator">*</span><span class="ruby-identifier">f</span>.<span class="ruby-identifier">two</span>
<span class="ruby-identifier">x</span>[<span class="ruby-identifier">i</span>] <span class="ruby-operator">+=</span> <span class="ruby-identifier">dx</span>
<span class="ruby-identifier">fxNew</span> = <span class="ruby-identifier">f</span>.<span class="ruby-identifier">values</span>(<span class="ruby-identifier">x</span>)
<span class="ruby-keyword">for</span> <span class="ruby-identifier">j</span> <span class="ruby-keyword">in</span> <span class="ruby-value">0</span><span class="ruby-operator">...</span><span class="ruby-identifier">n</span> <span class="ruby-keyword">do</span>
<span class="ruby-keyword">if</span> <span class="ruby-operator">!</span><span class="ruby-identifier">isEqual</span>(<span class="ruby-identifier">fxNew</span>[<span class="ruby-identifier">j</span>],<span class="ruby-identifier">fx</span>[<span class="ruby-identifier">j</span>],<span class="ruby-identifier">f</span>.<span class="ruby-identifier">zero</span>,<span class="ruby-identifier">f</span>.<span class="ruby-identifier">eps</span>) <span class="ruby-keyword">then</span>
<span class="ruby-identifier">ok</span> <span class="ruby-operator">+=</span> <span class="ruby-value">1</span>
<span class="ruby-identifier">deriv</span> <span class="ruby-operator"><<=</span> (<span class="ruby-identifier">fxNew</span>[<span class="ruby-identifier">j</span>]<span class="ruby-operator">-</span><span class="ruby-identifier">fx</span>[<span class="ruby-identifier">j</span>])<span class="ruby-operator">/</span><span class="ruby-identifier">dx</span>
<span class="ruby-keyword">else</span>
<span class="ruby-identifier">deriv</span> <span class="ruby-operator"><<=</span> <span class="ruby-identifier">f</span>.<span class="ruby-identifier">zero</span>
<span class="ruby-keyword">end</span>
<span class="ruby-keyword">end</span>
<span class="ruby-identifier">x</span>[<span class="ruby-identifier">i</span>] = <span class="ruby-identifier">xSave</span>
<span class="ruby-keyword">end</span>
<span class="ruby-identifier">deriv</span>
<span class="ruby-keyword">end</span></pre>
</div>
</div>
</div>
<div id="method-i-isEqual" class="method-detail ">
<div class="method-heading">
<span class="method-name">isEqual</span><span
class="method-args">(a,b,zero=0.0,e=1.0e-8)</span>
<span class="method-click-advice">click to toggle source</span>
</div>
<div class="method-description">
<p>Determines the equality of two numbers by comparing to zero, or using the epsilon value</p>
<div class="method-source-code" id="isEqual-source">
<pre><span class="ruby-comment"># File ext/bigdecimal/lib/bigdecimal/jacobian.rb, line 30</span>
<span class="ruby-keyword">def</span> <span class="ruby-identifier ruby-title">isEqual</span>(<span class="ruby-identifier">a</span>,<span class="ruby-identifier">b</span>,<span class="ruby-identifier">zero</span>=<span class="ruby-value">0.0</span>,<span class="ruby-identifier">e</span>=<span class="ruby-value">1.0e-8</span>)
<span class="ruby-identifier">aa</span> = <span class="ruby-identifier">a</span>.<span class="ruby-identifier">abs</span>
<span class="ruby-identifier">bb</span> = <span class="ruby-identifier">b</span>.<span class="ruby-identifier">abs</span>
<span class="ruby-keyword">if</span> <span class="ruby-identifier">aa</span> <span class="ruby-operator">==</span> <span class="ruby-identifier">zero</span> <span class="ruby-operator">&&</span> <span class="ruby-identifier">bb</span> <span class="ruby-operator">==</span> <span class="ruby-identifier">zero</span> <span class="ruby-keyword">then</span>
<span class="ruby-keyword">true</span>
<span class="ruby-keyword">else</span>
<span class="ruby-keyword">if</span> ((<span class="ruby-identifier">a</span><span class="ruby-operator">-</span><span class="ruby-identifier">b</span>)<span class="ruby-operator">/</span>(<span class="ruby-identifier">aa</span><span class="ruby-operator">+</span><span class="ruby-identifier">bb</span>)).<span class="ruby-identifier">abs</span> <span class="ruby-operator"><</span> <span class="ruby-identifier">e</span> <span class="ruby-keyword">then</span>
<span class="ruby-keyword">true</span>
<span class="ruby-keyword">else</span>
<span class="ruby-keyword">false</span>
<span class="ruby-keyword">end</span>
<span class="ruby-keyword">end</span>
<span class="ruby-keyword">end</span></pre>
</div>
</div>
</div>
<div id="method-i-jacobian" class="method-detail ">
<div class="method-heading">
<span class="method-name">jacobian</span><span
class="method-args">(f,fx,x)</span>
<span class="method-click-advice">click to toggle source</span>
</div>
<div class="method-description">
<p>Computes the <a href="Jacobian.html"><code>Jacobian</code></a> of f at x. fx is the value of f at x.</p>
<div class="method-source-code" id="jacobian-source">
<pre><span class="ruby-comment"># File ext/bigdecimal/lib/bigdecimal/jacobian.rb, line 79</span>
<span class="ruby-keyword">def</span> <span class="ruby-identifier ruby-title">jacobian</span>(<span class="ruby-identifier">f</span>,<span class="ruby-identifier">fx</span>,<span class="ruby-identifier">x</span>)
<span class="ruby-identifier">n</span> = <span class="ruby-identifier">x</span>.<span class="ruby-identifier">size</span>
<span class="ruby-identifier">dfdx</span> = <span class="ruby-constant">Array</span>.<span class="ruby-identifier">new</span>(<span class="ruby-identifier">n</span><span class="ruby-operator">*</span><span class="ruby-identifier">n</span>)
<span class="ruby-keyword">for</span> <span class="ruby-identifier">i</span> <span class="ruby-keyword">in</span> <span class="ruby-value">0</span><span class="ruby-operator">...</span><span class="ruby-identifier">n</span> <span class="ruby-keyword">do</span>
<span class="ruby-identifier">df</span> = <span class="ruby-identifier">dfdxi</span>(<span class="ruby-identifier">f</span>,<span class="ruby-identifier">fx</span>,<span class="ruby-identifier">x</span>,<span class="ruby-identifier">i</span>)
<span class="ruby-keyword">for</span> <span class="ruby-identifier">j</span> <span class="ruby-keyword">in</span> <span class="ruby-value">0</span><span class="ruby-operator">...</span><span class="ruby-identifier">n</span> <span class="ruby-keyword">do</span>
<span class="ruby-identifier">dfdx</span>[<span class="ruby-identifier">j</span><span class="ruby-operator">*</span><span class="ruby-identifier">n</span><span class="ruby-operator">+</span><span class="ruby-identifier">i</span>] = <span class="ruby-identifier">df</span>[<span class="ruby-identifier">j</span>]
<span class="ruby-keyword">end</span>
<span class="ruby-keyword">end</span>
<span class="ruby-identifier">dfdx</span>
<span class="ruby-keyword">end</span></pre>
</div>
</div>
</div>
</section>
</section>
</main>
<footer id="validator-badges" role="contentinfo">
<p><a href="https://validator.w3.org/check/referer">Validate</a>
<p>Generated by <a href="https://ruby.github.io/rdoc/">RDoc</a> 6.2.1.1.
<p>Based on <a href="http://deveiate.org/projects/Darkfish-RDoc/">Darkfish</a> by <a href="http://deveiate.org">Michael Granger</a>.
</footer>