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<h3>Included Modules</h3>
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<li><a class="include" href="LUSolve.html">LUSolve</a>
<li><a class="include" href="Jacobian.html">Jacobian</a>
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<h3>Methods</h3>
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<li ><a href="#method-i-nlsolve">#nlsolve</a>
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<h1 id="module-Newton" class="module">
module Newton
</h1>
<section class="description">
<p>newton.rb</p>
<p>Solves the nonlinear algebraic equation system f = 0 by Newton's method. This program is not dependent on <a href="BigDecimal.html"><code>BigDecimal</code></a>.</p>
<p>To call:</p>
<pre> n = nlsolve(f,x)
where n is the number of iterations required,
x is the initial value vector
f is an Object which is used to compute the values of the equations to be solved.</pre>
<p>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>On exit, x is the solution vector.</p>
</section>
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<header>
<h3>Public Instance Methods</h3>
</header>
<div id="method-i-nlsolve" class="method-detail ">
<div class="method-heading">
<span class="method-name">nlsolve</span><span
class="method-args">(f,x)</span>
<span class="method-click-advice">click to toggle source</span>
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<div class="method-description">
<p>See also <a href="Newton.html"><code>Newton</code></a></p>
<div class="method-source-code" id="nlsolve-source">
<pre><span class="ruby-comment"># File ext/bigdecimal/lib/bigdecimal/newton.rb, line 44</span>
<span class="ruby-keyword">def</span> <span class="ruby-identifier ruby-title">nlsolve</span>(<span class="ruby-identifier">f</span>,<span class="ruby-identifier">x</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">f0</span> = <span class="ruby-identifier">f</span>.<span class="ruby-identifier">values</span>(<span class="ruby-identifier">x</span>)
<span class="ruby-identifier">zero</span> = <span class="ruby-identifier">f</span>.<span class="ruby-identifier">zero</span>
<span class="ruby-identifier">one</span> = <span class="ruby-identifier">f</span>.<span class="ruby-identifier">one</span>
<span class="ruby-identifier">two</span> = <span class="ruby-identifier">f</span>.<span class="ruby-identifier">two</span>
<span class="ruby-identifier">p5</span> = <span class="ruby-identifier">one</span><span class="ruby-operator">/</span><span class="ruby-identifier">two</span>
<span class="ruby-identifier">d</span> = <span class="ruby-identifier">norm</span>(<span class="ruby-identifier">f0</span>,<span class="ruby-identifier">zero</span>)
<span class="ruby-identifier">minfact</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">minfact</span> = <span class="ruby-identifier">one</span><span class="ruby-operator">/</span><span class="ruby-identifier">minfact</span>
<span class="ruby-identifier">e</span> = <span class="ruby-identifier">f</span>.<span class="ruby-identifier">eps</span>
<span class="ruby-keyword">while</span> <span class="ruby-identifier">d</span> <span class="ruby-operator">>=</span> <span class="ruby-identifier">e</span> <span class="ruby-keyword">do</span>
<span class="ruby-identifier">nRetry</span> <span class="ruby-operator">+=</span> <span class="ruby-value">1</span>
<span class="ruby-comment"># Not yet converged. => Compute Jacobian matrix</span>
<span class="ruby-identifier">dfdx</span> = <span class="ruby-identifier">jacobian</span>(<span class="ruby-identifier">f</span>,<span class="ruby-identifier">f0</span>,<span class="ruby-identifier">x</span>)
<span class="ruby-comment"># Solve dfdx*dx = -f0 to estimate dx</span>
<span class="ruby-identifier">dx</span> = <span class="ruby-identifier">lusolve</span>(<span class="ruby-identifier">dfdx</span>,<span class="ruby-identifier">f0</span>,<span class="ruby-identifier">ludecomp</span>(<span class="ruby-identifier">dfdx</span>,<span class="ruby-identifier">n</span>,<span class="ruby-identifier">zero</span>,<span class="ruby-identifier">one</span>),<span class="ruby-identifier">zero</span>)
<span class="ruby-identifier">fact</span> = <span class="ruby-identifier">two</span>
<span class="ruby-identifier">xs</span> = <span class="ruby-identifier">x</span>.<span class="ruby-identifier">dup</span>
<span class="ruby-keyword">begin</span>
<span class="ruby-identifier">fact</span> <span class="ruby-operator">*=</span> <span class="ruby-identifier">p5</span>
<span class="ruby-keyword">if</span> <span class="ruby-identifier">fact</span> <span class="ruby-operator"><</span> <span class="ruby-identifier">minfact</span> <span class="ruby-keyword">then</span>
<span class="ruby-identifier">raise</span> <span class="ruby-string">"Failed to reduce function values."</span>
<span class="ruby-keyword">end</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">x</span>[<span class="ruby-identifier">i</span>] = <span class="ruby-identifier">xs</span>[<span class="ruby-identifier">i</span>] <span class="ruby-operator">-</span> <span class="ruby-identifier">dx</span>[<span class="ruby-identifier">i</span>]<span class="ruby-operator">*</span><span class="ruby-identifier">fact</span>
<span class="ruby-keyword">end</span>
<span class="ruby-identifier">f0</span> = <span class="ruby-identifier">f</span>.<span class="ruby-identifier">values</span>(<span class="ruby-identifier">x</span>)
<span class="ruby-identifier">dn</span> = <span class="ruby-identifier">norm</span>(<span class="ruby-identifier">f0</span>,<span class="ruby-identifier">zero</span>)
<span class="ruby-keyword">end</span> <span class="ruby-keyword">while</span>(<span class="ruby-identifier">dn</span><span class="ruby-operator">>=</span><span class="ruby-identifier">d</span>)
<span class="ruby-identifier">d</span> = <span class="ruby-identifier">dn</span>
<span class="ruby-keyword">end</span>
<span class="ruby-identifier">nRetry</span>
<span class="ruby-keyword">end</span></pre>
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