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<title>Root Finding Without Derivatives: Bisection, Bracket and TOMS748</title>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.internals1.roots2"></a><a class="link" href="roots2.html" title="Root Finding Without Derivatives: Bisection, Bracket and TOMS748">Root Finding Without
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Derivatives: Bisection, Bracket and TOMS748</a>
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</h3></div></div></div>
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<h5>
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<a name="math_toolkit.internals1.roots2.h0"></a>
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<span class="phrase"><a name="math_toolkit.internals1.roots2.synopsis"></a></span><a class="link" href="roots2.html#math_toolkit.internals1.roots2.synopsis">Synopsis</a>
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</h5>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">roots</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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</pre>
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<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
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<span class="keyword">namespace</span> <span class="identifier">tools</span><span class="special">{</span>
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<span class="comment">// Bisection</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
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<span class="identifier">bisect</span><span class="special">(</span>
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<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
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<span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span>
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<span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span>
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<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
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<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">);</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
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<span class="identifier">bisect</span><span class="special">(</span>
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<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
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<span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span>
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<span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span>
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<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">);</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 13. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
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<span class="identifier">bisect</span><span class="special">(</span>
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<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
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<span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span>
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<span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span>
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<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
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<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">,</span>
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<span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 13. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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<span class="comment">// Bracket and Solve Root</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
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<span class="identifier">bracket_and_solve_root</span><span class="special">(</span>
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<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">guess</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">factor</span><span class="special">,</span>
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<span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span>
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<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
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<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">);</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 13. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
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<span class="identifier">bracket_and_solve_root</span><span class="special">(</span>
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<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">guess</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">factor</span><span class="special">,</span>
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<span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span>
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<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
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<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">,</span>
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<span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 13. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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<span class="comment">// TOMS 748 algorithm</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
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<span class="identifier">toms748_solve</span><span class="special">(</span>
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<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span>
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<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
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<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">);</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 13. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
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<span class="identifier">toms748_solve</span><span class="special">(</span>
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<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span>
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<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
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<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">,</span>
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<span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 13. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
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<span class="identifier">toms748_solve</span><span class="special">(</span>
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<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fa</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fb</span><span class="special">,</span>
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<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
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<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">);</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 13. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
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<span class="identifier">toms748_solve</span><span class="special">(</span>
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<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fa</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fb</span><span class="special">,</span>
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<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
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<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">,</span>
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<span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 13. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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<span class="comment">// Termination conditions:</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
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<span class="keyword">struct</span> <span class="identifier">eps_tolerance</span><span class="special">;</span>
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<span class="keyword">struct</span> <span class="identifier">equal_floor</span><span class="special">;</span>
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<span class="keyword">struct</span> <span class="identifier">equal_ceil</span><span class="special">;</span>
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<span class="keyword">struct</span> <span class="identifier">equal_nearest_integer</span><span class="special">;</span>
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<span class="special">}}}</span> <span class="comment">// namespaces</span>
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</pre>
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<h5>
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<a name="math_toolkit.internals1.roots2.h1"></a>
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<span class="phrase"><a name="math_toolkit.internals1.roots2.description"></a></span><a class="link" href="roots2.html#math_toolkit.internals1.roots2.description">Description</a>
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</h5>
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<p>
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These functions solve the root of some function <span class="emphasis"><em>f(x)</em></span>
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<span class="emphasis"><em>without the need for any derivatives of <span class="emphasis"><em>f(x)</em></span></em></span>.
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</p>
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<p>
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The <code class="computeroutput"><span class="identifier">bracket_and_solve_root</span></code>
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functions use TOMS Algorithm 748 that is asymptotically the most efficient
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known, and have been shown to be optimal for a certain classes of smooth
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functions. Variants with and without __policies are provided.
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</p>
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<p>
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Alternatively, there is a simple bisection routine which can be useful in
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its own right in some situations, or alternatively for narrowing down the
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range containing the root, prior to calling a more advanced algorithm.
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</p>
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<p>
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All the algorithms in this section reduce the diameter of the enclosing interval
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with the same asymptotic efficiency with which they locate the root. This
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is in contrast to the derivative based methods which may <span class="emphasis"><em>never</em></span>
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significantly reduce the enclosing interval, even though they rapidly approach
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the root. This is also in contrast to some other derivative-free methods
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(for example the methods of <a href="http://en.wikipedia.org/wiki/Brent%27s_method" target="_top">Brent
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or Dekker)</a> which only reduce the enclosing interval on the final
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step. Therefore these methods return a std::pair containing the enclosing
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interval found, and accept a function object specifying the termination condition.
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Three function objects are provided for ready-made termination conditions:
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<span class="emphasis"><em>eps_tolerance</em></span> causes termination when the relative error
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in the enclosing interval is below a certain threshold, while <span class="emphasis"><em>equal_floor</em></span>
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and <span class="emphasis"><em>equal_ceil</em></span> are useful for certain statistical applications
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where the result is known to be an integer. Other user-defined termination
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conditions are likely to be used only rarely, but may be useful in some specific
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circumstances.
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</p>
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<h4>
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<a name="math_toolkit.internals1.roots2.h2"></a>
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<span class="phrase"><a name="math_toolkit.internals1.roots2.bisection"></a></span><a class="link" href="roots2.html#math_toolkit.internals1.roots2.bisection">Bisection</a>
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</h4>
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<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">></span>
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|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
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<span class="identifier">bisect</span><span class="special">(</span>
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<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
|
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<span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span>
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<span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span>
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<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
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|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">);</span>
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|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">></span>
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|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
|
|
<span class="identifier">bisect</span><span class="special">(</span>
|
|
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
|
|
<span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span>
|
|
<span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span>
|
|
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">);</span>
|
|
|
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 13. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
|
|
<span class="identifier">bisect</span><span class="special">(</span>
|
|
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
|
|
<span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span>
|
|
<span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span>
|
|
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
|
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">,</span>
|
|
<span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 13. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
|
</pre>
|
|
<p>
|
|
These functions locate the root using bisection: function arguments are:
|
|
</p>
|
|
<div class="variablelist">
|
|
<p class="title"><b></b></p>
|
|
<dl class="variablelist">
|
|
<dt><span class="term">f</span></dt>
|
|
<dd><p>
|
|
A unary functor which is the function whose root is to be found.
|
|
</p></dd>
|
|
<dt><span class="term">min</span></dt>
|
|
<dd><p>
|
|
The left bracket of the interval known to contain the root.
|
|
</p></dd>
|
|
<dt><span class="term">max</span></dt>
|
|
<dd><p>
|
|
The right bracket of the interval known to contain the root. It is
|
|
a precondition that <span class="emphasis"><em>min < max</em></span> and <span class="emphasis"><em>f(min)*f(max)
|
|
<= 0</em></span>, the function signals evaluation error if these
|
|
preconditions are violated. The action taken is controlled by the evaluation
|
|
error policy. A best guess may be returned, perhaps significantly wrong.
|
|
</p></dd>
|
|
<dt><span class="term">tol</span></dt>
|
|
<dd><p>
|
|
A binary functor that specifies the termination condition: the function
|
|
will return the current brackets enclosing the root when <span class="emphasis"><em>tol(min,max)</em></span>
|
|
becomes true.
|
|
</p></dd>
|
|
<dt><span class="term">max_iter</span></dt>
|
|
<dd><p>
|
|
The maximum number of invocations of <span class="emphasis"><em>f(x)</em></span> to make
|
|
while searching for the root.
|
|
</p></dd>
|
|
</dl>
|
|
</div>
|
|
<p>
|
|
The final <a class="link" href="../../policy.html" title="Chapter 13. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
|
|
be used to control the behaviour of the function: how it handles errors,
|
|
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 13. Policies: Controlling Precision, Error Handling etc">policy
|
|
documentation for more details</a>.
|
|
</p>
|
|
<p>
|
|
Returns: a pair of values <span class="emphasis"><em>r</em></span> that bracket the root so
|
|
that:
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">f</span><span class="special">(</span><span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">f</span><span class="special">(</span><span class="identifier">r</span><span class="special">.</span><span class="identifier">second</span><span class="special">)</span> <span class="special"><=</span> <span class="number">0</span>
|
|
</pre>
|
|
<p>
|
|
and either
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">tol</span><span class="special">(</span><span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span><span class="special">,</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">second</span><span class="special">)</span> <span class="special">==</span> <span class="keyword">true</span>
|
|
</pre>
|
|
<p>
|
|
or
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">max_iter</span> <span class="special">>=</span> <span class="identifier">m</span>
|
|
</pre>
|
|
<p>
|
|
where <span class="emphasis"><em>m</em></span> is the initial value of <span class="emphasis"><em>max_iter</em></span>
|
|
passed to the function.
|
|
</p>
|
|
<p>
|
|
In other words, it's up to the caller to verify whether termination occurred
|
|
as a result of exceeding <span class="emphasis"><em>max_iter</em></span> function invocations
|
|
(easily done by checking the updated value of <span class="emphasis"><em>max_iter</em></span>
|
|
when the function returns), rather than because the termination condition
|
|
<span class="emphasis"><em>tol</em></span> was satisfied.
|
|
</p>
|
|
<h4>
|
|
<a name="math_toolkit.internals1.roots2.h3"></a>
|
|
<span class="phrase"><a name="math_toolkit.internals1.roots2.bracket_and_solve"></a></span><a class="link" href="roots2.html#math_toolkit.internals1.roots2.bracket_and_solve">Bracket
|
|
and solve</a>
|
|
</h4>
|
|
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">></span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
|
|
<span class="identifier">bracket_and_solve_root</span><span class="special">(</span>
|
|
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
|
|
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">guess</span><span class="special">,</span>
|
|
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">factor</span><span class="special">,</span>
|
|
<span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span>
|
|
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
|
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">);</span>
|
|
|
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 13. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
|
|
<span class="identifier">bracket_and_solve_root</span><span class="special">(</span>
|
|
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
|
|
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">guess</span><span class="special">,</span>
|
|
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">factor</span><span class="special">,</span>
|
|
<span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span>
|
|
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
|
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">,</span>
|
|
<span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 13. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
|
</pre>
|
|
<p>
|
|
This is a convenience function that calls <span class="emphasis"><em>toms748_solve</em></span>
|
|
internally to find the root of <span class="emphasis"><em>f(x)</em></span>. It's usable only
|
|
when <span class="emphasis"><em>f(x)</em></span> is a monotonic function, and the location
|
|
of the root is known approximately, and in particular it is known whether
|
|
the root is occurs for positive or negative <span class="emphasis"><em>x</em></span>. The parameters
|
|
are:
|
|
</p>
|
|
<div class="variablelist">
|
|
<p class="title"><b></b></p>
|
|
<dl class="variablelist">
|
|
<dt><span class="term">f</span></dt>
|
|
<dd><p>
|
|
A unary functor that is the function whose root is to be solved. f(x)
|
|
must be uniformly increasing or decreasing on <span class="emphasis"><em>x</em></span>.
|
|
</p></dd>
|
|
<dt><span class="term">guess</span></dt>
|
|
<dd><p>
|
|
An initial approximation to the root
|
|
</p></dd>
|
|
<dt><span class="term">factor</span></dt>
|
|
<dd><p>
|
|
A scaling factor that is used to bracket the root: the value <span class="emphasis"><em>guess</em></span>
|
|
is multiplied (or divided as appropriate) by <span class="emphasis"><em>factor</em></span>
|
|
until two values are found that bracket the root. A value such as 2
|
|
is a typical choice for <span class="emphasis"><em>factor</em></span>.
|
|
</p></dd>
|
|
<dt><span class="term">rising</span></dt>
|
|
<dd><p>
|
|
Set to <span class="emphasis"><em>true</em></span> if <span class="emphasis"><em>f(x)</em></span> is rising
|
|
on <span class="emphasis"><em>x</em></span> and <span class="emphasis"><em>false</em></span> if <span class="emphasis"><em>f(x)</em></span>
|
|
is falling on <span class="emphasis"><em>x</em></span>. This value is used along with
|
|
the result of <span class="emphasis"><em>f(guess)</em></span> to determine if <span class="emphasis"><em>guess</em></span>
|
|
is above or below the root.
|
|
</p></dd>
|
|
<dt><span class="term">tol</span></dt>
|
|
<dd><p>
|
|
A binary functor that determines the termination condition for the
|
|
search for the root. <span class="emphasis"><em>tol</em></span> is passed the current
|
|
brackets at each step, when it returns true then the current brackets
|
|
are returned as the result.
|
|
</p></dd>
|
|
<dt><span class="term">max_iter</span></dt>
|
|
<dd><p>
|
|
The maximum number of function invocations to perform in the search
|
|
for the root.
|
|
</p></dd>
|
|
</dl>
|
|
</div>
|
|
<p>
|
|
The final <a class="link" href="../../policy.html" title="Chapter 13. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
|
|
be used to control the behaviour of the function: how it handles errors,
|
|
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 13. Policies: Controlling Precision, Error Handling etc">policy
|
|
documentation for more details</a>.
|
|
</p>
|
|
<p>
|
|
Returns: a pair of values <span class="emphasis"><em>r</em></span> that bracket the root so
|
|
that:
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">f</span><span class="special">(</span><span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">f</span><span class="special">(</span><span class="identifier">r</span><span class="special">.</span><span class="identifier">second</span><span class="special">)</span> <span class="special"><=</span> <span class="number">0</span>
|
|
</pre>
|
|
<p>
|
|
and either
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">tol</span><span class="special">(</span><span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span><span class="special">,</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">second</span><span class="special">)</span> <span class="special">==</span> <span class="keyword">true</span>
|
|
</pre>
|
|
<p>
|
|
or
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">max_iter</span> <span class="special">>=</span> <span class="identifier">m</span>
|
|
</pre>
|
|
<p>
|
|
where <span class="emphasis"><em>m</em></span> is the initial value of <span class="emphasis"><em>max_iter</em></span>
|
|
passed to the function.
|
|
</p>
|
|
<p>
|
|
In other words, it's up to the caller to verify whether termination occurred
|
|
as a result of exceeding <span class="emphasis"><em>max_iter</em></span> function invocations
|
|
(easily done by checking the value of <span class="emphasis"><em>max_iter</em></span> when
|
|
the function returns), rather than because the termination condition <span class="emphasis"><em>tol</em></span>
|
|
was satisfied.
|
|
</p>
|
|
<h4>
|
|
<a name="math_toolkit.internals1.roots2.h4"></a>
|
|
<span class="phrase"><a name="math_toolkit.internals1.roots2.algorithm_toms_748_alefeld_potra"></a></span><a class="link" href="roots2.html#math_toolkit.internals1.roots2.algorithm_toms_748_alefeld_potra">Algorithm
|
|
TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions</a>
|
|
</h4>
|
|
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">></span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
|
|
<span class="identifier">toms748_solve</span><span class="special">(</span>
|
|
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
|
|
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span>
|
|
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span>
|
|
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
|
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">);</span>
|
|
|
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 13. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
|
|
<span class="identifier">toms748_solve</span><span class="special">(</span>
|
|
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
|
|
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span>
|
|
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span>
|
|
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
|
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">,</span>
|
|
<span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 13. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
|
|
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">></span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
|
|
<span class="identifier">toms748_solve</span><span class="special">(</span>
|
|
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
|
|
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span>
|
|
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span>
|
|
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fa</span><span class="special">,</span>
|
|
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fb</span><span class="special">,</span>
|
|
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
|
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">);</span>
|
|
|
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 13. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
|
|
<span class="identifier">toms748_solve</span><span class="special">(</span>
|
|
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
|
|
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span>
|
|
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span>
|
|
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fa</span><span class="special">,</span>
|
|
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fb</span><span class="special">,</span>
|
|
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
|
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">,</span>
|
|
<span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 13. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
|
</pre>
|
|
<p>
|
|
These two functions implement TOMS Algorithm 748: it uses a mixture of cubic,
|
|
quadratic and linear (secant) interpolation to locate the root of <span class="emphasis"><em>f(x)</em></span>.
|
|
The two functions differ only by whether values for <span class="emphasis"><em>f(a)</em></span>
|
|
and <span class="emphasis"><em>f(b)</em></span> are already available. The toms748_solve parameters
|
|
are:
|
|
</p>
|
|
<div class="variablelist">
|
|
<p class="title"><b></b></p>
|
|
<dl class="variablelist">
|
|
<dt><span class="term">f</span></dt>
|
|
<dd><p>
|
|
A unary functor that is the function whose root is to be solved. f(x)
|
|
need not be uniformly increasing or decreasing on <span class="emphasis"><em>x</em></span>
|
|
and may have multiple roots.
|
|
</p></dd>
|
|
<dt><span class="term">a</span></dt>
|
|
<dd><p>
|
|
The lower bound for the initial bracket of the root.
|
|
</p></dd>
|
|
<dt><span class="term">b</span></dt>
|
|
<dd><p>
|
|
The upper bound for the initial bracket of the root. It is a precondition
|
|
that <span class="emphasis"><em>a < b</em></span> and that <span class="emphasis"><em>a</em></span>
|
|
and <span class="emphasis"><em>b</em></span> bracket the root to find so that <span class="emphasis"><em>f(a)*f(b)
|
|
< 0</em></span>.
|
|
</p></dd>
|
|
<dt><span class="term">fa</span></dt>
|
|
<dd><p>
|
|
Optional: the value of <span class="emphasis"><em>f(a)</em></span>.
|
|
</p></dd>
|
|
<dt><span class="term">fb</span></dt>
|
|
<dd><p>
|
|
Optional: the value of <span class="emphasis"><em>f(b)</em></span>.
|
|
</p></dd>
|
|
<dt><span class="term">tol</span></dt>
|
|
<dd><p>
|
|
A binary functor that determines the termination condition for the
|
|
search for the root. <span class="emphasis"><em>tol</em></span> is passed the current
|
|
brackets at each step, when it returns true, then the current brackets
|
|
are returned as the result.
|
|
</p></dd>
|
|
<dt><span class="term">max_iter</span></dt>
|
|
<dd><p>
|
|
The maximum number of function invocations to perform in the search
|
|
for the root. On exit <span class="emphasis"><em>max_iter</em></span> is set to actual
|
|
number of function invocations used.
|
|
</p></dd>
|
|
</dl>
|
|
</div>
|
|
<p>
|
|
The final <a class="link" href="../../policy.html" title="Chapter 13. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
|
|
be used to control the behaviour of the function: how it handles errors,
|
|
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 13. Policies: Controlling Precision, Error Handling etc">policy
|
|
documentation for more details</a>.
|
|
</p>
|
|
<p>
|
|
toms748_solve returns: a pair of values <span class="emphasis"><em>r</em></span> that bracket
|
|
the root so that: f(r.first) * f(r.second) <= 0 and either tol(r.first,
|
|
r.second) == true or max_iter >= m where <span class="emphasis"><em>m</em></span> is the
|
|
initial value of <span class="emphasis"><em>max_iter</em></span> passed to the function.
|
|
</p>
|
|
<p>
|
|
In other words, it's up to the caller to verify whether termination occurred
|
|
as a result of exceeding <span class="emphasis"><em>max_iter</em></span> function invocations
|
|
(easily done by checking the updated value of <span class="emphasis"><em>max_iter</em></span>
|
|
against its previous value passed as parameter), rather than because the
|
|
termination condition <span class="emphasis"><em>tol</em></span> was satisfied.
|
|
</p>
|
|
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
|
<span class="keyword">struct</span> <span class="identifier">eps_tolerance</span>
|
|
<span class="special">{</span>
|
|
<span class="identifier">eps_tolerance</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">bits</span><span class="special">);</span>
|
|
<span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">()(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
|
|
<span class="special">};</span>
|
|
</pre>
|
|
<p>
|
|
<code class="computeroutput"><span class="identifier">eps_tolerance</span></code> is the usual
|
|
termination condition used with these root finding functions. Its operator()
|
|
will return true when the relative distance between <span class="emphasis"><em>a</em></span>
|
|
and <span class="emphasis"><em>b</em></span> is less than twice the machine epsilon for T,
|
|
or 2<sup>1-bits</sup>, whichever is the larger. In other words, you set <span class="emphasis"><em>bits</em></span>
|
|
to the number of bits of precision you want in the result. The minimal tolerance
|
|
of twice the machine epsilon of T is required to ensure that we get back
|
|
a bracketing interval: since this must clearly be at least 1 epsilon in size.
|
|
</p>
|
|
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">equal_floor</span>
|
|
<span class="special">{</span>
|
|
<span class="identifier">equal_floor</span><span class="special">();</span>
|
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> <span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">()(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
|
|
<span class="special">};</span>
|
|
</pre>
|
|
<p>
|
|
This termination condition is used when you want to find an integer result
|
|
that is the <span class="emphasis"><em>floor</em></span> of the true root. It will terminate
|
|
as soon as both ends of the interval have the same <span class="emphasis"><em>floor</em></span>.
|
|
</p>
|
|
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">equal_ceil</span>
|
|
<span class="special">{</span>
|
|
<span class="identifier">equal_ceil</span><span class="special">();</span>
|
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> <span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">()(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
|
|
<span class="special">};</span>
|
|
</pre>
|
|
<p>
|
|
This termination condition is used when you want to find an integer result
|
|
that is the <span class="emphasis"><em>ceil</em></span> of the true root. It will terminate
|
|
as soon as both ends of the interval have the same <span class="emphasis"><em>ceil</em></span>.
|
|
</p>
|
|
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">equal_nearest_integer</span>
|
|
<span class="special">{</span>
|
|
<span class="identifier">equal_nearest_integer</span><span class="special">();</span>
|
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> <span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">()(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
|
|
<span class="special">};</span>
|
|
</pre>
|
|
<p>
|
|
This termination condition is used when you want to find an integer result
|
|
that is the <span class="emphasis"><em>closest</em></span> to the true root. It will terminate
|
|
as soon as both ends of the interval round to the same nearest integer.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.internals1.roots2.h5"></a>
|
|
<span class="phrase"><a name="math_toolkit.internals1.roots2.implementation"></a></span><a class="link" href="roots2.html#math_toolkit.internals1.roots2.implementation">Implementation</a>
|
|
</h5>
|
|
<p>
|
|
The implementation of the bisection algorithm is extremely straightforward
|
|
and not detailed here. <a href="http://portal.acm.org/citation.cfm?id=210111" target="_top">TOMS
|
|
Algorithm 748: enclosing zeros of continuous functions</a> is described
|
|
in detail in:
|
|
</p>
|
|
<p>
|
|
<span class="emphasis"><em>Algorithm 748: Enclosing Zeros of Continuous Functions, G. E. Alefeld,
|
|
F. A. Potra and Yixun Shi, ACM Transactions on Mathematica1 Software, Vol.
|
|
21. No. 3. September 1995. Pages 327-344.</em></span>
|
|
</p>
|
|
<p>
|
|
The implementation here is a faithful translation of this paper into C++.
|
|
</p>
|
|
</div>
|
|
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
|
<td align="left"></td>
|
|
<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012, 2013 Paul A. Bristow, Christopher Kormanyos,
|
|
Hubert Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin
|
|
Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
|
|
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
|
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
|
</p>
|
|
</div></td>
|
|
</tr></table>
|
|
<hr>
|
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