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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="math_toolkit.roots_noderiv"></a><a class="link" href="roots_noderiv.html" title="Root Finding Without Derivatives">Root Finding Without Derivatives</a>
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</h2></div></div></div>
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<div class="toc"><dl class="toc">
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<dt><span class="section"><a href="roots_noderiv/bisect.html">Bisection</a></span></dt>
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<dt><span class="section"><a href="roots_noderiv/bracket_solve.html">Bracket and
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Solve Root</a></span></dt>
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<dt><span class="section"><a href="roots_noderiv/TOMS748.html">Algorithm TOMS 748:
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Alefeld, Potra and Shi: Enclosing zeros of continuous functions</a></span></dt>
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<dt><span class="section"><a href="roots_noderiv/brent.html">Brent-Decker Algorithm</a></span></dt>
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<dt><span class="section"><a href="roots_noderiv/root_termination.html">Termination
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Condition Functors</a></span></dt>
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<dt><span class="section"><a href="roots_noderiv/implementation.html">Implementation</a></span></dt>
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</dl></div>
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<h5>
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<a name="math_toolkit.roots_noderiv.h0"></a>
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<span class="phrase"><a name="math_toolkit.roots_noderiv.synopsis"></a></span><a class="link" href="roots_noderiv.html#math_toolkit.roots_noderiv.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> <span class="comment">// Note namespace boost::math::tools.</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 20. 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 20. 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 20. 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 20. 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 20. 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 20. 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 20. 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 20. 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">// boost::math::tools namespaces</span>
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</pre>
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<h5>
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<a name="math_toolkit.roots_noderiv.h1"></a>
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<span class="phrase"><a name="math_toolkit.roots_noderiv.description"></a></span><a class="link" href="roots_noderiv.html#math_toolkit.roots_noderiv.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 <a class="link" href="roots_noderiv/TOMS748.html" title="Algorithm TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions">TOMS 748 algorithm</a>
|
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by Alefeld, Potra and Shi that is asymptotically the most efficient known,
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and has been shown to be optimal for a certain classes of smooth functions.
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Variants with and without <a class="link" href="../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policies</a> are provided.
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</p>
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<p>
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Alternatively, <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisect</a>
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is a simple <a href="https://en.wikipedia.org/wiki/Bisection" target="_top">bisection</a>
|
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routine which can be useful in its own right in some situations, or alternatively
|
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for narrowing down the range containing the root, prior to calling a more advanced
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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 is
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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 (for
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example, Brent's method described at <a href="http://en.wikipedia.org/wiki/Brent%27s_method" target="_top">Brent-Dekker)</a>
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which only reduces the enclosing interval on the final step. Therefore these
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methods return a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span></code> containing the enclosing interval found,
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and accept a function object specifying the termination condition.
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</p>
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<p>
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Three function objects are provided for ready-made termination conditions:
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</p>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem">
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<span class="emphasis"><em>eps_tolerance</em></span> causes termination when the relative
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error in the enclosing interval is below a certain threshold.
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</li>
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<li class="listitem">
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<span class="emphasis"><em>equal_floor</em></span> and <span class="emphasis"><em>equal_ceil</em></span> are
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useful for certain statistical applications where the result is known to
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be an integer.
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</li>
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<li class="listitem">
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Other user-defined termination conditions are likely to be used only rarely,
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but may be useful in some specific circumstances.
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</li>
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</ul></div>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
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Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
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Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
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Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
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Daryle Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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>)
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