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<div class="section">
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="math_toolkit.roots_deriv"></a><a class="link" href="roots_deriv.html" title="Root Finding With Derivatives: Newton-Raphson, Halley & Schröder">Root Finding With Derivatives:
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Newton-Raphson, Halley & Schröder</a>
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</h2></div></div></div>
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<h5>
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<a name="math_toolkit.roots_deriv.h0"></a>
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<span class="phrase"><a name="math_toolkit.roots_deriv.synopsis"></a></span><a class="link" href="roots_deriv.html#math_toolkit.roots_deriv.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">// Newton-Raphson</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>
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<span class="identifier">T</span> <span class="identifier">newton_raphson_iterate</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">guess</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="keyword">int</span> <span class="identifier">digits</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>
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<span class="identifier">T</span> <span class="identifier">newton_raphson_iterate</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">guess</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="keyword">int</span> <span class="identifier">digits</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>
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<span class="comment">// Halley</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>
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<span class="identifier">T</span> <span class="identifier">halley_iterate</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">guess</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="keyword">int</span> <span class="identifier">digits</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>
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<span class="identifier">T</span> <span class="identifier">halley_iterate</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">guess</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="keyword">int</span> <span class="identifier">digits</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>
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<span class="comment">// Schr'''&#xf6;'''der</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>
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<span class="identifier">T</span> <span class="identifier">schroder_iterate</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">guess</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="keyword">int</span> <span class="identifier">digits</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>
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<span class="identifier">T</span> <span class="identifier">schroder_iterate</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">guess</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="keyword">int</span> <span class="identifier">digits</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>
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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">Complex</span><span class="special">></span>
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<span class="identifier">Complex</span> <span class="identifier">complex_newton</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">Complex</span> <span class="identifier">guess</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">max_iterations</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">Complex</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">>::</span><span class="identifier">digits</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">T</span><span class="special">></span>
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<span class="keyword">auto</span> <span class="identifier">quadratic_roots</span><span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">c</span><span class="special">);</span>
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<span class="special">}}}</span> <span class="comment">// namespaces boost::math::tools.</span>
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</pre>
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<h5>
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<a name="math_toolkit.roots_deriv.h1"></a>
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<span class="phrase"><a name="math_toolkit.roots_deriv.description"></a></span><a class="link" href="roots_deriv.html#math_toolkit.roots_deriv.description">Description</a>
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</h5>
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<p>
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These functions all perform iterative root-finding <span class="bold"><strong>using
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derivatives</strong></span>:
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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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<code class="computeroutput"><span class="identifier">newton_raphson_iterate</span></code>
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performs second-order <a class="link" href="roots_deriv.html#math_toolkit.roots_deriv.newton">Newton-Raphson
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iteration</a>.
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</li>
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<li class="listitem">
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<code class="computeroutput"><span class="identifier">halley_iterate</span></code> and <code class="computeroutput"><span class="identifier">schroder_iterate</span></code> perform third-order
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<a class="link" href="roots_deriv.html#math_toolkit.roots_deriv.halley">Halley</a> and <a class="link" href="roots_deriv.html#math_toolkit.roots_deriv.schroder">Schröder</a> iteration.
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</li>
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<li class="listitem">
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<code class="computeroutput"><span class="identifier">complex_newton</span></code> performs
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Newton's method on complex-analytic functions.
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</li>
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<li class="listitem">
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<code class="computeroutput"><span class="identifier">solve_quadratic</span></code> solves
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quadratic equations using various tricks to keep catastrophic cancellation
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from occurring in computation of the discriminant.
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</li>
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</ul></div>
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<div class="variablelist">
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<p class="title"><b>Parameters of the real-valued root finding functions</b></p>
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<dl class="variablelist">
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<dt><span class="term">F f</span></dt>
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<dd>
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<p>
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Type F must be a callable function object (or C++ lambda) that accepts
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one parameter and returns a <a class="link" href="internals/tuples.html" title="Tuples">std::pair,
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std::tuple, boost::tuple or boost::fusion::tuple</a>:
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</p>
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<p>
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For second-order iterative method (<a href="http://en.wikipedia.org/wiki/Newton_Raphson" target="_top">Newton
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Raphson</a>) the <code class="computeroutput"><span class="identifier">tuple</span></code>
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should have <span class="bold"><strong>two</strong></span> elements containing
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the evaluation of the function and its first derivative.
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</p>
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<p>
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For the third-order methods (<a href="http://en.wikipedia.org/wiki/Halley%27s_method" target="_top">Halley</a>
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and Schröder) the <code class="computeroutput"><span class="identifier">tuple</span></code>
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should have <span class="bold"><strong>three</strong></span> elements containing
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the evaluation of the function and its first and second derivatives.
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</p>
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</dd>
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<dt><span class="term">T guess</span></dt>
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<dd><p>
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The initial starting value. A good guess is crucial to quick convergence!
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</p></dd>
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<dt><span class="term">T min</span></dt>
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<dd><p>
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The minimum possible value for the result, this is used as an initial
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lower bracket.
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</p></dd>
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<dt><span class="term">T max</span></dt>
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<dd><p>
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The maximum possible value for the result, this is used as an initial
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upper bracket.
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</p></dd>
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<dt><span class="term">int digits</span></dt>
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<dd><p>
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The desired number of binary digits precision.
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</p></dd>
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<dt><span class="term">uintmax_t& max_iter</span></dt>
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<dd><p>
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An optional maximum number of iterations to perform. On exit, this is
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updated to the actual number of iterations performed.
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</p></dd>
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</dl>
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</div>
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<p>
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When using these functions you should note that:
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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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Default <code class="computeroutput"><span class="identifier">max_iter</span> <span class="special">=</span>
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<span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</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</span><span class="special">)()</span></code> is effectively 'iterate for ever'.
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</li>
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<li class="listitem">
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They may be very sensitive to the initial guess, typically they converge
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very rapidly if the initial guess has two or three decimal digits correct.
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However convergence can be no better than <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisect</a>,
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or in some rare cases, even worse than <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisect</a>
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if the initial guess is a long way from the correct value and the derivatives
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are close to zero.
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</li>
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<li class="listitem">
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These functions include special cases to handle zero first (and second
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where appropriate) derivatives, and fall back to <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisect</a>
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in this case. However, it is helpful if functor F is defined to return
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an arbitrarily small value <span class="emphasis"><em>of the correct sign</em></span> rather
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than zero.
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</li>
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<li class="listitem">
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The functions will raise an <a class="link" href="error_handling.html#math_toolkit.error_handling.evaluation_error">evaluation_error</a>
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if arguments <code class="computeroutput"><span class="identifier">min</span></code> and <code class="computeroutput"><span class="identifier">max</span></code> are the wrong way around or if they
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converge to a local minima.
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</li>
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<li class="listitem">
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If the derivative at the current best guess for the result is infinite
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(or very close to being infinite) then these functions may terminate prematurely.
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A large first derivative leads to a very small next step, triggering the
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termination condition. Derivative based iteration may not be appropriate
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in such cases.
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</li>
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<li class="listitem">
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If the function is 'Really Well Behaved' (is monotonic and has only one
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root) the bracket bounds <span class="emphasis"><em>min</em></span> and <span class="emphasis"><em>max</em></span>
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may as well be set to the widest limits like zero and <code class="computeroutput"><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">max</span><span class="special">()</span></code>.
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</li>
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<li class="listitem">
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But if the function more complex and may have more than one root or a pole,
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the choice of bounds is protection against jumping out to seek the 'wrong'
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root.
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</li>
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<li class="listitem">
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These functions fall back to <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisect</a>
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if the next computed step would take the next value out of bounds. The
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bounds are updated after each step to ensure this leads to convergence.
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However, a good initial guess backed up by asymptotically-tight bounds
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will improve performance no end - rather than relying on <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisection</a>.
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</li>
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<li class="listitem">
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The value of <span class="emphasis"><em>digits</em></span> is crucial to good performance
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of these functions, if it is set too high then at best you will get one
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extra (unnecessary) iteration, and at worst the last few steps will proceed
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by <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisection</a>.
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Remember that the returned value can never be more accurate than <span class="emphasis"><em>f(x)</em></span>
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can be evaluated, and that if <span class="emphasis"><em>f(x)</em></span> suffers from cancellation
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errors as it tends to zero then the computed steps will be effectively
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random. The value of <span class="emphasis"><em>digits</em></span> should be set so that
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iteration terminates before this point: remember that for second and third
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order methods the number of correct digits in the result is increasing
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quite substantially with each iteration, <span class="emphasis"><em>digits</em></span> should
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be set by experiment so that the final iteration just takes the next value
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into the zone where <span class="emphasis"><em>f(x)</em></span> becomes inaccurate. A good
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starting point for <span class="emphasis"><em>digits</em></span> would be 0.6*D for Newton
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and 0.4*D for Halley or Shröder iteration, where D is <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">digits</span></code>.
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</li>
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<li class="listitem">
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If you need some diagnostic output to see what is going on, you can <code class="computeroutput"><span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_INSTRUMENT</span></code>
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before the <code class="computeroutput"><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></code>, and also ensure that display of all
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the significant digits with <code class="computeroutput"> <span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">)</span></code>: or even possibly significant digits with
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<code class="computeroutput"> <span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">max_digits10</span><span class="special">)</span></code>:
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but be warned, this may produce copious output!
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</li>
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<li class="listitem">
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Finally: you may well be able to do better than these functions by hand-coding
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the heuristics used so that they are tailored to a specific function. You
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may also be able to compute the ratio of derivatives used by these methods
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more efficiently than computing the derivatives themselves. As ever, algebraic
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simplification can be a big win.
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</li>
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</ul></div>
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<h5>
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<a name="math_toolkit.roots_deriv.h2"></a>
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<span class="phrase"><a name="math_toolkit.roots_deriv.newton"></a></span><a class="link" href="roots_deriv.html#math_toolkit.roots_deriv.newton">Newton
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Raphson Method</a>
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</h5>
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<p>
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Given an initial guess <span class="emphasis"><em>x0</em></span> the subsequent values are computed
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using:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../equations/roots1.svg"></span>
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</p></blockquote></div>
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<p>
|
|
Out-of-bounds steps revert to <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisection</a>
|
|
of the current bounds.
|
|
</p>
|
|
<p>
|
|
Under ideal conditions, the number of correct digits doubles with each iteration.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.roots_deriv.h3"></a>
|
|
<span class="phrase"><a name="math_toolkit.roots_deriv.halley"></a></span><a class="link" href="roots_deriv.html#math_toolkit.roots_deriv.halley">Halley's
|
|
Method</a>
|
|
</h5>
|
|
<p>
|
|
Given an initial guess <span class="emphasis"><em>x0</em></span> the subsequent values are computed
|
|
using:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../equations/roots2.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<p>
|
|
Over-compensation by the second derivative (one which would proceed in the
|
|
wrong direction) causes the method to revert to a Newton-Raphson step.
|
|
</p>
|
|
<p>
|
|
Out of bounds steps revert to bisection of the current bounds.
|
|
</p>
|
|
<p>
|
|
Under ideal conditions, the number of correct digits trebles with each iteration.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.roots_deriv.h4"></a>
|
|
<span class="phrase"><a name="math_toolkit.roots_deriv.schroder"></a></span><a class="link" href="roots_deriv.html#math_toolkit.roots_deriv.schroder">Schröder's
|
|
Method</a>
|
|
</h5>
|
|
<p>
|
|
Given an initial guess x0 the subsequent values are computed using:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../equations/roots3.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<p>
|
|
Over-compensation by the second derivative (one which would proceed in the
|
|
wrong direction) causes the method to revert to a Newton-Raphson step. Likewise
|
|
a Newton step is used whenever that Newton step would change the next value
|
|
by more than 10%.
|
|
</p>
|
|
<p>
|
|
Out of bounds steps revert to <a href="https://en.wikipedia.org/wiki/Bisection" target="_top">bisection</a>
|
|
of the current bounds.
|
|
</p>
|
|
<p>
|
|
Under ideal conditions, the number of correct digits trebles with each iteration.
|
|
</p>
|
|
<p>
|
|
This is Schröder's general result (equation 18 from <a href="http://drum.lib.umd.edu/handle/1903/577" target="_top">Stewart,
|
|
G. W. "On Infinitely Many Algorithms for Solving Equations." English
|
|
translation of Schröder's original paper. College Park, MD: University of Maryland,
|
|
Institute for Advanced Computer Studies, Department of Computer Science, 1993</a>.)
|
|
</p>
|
|
<p>
|
|
This method guarantees at least quadratic convergence (the same as Newton's
|
|
method), and is known to work well in the presence of multiple roots: something
|
|
that neither Newton nor Halley can do.
|
|
</p>
|
|
<p>
|
|
The complex Newton method works slightly differently than the rest of the methods:
|
|
Since there is no way to bracket roots in the complex plane, the <code class="computeroutput"><span class="identifier">min</span></code> and <code class="computeroutput"><span class="identifier">max</span></code>
|
|
arguments are not accepted. Failure to reach a root is communicated by returning
|
|
<code class="computeroutput"><span class="identifier">nan</span></code>s. Remember that if a function
|
|
has many roots, then which root the complex Newton's method converges to is
|
|
essentially impossible to predict a priori; see the Newton's fractal for more
|
|
information.
|
|
</p>
|
|
<p>
|
|
Finally, the derivative of <span class="emphasis"><em>f</em></span> must be continuous at the
|
|
root or else non-roots can be found; see <a href="https://math.stackexchange.com/questions/3017766/constructing-newton-iteration-converging-to-non-root" target="_top">here</a>
|
|
for an example.
|
|
</p>
|
|
<p>
|
|
An example usage of <code class="computeroutput"><span class="identifier">complex_newton</span></code>
|
|
is given in <code class="computeroutput"><span class="identifier">examples</span><span class="special">/</span><span class="identifier">daubechies_coefficients</span><span class="special">.</span><span class="identifier">cpp</span></code>.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.roots_deriv.h5"></a>
|
|
<span class="phrase"><a name="math_toolkit.roots_deriv.quadratics"></a></span><a class="link" href="roots_deriv.html#math_toolkit.roots_deriv.quadratics">Quadratics</a>
|
|
</h5>
|
|
<p>
|
|
To solve a quadratic <span class="emphasis"><em>ax</em></span><sup>2</sup> + <span class="emphasis"><em>bx</em></span> + <span class="emphasis"><em>c</em></span>
|
|
= 0, we may use
|
|
</p>
|
|
<pre class="programlisting"><span class="keyword">auto</span> <span class="special">[</span><span class="identifier">x0</span><span class="special">,</span> <span class="identifier">x1</span><span class="special">]</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">quadratic_roots</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">c</span><span class="special">);</span>
|
|
</pre>
|
|
<p>
|
|
If the roots are real, they are arranged so that <code class="computeroutput"><span class="identifier">x0</span></code>
|
|
≤ <code class="computeroutput"><span class="identifier">x1</span></code>. If the roots are
|
|
complex and the inputs are real, <code class="computeroutput"><span class="identifier">x0</span></code>
|
|
and <code class="computeroutput"><span class="identifier">x1</span></code> are both <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>::</span><span class="identifier">quiet_NaN</span><span class="special">()</span></code>. In this case we must cast <code class="computeroutput"><span class="identifier">a</span></code>, <code class="computeroutput"><span class="identifier">b</span></code>
|
|
and <code class="computeroutput"><span class="identifier">c</span></code> to a complex type to
|
|
extract the complex roots. If <code class="computeroutput"><span class="identifier">a</span></code>,
|
|
<code class="computeroutput"><span class="identifier">b</span></code> and <code class="computeroutput"><span class="identifier">c</span></code>
|
|
are integral, then the roots are of type double. The routine is much faster
|
|
if the fused-multiply-add instruction is available on your architecture. If
|
|
the fma is not available, the function resorts to slow emulation. Finally,
|
|
speed is improved if you compile for your particular architecture. For instance,
|
|
if you compile without any architecture flags, then the <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">fma</span></code> call
|
|
compiles down to <code class="computeroutput"><span class="identifier">call</span> <span class="identifier">_fma</span></code>,
|
|
which dynamically chooses to emulate or execute the <code class="computeroutput"><span class="identifier">vfmadd132sd</span></code>
|
|
instruction based on the capabilities of the architecture. If instead, you
|
|
compile with (say) <code class="computeroutput"><span class="special">-</span><span class="identifier">march</span><span class="special">=</span><span class="identifier">native</span></code> then
|
|
no dynamic choice is made: The <code class="computeroutput"><span class="identifier">vfmadd132sd</span></code>
|
|
instruction is always executed if available and emulation is used if not.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.roots_deriv.h6"></a>
|
|
<span class="phrase"><a name="math_toolkit.roots_deriv.examples"></a></span><a class="link" href="roots_deriv.html#math_toolkit.roots_deriv.examples">Examples</a>
|
|
</h5>
|
|
<p>
|
|
See <a class="link" href="root_finding_examples.html" title="Examples of Root-Finding (with and without derivatives)">root-finding examples</a>.
|
|
</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 © 2006-2019 Nikhar
|
|
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
|
|
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
|
|
Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, 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>
|
|
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