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<title>Root Finding With Derivatives: Newton-Raphson, Halley & Schröder</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.roots.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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</h3></div></div></div>
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<h5>
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<a name="math_toolkit.roots.roots_deriv.h0"></a>
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<span class="phrase"><a name="math_toolkit.roots.roots_deriv.synopsis"></a></span><a class="link" href="roots_deriv.html#math_toolkit.roots.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="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.roots_deriv.h1"></a>
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<span class="phrase"><a name="math_toolkit.roots.roots_deriv.description"></a></span><a class="link" href="roots_deriv.html#math_toolkit.roots.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.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.roots_deriv.halley">Halley</a> and
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<a class="link" href="roots_deriv.html#math_toolkit.roots.roots_deriv.schroder">Schröder</a>
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iteration.
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</li>
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</ul></div>
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<p>
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The functions all take the same parameters:
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</p>
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<div class="variablelist">
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<p class="title"><b>Parameters of the 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 that accepts one parameter
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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
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is 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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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
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prematurely. A large first derivative leads to a very small next step,
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triggering the termination condition. Derivative based iteration may
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not be appropriate 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
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pole, the choice of bounds is protection against jumping out to seek
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the 'wrong' 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
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cancellation errors as it tends to zero then the computed steps will
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be effectively random. The value of <span class="emphasis"><em>digits</em></span> should
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be set so that iteration terminates before this point: remember that
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for second and third order methods the number of correct digits in the
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result is increasing quite substantially with each iteration, <span class="emphasis"><em>digits</em></span>
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should be set by experiment so that the final iteration just takes the
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next value into the zone where <span class="emphasis"><em>f(x)</em></span> becomes inaccurate.
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A good starting point for <span class="emphasis"><em>digits</em></span> would be 0.6*D
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for Newton 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
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<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
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all 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
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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">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.
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You may also be able to compute the ratio of derivatives used by these
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methods more efficiently than computing the derivatives themselves. As
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ever, algebraic 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.roots_deriv.h2"></a>
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<span class="phrase"><a name="math_toolkit.roots.roots_deriv.newton"></a></span><a class="link" href="roots_deriv.html#math_toolkit.roots.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
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computed using:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/roots1.svg"></span>
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</p>
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<p>
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Out of bounds steps revert to <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisection</a>
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of the current bounds.
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</p>
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<p>
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Under ideal conditions, the number of correct digits doubles with each iteration.
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</p>
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<h5>
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<a name="math_toolkit.roots.roots_deriv.h3"></a>
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<span class="phrase"><a name="math_toolkit.roots.roots_deriv.halley"></a></span><a class="link" href="roots_deriv.html#math_toolkit.roots.roots_deriv.halley">Halley's
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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
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computed using:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/roots2.svg"></span>
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</p>
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<p>
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Over-compensation by the second derivative (one which would proceed in the
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wrong direction) causes the method to revert to a Newton-Raphson step.
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</p>
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<p>
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Out of bounds steps revert to bisection of the current bounds.
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</p>
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<p>
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Under ideal conditions, the number of correct digits trebles with each iteration.
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</p>
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<h5>
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<a name="math_toolkit.roots.roots_deriv.h4"></a>
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<span class="phrase"><a name="math_toolkit.roots.roots_deriv.schroder"></a></span><a class="link" href="roots_deriv.html#math_toolkit.roots.roots_deriv.schroder">Schröder's
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Method</a>
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</h5>
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<p>
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Given an initial guess x0 the subsequent values are computed using:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/roots3.svg"></span>
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</p>
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<p>
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Over-compensation by the second derivative (one which would proceed in the
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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>
|
|
<h5>
|
|
<a name="math_toolkit.roots.roots_deriv.h5"></a>
|
|
<span class="phrase"><a name="math_toolkit.roots.roots_deriv.examples"></a></span><a class="link" href="roots_deriv.html#math_toolkit.roots.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 © 2006-2010, 2012-2014 Nikhar Agrawal,
|
|
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
|
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, 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>
|
|
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