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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.internals1.roots"></a><a class="link" href="roots.html" title="Root Finding With Derivatives: Newton-Raphson, Halley & Schroeder">Root Finding With Derivatives:
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Newton-Raphson, Halley & Schroeder</a>
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</h3></div></div></div>
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<h5>
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<a name="math_toolkit.internals1.roots.h0"></a>
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<span class="phrase"><a name="math_toolkit.internals1.roots.synopsis"></a></span><a class="link" href="roots.html#math_toolkit.internals1.roots.synopsis">Synopsis</a>
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</h5>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">roots</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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</pre>
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<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
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<span class="keyword">namespace</span> <span class="identifier">tools</span><span class="special">{</span>
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<span class="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="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="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">schroeder_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">schroeder_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</span>
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</pre>
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<h5>
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<a name="math_toolkit.internals1.roots.h1"></a>
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<span class="phrase"><a name="math_toolkit.internals1.roots.description"></a></span><a class="link" href="roots.html#math_toolkit.internals1.roots.description">Description</a>
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</h5>
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<p>
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These functions all perform iterative root finding using derivatives:
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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.html#math_toolkit.internals1.roots.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">schroeder_iterate</span></code> perform third order
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<a class="link" href="roots.html#math_toolkit.internals1.roots.halley">Halley</a> and
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<a class="link" href="roots.html#math_toolkit.internals1.roots.schroeder">Schroeder</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="tuples.html" title="Tuples">boost::math::tuple</a>:
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</p>
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<p>
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For the second order iterative methods (<a href="http://en.wikipedia.org/wiki/Newton_Raphson" target="_top">Newton
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Raphson</a>) the <a class="link" href="tuples.html" title="Tuples">boost::math::tuple</a>
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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 Schroeder) the <a class="link" href="tuples.html" title="Tuples">boost::math::tuple</a>
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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.
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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.
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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 max_iter = <code class="computeroutput"><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 bisection, or in some rare
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cases, even worse than bisection if the initial guess is a long way from
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the correct value and the derivatives 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 bisection in this case.
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However, it is helpful if functor F is defined to return an arbitrarily
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small value <span class="emphasis"><em>of the correct sign</em></span> rather 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' (monotonic and has only one
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root) the bracket bounds min and max may as well be set to the widest
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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 bisection if the next computed step would
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take the next value out of bounds. The bounds are updated after each
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step to ensure this leads to convergence. However, a good initial guess
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backed up by asymptotically-tight bounds will improve performance no
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end - rather than relying on bisection.
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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 bisection. Remember that the returned value can never be more accurate
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than f(x) can be evaluated, and that if f(x) 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
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third 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>
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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 f(x) becomes inaccurate.
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</li>
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<li class="listitem">
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To get the binary digits of accuracy, use policies::get_max_root_iterations<Policy>()).
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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 possibly 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">max_digits10</span><span class="special">)</span></code>: but be warned, this may produce copious
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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.internals1.roots.h2"></a>
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<span class="phrase"><a name="math_toolkit.internals1.roots.newton"></a></span><a class="link" href="roots.html#math_toolkit.internals1.roots.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 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/roots1.png"></span>
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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 doubles with each iteration.
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</p>
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<h5>
|
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|
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<a name="math_toolkit.internals1.roots.h3"></a>
|
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<span class="phrase"><a name="math_toolkit.internals1.roots.halley"></a></span><a class="link" href="roots.html#math_toolkit.internals1.roots.halley">Halley's
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|
|
Method</a>
|
||
|
|
</h5>
|
||
|
|
<p>
|
||
|
|
Given an initial guess x0 the subsequent values are computed using:
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<span class="inlinemediaobject"><img src="../../../equations/roots2.png"></span>
|
||
|
|
</p>
|
||
|
|
<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.internals1.roots.h4"></a>
|
||
|
|
<span class="phrase"><a name="math_toolkit.internals1.roots.schroeder"></a></span><a class="link" href="roots.html#math_toolkit.internals1.roots.schroeder">Schroeder's
|
||
|
|
Method</a>
|
||
|
|
</h5>
|
||
|
|
<p>
|
||
|
|
Given an initial guess x0 the subsequent values are computed using:
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<span class="inlinemediaobject"><img src="../../../equations/roots3.png"></span>
|
||
|
|
</p>
|
||
|
|
<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 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.internals1.roots.h5"></a>
|
||
|
|
<span class="phrase"><a name="math_toolkit.internals1.roots.example"></a></span><a class="link" href="roots.html#math_toolkit.internals1.roots.example">Example</a>
|
||
|
|
</h5>
|
||
|
|
<p>
|
||
|
|
Let's suppose we want to find the cube root of a number: the equation we
|
||
|
|
want to solve along with its derivatives are:
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<span class="inlinemediaobject"><img src="../../../equations/roots4.png"></span>
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
To begin with lets solve the problem using Newton-Raphson iterations, we'll
|
||
|
|
begin by defining a function object (functor) that returns the evaluation
|
||
|
|
of the function to solve, along with its first derivative f'(x):
|
||
|
|
</p>
|
||
|
|
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||
|
|
<span class="keyword">struct</span> <span class="identifier">cbrt_functor</span>
|
||
|
|
<span class="special">{</span>
|
||
|
|
<span class="identifier">cbrt_functor</span><span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span><span class="special">&</span> <span class="identifier">target</span><span class="special">)</span> <span class="special">:</span> <span class="identifier">a</span><span class="special">(</span><span class="identifier">target</span><span class="special">)</span>
|
||
|
|
<span class="special">{</span> <span class="comment">// Constructor stores value to be 'cube-rooted'.</span>
|
||
|
|
<span class="special">}</span>
|
||
|
|
<a class="link" href="tuples.html" title="Tuples">boost::math::tuple</a><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">T</span> <span class="keyword">const</span><span class="special">&</span> <span class="identifier">z</span><span class="special">)</span>
|
||
|
|
<span class="special">{</span> <span class="comment">// z is estimate so far.</span>
|
||
|
|
<span class="keyword">return</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">make_tuple</span><span class="special">(</span>
|
||
|
|
<span class="identifier">z</span><span class="special">*</span><span class="identifier">z</span><span class="special">*</span><span class="identifier">z</span> <span class="special">-</span> <span class="identifier">a</span><span class="special">,</span> <span class="comment">// return both f(x)</span>
|
||
|
|
<span class="number">3</span> <span class="special">*</span> <span class="identifier">z</span><span class="special">*</span><span class="identifier">z</span><span class="special">);</span> <span class="comment">// and f'(x)</span>
|
||
|
|
<span class="special">}</span>
|
||
|
|
<span class="keyword">private</span><span class="special">:</span>
|
||
|
|
<span class="identifier">T</span> <span class="identifier">a</span><span class="special">;</span> <span class="comment">// to be 'cube-rooted'.</span>
|
||
|
|
<span class="special">};</span>
|
||
|
|
</pre>
|
||
|
|
<p>
|
||
|
|
Implementing the cube root is fairly trivial now, the hardest part is finding
|
||
|
|
a good approximation to begin with: in this case we'll just divide the exponent
|
||
|
|
by three:
|
||
|
|
</p>
|
||
|
|
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||
|
|
<span class="identifier">T</span> <span class="identifier">cbrt</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">)</span>
|
||
|
|
<span class="special">{</span>
|
||
|
|
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span> <span class="comment">// for frexp, ldexp, numeric_limits.</span>
|
||
|
|
<span class="keyword">using</span> <span class="keyword">namespace</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="keyword">int</span> <span class="identifier">exp</span><span class="special">;</span>
|
||
|
|
<span class="identifier">frexp</span><span class="special">(</span><span class="identifier">z</span><span class="special">,</span> <span class="special">&</span><span class="identifier">exp</span><span class="special">);</span> <span class="comment">// Get exponent of z (ignore mantissa).</span>
|
||
|
|
<span class="identifier">T</span> <span class="identifier">min</span> <span class="special">=</span> <span class="identifier">ldexp</span><span class="special">(</span><span class="number">0.5</span><span class="special">,</span> <span class="identifier">exp</span><span class="special">/</span><span class="number">3</span><span class="special">);</span>
|
||
|
|
<span class="identifier">T</span> <span class="identifier">max</span> <span class="special">=</span> <span class="identifier">ldexp</span><span class="special">(</span><span class="number">2.0</span><span class="special">,</span> <span class="identifier">exp</span><span class="special">/</span><span class="number">3</span><span class="special">);</span>
|
||
|
|
<span class="identifier">T</span> <span class="identifier">guess</span> <span class="special">=</span> <span class="identifier">ldexp</span><span class="special">(</span><span class="number">1.0</span><span class="special">,</span> <span class="identifier">exp</span><span class="special">/</span><span class="number">3</span><span class="special">);</span> <span class="comment">// Rough guess is to divide the exponent by three.</span>
|
||
|
|
<span class="keyword">int</span> <span class="identifier">digits</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="identifier">T</span><span class="special">>::</span><span class="identifier">digits</span><span class="special">;</span> <span class="comment">// Maximum possible binary digits accuracy for type T.</span>
|
||
|
|
<span class="keyword">return</span> <span class="identifier">newton_raphson_iterate</span><span class="special">(</span><span class="identifier">detail</span><span class="special">::</span><span class="identifier">cbrt_functor</span><span class="special"><</span><span class="identifier">T</span><span class="special">>(</span><span class="identifier">z</span><span class="special">),</span> <span class="identifier">guess</span><span class="special">,</span> <span class="identifier">min</span><span class="special">,</span> <span class="identifier">max</span><span class="special">,</span> <span class="identifier">digits</span><span class="special">);</span>
|
||
|
|
<span class="special">}</span>
|
||
|
|
</pre>
|
||
|
|
<p>
|
||
|
|
Using the test data in <code class="computeroutput"><span class="identifier">libs</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">test</span><span class="special">/</span><span class="identifier">cbrt_test</span><span class="special">.</span><span class="identifier">cpp</span></code> this
|
||
|
|
found the cube root exact to the last digit in every case, and in no more
|
||
|
|
than 6 iterations at double precision. However, you will note that a high
|
||
|
|
precision was used in this example, exactly what was warned against earlier
|
||
|
|
on in these docs! In this particular case it is possible to compute f(x)
|
||
|
|
exactly and without undue cancellation error, so a high limit is not too
|
||
|
|
much of an issue. However, reducing the limit to <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> <span class="special">*</span> <span class="number">2</span> <span class="special">/</span> <span class="number">3</span></code>
|
||
|
|
gave full precision in all but one of the test cases (and that one was out
|
||
|
|
by just one bit). The maximum number of iterations remained 6, but in most
|
||
|
|
cases was reduced by one.
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
Note also that the above code omits a probably optimization by computing
|
||
|
|
z², and reusing it, omits error handling, and does not handle negative values
|
||
|
|
of z correctly. (These are left as an exercise for the reader!)
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
The <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cbrt</span></code> function also includes these and other
|
||
|
|
improvements.
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
Now let's adapt the functor slightly to return the second derivative as well:
|
||
|
|
</p>
|
||
|
|
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||
|
|
<span class="keyword">struct</span> <span class="identifier">cbrt_functor</span>
|
||
|
|
<span class="special">{</span>
|
||
|
|
<span class="identifier">cbrt_functor</span><span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span><span class="special">&</span> <span class="identifier">target</span><span class="special">)</span> <span class="special">:</span> <span class="identifier">a</span><span class="special">(</span><span class="identifier">target</span><span class="special">){}</span>
|
||
|
|
<a class="link" href="tuples.html" title="Tuples">boost::math::tuple</a><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">T</span> <span class="keyword">const</span><span class="special">&</span> <span class="identifier">z</span><span class="special">)</span>
|
||
|
|
<span class="special">{</span>
|
||
|
|
<span class="keyword">return</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">make_tuple</span><span class="special">(</span>
|
||
|
|
<span class="identifier">z</span><span class="special">*</span><span class="identifier">z</span><span class="special">*</span><span class="identifier">z</span> <span class="special">-</span> <span class="identifier">a</span><span class="special">,</span>
|
||
|
|
<span class="number">3</span> <span class="special">*</span> <span class="identifier">z</span><span class="special">*</span><span class="identifier">z</span><span class="special">,</span>
|
||
|
|
<span class="number">6</span> <span class="special">*</span> <span class="identifier">z</span><span class="special">);</span>
|
||
|
|
<span class="special">}</span>
|
||
|
|
<span class="keyword">private</span><span class="special">:</span>
|
||
|
|
<span class="identifier">T</span> <span class="identifier">a</span><span class="special">;</span>
|
||
|
|
<span class="special">};</span>
|
||
|
|
</pre>
|
||
|
|
<p>
|
||
|
|
And then adapt the <code class="computeroutput"><span class="identifier">cbrt</span></code> function
|
||
|
|
to use Halley iterations:
|
||
|
|
</p>
|
||
|
|
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||
|
|
<span class="identifier">T</span> <span class="identifier">cbrt</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">)</span>
|
||
|
|
<span class="special">{</span>
|
||
|
|
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>
|
||
|
|
<span class="keyword">using</span> <span class="keyword">namespace</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="keyword">int</span> <span class="identifier">exp</span><span class="special">;</span>
|
||
|
|
<span class="identifier">frexp</span><span class="special">(</span><span class="identifier">z</span><span class="special">,</span> <span class="special">&</span><span class="identifier">exp</span><span class="special">);</span>
|
||
|
|
<span class="identifier">T</span> <span class="identifier">min</span> <span class="special">=</span> <span class="identifier">ldexp</span><span class="special">(</span><span class="number">0.5</span><span class="special">,</span> <span class="identifier">exp</span><span class="special">/</span><span class="number">3</span><span class="special">);</span>
|
||
|
|
<span class="identifier">T</span> <span class="identifier">max</span> <span class="special">=</span> <span class="identifier">ldexp</span><span class="special">(</span><span class="number">2.0</span><span class="special">,</span> <span class="identifier">exp</span><span class="special">/</span><span class="number">3</span><span class="special">);</span>
|
||
|
|
<span class="identifier">T</span> <span class="identifier">guess</span> <span class="special">=</span> <span class="identifier">ldexp</span><span class="special">(</span><span class="number">1.0</span><span class="special">,</span> <span class="identifier">exp</span><span class="special">/</span><span class="number">3</span><span class="special">);</span>
|
||
|
|
<span class="keyword">int</span> <span class="identifier">digits</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="identifier">T</span><span class="special">>::</span><span class="identifier">digits</span> <span class="special">/</span> <span class="number">2</span><span class="special">;</span>
|
||
|
|
<span class="keyword">return</span> <span class="identifier">halley_iterate</span><span class="special">(</span><span class="identifier">detail</span><span class="special">::</span><span class="identifier">cbrt_functor</span><span class="special"><</span><span class="identifier">T</span><span class="special">>(</span><span class="identifier">z</span><span class="special">),</span> <span class="identifier">guess</span><span class="special">,</span> <span class="identifier">min</span><span class="special">,</span> <span class="identifier">max</span><span class="special">,</span> <span class="identifier">digits</span><span class="special">);</span>
|
||
|
|
<span class="special">}</span>
|
||
|
|
</pre>
|
||
|
|
<p>
|
||
|
|
Note that the iterations are set to stop at just one-half of full precision,
|
||
|
|
and yet, even so, not one of the test cases had a single bit wrong. What's
|
||
|
|
more, the maximum number of iterations was now just 4.
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
Just to complete the picture, we could have called <code class="computeroutput"><span class="identifier">schroeder_iterate</span></code>
|
||
|
|
in the last example: and in fact it makes no difference to the accuracy or
|
||
|
|
number of iterations in this particular case. However, the relative performance
|
||
|
|
of these two methods may vary depending upon the nature of f(x), and the
|
||
|
|
accuracy to which the initial guess can be computed. There appear to be no
|
||
|
|
generalisations that can be made except "try them and see".
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
Finally, had we called <code class="computeroutput"><span class="identifier">cbrt</span></code>
|
||
|
|
with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a> set to
|
||
|
|
1000 bit precision, then full precision can be obtained with just 7 iterations.
|
||
|
|
To put that in perspective, an increase in precision by a factor of 20, has
|
||
|
|
less than doubled the number of iterations. That just goes to emphasise that
|
||
|
|
most of the iterations are used up getting the first few digits correct:
|
||
|
|
after that these methods can churn out further digits with remarkable efficiency.
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
Or to put it another way: <span class="emphasis"><em>nothing beats a really good initial guess!</em></span>
|
||
|
|
</p>
|
||
|
|
</div>
|
||
|
|
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||
|
|
<td align="left"></td>
|
||
|
|
<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012, 2013 Paul A. Bristow, Christopher Kormanyos,
|
||
|
|
Hubert Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin
|
||
|
|
Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
|
||
|
|
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||
|
|
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||
|
|
</p>
|
||
|
|
</div></td>
|
||
|
|
</tr></table>
|
||
|
|
<hr>
|
||
|
|
<div class="spirit-nav">
|
||
|
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<a accesskey="p" href="rational.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../internals1.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="roots2.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
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|
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