| <html> |
| <head> |
| <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> |
| <title>Root Finding With Derivatives: Newton-Raphson, Halley & Schroeder</title> |
| <link rel="stylesheet" href="../../math.css" type="text/css"> |
| <meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> |
| <link rel="home" href="../../index.html" title="Math Toolkit 2.2.0"> |
| <link rel="up" href="../internals1.html" title="Utilities & internals"> |
| <link rel="prev" href="rational.html" title="Polynomial and Rational Function Evaluation"> |
| <link rel="next" href="roots2.html" title="Root Finding Without Derivatives: Bisection, Bracket and TOMS748"> |
| </head> |
| <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> |
| <table cellpadding="2" width="100%"><tr> |
| <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> |
| <td align="center"><a href="../../../../../../index.html">Home</a></td> |
| <td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> |
| <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> |
| <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> |
| <td align="center"><a href="../../../../../../more/index.htm">More</a></td> |
| </tr></table> |
| <hr> |
| <div class="spirit-nav"> |
| <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> |
| </div> |
| <div class="section"> |
| <div class="titlepage"><div><div><h3 class="title"> |
| <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: |
| Newton-Raphson, Halley & Schroeder</a> |
| </h3></div></div></div> |
| <h5> |
| <a name="math_toolkit.internals1.roots.h0"></a> |
| <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> |
| </h5> |
| <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> |
| </pre> |
| <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> |
| <span class="keyword">namespace</span> <span class="identifier">tools</span><span class="special">{</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> |
| <span class="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="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> |
| <span class="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> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> |
| <span class="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="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> |
| <span class="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> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> |
| <span class="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="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> |
| <span class="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> |
| |
| <span class="special">}}}</span> <span class="comment">// namespaces</span> |
| </pre> |
| <h5> |
| <a name="math_toolkit.internals1.roots.h1"></a> |
| <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> |
| </h5> |
| <p> |
| These functions all perform iterative root finding using derivatives: |
| </p> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| <code class="computeroutput"><span class="identifier">newton_raphson_iterate</span></code> |
| performs second order <a class="link" href="roots.html#math_toolkit.internals1.roots.newton">Newton-Raphson |
| iteration</a>, |
| </li> |
| <li class="listitem"> |
| <code class="computeroutput"><span class="identifier">halley_iterate</span></code> and<code class="computeroutput"><span class="identifier">schroeder_iterate</span></code> perform third order |
| <a class="link" href="roots.html#math_toolkit.internals1.roots.halley">Halley</a> and |
| <a class="link" href="roots.html#math_toolkit.internals1.roots.schroeder">Schroeder</a> |
| iteration. |
| </li> |
| </ul></div> |
| <p> |
| The functions all take the same parameters: |
| </p> |
| <div class="variablelist"> |
| <p class="title"><b>Parameters of the root finding functions</b></p> |
| <dl class="variablelist"> |
| <dt><span class="term">F f</span></dt> |
| <dd> |
| <p> |
| Type F must be a callable function object that accepts one parameter |
| and returns a <a class="link" href="tuples.html" title="Tuples">boost::math::tuple</a>: |
| </p> |
| <p> |
| For the second order iterative methods (<a href="http://en.wikipedia.org/wiki/Newton_Raphson" target="_top">Newton |
| Raphson</a>) the <a class="link" href="tuples.html" title="Tuples">boost::math::tuple</a> |
| should have <span class="bold"><strong>two</strong></span> elements containing |
| the evaluation of the function and its first derivative. |
| </p> |
| <p> |
| For the third order methods (<a href="http://en.wikipedia.org/wiki/Halley%27s_method" target="_top">Halley</a> |
| and Schroeder) the <a class="link" href="tuples.html" title="Tuples">boost::math::tuple</a> |
| should have <span class="bold"><strong>three</strong></span> elements containing |
| the evaluation of the function and its first and second derivatives. |
| </p> |
| </dd> |
| <dt><span class="term">T guess</span></dt> |
| <dd><p> |
| The initial starting value. A good guess is crucial to quick convergence! |
| </p></dd> |
| <dt><span class="term">T min</span></dt> |
| <dd><p> |
| The minimum possible value for the result, this is used as an initial |
| lower bracket. |
| </p></dd> |
| <dt><span class="term">T max</span></dt> |
| <dd><p> |
| The maximum possible value for the result, this is used as an initial |
| upper bracket. |
| </p></dd> |
| <dt><span class="term">int digits</span></dt> |
| <dd><p> |
| The desired number of binary digits. |
| </p></dd> |
| <dt><span class="term">uintmax_t& max_iter</span></dt> |
| <dd><p> |
| An optional maximum number of iterations to perform. On exit this is |
| set to the actual number of iterations performed. |
| </p></dd> |
| </dl> |
| </div> |
| <p> |
| When using these functions you should note that: |
| </p> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| 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'!. |
| </li> |
| <li class="listitem"> |
| They may be very sensitive to the initial guess, typically they converge |
| very rapidly if the initial guess has two or three decimal digits correct. |
| However convergence can be no better than bisection, or in some rare |
| cases, even worse than bisection if the initial guess is a long way from |
| the correct value and the derivatives are close to zero. |
| </li> |
| <li class="listitem"> |
| These functions include special cases to handle zero first (and second |
| where appropriate) derivatives, and fall back to bisection in this case. |
| However, it is helpful if functor F is defined to return an arbitrarily |
| small value <span class="emphasis"><em>of the correct sign</em></span> rather than zero. |
| </li> |
| <li class="listitem"> |
| If the derivative at the current best guess for the result is infinite |
| (or very close to being infinite) then these functions may terminate |
| prematurely. A large first derivative leads to a very small next step, |
| triggering the termination condition. Derivative based iteration may |
| not be appropriate in such cases. |
| </li> |
| <li class="listitem"> |
| If the function is 'Really Well Behaved' (monotonic and has only one |
| root) the bracket bounds min and max 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>. |
| </li> |
| <li class="listitem"> |
| But if the function more complex and may have more than one root or a |
| pole, the choice of bounds is protection against jumping out to seek |
| the 'wrong' root. |
| </li> |
| <li class="listitem"> |
| These functions fall back to bisection if the next computed step would |
| take the next value out of bounds. The bounds are updated after each |
| step to ensure this leads to convergence. However, a good initial guess |
| backed up by asymptotically-tight bounds will improve performance no |
| end - rather than relying on bisection. |
| </li> |
| <li class="listitem"> |
| The value of <span class="emphasis"><em>digits</em></span> is crucial to good performance |
| of these functions, if it is set too high then at best you will get one |
| extra (unnecessary) iteration, and at worst the last few steps will proceed |
| by bisection. Remember that the returned value can never be more accurate |
| than f(x) can be evaluated, and that if f(x) suffers from cancellation |
| errors as it tends to zero then the computed steps will be effectively |
| random. The value of <span class="emphasis"><em>digits</em></span> should be set so that |
| iteration terminates before this point: remember that for second and |
| third order methods the number of correct digits in the result is increasing |
| quite substantially with each iteration, <span class="emphasis"><em>digits</em></span> |
| should be set by experiment so that the final iteration just takes the |
| next value into the zone where f(x) becomes inaccurate. |
| </li> |
| <li class="listitem"> |
| To get the binary digits of accuracy, use policies::get_max_root_iterations<Policy>()). |
| </li> |
| <li class="listitem"> |
| 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> |
| 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 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 |
| output! |
| </li> |
| <li class="listitem"> |
| Finally: you may well be able to do better than these functions by hand-coding |
| the heuristics used so that they are tailored to a specific function. |
| You may also be able to compute the ratio of derivatives used by these |
| methods more efficiently than computing the derivatives themselves. As |
| ever, algebraic simplification can be a big win. |
| </li> |
| </ul></div> |
| <h5> |
| <a name="math_toolkit.internals1.roots.h2"></a> |
| <span class="phrase"><a name="math_toolkit.internals1.roots.newton"></a></span><a class="link" href="roots.html#math_toolkit.internals1.roots.newton">Newton |
| Raphson Method</a> |
| </h5> |
| <p> |
| Given an initial guess x0 the subsequent values are computed using: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/roots1.svg"></span> |
| </p> |
| <p> |
| Out of bounds steps revert to bisection of the current bounds. |
| </p> |
| <p> |
| Under ideal conditions, the number of correct digits doubles with each iteration. |
| </p> |
| <h5> |
| <a name="math_toolkit.internals1.roots.h3"></a> |
| <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 |
| Method</a> |
| </h5> |
| <p> |
| Given an initial guess x0 the subsequent values are computed using: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/roots2.svg"></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.svg"></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.svg"></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, 2012-2014 Nikhar Agrawal, |
| Anton Bikineev, Paul A. Bristow, Marco Guazzone, 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"> |
| <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> |
| </div> |
| </body> |
| </html> |