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1 | <html> |
| 2 | <head> | |
| 3 | <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> | |
| 4 | <title>Root Finding Without Derivatives</title> | |
| 5 | <link rel="stylesheet" href="../../math.css" type="text/css"> | |
| 6 | <meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> | |
| 7 | <link rel="home" href="../../index.html" title="Math Toolkit 2.5.1"> | |
| 8 | <link rel="up" href="../roots.html" title="Root finding"> | |
| 9 | <link rel="prev" href="../roots.html" title="Root finding"> | |
| 10 | <link rel="next" href="roots_noderiv/bisect.html" title="Bisection"> | |
| 11 | </head> | |
| 12 | <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> | |
| 13 | <table cellpadding="2" width="100%"><tr> | |
| 14 | <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> | |
| 15 | <td align="center"><a href="../../../../../../index.html">Home</a></td> | |
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| 17 | <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> | |
| 18 | <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> | |
| 19 | <td align="center"><a href="../../../../../../more/index.htm">More</a></td> | |
| 20 | </tr></table> | |
| 21 | <hr> | |
| 22 | <div class="spirit-nav"> | |
| 23 | <a accesskey="p" href="../roots.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../roots.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="roots_noderiv/bisect.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> | |
| 24 | </div> | |
| 25 | <div class="section"> | |
| 26 | <div class="titlepage"><div><div><h3 class="title"> | |
| 27 | <a name="math_toolkit.roots.roots_noderiv"></a><a class="link" href="roots_noderiv.html" title="Root Finding Without Derivatives">Root Finding Without | |
| 28 | Derivatives</a> | |
| 29 | </h3></div></div></div> | |
| 30 | <div class="toc"><dl> | |
| 31 | <dt><span class="section"><a href="roots_noderiv/bisect.html">Bisection</a></span></dt> | |
| 32 | <dt><span class="section"><a href="roots_noderiv/bracket_solve.html">Bracket | |
| 33 | and Solve Root</a></span></dt> | |
| 34 | <dt><span class="section"><a href="roots_noderiv/TOMS748.html">Algorithm | |
| 35 | TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions</a></span></dt> | |
| 36 | <dt><span class="section"><a href="roots_noderiv/brent.html">Brent-Decker | |
| 37 | Algorithm</a></span></dt> | |
| 38 | <dt><span class="section"><a href="roots_noderiv/root_termination.html">Termination | |
| 39 | Condition Functors</a></span></dt> | |
| 40 | <dt><span class="section"><a href="roots_noderiv/implementation.html">Implementation</a></span></dt> | |
| 41 | </dl></div> | |
| 42 | <h5> | |
| 43 | <a name="math_toolkit.roots.roots_noderiv.h0"></a> | |
| 44 | <span class="phrase"><a name="math_toolkit.roots.roots_noderiv.synopsis"></a></span><a class="link" href="roots_noderiv.html#math_toolkit.roots.roots_noderiv.synopsis">Synopsis</a> | |
| 45 | </h5> | |
| 46 | <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> | |
| 47 | </pre> | |
| 48 | <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> | |
| 49 | <span class="keyword">namespace</span> <span class="identifier">tools</span> <span class="special">{</span> <span class="comment">// Note namespace boost::math::tools.</span> | |
| 50 | <span class="comment">// Bisection</span> | |
| 51 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">></span> | |
| 52 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span> | |
| 53 | <span class="identifier">bisect</span><span class="special">(</span> | |
| 54 | <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> | |
| 55 | <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span> | |
| 56 | <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> | |
| 57 | <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> | |
| 58 | <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> | |
| 59 | ||
| 60 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">></span> | |
| 61 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span> | |
| 62 | <span class="identifier">bisect</span><span class="special">(</span> | |
| 63 | <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> | |
| 64 | <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span> | |
| 65 | <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> | |
| 66 | <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">);</span> | |
| 67 | ||
| 68 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
| 69 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span> | |
| 70 | <span class="identifier">bisect</span><span class="special">(</span> | |
| 71 | <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> | |
| 72 | <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span> | |
| 73 | <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> | |
| 74 | <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> | |
| 75 | <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> | |
| 76 | <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
| 77 | ||
| 78 | <span class="comment">// Bracket and Solve Root</span> | |
| 79 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">></span> | |
| 80 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span> | |
| 81 | <span class="identifier">bracket_and_solve_root</span><span class="special">(</span> | |
| 82 | <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> | |
| 83 | <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">guess</span><span class="special">,</span> | |
| 84 | <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">factor</span><span class="special">,</span> | |
| 85 | <span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span> | |
| 86 | <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> | |
| 87 | <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> | |
| 88 | ||
| 89 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
| 90 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span> | |
| 91 | <span class="identifier">bracket_and_solve_root</span><span class="special">(</span> | |
| 92 | <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> | |
| 93 | <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">guess</span><span class="special">,</span> | |
| 94 | <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">factor</span><span class="special">,</span> | |
| 95 | <span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span> | |
| 96 | <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> | |
| 97 | <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> | |
| 98 | <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
| 99 | ||
| 100 | <span class="comment">// TOMS 748 algorithm</span> | |
| 101 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">></span> | |
| 102 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span> | |
| 103 | <span class="identifier">toms748_solve</span><span class="special">(</span> | |
| 104 | <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> | |
| 105 | <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> | |
| 106 | <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span> | |
| 107 | <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> | |
| 108 | <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> | |
| 109 | ||
| 110 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
| 111 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span> | |
| 112 | <span class="identifier">toms748_solve</span><span class="special">(</span> | |
| 113 | <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> | |
| 114 | <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> | |
| 115 | <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span> | |
| 116 | <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> | |
| 117 | <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> | |
| 118 | <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
| 119 | ||
| 120 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">></span> | |
| 121 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span> | |
| 122 | <span class="identifier">toms748_solve</span><span class="special">(</span> | |
| 123 | <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> | |
| 124 | <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> | |
| 125 | <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span> | |
| 126 | <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fa</span><span class="special">,</span> | |
| 127 | <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fb</span><span class="special">,</span> | |
| 128 | <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> | |
| 129 | <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> | |
| 130 | ||
| 131 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
| 132 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span> | |
| 133 | <span class="identifier">toms748_solve</span><span class="special">(</span> | |
| 134 | <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> | |
| 135 | <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> | |
| 136 | <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span> | |
| 137 | <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fa</span><span class="special">,</span> | |
| 138 | <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fb</span><span class="special">,</span> | |
| 139 | <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> | |
| 140 | <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> | |
| 141 | <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
| 142 | ||
| 143 | <span class="comment">// Termination conditions:</span> | |
| 144 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
| 145 | <span class="keyword">struct</span> <span class="identifier">eps_tolerance</span><span class="special">;</span> | |
| 146 | ||
| 147 | <span class="keyword">struct</span> <span class="identifier">equal_floor</span><span class="special">;</span> | |
| 148 | <span class="keyword">struct</span> <span class="identifier">equal_ceil</span><span class="special">;</span> | |
| 149 | <span class="keyword">struct</span> <span class="identifier">equal_nearest_integer</span><span class="special">;</span> | |
| 150 | ||
| 151 | <span class="special">}}}</span> <span class="comment">// boost::math::tools namespaces</span> | |
| 152 | </pre> | |
| 153 | <h5> | |
| 154 | <a name="math_toolkit.roots.roots_noderiv.h1"></a> | |
| 155 | <span class="phrase"><a name="math_toolkit.roots.roots_noderiv.description"></a></span><a class="link" href="roots_noderiv.html#math_toolkit.roots.roots_noderiv.description">Description</a> | |
| 156 | </h5> | |
| 157 | <p> | |
| 158 | These functions solve the root of some function <span class="emphasis"><em>f(x)</em></span> | |
| 159 | - <span class="emphasis"><em>without the need for any derivatives of <span class="emphasis"><em>f(x)</em></span></em></span>. | |
| 160 | </p> | |
| 161 | <p> | |
| 162 | The <code class="computeroutput"><span class="identifier">bracket_and_solve_root</span></code> | |
| 163 | functions use <a class="link" href="roots_noderiv/TOMS748.html" title="Algorithm TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions">TOMS | |
| 164 | 748 algorithm</a> by Alefeld, Potra and Shi that is asymptotically the | |
| 165 | most efficient known, and has been shown to be optimal for a certain classes | |
| 166 | of smooth functions. Variants with and without <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policies</a> | |
| 167 | are provided. | |
| 168 | </p> | |
| 169 | <p> | |
| 170 | Alternatively, <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisect</a> | |
| 171 | is a simple <a href="https://en.wikipedia.org/wiki/Bisection" target="_top">bisection</a> | |
| 172 | routine which can be useful in its own right in some situations, or alternatively | |
| 173 | for narrowing down the range containing the root, prior to calling a more | |
| 174 | advanced algorithm. | |
| 175 | </p> | |
| 176 | <p> | |
| 177 | All the algorithms in this section reduce the diameter of the enclosing interval | |
| 178 | with the same asymptotic efficiency with which they locate the root. This | |
| 179 | is in contrast to the derivative based methods which may <span class="emphasis"><em>never</em></span> | |
| 180 | significantly reduce the enclosing interval, even though they rapidly approach | |
| 181 | the root. This is also in contrast to some other derivative-free methods | |
| 182 | (for example, Brent's method described at <a href="http://en.wikipedia.org/wiki/Brent%27s_method" target="_top">Brent-Dekker)</a> | |
| 183 | which only reduces the enclosing interval on the final step. Therefore these | |
| 184 | methods return a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span></code> containing the enclosing interval | |
| 185 | found, and accept a function object specifying the termination condition. | |
| 186 | </p> | |
| 187 | <p> | |
| 188 | Three function objects are provided for ready-made termination conditions: | |
| 189 | </p> | |
| 190 | <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> | |
| 191 | <li class="listitem"> | |
| 192 | <span class="emphasis"><em>eps_tolerance</em></span> causes termination when the relative | |
| 193 | error in the enclosing interval is below a certain threshold. | |
| 194 | </li> | |
| 195 | <li class="listitem"> | |
| 196 | <span class="emphasis"><em>equal_floor</em></span> and <span class="emphasis"><em>equal_ceil</em></span> | |
| 197 | are useful for certain statistical applications where the result is known | |
| 198 | to be an integer. | |
| 199 | </li> | |
| 200 | <li class="listitem"> | |
| 201 | Other user-defined termination conditions are likely to be used only | |
| 202 | rarely, but may be useful in some specific circumstances. | |
| 203 | </li> | |
| 204 | </ul></div> | |
| 205 | </div> | |
| 206 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
| 207 | <td align="left"></td> | |
| 208 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
| 209 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
| 210 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
| 211 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
| 212 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
| 213 | file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) | |
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