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  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.roots.roots_noderiv.bracket_solve"></a><a class="link" href="bracket_solve.html" title="Bracket and Solve Root">Bracket
  28         and Solve Root</a>
  29 </h4></div></div></div>
  30 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
  31 <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">&gt;</span>
  32    <span class="identifier">bracket_and_solve_root</span><span class="special">(</span>
  33       <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
  34       <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">guess</span><span class="special">,</span>
  35       <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">factor</span><span class="special">,</span>
  36       <span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span>
  37       <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
  38       <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&amp;</span> <span class="identifier">max_iter</span><span class="special">);</span>
  39
  40 <span class="keyword">template</span> <span class="special">&lt;</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&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  41 <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">&gt;</span>
  42    <span class="identifier">bracket_and_solve_root</span><span class="special">(</span>
  43       <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
  44       <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">guess</span><span class="special">,</span>
  45       <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">factor</span><span class="special">,</span>
  46       <span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span>
  47       <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
  48       <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&amp;</span> <span class="identifier">max_iter</span><span class="special">,</span>
  49       <span class="keyword">const</span> <a class="link" href="../../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  50 </pre>
  51 <p>
  52           <code class="computeroutput"><span class="identifier">bracket_and_solve_root</span></code>
  53           is a convenience function that calls <a class="link" href="TOMS748.html" title="Algorithm TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions">TOMS
  54           748 algorithm</a> internally to find the root of <span class="emphasis"><em>f(x)</em></span>.
  55           It is generally much easier to use this function rather than <a class="link" href="TOMS748.html" title="Algorithm TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions">TOMS
  56           748 algorithm</a>, since it does the hard work of bracketing the root
  57           for you. It's bracketing routines are quite robust and will usually be
  58           more foolproof than home-grown routines, unless the function can be analysed
  59           to yield tight brackets.
  60         </p>
  61 <p>
  62           Note that this routine can only be used when:
  63         </p>
  64 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
  65 <li class="listitem">
  66               <span class="emphasis"><em>f(x)</em></span> is monotonic in the half of the real axis
  67               containing <span class="emphasis"><em>guess</em></span>.
  68             </li>
  69 <li class="listitem">
  70               The value of the inital guess must have the same sign as the root:
  71               the function will <span class="emphasis"><em>never cross the origin</em></span> when
  72               searching for the root.
  73             </li>
  74 <li class="listitem">
  75               The location of the root should be known at least approximately, if
  76               the location of the root differs by many orders of magnitude from
  77               <span class="emphasis"><em>guess</em></span> then many iterations will be needed to bracket
  78               the root in spite of the special heuristics used to guard against this
  79               very situation. A typical example would be setting the initial guess
  80               to 0.1, when the root is at 1e-300.
  81             </li>
  82 </ul></div>
  83 <p>
  84           The <code class="computeroutput"><span class="identifier">bracket_and_solve_root</span></code>
  85           parameters are:
  86         </p>
  87 <div class="variablelist">
  88 <p class="title"><b></b></p>
  89 <dl class="variablelist">
  90 <dt><span class="term">f</span></dt>
  91 <dd><p>
  92                 A unary functor that is the function whose root is to be solved.
  93                 <span class="emphasis"><em>f(x)</em></span> must be uniformly increasing or decreasing
  94                 on <span class="emphasis"><em>x</em></span>.
  95               </p></dd>
  96 <dt><span class="term">guess</span></dt>
  97 <dd><p>
  98                 An initial approximation to the root.
  99               </p></dd>
 100 <dt><span class="term">factor</span></dt>
 101 <dd><p>
 102                 A scaling factor that is used to bracket the root: the value <span class="emphasis"><em>guess</em></span>
 103                 is multiplied (or divided as appropriate) by <span class="emphasis"><em>factor</em></span>
 104                 until two values are found that bracket the root. A value such as
 105                 2 is a typical choice for <span class="emphasis"><em>factor</em></span>. In addition
 106                 <span class="emphasis"><em>factor</em></span> will be multiplied by 2 every 32 iterations:
 107                 this is to guard against a really very bad initial guess, typically
 108                 these occur when it's known the result is very large or small, but
 109                 not the exact order of magnitude.
 110               </p></dd>
 111 <dt><span class="term">rising</span></dt>
 112 <dd><p>
 113                 Set to <span class="emphasis"><em>true</em></span> if <span class="emphasis"><em>f(x)</em></span> is
 114                 rising on <span class="emphasis"><em>x</em></span> and <span class="emphasis"><em>false</em></span> if
 115                 <span class="emphasis"><em>f(x)</em></span> is falling on <span class="emphasis"><em>x</em></span>. This
 116                 value is used along with the result of <span class="emphasis"><em>f(guess)</em></span>
 117                 to determine if <span class="emphasis"><em>guess</em></span> is above or below the
 118                 root.
 119               </p></dd>
 120 <dt><span class="term">tol</span></dt>
 121 <dd><p>
 122                 A binary functor that determines the termination condition for the
 123                 search for the root. <span class="emphasis"><em>tol</em></span> is passed the current
 124                 brackets at each step, when it returns true then the current brackets
 125                 are returned as the pair result. See also <a class="link" href="root_termination.html" title="Termination Condition Functors">predefined
 126                 termination functors</a>.
 127               </p></dd>
 128 <dt><span class="term">max_iter</span></dt>
 129 <dd><p>
 130                 The maximum number of function invocations to perform in the search
 131                 for the root. On exit is set to the actual number of invocations
 132                 performed.
 133               </p></dd>
 134 </dl>
 135 </div>
 136 <p>
 137           The final <a class="link" href="../../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and
 138           can be used to control the behaviour of the function: how it handles errors,
 139           what level of precision to use etc. Refer to the <a class="link" href="../../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">policy
 140           documentation for more details</a>.
 141         </p>
 142 <p>
 143           <span class="bold"><strong>Returns</strong></span>: a pair of values <span class="emphasis"><em>r</em></span>
 144           that bracket the root so that:
 145         </p>
 146 <pre class="programlisting"><span class="identifier">f</span><span class="special">(</span><span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">f</span><span class="special">(</span><span class="identifier">r</span><span class="special">.</span><span class="identifier">second</span><span class="special">)</span> <span class="special">&lt;=</span> <span class="number">0</span>
 147 </pre>
 148 <p>
 149           and either
 150         </p>
 151 <pre class="programlisting"><span class="identifier">tol</span><span class="special">(</span><span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span><span class="special">,</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">second</span><span class="special">)</span> <span class="special">==</span> <span class="keyword">true</span>
 152 </pre>
 153 <p>
 154           or
 155         </p>
 156 <pre class="programlisting"><span class="identifier">max_iter</span> <span class="special">&gt;=</span> <span class="identifier">m</span>
 157 </pre>
 158 <p>
 159           where <span class="emphasis"><em>m</em></span> is the initial value of <span class="emphasis"><em>max_iter</em></span>
 160           passed to the function.
 161         </p>
 162 <p>
 163           In other words, it's up to the caller to verify whether termination occurred
 164           as a result of exceeding <span class="emphasis"><em>max_iter</em></span> function invocations
 165           (easily done by checking the value of <span class="emphasis"><em>max_iter</em></span> when
 166           the function returns), rather than because the termination condition <span class="emphasis"><em>tol</em></span>
 167           was satisfied.
 168         </p>
 169 </div>
 170 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 171 <td align="left"></td>
 172 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 173       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 174       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 175       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 176         Distributed under the Boost Software License, Version 1.0. (See accompanying
 177         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>)
 178       </p>
 179 </div></td>
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