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  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.roots_noderiv.bracket_solve"></a><a class="link" href="bracket_solve.html" title="Bracket and Solve Root">Bracket and
  28       Solve Root</a>
  29 </h3></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;20.&#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;20.&#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> is
  53         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 more
  58         foolproof than home-grown routines, unless the function can be analysed to
  59         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 containing
  67             <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: the
  71             function will <span class="emphasis"><em>never cross the origin</em></span> when searching
  72             for the root.
  73           </li>
  74 <li class="listitem">
  75             The location of the root should be known at least approximately, if the
  76             location of the root differs by many orders of magnitude from <span class="emphasis"><em>guess</em></span>
  77             then many iterations will be needed to bracket the root in spite of the
  78             special heuristics used to guard against this very situation. A typical
  79             example would be setting the initial guess to 0.1, when the root is at
  80             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 (or C++ lambda) that is the function whose root is
  93               to be solved. <span class="emphasis"><em>f(x)</em></span> must be uniformly increasing
  94               or decreasing 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 2
 105               is a typical choice for <span class="emphasis"><em>factor</em></span>. In addition <span class="emphasis"><em>factor</em></span>
 106               will be multiplied by 2 every 32 iterations: this is to guard against
 107               a really very bad initial guess, typically these occur when it's known
 108               the result is very large or small, but not the exact order of magnitude.
 109             </p></dd>
 110 <dt><span class="term">rising</span></dt>
 111 <dd><p>
 112               Set to <span class="emphasis"><em>true</em></span> if <span class="emphasis"><em>f(x)</em></span> is rising
 113               on <span class="emphasis"><em>x</em></span> and <span class="emphasis"><em>false</em></span> if <span class="emphasis"><em>f(x)</em></span>
 114               is falling on <span class="emphasis"><em>x</em></span>. This value is used along with
 115               the result of <span class="emphasis"><em>f(guess)</em></span> to determine if <span class="emphasis"><em>guess</em></span>
 116               is above or below the root.
 117             </p></dd>
 118 <dt><span class="term">tol</span></dt>
 119 <dd><p>
 120               A binary functor (or C++ lambda) that determines the termination condition
 121               for the search for the root. <span class="emphasis"><em>tol</em></span> is passed the
 122               current brackets at each step, when it returns true then the current
 123               brackets are returned as the pair result. See also <a class="link" href="root_termination.html" title="Termination Condition Functors">predefined
 124               termination functors</a>.
 125             </p></dd>
 126 <dt><span class="term">max_iter</span></dt>
 127 <dd><p>
 128               The maximum number of function invocations to perform in the search
 129               for the root. On exit is set to the actual number of invocations performed.
 130             </p></dd>
 131 </dl>
 132 </div>
 133 <p>
 134         The final <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
 135         be used to control the behaviour of the function: how it handles errors,
 136         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">policy
 137         documentation for more details</a>.
 138       </p>
 139 <p>
 140         <span class="bold"><strong>Returns</strong></span>: a pair of values <span class="emphasis"><em>r</em></span>
 141         that bracket the root so that:
 142       </p>
 143 <div class="blockquote"><blockquote class="blockquote"><p>
 144           f(r.first) * f(r.second) &lt;= 0
 145         </p></blockquote></div>
 146 <p>
 147         and either
 148       </p>
 149 <div class="blockquote"><blockquote class="blockquote"><p>
 150           tol(r.first, r.second) == true
 151         </p></blockquote></div>
 152 <p>
 153         or
 154       </p>
 155 <div class="blockquote"><blockquote class="blockquote"><p>
 156           max_iter &gt;= m
 157         </p></blockquote></div>
 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; 2006-2019 Nikhar
 173       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 174       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 175       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 176       Daryle Walker and Xiaogang Zhang<p>
 177         Distributed under the Boost Software License, Version 1.0. (See accompanying
 178         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>)
 179       </p>
 180 </div></td>
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