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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.roots_noderiv.bracket_solve"></a><a class="link" href="bracket_solve.html" title="Bracket and Solve Root">Bracket and
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Solve Root</a>
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</h3></div></div></div>
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<pre class="programlisting"><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>
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<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>
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<span class="identifier">bracket_and_solve_root</span><span class="special">(</span>
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<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">guess</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">factor</span><span class="special">,</span>
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<span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span>
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<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
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<span class="identifier">std</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> <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 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<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>
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<span class="identifier">bracket_and_solve_root</span><span class="special">(</span>
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<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">guess</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">factor</span><span class="special">,</span>
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<span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span>
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<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
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<span class="identifier">std</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">const</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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</pre>
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<p>
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<code class="computeroutput"><span class="identifier">bracket_and_solve_root</span></code> is
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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
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748 algorithm</a> internally to find the root of <span class="emphasis"><em>f(x)</em></span>.
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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
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748 algorithm</a>, since it does the hard work of bracketing the root
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for you. It's bracketing routines are quite robust and will usually be more
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foolproof than home-grown routines, unless the function can be analysed to
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yield tight brackets.
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</p>
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<p>
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Note that this routine can only be used when:
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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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<span class="emphasis"><em>f(x)</em></span> is monotonic in the half of the real axis containing
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<span class="emphasis"><em>guess</em></span>.
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</li>
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<li class="listitem">
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The value of the initial guess must have the same sign as the root: the
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function will <span class="emphasis"><em>never cross the origin</em></span> when searching
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for the root.
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</li>
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<li class="listitem">
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The location of the root should be known at least approximately, if the
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location of the root differs by many orders of magnitude from <span class="emphasis"><em>guess</em></span>
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then many iterations will be needed to bracket the root in spite of the
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special heuristics used to guard against this very situation. A typical
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example would be setting the initial guess to 0.1, when the root is at
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1e-300.
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</li>
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</ul></div>
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<p>
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The <code class="computeroutput"><span class="identifier">bracket_and_solve_root</span></code>
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parameters are:
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</p>
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<div class="variablelist">
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<p class="title"><b></b></p>
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<dl class="variablelist">
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<dt><span class="term">f</span></dt>
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<dd><p>
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A unary functor (or C++ lambda) that is the function whose root is
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to be solved. <span class="emphasis"><em>f(x)</em></span> must be uniformly increasing
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or decreasing on <span class="emphasis"><em>x</em></span>.
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</p></dd>
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<dt><span class="term">guess</span></dt>
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<dd><p>
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An initial approximation to the root.
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</p></dd>
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<dt><span class="term">factor</span></dt>
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<dd><p>
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A scaling factor that is used to bracket the root: the value <span class="emphasis"><em>guess</em></span>
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is multiplied (or divided as appropriate) by <span class="emphasis"><em>factor</em></span>
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until two values are found that bracket the root. A value such as 2
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is a typical choice for <span class="emphasis"><em>factor</em></span>. In addition <span class="emphasis"><em>factor</em></span>
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will be multiplied by 2 every 32 iterations: this is to guard against
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a really very bad initial guess, typically these occur when it's known
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the result is very large or small, but not the exact order of magnitude.
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</p></dd>
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<dt><span class="term">rising</span></dt>
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<dd><p>
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Set to <span class="emphasis"><em>true</em></span> if <span class="emphasis"><em>f(x)</em></span> is rising
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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>
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is falling on <span class="emphasis"><em>x</em></span>. This value is used along with
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the result of <span class="emphasis"><em>f(guess)</em></span> to determine if <span class="emphasis"><em>guess</em></span>
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is above or below the root.
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</p></dd>
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<dt><span class="term">tol</span></dt>
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<dd><p>
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A binary functor (or C++ lambda) that determines the termination condition
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for the search for the root. <span class="emphasis"><em>tol</em></span> is passed the
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current brackets at each step, when it returns true then the current
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brackets are returned as the pair result. See also <a class="link" href="root_termination.html" title="Termination Condition Functors">predefined
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termination functors</a>.
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</p></dd>
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<dt><span class="term">max_iter</span></dt>
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<dd><p>
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The maximum number of function invocations to perform in the search
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for the root. On exit is set to the actual number of invocations performed.
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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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The final <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
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be used to control the behaviour of the function: how it handles errors,
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what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">policy
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documentation for more details</a>.
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</p>
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<p>
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<span class="bold"><strong>Returns</strong></span>: a pair of values <span class="emphasis"><em>r</em></span>
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that bracket the root so that:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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f(r.first) * f(r.second) <= 0
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</p></blockquote></div>
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<p>
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and either
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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tol(r.first, r.second) == true
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</p></blockquote></div>
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<p>
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or
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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max_iter >= m
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</p></blockquote></div>
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<p>
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where <span class="emphasis"><em>m</em></span> is the initial value of <span class="emphasis"><em>max_iter</em></span>
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passed to the function.
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</p>
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<p>
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In other words, it's up to the caller to verify whether termination occurred
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as a result of exceeding <span class="emphasis"><em>max_iter</em></span> function invocations
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(easily done by checking the value of <span class="emphasis"><em>max_iter</em></span> when
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the function returns), rather than because the termination condition <span class="emphasis"><em>tol</em></span>
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was satisfied.
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</p>
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</div>
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<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
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Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
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Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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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</p>
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</div>
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<hr>
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