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25<div class="section">
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>
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