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<title>Root Finding Without Derivatives: Bisection, Bracket and TOMS748</title>
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<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.internals1.roots2"></a><a class="link" href="roots2.html" title="Root Finding Without Derivatives: Bisection, Bracket and TOMS748">Root Finding Without
Derivatives: Bisection, Bracket and TOMS748</a>
</h3></div></div></div>
<h5>
<a name="math_toolkit.internals1.roots2.h0"></a>
<span class="phrase"><a name="math_toolkit.internals1.roots2.synopsis"></a></span><a class="link" href="roots2.html#math_toolkit.internals1.roots2.synopsis">Synopsis</a>
</h5>
<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>
</pre>
<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>
<span class="keyword">namespace</span> <span class="identifier">tools</span><span class="special">{</span>
<span class="comment">// Bisection</span>
<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="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>
<span class="identifier">bisect</span><span class="special">(</span>
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
<span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span>
<span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span>
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
<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>
<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="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>
<span class="identifier">bisect</span><span class="special">(</span>
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
<span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span>
<span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span>
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">);</span>
<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 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
<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>
<span class="identifier">bisect</span><span class="special">(</span>
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
<span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span>
<span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span>
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
<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>
<span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
<span class="comment">// Bracket and Solve Root</span>
<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="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>
<span class="identifier">bracket_and_solve_root</span><span class="special">(</span>
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">guess</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">factor</span><span class="special">,</span>
<span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span>
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
<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>
<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 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
<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>
<span class="identifier">bracket_and_solve_root</span><span class="special">(</span>
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">guess</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">factor</span><span class="special">,</span>
<span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span>
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
<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>
<span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
<span class="comment">// TOMS 748 algorithm</span>
<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="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>
<span class="identifier">toms748_solve</span><span class="special">(</span>
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span>
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
<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>
<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 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
<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>
<span class="identifier">toms748_solve</span><span class="special">(</span>
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span>
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
<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>
<span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
<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="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>
<span class="identifier">toms748_solve</span><span class="special">(</span>
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fa</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fb</span><span class="special">,</span>
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
<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>
<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 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
<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>
<span class="identifier">toms748_solve</span><span class="special">(</span>
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fa</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fb</span><span class="special">,</span>
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
<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>
<span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
<span class="comment">// Termination conditions:</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<span class="keyword">struct</span> <span class="identifier">eps_tolerance</span><span class="special">;</span>
<span class="keyword">struct</span> <span class="identifier">equal_floor</span><span class="special">;</span>
<span class="keyword">struct</span> <span class="identifier">equal_ceil</span><span class="special">;</span>
<span class="keyword">struct</span> <span class="identifier">equal_nearest_integer</span><span class="special">;</span>
<span class="special">}}}</span> <span class="comment">// namespaces</span>
</pre>
<h5>
<a name="math_toolkit.internals1.roots2.h1"></a>
<span class="phrase"><a name="math_toolkit.internals1.roots2.description"></a></span><a class="link" href="roots2.html#math_toolkit.internals1.roots2.description">Description</a>
</h5>
<p>
These functions solve the root of some function <span class="emphasis"><em>f(x)</em></span>
<span class="emphasis"><em>without the need for any derivatives of <span class="emphasis"><em>f(x)</em></span></em></span>.
</p>
<p>
The <code class="computeroutput"><span class="identifier">bracket_and_solve_root</span></code>
functions use TOMS Algorithm 748 that is asymptotically the most efficient
known, and have been shown to be optimal for a certain classes of smooth
functions. Variants with and without __policies are provided.
</p>
<p>
Alternatively, there is a simple bisection routine which can be useful in
its own right in some situations, or alternatively for narrowing down the
range containing the root, prior to calling a more advanced algorithm.
</p>
<p>
All the algorithms in this section reduce the diameter of the enclosing interval
with the same asymptotic efficiency with which they locate the root. This
is in contrast to the derivative based methods which may <span class="emphasis"><em>never</em></span>
significantly reduce the enclosing interval, even though they rapidly approach
the root. This is also in contrast to some other derivative-free methods
(for example the methods of <a href="http://en.wikipedia.org/wiki/Brent%27s_method" target="_top">Brent
or Dekker)</a> which only reduce the enclosing interval on the final
step. Therefore these methods return a std::pair containing the enclosing
interval found, and accept a function object specifying the termination condition.
Three function objects are provided for ready-made termination conditions:
<span class="emphasis"><em>eps_tolerance</em></span> causes termination when the relative error
in the enclosing interval is below a certain threshold, while <span class="emphasis"><em>equal_floor</em></span>
and <span class="emphasis"><em>equal_ceil</em></span> are useful for certain statistical applications
where the result is known to be an integer. Other user-defined termination
conditions are likely to be used only rarely, but may be useful in some specific
circumstances.
</p>
<h4>
<a name="math_toolkit.internals1.roots2.h2"></a>
<span class="phrase"><a name="math_toolkit.internals1.roots2.bisection"></a></span><a class="link" href="roots2.html#math_toolkit.internals1.roots2.bisection">Bisection</a>
</h4>
<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>
<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>
<span class="identifier">bisect</span><span class="special">(</span>
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
<span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span>
<span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span>
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
<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>
<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="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>
<span class="identifier">bisect</span><span class="special">(</span>
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
<span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span>
<span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span>
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">);</span>
<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 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
<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>
<span class="identifier">bisect</span><span class="special">(</span>
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
<span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span>
<span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span>
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
<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>
<span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
</pre>
<p>
These functions locate the root using bisection: function arguments are:
</p>
<div class="variablelist">
<p class="title"><b></b></p>
<dl class="variablelist">
<dt><span class="term">f</span></dt>
<dd><p>
A unary functor which is the function whose root is to be found.
</p></dd>
<dt><span class="term">min</span></dt>
<dd><p>
The left bracket of the interval known to contain the root.
</p></dd>
<dt><span class="term">max</span></dt>
<dd><p>
The right bracket of the interval known to contain the root. It is
a precondition that <span class="emphasis"><em>min < max</em></span> and <span class="emphasis"><em>f(min)*f(max)
<= 0</em></span>, the function signals evaluation error if these
preconditions are violated. The action taken is controlled by the evaluation
error policy. A best guess may be returned, perhaps significantly wrong.
</p></dd>
<dt><span class="term">tol</span></dt>
<dd><p>
A binary functor that specifies the termination condition: the function
will return the current brackets enclosing the root when <span class="emphasis"><em>tol(min,max)</em></span>
becomes true.
</p></dd>
<dt><span class="term">max_iter</span></dt>
<dd><p>
The maximum number of invocations of <span class="emphasis"><em>f(x)</em></span> to make
while searching for the root. On exist this is set to actual number
of invocations performed.
</p></dd>
</dl>
</div>
<p>
The final <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
be used to control the behaviour of the function: how it handles errors,
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<p>
Returns: a pair of values <span class="emphasis"><em>r</em></span> that bracket the root so
that:
</p>
<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"><=</span> <span class="number">0</span>
</pre>
<p>
and either
</p>
<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>
</pre>
<p>
or
</p>
<pre class="programlisting"><span class="identifier">max_iter</span> <span class="special">>=</span> <span class="identifier">m</span>
</pre>
<p>
where <span class="emphasis"><em>m</em></span> is the initial value of <span class="emphasis"><em>max_iter</em></span>
passed to the function.
</p>
<p>
In other words, it's up to the caller to verify whether termination occurred
as a result of exceeding <span class="emphasis"><em>max_iter</em></span> function invocations
(easily done by checking the updated value of <span class="emphasis"><em>max_iter</em></span>
when the function returns), rather than because the termination condition
<span class="emphasis"><em>tol</em></span> was satisfied.
</p>
<h4>
<a name="math_toolkit.internals1.roots2.h3"></a>
<span class="phrase"><a name="math_toolkit.internals1.roots2.bracket_and_solve"></a></span><a class="link" href="roots2.html#math_toolkit.internals1.roots2.bracket_and_solve">Bracket
and solve</a>
</h4>
<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>
<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>
<span class="identifier">bracket_and_solve_root</span><span class="special">(</span>
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">guess</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">factor</span><span class="special">,</span>
<span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span>
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
<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>
<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 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
<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>
<span class="identifier">bracket_and_solve_root</span><span class="special">(</span>
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">guess</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">factor</span><span class="special">,</span>
<span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span>
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
<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>
<span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
</pre>
<p>
This is a convenience function that calls <span class="emphasis"><em>toms748_solve</em></span>
internally to find the root of <span class="emphasis"><em>f(x)</em></span>. It's usable only
when <span class="emphasis"><em>f(x)</em></span> is a monotonic function, and the location
of the root is known approximately, and in particular it is known whether
the root is occurs for positive or negative <span class="emphasis"><em>x</em></span>. The parameters
are:
</p>
<div class="variablelist">
<p class="title"><b></b></p>
<dl class="variablelist">
<dt><span class="term">f</span></dt>
<dd><p>
A unary functor that is the function whose root is to be solved. f(x)
must be uniformly increasing or decreasing on <span class="emphasis"><em>x</em></span>.
</p></dd>
<dt><span class="term">guess</span></dt>
<dd><p>
An initial approximation to the root
</p></dd>
<dt><span class="term">factor</span></dt>
<dd><p>
A scaling factor that is used to bracket the root: the value <span class="emphasis"><em>guess</em></span>
is multiplied (or divided as appropriate) by <span class="emphasis"><em>factor</em></span>
until two values are found that bracket the root. A value such as 2
is a typical choice for <span class="emphasis"><em>factor</em></span>.
</p></dd>
<dt><span class="term">rising</span></dt>
<dd><p>
Set to <span class="emphasis"><em>true</em></span> if <span class="emphasis"><em>f(x)</em></span> is rising
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>
is falling on <span class="emphasis"><em>x</em></span>. This value is used along with
the result of <span class="emphasis"><em>f(guess)</em></span> to determine if <span class="emphasis"><em>guess</em></span>
is above or below the root.
</p></dd>
<dt><span class="term">tol</span></dt>
<dd><p>
A binary functor that determines the termination condition for the
search for the root. <span class="emphasis"><em>tol</em></span> is passed the current
brackets at each step, when it returns true then the current brackets
are returned as the result.
</p></dd>
<dt><span class="term">max_iter</span></dt>
<dd><p>
The maximum number of function invocations to perform in the search
for the root. On exit is set to the actual number of invocations performed.
</p></dd>
</dl>
</div>
<p>
The final <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
be used to control the behaviour of the function: how it handles errors,
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<p>
Returns: a pair of values <span class="emphasis"><em>r</em></span> that bracket the root so
that:
</p>
<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"><=</span> <span class="number">0</span>
</pre>
<p>
and either
</p>
<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>
</pre>
<p>
or
</p>
<pre class="programlisting"><span class="identifier">max_iter</span> <span class="special">>=</span> <span class="identifier">m</span>
</pre>
<p>
where <span class="emphasis"><em>m</em></span> is the initial value of <span class="emphasis"><em>max_iter</em></span>
passed to the function.
</p>
<p>
In other words, it's up to the caller to verify whether termination occurred
as a result of exceeding <span class="emphasis"><em>max_iter</em></span> function invocations
(easily done by checking the value of <span class="emphasis"><em>max_iter</em></span> when
the function returns), rather than because the termination condition <span class="emphasis"><em>tol</em></span>
was satisfied.
</p>
<h4>
<a name="math_toolkit.internals1.roots2.h4"></a>
<span class="phrase"><a name="math_toolkit.internals1.roots2.algorithm_toms_748_alefeld_potra"></a></span><a class="link" href="roots2.html#math_toolkit.internals1.roots2.algorithm_toms_748_alefeld_potra">Algorithm
TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions</a>
</h4>
<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>
<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>
<span class="identifier">toms748_solve</span><span class="special">(</span>
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span>
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
<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>
<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 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
<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>
<span class="identifier">toms748_solve</span><span class="special">(</span>
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span>
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
<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>
<span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
<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="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>
<span class="identifier">toms748_solve</span><span class="special">(</span>
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fa</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fb</span><span class="special">,</span>
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
<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>
<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 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
<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>
<span class="identifier">toms748_solve</span><span class="special">(</span>
<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fa</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fb</span><span class="special">,</span>
<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
<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>
<span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
</pre>
<p>
These two functions implement TOMS Algorithm 748: it uses a mixture of cubic,
quadratic and linear (secant) interpolation to locate the root of <span class="emphasis"><em>f(x)</em></span>.
The two functions differ only by whether values for <span class="emphasis"><em>f(a)</em></span>
and <span class="emphasis"><em>f(b)</em></span> are already available. The toms748_solve parameters
are:
</p>
<div class="variablelist">
<p class="title"><b></b></p>
<dl class="variablelist">
<dt><span class="term">f</span></dt>
<dd><p>
A unary functor that is the function whose root is to be solved. f(x)
need not be uniformly increasing or decreasing on <span class="emphasis"><em>x</em></span>
and may have multiple roots.
</p></dd>
<dt><span class="term">a</span></dt>
<dd><p>
The lower bound for the initial bracket of the root.
</p></dd>
<dt><span class="term">b</span></dt>
<dd><p>
The upper bound for the initial bracket of the root. It is a precondition
that <span class="emphasis"><em>a < b</em></span> and that <span class="emphasis"><em>a</em></span>
and <span class="emphasis"><em>b</em></span> bracket the root to find so that <span class="emphasis"><em>f(a)*f(b)
< 0</em></span>.
</p></dd>
<dt><span class="term">fa</span></dt>
<dd><p>
Optional: the value of <span class="emphasis"><em>f(a)</em></span>.
</p></dd>
<dt><span class="term">fb</span></dt>
<dd><p>
Optional: the value of <span class="emphasis"><em>f(b)</em></span>.
</p></dd>
<dt><span class="term">tol</span></dt>
<dd><p>
A binary functor that determines the termination condition for the
search for the root. <span class="emphasis"><em>tol</em></span> is passed the current
brackets at each step, when it returns true, then the current brackets
are returned as the result.
</p></dd>
<dt><span class="term">max_iter</span></dt>
<dd><p>
The maximum number of function invocations to perform in the search
for the root. On exit <span class="emphasis"><em>max_iter</em></span> is set to actual
number of function invocations used.
</p></dd>
</dl>
</div>
<p>
The final <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
be used to control the behaviour of the function: how it handles errors,
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<p>
toms748_solve returns: a pair of values <span class="emphasis"><em>r</em></span> that bracket
the root so that: f(r.first) * f(r.second) <= 0 and either tol(r.first,
r.second) == true or max_iter >= m where <span class="emphasis"><em>m</em></span> is the
initial value of <span class="emphasis"><em>max_iter</em></span> passed to the function.
</p>
<p>
In other words, it's up to the caller to verify whether termination occurred
as a result of exceeding <span class="emphasis"><em>max_iter</em></span> function invocations
(easily done by checking the updated value of <span class="emphasis"><em>max_iter</em></span>
against its previous value passed as parameter), rather than because the
termination condition <span class="emphasis"><em>tol</em></span> was satisfied.
</p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<span class="keyword">struct</span> <span class="identifier">eps_tolerance</span>
<span class="special">{</span>
<span class="identifier">eps_tolerance</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">bits</span><span class="special">);</span>
<span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">()(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
<span class="special">};</span>
</pre>
<p>
<code class="computeroutput"><span class="identifier">eps_tolerance</span></code> is the usual
termination condition used with these root finding functions. Its operator()
will return true when the relative distance between <span class="emphasis"><em>a</em></span>
and <span class="emphasis"><em>b</em></span> is less than twice the machine epsilon for T,
or 2<sup>1-bits</sup>, whichever is the larger. In other words, you set <span class="emphasis"><em>bits</em></span>
to the number of bits of precision you want in the result. The minimal tolerance
of twice the machine epsilon of T is required to ensure that we get back
a bracketing interval: since this must clearly be at least 1 epsilon in size.
</p>
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">equal_floor</span>
<span class="special">{</span>
<span class="identifier">equal_floor</span><span class="special">();</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> <span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">()(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
<span class="special">};</span>
</pre>
<p>
This termination condition is used when you want to find an integer result
that is the <span class="emphasis"><em>floor</em></span> of the true root. It will terminate
as soon as both ends of the interval have the same <span class="emphasis"><em>floor</em></span>.
</p>
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">equal_ceil</span>
<span class="special">{</span>
<span class="identifier">equal_ceil</span><span class="special">();</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> <span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">()(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
<span class="special">};</span>
</pre>
<p>
This termination condition is used when you want to find an integer result
that is the <span class="emphasis"><em>ceil</em></span> of the true root. It will terminate
as soon as both ends of the interval have the same <span class="emphasis"><em>ceil</em></span>.
</p>
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">equal_nearest_integer</span>
<span class="special">{</span>
<span class="identifier">equal_nearest_integer</span><span class="special">();</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> <span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">()(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
<span class="special">};</span>
</pre>
<p>
This termination condition is used when you want to find an integer result
that is the <span class="emphasis"><em>closest</em></span> to the true root. It will terminate
as soon as both ends of the interval round to the same nearest integer.
</p>
<h5>
<a name="math_toolkit.internals1.roots2.h5"></a>
<span class="phrase"><a name="math_toolkit.internals1.roots2.implementation"></a></span><a class="link" href="roots2.html#math_toolkit.internals1.roots2.implementation">Implementation</a>
</h5>
<p>
The implementation of the bisection algorithm is extremely straightforward
and not detailed here. <a href="http://portal.acm.org/citation.cfm?id=210111" target="_top">TOMS
Algorithm 748: enclosing zeros of continuous functions</a> is described
in detail in:
</p>
<p>
<span class="emphasis"><em>Algorithm 748: Enclosing Zeros of Continuous Functions, G. E. Alefeld,
F. A. Potra and Yixun Shi, ACM Transactions on Mathematica1 Software, Vol.
21. No. 3. September 1995. Pages 327-344.</em></span>
</p>
<p>
The implementation here is a faithful translation of this paper into C++.
</p>
</div>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta,
Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
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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