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<title>Finding Zeros of Bessel Functions of the First and Second Kinds</title>
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<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.bessel.bessel_root"></a><a class="link" href="bessel_root.html" title="Finding Zeros of Bessel Functions of the First and Second Kinds">Finding Zeros of Bessel
Functions of the First and Second Kinds</a>
</h3></div></div></div>
<h5>
<a name="math_toolkit.bessel.bessel_root.h0"></a>
<span class="phrase"><a name="math_toolkit.bessel.bessel_root.synopsis"></a></span><a class="link" href="bessel_root.html#math_toolkit.bessel.bessel_root.synopsis">Synopsis</a>
</h5>
<p>
<code class="computeroutput"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">bessel</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></code>
</p>
<p>
Functions for obtaining both a single zero or root of the Bessel function,
and placing multiple zeros into a container like <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span></code>
by providing an output iterator.
</p>
<p>
The signature of the single value functions are:
</p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<span class="identifier">T</span> <span class="identifier">cyl_bessel_j_zero</span><span class="special">(</span>
<span class="identifier">T</span> <span class="identifier">v</span><span class="special">,</span> <span class="comment">// Floating-point value for Jv.</span>
<span class="keyword">int</span> <span class="identifier">m</span><span class="special">);</span> <span class="comment">// 1-based index of zero.</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<span class="identifier">T</span> <span class="identifier">cyl_neumann_zero</span><span class="special">(</span>
<span class="identifier">T</span> <span class="identifier">v</span><span class="special">,</span> <span class="comment">// Floating-point value for Jv.</span>
<span class="keyword">int</span> <span class="identifier">m</span><span class="special">);</span> <span class="comment">// 1-based index of zero.</span>
</pre>
<p>
and for multiple zeros:
</p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">OutputIterator</span><span class="special">&gt;</span>
<span class="identifier">OutputIterator</span> <span class="identifier">cyl_bessel_j_zero</span><span class="special">(</span>
<span class="identifier">T</span> <span class="identifier">v</span><span class="special">,</span> <span class="comment">// Floating-point value for Jv.</span>
<span class="keyword">int</span> <span class="identifier">start_index</span><span class="special">,</span> <span class="comment">// 1-based index of first zero.</span>
<span class="keyword">unsigned</span> <span class="identifier">number_of_zeros</span><span class="special">,</span> <span class="comment">// How many zeros to generate.</span>
<span class="identifier">OutputIterator</span> <span class="identifier">out_it</span><span class="special">);</span> <span class="comment">// Destination for zeros.</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">OutputIterator</span><span class="special">&gt;</span>
<span class="identifier">OutputIterator</span> <span class="identifier">cyl_neumann_zero</span><span class="special">(</span>
<span class="identifier">T</span> <span class="identifier">v</span><span class="special">,</span> <span class="comment">// Floating-point value for Jv.</span>
<span class="keyword">int</span> <span class="identifier">start_index</span><span class="special">,</span> <span class="comment">// 1-based index of zero.</span>
<span class="keyword">unsigned</span> <span class="identifier">number_of_zeros</span><span class="special">,</span> <span class="comment">// How many zeros to generate</span>
<span class="identifier">OutputIterator</span> <span class="identifier">out_it</span><span class="special">);</span> <span class="comment">// Destination for zeros.</span>
</pre>
<p>
There are also versions which allow control of the <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policies</a>
for error handling and precision.
</p>
<pre class="programlisting"> <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<span class="identifier">T</span> <span class="identifier">cyl_bessel_j_zero</span><span class="special">(</span>
<span class="identifier">T</span> <span class="identifier">v</span><span class="special">,</span> <span class="comment">// Floating-point value for Jv.</span>
<span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="comment">// 1-based index of zero.</span>
<span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&amp;);</span> <span class="comment">// Policy to use.</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<span class="identifier">T</span> <span class="identifier">cyl_neumann_zero</span><span class="special">(</span>
<span class="identifier">T</span> <span class="identifier">v</span><span class="special">,</span> <span class="comment">// Floating-point value for Jv.</span>
<span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="comment">// 1-based index of zero.</span>
<span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&amp;);</span> <span class="comment">// Policy to use.</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">OutputIterator</span><span class="special">&gt;</span>
<span class="identifier">OutputIterator</span> <span class="identifier">cyl_bessel_j_zero</span><span class="special">(</span>
<span class="identifier">T</span> <span class="identifier">v</span><span class="special">,</span> <span class="comment">// Floating-point value for Jv.</span>
<span class="keyword">int</span> <span class="identifier">start_index</span><span class="special">,</span> <span class="comment">// 1-based index of first zero.</span>
<span class="keyword">unsigned</span> <span class="identifier">number_of_zeros</span><span class="special">,</span> <span class="comment">// How many zeros to generate.</span>
<span class="identifier">OutputIterator</span> <span class="identifier">out_it</span><span class="special">,</span> <span class="comment">// Destination for zeros.</span>
<span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&amp;</span> <span class="identifier">pol</span><span class="special">);</span> <span class="comment">// Policy to use.</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">OutputIterator</span><span class="special">&gt;</span>
<span class="identifier">OutputIterator</span> <span class="identifier">cyl_neumann_zero</span><span class="special">(</span>
<span class="identifier">T</span> <span class="identifier">v</span><span class="special">,</span> <span class="comment">// Floating-point value for Jv.</span>
<span class="keyword">int</span> <span class="identifier">start_index</span><span class="special">,</span> <span class="comment">// 1-based index of zero.</span>
<span class="keyword">unsigned</span> <span class="identifier">number_of_zeros</span><span class="special">,</span> <span class="comment">// How many zeros to generate.</span>
<span class="identifier">OutputIterator</span> <span class="identifier">out_it</span><span class="special">,</span> <span class="comment">// Destination for zeros.</span>
<span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&amp;</span> <span class="identifier">pol</span><span class="special">);</span> <span class="comment">// Policy to use.</span>
</pre>
<h5>
<a name="math_toolkit.bessel.bessel_root.h1"></a>
<span class="phrase"><a name="math_toolkit.bessel.bessel_root.description"></a></span><a class="link" href="bessel_root.html#math_toolkit.bessel.bessel_root.description">Description</a>
</h5>
<p>
Every real order &#957; cylindrical Bessel and Neumann functions have an infinite
number of zeros on the positive real axis. The real zeros on the positive
real axis can be found by solving for the roots of
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="emphasis"><em>J<sub>&#957;</sub>(j<sub>&#957;, m</sub>) = 0</em></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="emphasis"><em>Y<sub>&#957;</sub>(y<sub>&#957;, m</sub>) = 0</em></span>
</p></blockquote></div>
<p>
Here, <span class="emphasis"><em>j<sub>&#957;, m</sub></em></span> represents the <span class="emphasis"><em>m<sup>th</sup></em></span> root
of the cylindrical Bessel function of order <span class="emphasis"><em>&#957;</em></span>, and <span class="emphasis"><em>y<sub>&#957;,
m</sub></em></span> represents the <span class="emphasis"><em>m<sup>th</sup></em></span> root of the cylindrical
Neumann function of order <span class="emphasis"><em>&#957;</em></span>.
</p>
<p>
The zeros or roots (values of <code class="computeroutput"><span class="identifier">x</span></code>
where the function crosses the horizontal <code class="computeroutput"><span class="identifier">y</span>
<span class="special">=</span> <span class="number">0</span></code>
axis) of the Bessel and Neumann functions are computed by two functions,
<code class="computeroutput"><span class="identifier">cyl_bessel_j_zero</span></code> and <code class="computeroutput"><span class="identifier">cyl_neumann_zero</span></code>.
</p>
<p>
In each case the index or rank of the zero returned is 1-based, which is
to say:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
cyl_bessel_j_zero(v, 1);
</p></blockquote></div>
<p>
returns the first zero of Bessel J.
</p>
<p>
Passing an <code class="computeroutput"><span class="identifier">start_index</span> <span class="special">&lt;=</span>
<span class="number">0</span></code> results in a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">domain_error</span></code>
being raised.
</p>
<p>
For certain parameters, however, the zero'th root is defined and it has a
value of zero. For example, the zero'th root of <code class="computeroutput"><span class="identifier">J</span><span class="special">[</span><span class="identifier">sub</span> <span class="identifier">v</span><span class="special">](</span><span class="identifier">x</span><span class="special">)</span></code>
is defined and it has a value of zero for all values of <code class="computeroutput"><span class="identifier">v</span>
<span class="special">&gt;</span> <span class="number">0</span></code>
and for negative integer values of <code class="computeroutput"><span class="identifier">v</span>
<span class="special">=</span> <span class="special">-</span><span class="identifier">n</span></code>. Similar cases are described in the implementation
details below.
</p>
<p>
The order <code class="computeroutput"><span class="identifier">v</span></code> of <code class="computeroutput"><span class="identifier">J</span></code> can be positive, negative and zero for
the <code class="computeroutput"><span class="identifier">cyl_bessel_j</span></code> and <code class="computeroutput"><span class="identifier">cyl_neumann</span></code> functions, but not infinite
nor NaN.
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/bessel_j_zeros.svg" align="middle"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/neumann_y_zeros.svg" align="middle"></span>
</p></blockquote></div>
<h5>
<a name="math_toolkit.bessel.bessel_root.h2"></a>
<span class="phrase"><a name="math_toolkit.bessel.bessel_root.examples_of_finding_bessel_and_n"></a></span><a class="link" href="bessel_root.html#math_toolkit.bessel.bessel_root.examples_of_finding_bessel_and_n">Examples
of finding Bessel and Neumann zeros</a>
</h5>
<p>
This example demonstrates calculating zeros of the Bessel and Neumann functions.
It also shows how Boost.Math and Boost.Multiprecision can be combined to
provide a many decimal digit precision. For 50 decimal digit precision we
need to include
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">multiprecision</span><span class="special">/</span><span class="identifier">cpp_dec_float</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
</pre>
<p>
and a <code class="computeroutput"><span class="keyword">typedef</span></code> for <code class="computeroutput"><span class="identifier">float_type</span></code> may be convenient (allowing
a quick switch to re-compute at built-in <code class="computeroutput"><span class="keyword">double</span></code>
or other precision)
</p>
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_dec_float_50</span> <span class="identifier">float_type</span><span class="special">;</span>
</pre>
<p>
To use the functions for finding zeros of the functions we need
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">bessel</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
</pre>
<p>
This file includes the forward declaration signatures for the zero-finding
functions:
</p>
<pre class="programlisting"><span class="comment">// #include &lt;boost/math/special_functions/math_fwd.hpp&gt;</span>
</pre>
<p>
but more details are in the full documentation, for example at <a href="http://www.boost.org/doc/libs/1_53_0/libs/math/doc/sf_and_dist/html/math_toolkit/special/bessel/bessel_over.html" target="_top">Boost.Math
Bessel functions</a>.
</p>
<p>
This example shows obtaining both a single zero of the Bessel function, and
then placing multiple zeros into a container like <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span></code>
by providing an iterator.
</p>
<div class="tip"><table border="0" summary="Tip">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../doc/src/images/tip.png"></td>
<th align="left">Tip</th>
</tr>
<tr><td align="left" valign="top"><p>
It is always wise to place code using Boost.Math inside try'n'catch blocks;
this will ensure that helpful error messages are shown when exceptional
conditions arise.
</p></td></tr>
</table></div>
<p>
First, evaluate a single Bessel zero.
</p>
<p>
The precision is controlled by the float-point type of template parameter
<code class="computeroutput"><span class="identifier">T</span></code> of <code class="computeroutput"><span class="identifier">v</span></code>
so this example has <code class="computeroutput"><span class="keyword">double</span></code> precision,
at least 15 but up to 17 decimal digits (for the common 64-bit double).
</p>
<pre class="programlisting"><span class="comment">// double root = boost::math::cyl_bessel_j_zero(0.0, 1);</span>
<span class="comment">// // Displaying with default precision of 6 decimal digits:</span>
<span class="comment">// std::cout &lt;&lt; "boost::math::cyl_bessel_j_zero(0.0, 1) " &lt;&lt; root &lt;&lt; std::endl; // 2.40483</span>
<span class="comment">// // And with all the guaranteed (15) digits:</span>
<span class="comment">// std::cout.precision(std::numeric_limits&lt;double&gt;::digits10);</span>
<span class="comment">// std::cout &lt;&lt; "boost::math::cyl_bessel_j_zero(0.0, 1) " &lt;&lt; root &lt;&lt; std::endl; // 2.40482555769577</span>
</pre>
<p>
But note that because the parameter <code class="computeroutput"><span class="identifier">v</span></code>
controls the precision of the result, <code class="computeroutput"><span class="identifier">v</span></code>
<span class="bold"><strong>must be a floating-point type</strong></span>. So if you
provide an integer type, say 0, rather than 0.0, then it will fail to compile
thus:
</p>
<pre class="programlisting"><span class="identifier">root</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">cyl_bessel_j_zero</span><span class="special">(</span><span class="number">0</span><span class="special">,</span> <span class="number">1</span><span class="special">);</span>
</pre>
<p>
with this error message
</p>
<pre class="programlisting"><span class="identifier">error</span> <span class="identifier">C2338</span><span class="special">:</span> <span class="identifier">Order</span> <span class="identifier">must</span> <span class="identifier">be</span> <span class="identifier">a</span> <span class="identifier">floating</span><span class="special">-</span><span class="identifier">point</span> <span class="identifier">type</span><span class="special">.</span>
</pre>
<p>
Optionally, we can use a policy to ignore errors, C-style, returning some
value, perhaps infinity or NaN, or the best that can be done. (See <a class="link" href="../pol_tutorial/user_def_err_pol.html" title="Calling User Defined Error Handlers">user error handling</a>).
</p>
<p>
To create a (possibly unwise!) policy <code class="computeroutput"><span class="identifier">ignore_all_policy</span></code>
that ignores all errors:
</p>
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">policy</span><span class="special">&lt;</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">domain_error</span><span class="special">&lt;</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">ignore_error</span><span class="special">&gt;,</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">overflow_error</span><span class="special">&lt;</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">ignore_error</span><span class="special">&gt;,</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">underflow_error</span><span class="special">&lt;</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">ignore_error</span><span class="special">&gt;,</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">denorm_error</span><span class="special">&lt;</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">ignore_error</span><span class="special">&gt;,</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">pole_error</span><span class="special">&lt;</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">ignore_error</span><span class="special">&gt;,</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">evaluation_error</span><span class="special">&lt;</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">ignore_error</span><span class="special">&gt;</span>
<span class="special">&gt;</span> <span class="identifier">ignore_all_policy</span><span class="special">;</span>
</pre>
<p>
Examples of use of this <code class="computeroutput"><span class="identifier">ignore_all_policy</span></code>
are
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">inf</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;::</span><span class="identifier">infinity</span><span class="special">();</span>
<span class="keyword">double</span> <span class="identifier">nan</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;::</span><span class="identifier">quiet_NaN</span><span class="special">();</span>
<span class="keyword">double</span> <span class="identifier">dodgy_root</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">cyl_bessel_j_zero</span><span class="special">(-</span><span class="number">1.0</span><span class="special">,</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">ignore_all_policy</span><span class="special">());</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"boost::math::cyl_bessel_j_zero(-1.0, 1) "</span> <span class="special">&lt;&lt;</span> <span class="identifier">dodgy_root</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 1.#QNAN</span>
<span class="keyword">double</span> <span class="identifier">inf_root</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">cyl_bessel_j_zero</span><span class="special">(</span><span class="identifier">inf</span><span class="special">,</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">ignore_all_policy</span><span class="special">());</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"boost::math::cyl_bessel_j_zero(inf, 1) "</span> <span class="special">&lt;&lt;</span> <span class="identifier">inf_root</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 1.#QNAN</span>
<span class="keyword">double</span> <span class="identifier">nan_root</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">cyl_bessel_j_zero</span><span class="special">(</span><span class="identifier">nan</span><span class="special">,</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">ignore_all_policy</span><span class="special">());</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"boost::math::cyl_bessel_j_zero(nan, 1) "</span> <span class="special">&lt;&lt;</span> <span class="identifier">nan_root</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 1.#QNAN</span>
</pre>
<p>
Another version of <code class="computeroutput"><span class="identifier">cyl_bessel_j_zero</span></code>
allows calculation of multiple zeros with one call, placing the results in
a container, often <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span></code>. For example, generate and display
the first five <code class="computeroutput"><span class="keyword">double</span></code> roots
of J<sub>v</sub> for integral order 2, as column <span class="emphasis"><em>J<sub>2</sub>(x)</em></span> in table
1 of <a href="http://mathworld.wolfram.com/BesselFunctionZeros.html" target="_top">Wolfram
Bessel Function Zeros</a>.
</p>
<pre class="programlisting"><span class="keyword">unsigned</span> <span class="keyword">int</span> <span class="identifier">n_roots</span> <span class="special">=</span> <span class="number">5U</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">roots</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">cyl_bessel_j_zero</span><span class="special">(</span><span class="number">2.0</span><span class="special">,</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">n_roots</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">back_inserter</span><span class="special">(</span><span class="identifier">roots</span><span class="special">));</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">copy</span><span class="special">(</span><span class="identifier">roots</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span>
<span class="identifier">roots</span><span class="special">.</span><span class="identifier">end</span><span class="special">(),</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream_iterator</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">,</span> <span class="string">"\n"</span><span class="special">));</span>
</pre>
<p>
Or we can use Boost.Multiprecision to generate 50 decimal digit roots of
<span class="emphasis"><em>J<sub>v</sub></em></span> for non-integral order <code class="computeroutput"><span class="identifier">v</span><span class="special">=</span> <span class="number">71</span><span class="special">/</span><span class="number">19</span> <span class="special">==</span> <span class="number">3.736842</span></code>,
expressed as an exact-integer fraction to generate the most accurate value
possible for all floating-point types.
</p>
<p>
We set the precision of the output stream, and show trailing zeros to display
a fixed 50 decimal digits.
</p>
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">float_type</span><span class="special">&gt;::</span><span class="identifier">digits10</span><span class="special">);</span> <span class="comment">// 50 decimal digits.</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">showpoint</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// Show trailing zeros.</span>
<span class="identifier">float_type</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">float_type</span><span class="special">(</span><span class="number">71</span><span class="special">)</span> <span class="special">/</span> <span class="number">19</span><span class="special">;</span>
<span class="identifier">float_type</span> <span class="identifier">r</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">cyl_bessel_j_zero</span><span class="special">(</span><span class="identifier">x</span><span class="special">,</span> <span class="number">1</span><span class="special">);</span> <span class="comment">// 1st root.</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"x = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">x</span> <span class="special">&lt;&lt;</span> <span class="string">", r = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">r</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">r</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">cyl_bessel_j_zero</span><span class="special">(</span><span class="identifier">x</span><span class="special">,</span> <span class="number">20U</span><span class="special">);</span> <span class="comment">// 20th root.</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"x = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">x</span> <span class="special">&lt;&lt;</span> <span class="string">", r = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">r</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="identifier">float_type</span><span class="special">&gt;</span> <span class="identifier">zeros</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">cyl_bessel_j_zero</span><span class="special">(</span><span class="identifier">x</span><span class="special">,</span> <span class="number">1</span><span class="special">,</span> <span class="number">3</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">back_inserter</span><span class="special">(</span><span class="identifier">zeros</span><span class="special">));</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"cyl_bessel_j_zeros"</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// Print the roots to the output stream.</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">copy</span><span class="special">(</span><span class="identifier">zeros</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">zeros</span><span class="special">.</span><span class="identifier">end</span><span class="special">(),</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream_iterator</span><span class="special">&lt;</span><span class="identifier">float_type</span><span class="special">&gt;(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">,</span> <span class="string">"\n"</span><span class="special">));</span>
</pre>
<h6>
<a name="math_toolkit.bessel.bessel_root.h3"></a>
<span class="phrase"><a name="math_toolkit.bessel.bessel_root.using_output_iterator_to_sum_zer"></a></span><a class="link" href="bessel_root.html#math_toolkit.bessel.bessel_root.using_output_iterator_to_sum_zer">Using
Output Iterator to sum zeros of Bessel Functions</a>
</h6>
<p>
This example demonstrates summing zeros of the Bessel functions. To use the
functions for finding zeros of the functions we need
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">bessel</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
</pre>
<p>
We use the <code class="computeroutput"><span class="identifier">cyl_bessel_j_zero</span></code>
output iterator parameter <code class="computeroutput"><span class="identifier">out_it</span></code>
to create a sum of <span class="emphasis"><em>1/zeros<sup>2</sup></em></span> by defining a custom output
iterator:
</p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<span class="keyword">struct</span> <span class="identifier">output_summation_iterator</span>
<span class="special">{</span>
<span class="identifier">output_summation_iterator</span><span class="special">(</span><span class="identifier">T</span><span class="special">*</span> <span class="identifier">p</span><span class="special">)</span> <span class="special">:</span> <span class="identifier">p_sum</span><span class="special">(</span><span class="identifier">p</span><span class="special">)</span>
<span class="special">{}</span>
<span class="identifier">output_summation_iterator</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">*()</span>
<span class="special">{</span> <span class="keyword">return</span> <span class="special">*</span><span class="keyword">this</span><span class="special">;</span> <span class="special">}</span>
<span class="identifier">output_summation_iterator</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">++()</span>
<span class="special">{</span> <span class="keyword">return</span> <span class="special">*</span><span class="keyword">this</span><span class="special">;</span> <span class="special">}</span>
<span class="identifier">output_summation_iterator</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">++(</span><span class="keyword">int</span><span class="special">)</span>
<span class="special">{</span> <span class="keyword">return</span> <span class="special">*</span><span class="keyword">this</span><span class="special">;</span> <span class="special">}</span>
<span class="identifier">output_summation_iterator</span><span class="special">&amp;</span> <span class="keyword">operator</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">val</span><span class="special">)</span>
<span class="special">{</span>
<span class="special">*</span><span class="identifier">p_sum</span> <span class="special">+=</span> <span class="number">1.</span><span class="special">/</span> <span class="special">(</span><span class="identifier">val</span> <span class="special">*</span> <span class="identifier">val</span><span class="special">);</span> <span class="comment">// Summing 1/zero^2.</span>
<span class="keyword">return</span> <span class="special">*</span><span class="keyword">this</span><span class="special">;</span>
<span class="special">}</span>
<span class="keyword">private</span><span class="special">:</span>
<span class="identifier">T</span><span class="special">*</span> <span class="identifier">p_sum</span><span class="special">;</span>
<span class="special">};</span>
</pre>
<p>
The sum is calculated for many values, converging on the analytical exact
value of <code class="computeroutput"><span class="number">1</span><span class="special">/</span><span class="number">8</span></code>.
</p>
<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cyl_bessel_j_zero</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">nu</span> <span class="special">=</span> <span class="number">1.</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">sum</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span>
<span class="identifier">output_summation_iterator</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">it</span><span class="special">(&amp;</span><span class="identifier">sum</span><span class="special">);</span> <span class="comment">// sum of 1/zeros^2</span>
<span class="identifier">cyl_bessel_j_zero</span><span class="special">(</span><span class="identifier">nu</span><span class="special">,</span> <span class="number">1</span><span class="special">,</span> <span class="number">10000</span><span class="special">,</span> <span class="identifier">it</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">s</span> <span class="special">=</span> <span class="number">1</span><span class="special">/(</span><span class="number">4</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">nu</span> <span class="special">+</span> <span class="number">1</span><span class="special">));</span> <span class="comment">// 0.125 = 1/8 is exact analytical solution.</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">(</span><span class="number">6</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">"nu = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">nu</span> <span class="special">&lt;&lt;</span> <span class="string">", sum = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">sum</span>
<span class="special">&lt;&lt;</span> <span class="string">", exact = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">s</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// nu = 1.00000, sum = 0.124990, exact = 0.125000</span>
</pre>
<h6>
<a name="math_toolkit.bessel.bessel_root.h4"></a>
<span class="phrase"><a name="math_toolkit.bessel.bessel_root.calculating_zeros_of_the_neumann"></a></span><a class="link" href="bessel_root.html#math_toolkit.bessel.bessel_root.calculating_zeros_of_the_neumann">Calculating
zeros of the Neumann function.</a>
</h6>
<p>
This example also shows how Boost.Math and Boost.Multiprecision can be combined
to provide a many decimal digit precision. For 50 decimal digit precision
we need to include
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">multiprecision</span><span class="special">/</span><span class="identifier">cpp_dec_float</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
</pre>
<p>
and a <code class="computeroutput"><span class="keyword">typedef</span></code> for <code class="computeroutput"><span class="identifier">float_type</span></code> may be convenient (allowing
a quick switch to re-compute at built-in <code class="computeroutput"><span class="keyword">double</span></code>
or other precision)
</p>
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_dec_float_50</span> <span class="identifier">float_type</span><span class="special">;</span>
</pre>
<p>
To use the functions for finding zeros of the <code class="computeroutput"><span class="identifier">cyl_neumann</span></code>
function we need:
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">bessel</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
</pre>
<p>
The Neumann (Bessel Y) function zeros are evaluated very similarly:
</p>
<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cyl_neumann_zero</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">zn</span> <span class="special">=</span> <span class="identifier">cyl_neumann_zero</span><span class="special">(</span><span class="number">2.</span><span class="special">,</span> <span class="number">1</span><span class="special">);</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"cyl_neumann_zero(2., 1) = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">zn</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;</span> <span class="identifier">nzeros</span><span class="special">(</span><span class="number">3</span><span class="special">);</span> <span class="comment">// Space for 3 zeros.</span>
<span class="identifier">cyl_neumann_zero</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;(</span><span class="number">2.F</span><span class="special">,</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">nzeros</span><span class="special">.</span><span class="identifier">size</span><span class="special">(),</span> <span class="identifier">nzeros</span><span class="special">.</span><span class="identifier">begin</span><span class="special">());</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"cyl_neumann_zero&lt;float&gt;(2.F, 1, "</span><span class="special">;</span>
<span class="comment">// Print the zeros to the output stream.</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">copy</span><span class="special">(</span><span class="identifier">nzeros</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">nzeros</span><span class="special">.</span><span class="identifier">end</span><span class="special">(),</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream_iterator</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">,</span> <span class="string">", "</span><span class="special">));</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="string">"cyl_neumann_zero(static_cast&lt;float_type&gt;(220)/100, 1) = "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">cyl_neumann_zero</span><span class="special">(</span><span class="keyword">static_cast</span><span class="special">&lt;</span><span class="identifier">float_type</span><span class="special">&gt;(</span><span class="number">220</span><span class="special">)/</span><span class="number">100</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// 3.6154383428745996706772556069431792744372398748422</span>
</pre>
<h6>
<a name="math_toolkit.bessel.bessel_root.h5"></a>
<span class="phrase"><a name="math_toolkit.bessel.bessel_root.error_messages_from_bad_input"></a></span><a class="link" href="bessel_root.html#math_toolkit.bessel.bessel_root.error_messages_from_bad_input">Error
messages from 'bad' input</a>
</h6>
<p>
Another example demonstrates calculating zeros of the Bessel functions showing
the error messages from 'bad' input is handled by throwing exceptions.
</p>
<p>
To use the functions for finding zeros of the functions we need:
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">bessel</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">airy</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
</pre>
<div class="tip"><table border="0" summary="Tip">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../doc/src/images/tip.png"></td>
<th align="left">Tip</th>
</tr>
<tr><td align="left" valign="top"><p>
It is always wise to place all code using Boost.Math inside try'n'catch
blocks; this will ensure that helpful error messages can be shown when
exceptional conditions arise.
</p></td></tr>
</table></div>
<p>
Examples below show messages from several 'bad' arguments that throw a <code class="computeroutput"><span class="identifier">domain_error</span></code> exception.
</p>
<pre class="programlisting"><span class="keyword">try</span>
<span class="special">{</span> <span class="comment">// Try a zero order v.</span>
<span class="keyword">float</span> <span class="identifier">dodgy_root</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">cyl_bessel_j_zero</span><span class="special">(</span><span class="number">0.F</span><span class="special">,</span> <span class="number">0</span><span class="special">);</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"boost::math::cyl_bessel_j_zero(0.F, 0) "</span> <span class="special">&lt;&lt;</span> <span class="identifier">dodgy_root</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// Thrown exception Error in function boost::math::cyl_bessel_j_zero&lt;double&gt;(double, int):</span>
<span class="comment">// Requested the 0'th zero of J0, but the rank must be &gt; 0 !</span>
<span class="special">}</span>
<span class="keyword">catch</span> <span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">exception</span><span class="special">&amp;</span> <span class="identifier">ex</span><span class="special">)</span>
<span class="special">{</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Thrown exception "</span> <span class="special">&lt;&lt;</span> <span class="identifier">ex</span><span class="special">.</span><span class="identifier">what</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
</pre>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top"><p>
The type shown in the error message is the type <span class="bold"><strong>after
promotion</strong></span>, using <a class="link" href="../pol_ref/precision_pol.html" title="Precision Policies">precision
policy</a> and <a class="link" href="../pol_ref/internal_promotion.html" title="Internal Floating-point Promotion Policies">internal
promotion policy</a>, from <code class="computeroutput"><span class="keyword">float</span></code>
to <code class="computeroutput"><span class="keyword">double</span></code> in this case.
</p></td></tr>
</table></div>
<p>
In this example the promotion goes:
</p>
<div class="orderedlist"><ol class="orderedlist" type="1">
<li class="listitem">
Arguments are <code class="computeroutput"><span class="keyword">float</span></code> and
<code class="computeroutput"><span class="keyword">int</span></code>.
</li>
<li class="listitem">
Treat <code class="computeroutput"><span class="keyword">int</span></code> "as if"
it were a <code class="computeroutput"><span class="keyword">double</span></code>, so arguments
are <code class="computeroutput"><span class="keyword">float</span></code> and <code class="computeroutput"><span class="keyword">double</span></code>.
</li>
<li class="listitem">
Common type is <code class="computeroutput"><span class="keyword">double</span></code> -
so that's the precision we want (and the type that will be returned).
</li>
<li class="listitem">
Evaluate internally as <code class="computeroutput"><span class="keyword">double</span></code>
for full <code class="computeroutput"><span class="keyword">float</span></code> precision.
</li>
</ol></div>
<p>
See full code for other examples that promote from <code class="computeroutput"><span class="keyword">double</span></code>
to <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>.
</p>
<p>
Other examples of 'bad' inputs like infinity and NaN are below. Some compiler
warnings indicate that 'bad' values are detected at compile time.
</p>
<pre class="programlisting"><span class="keyword">try</span>
<span class="special">{</span> <span class="comment">// order v = inf</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"boost::math::cyl_bessel_j_zero(inf, 1) "</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">inf</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;::</span><span class="identifier">infinity</span><span class="special">();</span>
<span class="keyword">double</span> <span class="identifier">inf_root</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">cyl_bessel_j_zero</span><span class="special">(</span><span class="identifier">inf</span><span class="special">,</span> <span class="number">1</span><span class="special">);</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"boost::math::cyl_bessel_j_zero(inf, 1) "</span> <span class="special">&lt;&lt;</span> <span class="identifier">inf_root</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// Throw exception Error in function boost::math::cyl_bessel_j_zero&lt;long double&gt;(long double, unsigned):</span>
<span class="comment">// Order argument is 1.#INF, but must be finite &gt;= 0 !</span>
<span class="special">}</span>
<span class="keyword">catch</span> <span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">exception</span><span class="special">&amp;</span> <span class="identifier">ex</span><span class="special">)</span>
<span class="special">{</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Thrown exception "</span> <span class="special">&lt;&lt;</span> <span class="identifier">ex</span><span class="special">.</span><span class="identifier">what</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
<span class="keyword">try</span>
<span class="special">{</span> <span class="comment">// order v = NaN, rank m = 1</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"boost::math::cyl_bessel_j_zero(nan, 1) "</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">nan</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;::</span><span class="identifier">quiet_NaN</span><span class="special">();</span>
<span class="keyword">double</span> <span class="identifier">nan_root</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">cyl_bessel_j_zero</span><span class="special">(</span><span class="identifier">nan</span><span class="special">,</span> <span class="number">1</span><span class="special">);</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"boost::math::cyl_bessel_j_zero(nan, 1) "</span> <span class="special">&lt;&lt;</span> <span class="identifier">nan_root</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// Throw exception Error in function boost::math::cyl_bessel_j_zero&lt;long double&gt;(long double, unsigned):</span>
<span class="comment">// Order argument is 1.#QNAN, but must be finite &gt;= 0 !</span>
<span class="special">}</span>
<span class="keyword">catch</span> <span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">exception</span><span class="special">&amp;</span> <span class="identifier">ex</span><span class="special">)</span>
<span class="special">{</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Thrown exception "</span> <span class="special">&lt;&lt;</span> <span class="identifier">ex</span><span class="special">.</span><span class="identifier">what</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
</pre>
<p>
The output from other examples are shown appended to the full code listing.
</p>
<p>
The full code (and output) for these examples is at <a href="../../../../example/bessel_zeros_example_1.cpp" target="_top">Bessel
zeros</a>, <a href="../../../../example/bessel_zeros_interator_example.cpp" target="_top">Bessel
zeros iterator</a>, <a href="../../../../example/neumann_zeros_example_1.cpp" target="_top">Neumann
zeros</a>, <a href="../../../../example/bessel_errors_example.cpp" target="_top">Bessel
error messages</a>.
</p>
<h4>
<a name="math_toolkit.bessel.bessel_root.h6"></a>
<span class="phrase"><a name="math_toolkit.bessel.bessel_root.implementation"></a></span><a class="link" href="bessel_root.html#math_toolkit.bessel.bessel_root.implementation">Implementation</a>
</h4>
<p>
Various methods are used to compute initial estimates for <span class="emphasis"><em>j<sub>&#957;, m</sub></em></span>
and <span class="emphasis"><em>y<sub>&#957;, m</sub></em></span> ; these are described in detail below.
</p>
<p>
After finding the initial estimate of a given root, its precision is subsequently
refined to the desired level using Newton-Raphson iteration from Boost.Math's
<a class="link" href="../roots_deriv.html" title="Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schr&#246;der">root-finding with derivatives</a>
utilities combined with the functions <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a>
and <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a>.
</p>
<p>
Newton iteration requires both <span class="emphasis"><em>J<sub>&#957;</sub>(x)</em></span> or <span class="emphasis"><em>Y<sub>&#957;</sub>(x)</em></span>
as well as its derivative. The derivatives of <span class="emphasis"><em>J<sub>&#957;</sub>(x)</em></span> and
<span class="emphasis"><em>Y<sub>&#957;</sub>(x)</em></span> with respect to <span class="emphasis"><em>x</em></span> are given
by M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, NBS
(1964). In particular,
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">d/<sub>dx</sub> <span class="emphasis"><em>J<sub>&#957;</sub>(x)</em></span> = <span class="emphasis"><em>J<sub>&#957;-1</sub>(x)</em></span>
- &#957; J<sub>&#957;</sub>(x) / x</span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">d/<sub>dx</sub> <span class="emphasis"><em>Y<sub>&#957;</sub>(x)</em></span> = <span class="emphasis"><em>Y<sub>&#957;-1</sub>(x)</em></span>
- &#957; Y<sub>&#957;</sub>(x) / x</span>
</p></blockquote></div>
<p>
Enumeration of the rank of a root (in other words the index of a root) begins
with one and counts up, in other words <span class="emphasis"><em>m,=1,2,3,&#8230;</em></span> The
value of the first root is always greater than zero.
</p>
<p>
For certain special parameters, cylindrical Bessel functions and cylindrical
Neumann functions have a root at the origin. For example, <span class="emphasis"><em>J<sub>&#957;</sub>(x)</em></span>
has a root at the origin for every positive order <span class="emphasis"><em>&#957; &gt; 0</em></span>,
and for every negative integer order <span class="emphasis"><em>&#957; = -n</em></span> with <span class="emphasis"><em>n
&#8712; &#8469; <sup>+</sup></em></span> and <span class="emphasis"><em>n &#8800; 0</em></span>.
</p>
<p>
In addition, <span class="emphasis"><em>Y<sub>&#957;</sub>(x)</em></span> has a root at the origin for every
negative half-integer order <span class="emphasis"><em>&#957; = -n/2</em></span>, with <span class="emphasis"><em>n
&#8712; &#8469; <sup>+</sup></em></span> and and <span class="emphasis"><em>n &#8800; 0</em></span>.
</p>
<p>
For these special parameter values, the origin with a value of <span class="emphasis"><em>x
= 0</em></span> is provided as the <span class="emphasis"><em>0<sup>th</sup></em></span> root generated
by <code class="computeroutput"><span class="identifier">cyl_bessel_j_zero</span><span class="special">()</span></code>
and <code class="computeroutput"><span class="identifier">cyl_neumann_zero</span><span class="special">()</span></code>.
</p>
<p>
When calculating initial estimates for the roots of Bessel functions, a distinction
is made between positive order and negative order, and different methods
are used for these. In addition, different algorithms are used for the first
root <span class="emphasis"><em>m = 1</em></span> and for subsequent roots with higher rank
<span class="emphasis"><em>m &#8805; 2</em></span>. Furthermore, estimates of the roots for Bessel
functions with order above and below a cutoff at <span class="emphasis"><em>&#957; = 2.2</em></span>
are calculated with different methods.
</p>
<p>
Calculations of the estimates of <span class="emphasis"><em>j<sub>&#957;,1</sub></em></span> and <span class="emphasis"><em>y<sub>&#957;,1</sub></em></span>
with <span class="emphasis"><em>0 &#8804; &#957; &lt; 2.2</em></span> use empirically tabulated values. The
coefficients for these have been generated by a computer algebra system.
</p>
<p>
Calculations of the estimates of <span class="emphasis"><em>j<sub>&#957;,1</sub></em></span> and <span class="emphasis"><em>y<sub>&#957;,1</sub></em></span>
with <span class="emphasis"><em>&#957;&#8805; 2.2</em></span> use Eqs.9.5.14 and 9.5.15 in M. Abramowitz
and I. A. Stegun, Handbook of Mathematical Functions, NBS (1964).
</p>
<p>
In particular,
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">j<sub>&#957;,1</sub> &#8773; &#957; + 1.85575 &#957;<sup>&#8531;</sup> + 1.033150 &#957;<sup>-&#8531;</sup> - 0.00397 &#957;<sup>-1</sup> - 0.0908
&#957;<sup>-5/3</sup> + 0.043 &#957;<sup>-7/3</sup> + &#8230;</span>
</p></blockquote></div>
<p>
and
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">y<sub>&#957;,1</sub> &#8773; &#957; + 0.93157 &#957;<sup>&#8531;</sup> + 0.26035 &#957;<sup>-&#8531;</sup> + 0.01198 &#957;<sup>-1</sup> - 0.0060
&#957;<sup>-5/3</sup> - 0.001 &#957;<sup>-7/3</sup> + &#8230;</span>
</p></blockquote></div>
<p>
Calculations of the estimates of <span class="emphasis"><em>j<sub>&#957;, m</sub></em></span> and <span class="emphasis"><em>y<sub>&#957;,
m</sub></em></span> with rank <span class="emphasis"><em>m &gt; 2</em></span> and <span class="emphasis"><em>0 &#8804; &#957; &lt;
2.2</em></span> use McMahon's approximation, as described in M. Abramowitz
and I. A. Stegan, Section 9.5 and 9.5.12. In particular,
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="emphasis"><em>j<sub>&#957;,m</sub>, y<sub>&#957;,m</sub> &#8773;</em></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><div class="blockquote"><blockquote class="blockquote"><p>
&#946; - (&#956;-1) / 8&#946;
</p></blockquote></div></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><div class="blockquote"><blockquote class="blockquote"><p>
<span class="emphasis"><em>- 4(&#956;-1)(7&#956; - 31) / 3(8&#946;)<sup>3</sup></em></span>
</p></blockquote></div></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><div class="blockquote"><blockquote class="blockquote"><p>
<span class="emphasis"><em>-32(&#956;-1)(83&#956;&#178; - 982&#956; + 3779) / 15(8&#946;)<sup>5</sup></em></span>
</p></blockquote></div></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><div class="blockquote"><blockquote class="blockquote"><p>
<span class="emphasis"><em>-64(&#956;-1)(6949&#956;<sup>3</sup> - 153855&#956;&#178; + 1585743&#956;- 6277237) / 105(8a)<sup>7</sup></em></span>
</p></blockquote></div></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><div class="blockquote"><blockquote class="blockquote"><p>
<span class="emphasis"><em>- &#8230;</em></span> &#8193; (5)
</p></blockquote></div></blockquote></div>
<p>
where <span class="emphasis"><em>&#956; = 4&#957;<sup>2</sup></em></span> and <span class="emphasis"><em>&#946; = (m + &#189;&#957; - &#188;)&#960;</em></span> for
<span class="emphasis"><em>j<sub>&#957;,m</sub></em></span> and <span class="emphasis"><em>&#946; = (m + &#189;&#957; -&#190;)&#960; for <span class="emphasis"><em>y<sub>&#957;,m</sub></em></span></em></span>.
</p>
<p>
Calculations of the estimates of <span class="emphasis"><em>j<sub>&#957;, m</sub></em></span> and <span class="emphasis"><em>y<sub>&#957;,
m</sub></em></span> with <span class="emphasis"><em>&#957; &#8805; 2.2</em></span> use one term in the asymptotic
expansion given in Eq.9.5.22 and top line of Eq.9.5.26 combined with Eq.
9.3.39, all in M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions,
NBS (1964) explicit and easy-to-understand treatment for asymptotic expansion
of zeros. The latter two equations are expressed for argument <span class="emphasis"><em>(x)</em></span>
greater than one. (Olver also gives the series form of the equations in
<a href="http://dlmf.nist.gov/10.21#vi" target="_top">&#167;10.21(vi) McMahon's Asymptotic
Expansions for Large Zeros</a> - using slightly different variable names).
</p>
<p>
In summary,
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">j<sub>&#957;, m</sub> &#8764; &#957;x(-&#950;) + f<sub>1</sub>(-&#950;/&#957;)</span>
</p></blockquote></div>
<p>
where <span class="emphasis"><em>-&#950; = &#957;<sup>-2/3</sup>a<sub>m</sub></em></span> and <span class="emphasis"><em>a<sub>m</sub></em></span> is the absolute
value of the <span class="emphasis"><em>m<sup>th</sup></em></span> root of <span class="emphasis"><em>Ai(x)</em></span>
on the negative real axis.
</p>
<p>
Here <span class="emphasis"><em>x = x(-&#950;)</em></span> is the inverse of the function
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">&#8532;(-&#950;)<sup>3/2</sup> = &#8730;(x&#178; - 1) - cos&#8315;&#185;(1/x)</span>
</p></blockquote></div>
<p>
(7)
</p>
<p>
Furthermore,
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">f<sub>1</sub>(-&#950;) = &#189;x(-&#950;) {h(-&#950;)}&#178; &#8901; b<sub>0</sub>(-&#950;)</span>
</p></blockquote></div>
<p>
where
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">h(-&#950;) = {4(-&#950;) / (x&#178; - 1)}<sup>4</sup></span>
</p></blockquote></div>
<p>
and
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">b<sub>0</sub>(-&#950;) = -5/(48&#950;&#178;) + 1/(-&#950;)<sup>&#189;</sup> &#8901; { 5/(24(x<sup>2</sup>-1)<sup>3/2</sup>) +
1/(8(x<sup>2</sup>-1)<sup>&#189;)</sup>}</span>
</p></blockquote></div>
<p>
When solving for <span class="emphasis"><em>x(-&#950;)</em></span> in Eq. 7 above, the right-hand-side
is expanded to order 2 in a Taylor series for large <span class="emphasis"><em>x</em></span>.
This results in
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">&#8532;(-&#950;)<sup>3/2</sup> &#8776; x + 1/2x - &#960;/2</span>
</p></blockquote></div>
<p>
The positive root of the resulting quadratic equation is used to find an
initial estimate <span class="emphasis"><em>x(-&#950;)</em></span>. This initial estimate is subsequently
refined with several steps of Newton-Raphson iteration in Eq. 7.
</p>
<p>
Estimates of the roots of cylindrical Bessel functions of negative order
on the positive real axis are found using interlacing relations. For example,
the <span class="emphasis"><em>m<sup>th</sup></em></span> root of the cylindrical Bessel function <span class="emphasis"><em>j<sub>-&#957;,m</sub></em></span>
is bracketed by the <span class="emphasis"><em>m<sup>th</sup></em></span> root and the <span class="emphasis"><em>(m+1)<sup>th</sup></em></span>
root of the Bessel function of corresponding positive integer order. In other
words,
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">j<sub>n&#957;, m</sub> &lt; j<sub>-&#957;, m</sub> &lt; j<sub>n&#957;, m+1</sub></span>
</p></blockquote></div>
<p>
where <span class="emphasis"><em>m &gt; 1</em></span> and <span class="emphasis"><em>n<sub>&#957;</sub></em></span> represents
the integral floor of the absolute value of <span class="emphasis"><em>|-&#957;|</em></span>.
</p>
<p>
Similar bracketing relations are used to find estimates of the roots of Neumann
functions of negative order, whereby a discontinuity at every negative half-integer
order needs to be handled.
</p>
<p>
Bracketing relations do not hold for the first root of cylindrical Bessel
functions and cylindrical Neumann functions with negative order. Therefore,
iterative algorithms combined with root-finding via bisection are used to
localize <span class="emphasis"><em>j<sub>-&#957;,1</sub></em></span> and <span class="emphasis"><em>y<sub>-&#957;,1</sub></em></span>.
</p>
<h4>
<a name="math_toolkit.bessel.bessel_root.h7"></a>
<span class="phrase"><a name="math_toolkit.bessel.bessel_root.testing"></a></span><a class="link" href="bessel_root.html#math_toolkit.bessel.bessel_root.testing">Testing</a>
</h4>
<p>
The precision of evaluation of zeros was tested at 50 decimal digits using
<code class="computeroutput"><span class="identifier">cpp_dec_float_50</span></code> and found
identical with spot values computed by <a href="http://www.wolframalpha.com/" target="_top">Wolfram
Alpha</a>.
</p>
</div>
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<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, 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>)
</p>
</div></td>
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