<title>Bessel Functions of the First and Second Kinds</title>
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<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">cyl_bessel_j</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">);</span>
-<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
-<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">cyl_bessel_j</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 19. 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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
+<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">cyl_bessel_j</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 20. 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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">cyl_neumann</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">);</span>
-<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
-<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">cyl_neumann</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 19. 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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
+<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">cyl_neumann</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
</pre>
<h5>
<a name="math_toolkit.bessel.bessel_first.h1"></a>
and <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a> return
the result of the Bessel functions of the first and second kinds respectively:
</p>
-<p>
- cyl_bessel_j(v, x) = J<sub>v</sub>(x)
- </p>
-<p>
- cyl_neumann(v, x) = Y<sub>v</sub>(x) = N<sub>v</sub>(x)
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">cyl_bessel_j(v, x) = J<sub>v</sub>(x)</span>
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">cyl_neumann(v, x) = Y<sub>v</sub>(x) = N<sub>v</sub>(x)</span>
+ </p></blockquote></div>
<p>
where:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel2.svg"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel3.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel2.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel3.svg"></span>
+
+ </p></blockquote></div>
<p>
The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
type calculation rules</em></span></a> when T1 and T2 are different types.
an integer.
</p>
<p>
- The final <a class="link" href="../../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
+ The final <a class="link" href="../../policy.html" title="Chapter 20. 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 19. Policies: Controlling Precision, Error Handling etc">policy
+ what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<p>
<p>
The following graph illustrates the cyclic nature of J<sub>v</sub>:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/cyl_bessel_j.svg" align="middle"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/cyl_bessel_j.svg" align="middle"></span>
+
+ </p></blockquote></div>
<p>
The following graph shows the behaviour of Y<sub>v</sub>: this is also cyclic for large
- <span class="emphasis"><em>x</em></span>, but tends to -∞   for small <span class="emphasis"><em>x</em></span>:
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/cyl_neumann.svg" align="middle"></span>
+ <span class="emphasis"><em>x</em></span>, but tends to -∞ for small <span class="emphasis"><em>x</em></span>:
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/cyl_neumann.svg" align="middle"></span>
+
+ </p></blockquote></div>
<h5>
<a name="math_toolkit.bessel.bessel_first.h2"></a>
<span class="phrase"><a name="math_toolkit.bessel.bessel_first.testing"></a></span><a class="link" href="bessel_first.html#math_toolkit.bessel.bessel_first.testing">Testing</a>
more information on this.
</p>
<div class="table">
-<a name="math_toolkit.bessel.bessel_first.table_cyl_bessel_j_integer_orders_"></a><p class="title"><b>Table 7.40. Error rates for cyl_bessel_j (integer orders)</b></p>
+<a name="math_toolkit.bessel.bessel_first.table_cyl_bessel_j_integer_orders_"></a><p class="title"><b>Table 8.40. Error rates for cyl_bessel_j (integer orders)</b></p>
<div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_j (integer orders)">
<colgroup>
<col>
</table></div>
</div>
<br class="table-break"><div class="table">
-<a name="math_toolkit.bessel.bessel_first.table_cyl_bessel_j"></a><p class="title"><b>Table 7.41. Error rates for cyl_bessel_j</b></p>
+<a name="math_toolkit.bessel.bessel_first.table_cyl_bessel_j"></a><p class="title"><b>Table 8.41. Error rates for cyl_bessel_j</b></p>
<div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_j">
<colgroup>
<col>
</table></div>
</div>
<br class="table-break"><div class="table">
-<a name="math_toolkit.bessel.bessel_first.table_cyl_neumann_integer_orders_"></a><p class="title"><b>Table 7.42. Error rates for cyl_neumann (integer orders)</b></p>
+<a name="math_toolkit.bessel.bessel_first.table_cyl_neumann_integer_orders_"></a><p class="title"><b>Table 8.42. Error rates for cyl_neumann (integer orders)</b></p>
<div class="table-contents"><table class="table" summary="Error rates for cyl_neumann (integer orders)">
<colgroup>
<col>
</table></div>
</div>
<br class="table-break"><div class="table">
-<a name="math_toolkit.bessel.bessel_first.table_cyl_neumann"></a><p class="title"><b>Table 7.43. Error rates for cyl_neumann</b></p>
+<a name="math_toolkit.bessel.bessel_first.table_cyl_neumann"></a><p class="title"><b>Table 8.43. Error rates for cyl_neumann</b></p>
<div class="table-contents"><table class="table" summary="Error rates for cyl_neumann">
<colgroup>
<col>
illustrate the relatively large errors as you approach a zero, and the very
low errors elsewhere.
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/j0__double.svg" align="middle"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/y0__double.svg" align="middle"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/j0__double.svg" align="middle"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/y0__double.svg" align="middle"></span>
+
+ </p></blockquote></div>
<h5>
<a name="math_toolkit.bessel.bessel_first.h4"></a>
<span class="phrase"><a name="math_toolkit.bessel.bessel_first.implementation"></a></span><a class="link" href="bessel_first.html#math_toolkit.bessel.bessel_first.implementation">Implementation</a>
When the order <span class="emphasis"><em>v</em></span> is negative then the reflection formulae
can be used to move to <span class="emphasis"><em>v > 0</em></span>:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel9.svg"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel10.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel9.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel10.svg"></span>
+
+ </p></blockquote></div>
<p>
Note that if the order is an integer, then these formulae reduce to:
</p>
-<p>
- J<sub>-n</sub> = (-1)<sup>n</sup>J<sub>n</sub>
- </p>
-<p>
- Y<sub>-n</sub> = (-1)<sup>n</sup>Y<sub>n</sub>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">J<sub>-n</sub> = (-1)<sup>n</sup>J<sub>n</sub></span>
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">Y<sub>-n</sub> = (-1)<sup>n</sup>Y<sub>n</sub></span>
+ </p></blockquote></div>
<p>
However, in general, a negative order implies that we will need to compute
both J and Y.
</p>
<p>
When the order <span class="emphasis"><em>v</em></span> is an integer the method first relates
- the result to J<sub>0</sub>, J<sub>1</sub>, Y<sub>0</sub>   and Y<sub>1</sub>   using either forwards or backwards recurrence
+ the result to J<sub>0</sub>, J<sub>1</sub>, Y<sub>0</sub> and Y<sub>1</sub> using either forwards or backwards recurrence
(Miller's algorithm) depending upon which is stable. The values for J<sub>0</sub>, J<sub>1</sub>,
- Y<sub>0</sub>   and Y<sub>1</sub>   are calculated using the rational minimax approximations on root-bracketing
+ Y<sub>0</sub> and Y<sub>1</sub> are calculated using the rational minimax approximations on root-bracketing
intervals for small <span class="emphasis"><em>|x|</em></span> and Hankel asymptotic expansion
for large <span class="emphasis"><em>|x|</em></span>. The coefficients are from:
</p>
-<p>
- W.J. Cody, <span class="emphasis"><em>ALGORITHM 715: SPECFUN - A Portable FORTRAN Package
- of Special Function Routines and Test Drivers</em></span>, ACM Transactions
- on Mathematical Software, vol 19, 22 (1993).
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ W.J. Cody, <span class="emphasis"><em>ALGORITHM 715: SPECFUN - A Portable FORTRAN Package
+ of Special Function Routines and Test Drivers</em></span>, ACM Transactions
+ on Mathematical Software, vol 19, 22 (1993).
+ </p></blockquote></div>
<p>
and
</p>
-<p>
- J.F. Hart et al, <span class="emphasis"><em>Computer Approximations</em></span>, John Wiley
- & Sons, New York, 1968.
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ J.F. Hart et al, <span class="emphasis"><em>Computer Approximations</em></span>, John Wiley
+ & Sons, New York, 1968.
+ </p></blockquote></div>
<p>
These approximations are accurate to around 19 decimal digits: therefore
these methods are not used when type T has more than 64 binary digits.
<a href="http://functions.wolfram.com/03.03.06.0039.01" target="_top">http://functions.wolfram.com/03.03.06.0039.01</a>
and <a href="http://functions.wolfram.com/03.03.06.0040.01" target="_top">http://functions.wolfram.com/03.03.06.0040.01</a>):
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel_y0_small_z.svg"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel_y1_small_z.svg"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel_y2_small_z.svg"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel_yn_small_z.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel_y0_small_z.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel_y1_small_z.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel_y2_small_z.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel_yn_small_z.svg"></span>
+
+ </p></blockquote></div>
<p>
When <span class="emphasis"><em>x</em></span> is small compared to <span class="emphasis"><em>v</em></span> and
<span class="emphasis"><em>v</em></span> is not an integer, then the following series approximation
can be used for Y<sub>v</sub>(x), this is also an area where other approximations are
often too slow to converge to be used (see <a href="http://functions.wolfram.com/03.03.06.0034.01" target="_top">http://functions.wolfram.com/03.03.06.0034.01</a>):
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel_yv_small_z.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel_yv_small_z.svg"></span>
+
+ </p></blockquote></div>
<p>
When <span class="emphasis"><em>x</em></span> is small compared to <span class="emphasis"><em>v</em></span>,
- J<sub>v</sub>x   is best computed directly from the series:
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel2.svg"></span>
+ J<sub>v</sub>x is best computed directly from the series:
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel2.svg"></span>
+
+ </p></blockquote></div>
<p>
- In the general case we compute J<sub>v</sub>   and Y<sub>v</sub>   simultaneously.
+ In the general case we compute J<sub>v</sub> and Y<sub>v</sub> simultaneously.
</p>
<p>
- To get the initial values, let μ   = ν - floor(ν + 1/2), then μ   is the fractional part
- of ν   such that |μ| <= 1/2 (we need this for convergence later). The idea
+ To get the initial values, let μ = ν - floor(ν + 1/2), then μ is the fractional part
+ of ν such that |μ| <= 1/2 (we need this for convergence later). The idea
is to calculate J<sub>μ</sub>(x), J<sub>μ+1</sub>(x), Y<sub>μ</sub>(x), Y<sub>μ+1</sub>(x) and use them to obtain J<sub>ν</sub>(x), Y<sub>ν</sub>(x).
</p>
<p>
The algorithm is called Steed's method, which needs two continued fractions
as well as the Wronskian:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel8.svg"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel11.svg"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel12.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel8.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel11.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel12.svg"></span>
+
+ </p></blockquote></div>
<p>
See: F.S. Acton, <span class="emphasis"><em>Numerical Methods that Work</em></span>, The Mathematical
Association of America, Washington, 1997.
Lentz, <span class="emphasis"><em>Generating Bessel functions in Mie scattering calculations
using continued fractions</em></span>, Applied Optics, vol 15, 668 (1976)).
Their convergence rates depend on <span class="emphasis"><em>x</em></span>, therefore we need
- different strategies for large <span class="emphasis"><em>x</em></span> and small <span class="emphasis"><em>x</em></span>.
- </p>
-<p>
- <span class="emphasis"><em>x > v</em></span>, CF1 needs O(<span class="emphasis"><em>x</em></span>) iterations
- to converge, CF2 converges rapidly
- </p>
-<p>
- <span class="emphasis"><em>x <= v</em></span>, CF1 converges rapidly, CF2 fails to converge
- when <span class="emphasis"><em>x</em></span> <code class="literal">-></code> 0
+ different strategies for large <span class="emphasis"><em>x</em></span> and small <span class="emphasis"><em>x</em></span>:
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="emphasis"><em>x > v</em></span>, CF1 needs O(<span class="emphasis"><em>x</em></span>) iterations
+ to converge, CF2 converges rapidly
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="emphasis"><em>x <= v</em></span>, CF1 converges rapidly, CF2 fails to converge
+ when <span class="emphasis"><em>x</em></span> <code class="literal">-></code> 0
+ </p></blockquote></div>
<p>
When <span class="emphasis"><em>x</em></span> is large (<span class="emphasis"><em>x</em></span> > 2), both
continued fractions converge (CF1 may be slow for really large <span class="emphasis"><em>x</em></span>).
J<sub>μ</sub>, J<sub>μ+1</sub>, Y<sub>μ</sub>, Y<sub>μ+1</sub> can be calculated by
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel13.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel13.svg"></span>
+
+ </p></blockquote></div>
<p>
where
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel14.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel14.svg"></span>
+
+ </p></blockquote></div>
<p>
J<sub>ν</sub> and Y<sub>μ</sub> are then calculated using backward (Miller's algorithm) and forward
recurrence respectively.
convergence may fail (but CF1 works very well). The solution here is Temme's
series:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel15.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel15.svg"></span>
+
+ </p></blockquote></div>
<p>
where
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel16.svg"></span>
+
+ </p></blockquote></div>
<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel16.svg"></span>
- </p>
-<p>
- g<sub>k</sub>   and h<sub>k</sub>  
+ g<sub>k</sub> and h<sub>k</sub>
are also computed by recursions (involving gamma functions), but
the formulas are a little complicated, readers are refered to N.M. Temme,
<span class="emphasis"><em>On the numerical evaluation of the ordinary Bessel function of
(1976). Note Temme's series converge only for |μ| <= 1/2.
</p>
<p>
- As the previous case, Y<sub>ν</sub>   is calculated from the forward recurrence, so is Y<sub>ν+1</sub>.
+ As the previous case, Y<sub>ν</sub> is calculated from the forward recurrence, so is Y<sub>ν+1</sub>.
With these two values and f<sub>ν</sub>, the Wronskian yields J<sub>ν</sub>(x) directly without backward
recurrence.
</p>
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