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26 <div class=
"titlepage"><div><div><h3 class=
"title">
27 <a name=
"math_toolkit.bessel.bessel_derivatives"></a><a class=
"link" href=
"bessel_derivatives.html" title=
"Derivatives of the Bessel Functions">Derivatives of
28 the Bessel Functions
</a>
29 </h3></div></div></div>
31 <a name=
"math_toolkit.bessel.bessel_derivatives.h0"></a>
32 <span class=
"phrase"><a name=
"math_toolkit.bessel.bessel_derivatives.synopsis"></a></span><a class=
"link" href=
"bessel_derivatives.html#math_toolkit.bessel.bessel_derivatives.synopsis">Synopsis
</a>
35 <code class=
"computeroutput"><span class=
"preprocessor">#include
</span> <span class=
"special"><</span><span class=
"identifier">boost
</span><span class=
"special">/
</span><span class=
"identifier">math
</span><span class=
"special">/
</span><span class=
"identifier">special_functions
</span><span class=
"special">/
</span><span class=
"identifier">bessel_prime
</span><span class=
"special">.
</span><span class=
"identifier">hpp
</span><span class=
"special">></span></code>
37 <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>
38 <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_prime
</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>
40 <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 15. Policies: Controlling Precision, Error Handling etc">Policy
</a><span class=
"special">></span>
41 <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_prime
</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 15. Policies: Controlling Precision, Error Handling etc">Policy
</a><span class=
"special">&);
</span>
43 <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>
44 <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_prime
</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>
46 <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 15. Policies: Controlling Precision, Error Handling etc">Policy
</a><span class=
"special">></span>
47 <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_prime
</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 15. Policies: Controlling Precision, Error Handling etc">Policy
</a><span class=
"special">&);
</span>
49 <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>
50 <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_i_prime
</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>
52 <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 15. Policies: Controlling Precision, Error Handling etc">Policy
</a><span class=
"special">></span>
53 <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_i_prime
</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 15. Policies: Controlling Precision, Error Handling etc">Policy
</a><span class=
"special">&);
</span>
55 <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>
56 <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_k_prime
</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>
58 <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 15. Policies: Controlling Precision, Error Handling etc">Policy
</a><span class=
"special">></span>
59 <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_k_prime
</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 15. Policies: Controlling Precision, Error Handling etc">Policy
</a><span class=
"special">&);
</span>
61 <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>
62 <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">sph_bessel_prime
</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>
64 <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 15. Policies: Controlling Precision, Error Handling etc">Policy
</a><span class=
"special">></span>
65 <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">sph_bessel_prime
</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 15. Policies: Controlling Precision, Error Handling etc">Policy
</a><span class=
"special">&);
</span>
67 <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>
68 <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">sph_neumann_prime
</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>
70 <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 15. Policies: Controlling Precision, Error Handling etc">Policy
</a><span class=
"special">></span>
71 <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">sph_neumann_prime
</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 15. Policies: Controlling Precision, Error Handling etc">Policy
</a><span class=
"special">&);
</span>
74 <a name=
"math_toolkit.bessel.bessel_derivatives.h1"></a>
75 <span class=
"phrase"><a name=
"math_toolkit.bessel.bessel_derivatives.description"></a></span><a class=
"link" href=
"bessel_derivatives.html#math_toolkit.bessel.bessel_derivatives.description">Description
</a>
78 These functions return the first derivative with respect to
<span class=
"emphasis"><em>x
</em></span>
79 of the corresponding Bessel function.
82 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
83 type calculation rules
</em></span></a> when T1 and T2 are different types.
84 The functions are also optimised for the relatively common case that T1 is
88 The final
<a class=
"link" href=
"../../policy.html" title=
"Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy
</a> argument is optional and can
89 be used to control the behaviour of the function: how it handles errors,
90 what level of precision to use etc. Refer to the
<a class=
"link" href=
"../../policy.html" title=
"Chapter 15. Policies: Controlling Precision, Error Handling etc">policy
91 documentation for more details
</a>.
94 The functions return the result of
<a class=
"link" href=
"../error_handling.html#math_toolkit.error_handling.domain_error">domain_error
</a>
95 whenever the result is undefined or complex.
98 <a name=
"math_toolkit.bessel.bessel_derivatives.h2"></a>
99 <span class=
"phrase"><a name=
"math_toolkit.bessel.bessel_derivatives.testing"></a></span><a class=
"link" href=
"bessel_derivatives.html#math_toolkit.bessel.bessel_derivatives.testing">Testing
</a>
102 There are two sets of test values: spot values calculated using
<a href=
"http://www.wolframalpha.com/" target=
"_top">wolframalpha.com
</a>,
103 and a much larger set of tests computed using a relation to the underlying
104 Bessel functions that the implementation does not use.
107 <a name=
"math_toolkit.bessel.bessel_derivatives.h3"></a>
108 <span class=
"phrase"><a name=
"math_toolkit.bessel.bessel_derivatives.accuracy"></a></span><a class=
"link" href=
"bessel_derivatives.html#math_toolkit.bessel.bessel_derivatives.accuracy">Accuracy
</a>
111 The accuracy of these functions is broadly similar to the underlying Bessel
115 <a name=
"math_toolkit.bessel.bessel_derivatives.table_cyl_bessel_i_prime_integer_orders_"></a><p class=
"title"><b>Table
 6.50.
 Error rates for cyl_bessel_i_prime (integer orders)
</b></p>
116 <div class=
"table-contents"><table class=
"table" summary=
"Error rates for cyl_bessel_i_prime (integer orders)">
129 Microsoft Visual C++ version
12.0<br> Win32
<br> double
134 GNU C++ version
5.1.0<br> linux
<br> double
139 GNU C++ version
5.1.0<br> linux
<br> long double
144 Sun compiler version
0x5130<br> Sun Solaris
<br> long double
152 Bessel I'
0: Mathworld Data (Integer Version)
157 <span class=
"blue">Max =
0.885ε (Mean =
0.567ε)
</span>
162 <span class=
"blue">Max =
0ε (Mean =
0ε)
</span>
167 <span class=
"blue">Max =
7.83ε (Mean =
2.84ε)
</span>
172 <span class=
"blue">Max =
6.52ε (Mean =
2.26ε)
</span>
179 Bessel I'
1: Mathworld Data (Integer Version)
184 <span class=
"blue">Max =
1.61ε (Mean =
0.786ε)
</span>
189 <span class=
"blue">Max =
0ε (Mean =
0ε)
</span>
194 <span class=
"blue">Max =
2.3ε (Mean =
1.28ε)
</span>
199 <span class=
"blue">Max =
2.3ε (Mean =
1.29ε)
</span>
206 Bessel I'n: Mathworld Data (Integer Version)
211 <span class=
"blue">Max =
3.61ε (Mean =
1.22ε)
</span>
216 <span class=
"blue">Max =
0ε (Mean =
0ε)
</span>
221 <span class=
"blue">Max =
2.31ε (Mean =
1.41ε)
</span>
226 <span class=
"blue">Max =
701ε (Mean =
212ε)
</span>
233 <br class=
"table-break"><div class=
"table">
234 <a name=
"math_toolkit.bessel.bessel_derivatives.table_cyl_bessel_i_prime"></a><p class=
"title"><b>Table
 6.51.
 Error rates for cyl_bessel_i_prime
</b></p>
235 <div class=
"table-contents"><table class=
"table" summary=
"Error rates for cyl_bessel_i_prime">
248 Microsoft Visual C++ version
12.0<br> Win32
<br> double
253 GNU C++ version
5.1.0<br> linux
<br> double
258 GNU C++ version
5.1.0<br> linux
<br> long double
263 Sun compiler version
0x5130<br> Sun Solaris
<br> long double
271 Bessel I'
0: Mathworld Data
276 <span class=
"blue">Max =
0.885ε (Mean =
0.567ε)
</span>
281 <span class=
"blue">Max =
0ε (Mean =
0ε)
</span>
286 <span class=
"blue">Max =
7.83ε (Mean =
2.84ε)
</span>
291 <span class=
"blue">Max =
6.52ε (Mean =
2.26ε)
</span>
298 Bessel I'
1: Mathworld Data
303 <span class=
"blue">Max =
1.61ε (Mean =
0.786ε)
</span>
308 <span class=
"blue">Max =
0ε (Mean =
0ε)
</span>
313 <span class=
"blue">Max =
2.3ε (Mean =
1.28ε)
</span>
318 <span class=
"blue">Max =
2.3ε (Mean =
1.29ε)
</span>
325 Bessel I'n: Mathworld Data
330 <span class=
"blue">Max =
3.61ε (Mean =
1.22ε)
</span>
335 <span class=
"blue">Max =
0ε (Mean =
0ε)
</span>
340 <span class=
"blue">Max =
2.31ε (Mean =
1.41ε)
</span>
345 <span class=
"blue">Max =
701ε (Mean =
212ε)
</span>
352 Bessel I'v: Mathworld Data
357 <span class=
"blue">Max =
3.76e+003ε (Mean =
1.19e+003ε)
</span>
362 <span class=
"blue">Max =
1.62ε (Mean =
0.512ε)
</span>
367 <span class=
"blue">Max =
2.89e+03ε (Mean =
914ε)
</span>
372 <span class=
"blue">Max =
2.89e+03ε (Mean =
914ε)
</span>
379 Bessel I'n: Random Data
384 <span class=
"blue">Max =
9.85ε (Mean =
1.83ε)
</span>
389 <span class=
"blue">Max =
0ε (Mean =
0ε)
</span>
394 <span class=
"blue">Max =
6.79ε (Mean =
1.13ε)
</span>
399 <span class=
"blue">Max =
195ε (Mean =
37.1ε)
</span>
406 Bessel I'v: Random Data
411 <span class=
"blue">Max =
14ε (Mean =
2.46ε)
</span>
416 <span class=
"blue">Max =
0ε (Mean =
0ε)
</span>
421 <span class=
"blue">Max =
14.1ε (Mean =
2.93ε)
</span>
426 <span class=
"blue">Max =
336ε (Mean =
68.5ε)
</span>
433 Bessel I'v: Mathworld Data (large values)
438 <span class=
"blue">Max =
59.5ε (Mean =
26.8ε)
</span>
443 <span class=
"blue">Max =
0ε (Mean =
0ε)
</span>
448 <span class=
"blue">Max =
42.6ε (Mean =
20.2ε)
</span>
453 <span class=
"blue">Max =
43.9ε (Mean =
20.7ε)
</span>
460 <br class=
"table-break"><div class=
"table">
461 <a name=
"math_toolkit.bessel.bessel_derivatives.table_cyl_bessel_j_prime_integer_orders_"></a><p class=
"title"><b>Table
 6.52.
 Error rates for cyl_bessel_j_prime (integer orders)
</b></p>
462 <div class=
"table-contents"><table class=
"table" summary=
"Error rates for cyl_bessel_j_prime (integer orders)">
475 Microsoft Visual C++ version
12.0<br> Win32
<br> double
480 GNU C++ version
5.1.0<br> linux
<br> double
485 GNU C++ version
5.1.0<br> linux
<br> long double
490 Sun compiler version
0x5130<br> Sun Solaris
<br> long double
498 Bessel J0': Mathworld Data (Integer Version)
503 <span class=
"blue">Max =
6.62ε (Mean =
2.55ε)
</span>
508 <span class=
"blue">Max =
0ε (Mean =
0ε)
</span>
513 <span class=
"blue">Max =
18.9ε (Mean =
6.81ε)
</span>
518 <span class=
"blue">Max =
18.9ε (Mean =
6.72ε)
</span>
525 Bessel J0': Mathworld Data (Tricky cases) (Integer Version)
530 <span class=
"blue">Max =
3.67ε (Mean =
1.74ε)
</span>
535 <span class=
"blue">Max =
0ε (Mean =
0ε)
</span>
540 <span class=
"blue">Max =
8.08ε (Mean =
4.57ε)
</span>
545 <span class=
"blue">Max =
7.44ε (Mean =
3.31ε)
</span>
552 Bessel J1': Mathworld Data (Integer Version)
557 <span class=
"blue">Max =
0.999ε (Mean =
0.627ε)
</span>
562 <span class=
"blue">Max =
0ε (Mean =
0ε)
</span>
567 <span class=
"blue">Max =
7.9ε (Mean =
3.37ε)
</span>
572 <span class=
"blue">Max =
7.9ε (Mean =
3.37ε)
</span>
579 Bessel J1': Mathworld Data (tricky cases) (Integer Version)
584 <span class=
"blue">Max =
288ε (Mean =
129ε)
</span>
589 <span class=
"blue">Max =
287ε (Mean =
129ε)
</span>
594 <span class=
"blue">Max =
5.88e+05ε (Mean =
2.63e+05ε)
</span>
599 <span class=
"blue">Max =
5.88e+05ε (Mean =
2.63e+05ε)
</span>
606 Bessel JN': Mathworld Data (Integer Version)
611 <span class=
"blue">Max =
14ε (Mean =
6.13ε)
</span>
616 <span class=
"blue">Max =
0.527ε (Mean =
0.128ε)
</span>
621 <span class=
"blue">Max =
1.29e+03ε (Mean =
312ε)
</span>
626 <span class=
"blue">Max =
1.29e+03ε (Mean =
355ε)
</span>
633 <br class=
"table-break"><div class=
"table">
634 <a name=
"math_toolkit.bessel.bessel_derivatives.table_cyl_bessel_j_prime"></a><p class=
"title"><b>Table
 6.53.
 Error rates for cyl_bessel_j_prime
</b></p>
635 <div class=
"table-contents"><table class=
"table" summary=
"Error rates for cyl_bessel_j_prime">
648 Microsoft Visual C++ version
12.0<br> Win32
<br> double
653 GNU C++ version
5.1.0<br> linux
<br> double
658 GNU C++ version
5.1.0<br> linux
<br> long double
663 Sun compiler version
0x5130<br> Sun Solaris
<br> long double
671 Bessel J0': Mathworld Data
676 <span class=
"blue">Max =
6.62ε (Mean =
2.55ε)
</span>
681 <span class=
"blue">Max =
0ε (Mean =
0ε)
</span>
686 <span class=
"blue">Max =
18.9ε (Mean =
6.81ε)
</span>
691 <span class=
"blue">Max =
18.9ε (Mean =
6.72ε)
</span>
698 Bessel J0': Mathworld Data (Tricky cases)
703 <span class=
"blue">Max =
3.67ε (Mean =
1.74ε)
</span>
708 <span class=
"blue">Max =
0ε (Mean =
0ε)
</span>
713 <span class=
"blue">Max =
8.08ε (Mean =
4.57ε)
</span>
718 <span class=
"blue">Max =
7.44ε (Mean =
3.31ε)
</span>
725 Bessel J1': Mathworld Data
730 <span class=
"blue">Max =
0.999ε (Mean =
0.627ε)
</span>
735 <span class=
"blue">Max =
0ε (Mean =
0ε)
</span>
740 <span class=
"blue">Max =
7.9ε (Mean =
3.37ε)
</span>
745 <span class=
"blue">Max =
7.9ε (Mean =
3.37ε)
</span>
752 Bessel J1': Mathworld Data (tricky cases)
757 <span class=
"blue">Max =
288ε (Mean =
129ε)
</span>
762 <span class=
"blue">Max =
287ε (Mean =
129ε)
</span>
767 <span class=
"blue">Max =
5.88e+05ε (Mean =
2.63e+05ε)
</span>
772 <span class=
"blue">Max =
5.88e+05ε (Mean =
2.63e+05ε)
</span>
779 Bessel JN': Mathworld Data
784 <span class=
"blue">Max =
14ε (Mean =
6.13ε)
</span>
789 <span class=
"blue">Max =
0.527ε (Mean =
0.128ε)
</span>
794 <span class=
"blue">Max =
1.29e+03ε (Mean =
312ε)
</span>
799 <span class=
"blue">Max =
1.29e+03ε (Mean =
355ε)
</span>
806 Bessel J': Mathworld Data
811 <span class=
"blue">Max =
23.7ε (Mean =
8.01ε)
</span>
816 <span class=
"blue">Max =
21.5ε (Mean =
4.7ε)
</span>
821 <span class=
"blue">Max =
42.5ε (Mean =
9.31ε)
</span>
826 <span class=
"blue">Max =
42.5ε (Mean =
9.32ε)
</span>
833 Bessel J': Mathworld Data (large values)
838 <span class=
"blue">Max =
2.9ε (Mean =
1.61ε)
</span>
843 <span class=
"blue">Max =
0ε (Mean =
0ε)
</span>
848 <span class=
"blue">Max =
989ε (Mean =
495ε)
</span>
853 <span class=
"blue">Max =
989ε (Mean =
495ε)
</span>
860 Bessel JN': Random Data
865 <span class=
"blue">Max =
6.34ε (Mean =
0.997ε)
</span>
870 <span class=
"blue">Max =
0.593ε (Mean =
0.0396ε)
</span>
875 <span class=
"blue">Max =
17.3ε (Mean =
2.47ε)
</span>
880 <span class=
"blue">Max =
79.4ε (Mean =
16.2ε)
</span>
887 Bessel J': Random Data
892 <span class=
"blue">Max =
176ε (Mean =
9.76ε)
</span>
897 <span class=
"blue">Max =
0.885ε (Mean =
0.0457ε)
</span>
902 <span class=
"blue">Max =
139ε (Mean =
6.47ε)
</span>
907 <span class=
"blue">Max =
279ε (Mean =
27.2ε)
</span>
914 Bessel J': Random Data (Tricky large values)
919 <span class=
"blue">Max =
379ε (Mean =
45.4ε)
</span>
924 <span class=
"blue">Max =
0ε (Mean =
0ε)
</span>
929 <span class=
"blue">Max =
474ε (Mean =
62.2ε)
</span>
934 <span class=
"blue">Max =
474ε (Mean =
64.5ε)
</span>
941 <br class=
"table-break"><div class=
"table">
942 <a name=
"math_toolkit.bessel.bessel_derivatives.table_cyl_bessel_k_prime_integer_orders_"></a><p class=
"title"><b>Table
 6.54.
 Error rates for cyl_bessel_k_prime (integer orders)
</b></p>
943 <div class=
"table-contents"><table class=
"table" summary=
"Error rates for cyl_bessel_k_prime (integer orders)">
956 Microsoft Visual C++ version
12.0<br> Win32
<br> double
961 GNU C++ version
5.1.0<br> linux
<br> double
966 GNU C++ version
5.1.0<br> linux
<br> long double
971 Sun compiler version
0x5130<br> Sun Solaris
<br> long double
979 Bessel K'
0: Mathworld Data (Integer Version)
984 <span class=
"blue">Max =
1ε (Mean =
0.573ε)
</span>
989 <span class=
"blue">Max =
0ε (Mean =
0ε)
</span>
994 <span class=
"blue">Max =
1.8ε (Mean =
1.02ε)
</span>
999 <span class=
"blue">Max =
1.8ε (Mean =
1.02ε)
</span>
1006 Bessel K'
1: Mathworld Data (Integer Version)
1011 <span class=
"blue">Max =
1.09ε (Mean =
0.607ε)
</span>
1016 <span class=
"blue">Max =
0ε (Mean =
0ε)
</span>
1021 <span class=
"blue">Max =
1.96ε (Mean =
1ε)
</span>
1026 <span class=
"blue">Max =
1.96ε (Mean =
1.04ε)
</span>
1033 Bessel K'n: Mathworld Data (Integer Version)
1038 <span class=
"blue">Max =
4.17ε (Mean =
1.74ε)
</span>
1043 <span class=
"blue">Max =
0ε (Mean =
0ε)
</span>
1048 <span class=
"blue">Max =
4.47ε (Mean =
2.05ε)
</span>
1053 <span class=
"blue">Max =
4.47ε (Mean =
1.93ε)
</span>
1060 <br class=
"table-break"><div class=
"table">
1061 <a name=
"math_toolkit.bessel.bessel_derivatives.table_cyl_bessel_k_prime"></a><p class=
"title"><b>Table
 6.55.
 Error rates for cyl_bessel_k_prime
</b></p>
1062 <div class=
"table-contents"><table class=
"table" summary=
"Error rates for cyl_bessel_k_prime">
1075 Microsoft Visual C++ version
12.0<br> Win32
<br> double
1080 GNU C++ version
5.1.0<br> linux
<br> double
1085 GNU C++ version
5.1.0<br> linux
<br> long double
1090 Sun compiler version
0x5130<br> Sun Solaris
<br> long double
1098 Bessel K'
0: Mathworld Data
1103 <span class=
"blue">Max =
1ε (Mean =
0.573ε)
</span>
1108 <span class=
"blue">Max =
0ε (Mean =
0ε)
</span>
1113 <span class=
"blue">Max =
1.8ε (Mean =
1.02ε)
</span>
1118 <span class=
"blue">Max =
1.8ε (Mean =
1.02ε)
</span>
1125 Bessel K'
1: Mathworld Data
1130 <span class=
"blue">Max =
1.09ε (Mean =
0.607ε)
</span>
1135 <span class=
"blue">Max =
0ε (Mean =
0ε)
</span>
1140 <span class=
"blue">Max =
1.96ε (Mean =
1ε)
</span>
1145 <span class=
"blue">Max =
1.96ε (Mean =
1.04ε)
</span>
1152 Bessel K'n: Mathworld Data
1157 <span class=
"blue">Max =
4.17ε (Mean =
1.74ε)
</span>
1162 <span class=
"blue">Max =
0ε (Mean =
0ε)
</span>
1167 <span class=
"blue">Max =
4.47ε (Mean =
2.05ε)
</span>
1172 <span class=
"blue">Max =
4.47ε (Mean =
1.93ε)
</span>
1179 Bessel K'v: Mathworld Data
1184 <span class=
"blue">Max =
3.94ε (Mean =
1.43ε)
</span>
1189 <span class=
"blue">Max =
0ε (Mean =
0ε)
</span>
1194 <span class=
"blue">Max =
3.94ε (Mean =
2.49ε)
</span>
1199 <span class=
"blue">Max =
3.94ε (Mean =
2.35ε)
</span>
1206 Bessel K'v: Mathworld Data (large values)
1211 <span class=
"blue">Max =
18.6ε (Mean =
12.1ε)
</span>
1216 <span class=
"blue">Max =
0ε (Mean =
0ε)
</span>
1221 <span class=
"blue">Max =
59.2ε (Mean =
42.9ε)
</span>
1226 <span class=
"blue">Max =
59.8ε (Mean =
43.4ε)
</span>
1233 Bessel K'n: Random Data
1238 <span class=
"blue">Max =
8.18ε (Mean =
1.45ε)
</span>
1243 <span class=
"blue">Max =
0ε (Mean =
0ε)
</span>
1248 <span class=
"blue">Max =
4.53ε (Mean =
1.12ε)
</span>
1253 <span class=
"blue">Max =
4.45ε (Mean =
1.23ε)
</span>
1260 Bessel K'v: Random Data
1265 <span class=
"blue">Max =
8.32ε (Mean =
1.65ε)
</span>
1270 <span class=
"blue">Max =
0ε (Mean =
0ε)
</span>
1275 <span class=
"blue">Max =
7.95ε (Mean =
1.53ε)
</span>
1280 <span class=
"blue">Max =
7.95ε (Mean =
1.55ε)
</span>
1287 <br class=
"table-break"><div class=
"table">
1288 <a name=
"math_toolkit.bessel.bessel_derivatives.table_sph_bessel_prime"></a><p class=
"title"><b>Table
 6.56.
 Error rates for sph_bessel_prime
</b></p>
1289 <div class=
"table-contents"><table class=
"table" summary=
"Error rates for sph_bessel_prime">
1302 Microsoft Visual C++ version
12.0<br> Win32
<br> double
1307 GNU C++ version
5.1.0<br> linux
<br> double
1312 GNU C++ version
5.1.0<br> linux
<br> long double
1317 Sun compiler version
0x5130<br> Sun Solaris
<br> long double
1324 Bessel j': Random Data
1329 <span class=
"blue">Max =
307ε (Mean =
25.2ε)
</span>
1334 <span class=
"blue">Max =
0.753ε (Mean =
0.0343ε)
</span>
1339 <span class=
"blue">Max =
167ε (Mean =
12ε)
</span>
1344 <span class=
"blue">Max =
167ε (Mean =
33.2ε)
</span>
1350 <br class=
"table-break"><div class=
"table">
1351 <a name=
"math_toolkit.bessel.bessel_derivatives.table_sph_neumann_prime"></a><p class=
"title"><b>Table
 6.57.
 Error rates for sph_neumann_prime
</b></p>
1352 <div class=
"table-contents"><table class=
"table" summary=
"Error rates for sph_neumann_prime">
1365 Microsoft Visual C++ version
12.0<br> Win32
<br> double
1370 GNU C++ version
5.1.0<br> linux
<br> double
1375 GNU C++ version
5.1.0<br> linux
<br> long double
1380 Sun compiler version
0x5130<br> Sun Solaris
<br> long double
1392 <span class=
"blue">Max =
296ε (Mean =
25.6ε)
</span>
1397 <span class=
"blue">Max =
0.752ε (Mean =
0.062ε)
</span>
1402 <span class=
"blue">Max =
158ε (Mean =
18.8ε)
</span>
1407 <span class=
"blue">Max =
158ε (Mean =
20.2ε)
</span>
1413 <br class=
"table-break"><h5>
1414 <a name=
"math_toolkit.bessel.bessel_derivatives.h4"></a>
1415 <span class=
"phrase"><a name=
"math_toolkit.bessel.bessel_derivatives.implementation"></a></span><a class=
"link" href=
"bessel_derivatives.html#math_toolkit.bessel.bessel_derivatives.implementation">Implementation
</a>
1418 In the general case, the derivatives are calculated using the relations:
1421 <span class=
"inlinemediaobject"><img src=
"../../../equations/bessel_derivatives1.svg"></span>
1424 There are also a number of special cases, for large x we have:
1427 <span class=
"inlinemediaobject"><img src=
"../../../equations/bessel_derivatives4.svg"></span>
1433 <span class=
"inlinemediaobject"><img src=
"../../../equations/bessel_derivatives5.svg"></span>
1436 <table xmlns:
rev=
"http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width=
"100%"><tr>
1437 <td align=
"left"></td>
1438 <td align=
"right"><div class=
"copyright-footer">Copyright
© 2006-
2010,
2012-
2014 Nikhar Agrawal,
1439 Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
1440 Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R
åde, Gautam Sewani,
1441 Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang
<p>
1442 Distributed under the Boost Software License, Version
1.0. (See accompanying
1443 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>)
1448 <div class=
"spirit-nav">
1449 <a accesskey=
"p" href=
"sph_bessel.html"><img src=
"../../../../../../doc/src/images/prev.png" alt=
"Prev"></a><a accesskey=
"u" href=
"../bessel.html"><img src=
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