]> git.proxmox.com Git - ceph.git/blame - ceph/src/boost/libs/math/doc/html/math_toolkit/bessel/bessel_derivatives.html
projects / ceph.git / blame
? search:
summary | shortlog | log | commit | commitdiff | tree
blob | blame (incremental) | history | HEAD
bump version to 12.2.2-pve1
[ceph.git] / ceph / src / boost / libs / math / doc / html / math_toolkit / bessel / bessel_derivatives.html
CommitLineData
7c673cae
FG		 1<html>
2<head>
3<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
4<title>Derivatives of the Bessel Functions</title>
5<link rel="stylesheet" href="../../math.css" type="text/css">
6<meta name="generator" content="DocBook XSL Stylesheets V1.77.1">
7<link rel="home" href="../../index.html" title="Math Toolkit 2.5.1">
8<link rel="up" href="../bessel.html" title="Bessel Functions">
9<link rel="prev" href="sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds">
10<link rel="next" href="../hankel.html" title="Hankel Functions">
11</head>
12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
13<table cellpadding="2" width="100%"><tr>
14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td>
15<td align="center"><a href="../../../../../../index.html">Home</a></td>
16<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td>
17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
19<td align="center"><a href="../../../../../../more/index.htm">More</a></td>
20</tr></table>
21<hr>
22<div class="spirit-nav">
23<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="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="../hankel.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
24</div>
25<div class="section">
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>
30<h5>
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>
33 </h5>
34<p>
35 <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_prime</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></code>
36 </p>
37<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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">&gt;</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>
39
40<span class="keyword">template</span> <span class="special">&lt;</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&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
42
43<span class="keyword">template</span> <span class="special">&lt;</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">&gt;</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>
45
46<span class="keyword">template</span> <span class="special">&lt;</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&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
48
49<span class="keyword">template</span> <span class="special">&lt;</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">&gt;</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>
51
52<span class="keyword">template</span> <span class="special">&lt;</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&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
54
55<span class="keyword">template</span> <span class="special">&lt;</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">&gt;</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>
57
58<span class="keyword">template</span> <span class="special">&lt;</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&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
60
61<span class="keyword">template</span> <span class="special">&lt;</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">&gt;</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>
63
64<span class="keyword">template</span> <span class="special">&lt;</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&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
66
67<span class="keyword">template</span> <span class="special">&lt;</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">&gt;</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>
69
70<span class="keyword">template</span> <span class="special">&lt;</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&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
72</pre>
73<h5>
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>
76 </h5>
77<p>
78 These functions return the first derivative with respect to <span class="emphasis"><em>x</em></span>
79 of the corresponding Bessel function.
80 </p>
81<p>
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
85 an integer.
86 </p>
87<p>
88 The final <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and 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&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">policy
91 documentation for more details</a>.
92 </p>
93<p>
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.
96 </p>
97<h5>
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>
100 </h5>
101<p>
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.
105 </p>
106<h5>
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>
109 </h5>
110<p>
111 The accuracy of these functions is broadly similar to the underlying Bessel
112 functions.
113 </p>
114<div class="table">
115<a name="math_toolkit.bessel.bessel_derivatives.table_cyl_bessel_i_prime_integer_orders_"></a><p class="title"><b>Table&#160;6.50.&#160;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)">
117<colgroup>
118<col>
119<col>
120<col>
121<col>
122<col>
123</colgroup>
124<thead><tr>
125<th>
126 </th>
127<th>
128 <p>
129 Microsoft Visual C++ version 12.0<br> Win32<br> double
130 </p>
131 </th>
132<th>
133 <p>
134 GNU C++ version 5.1.0<br> linux<br> double
135 </p>
136 </th>
137<th>
138 <p>
139 GNU C++ version 5.1.0<br> linux<br> long double
140 </p>
141 </th>
142<th>
143 <p>
144 Sun compiler version 0x5130<br> Sun Solaris<br> long double
145 </p>
146 </th>
147</tr></thead>
148<tbody>
149<tr>
150<td>
151 <p>
152 Bessel I'0: Mathworld Data (Integer Version)
153 </p>
154 </td>
155<td>
156 <p>
157 <span class="blue">Max = 0.885&#949; (Mean = 0.567&#949;)</span>
158 </p>
159 </td>
160<td>
161 <p>
162 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
163 </p>
164 </td>
165<td>
166 <p>
167 <span class="blue">Max = 7.83&#949; (Mean = 2.84&#949;)</span>
168 </p>
169 </td>
170<td>
171 <p>
172 <span class="blue">Max = 6.52&#949; (Mean = 2.26&#949;)</span>
173 </p>
174 </td>
175</tr>
176<tr>
177<td>
178 <p>
179 Bessel I'1: Mathworld Data (Integer Version)
180 </p>
181 </td>
182<td>
183 <p>
184 <span class="blue">Max = 1.61&#949; (Mean = 0.786&#949;)</span>
185 </p>
186 </td>
187<td>
188 <p>
189 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
190 </p>
191 </td>
192<td>
193 <p>
194 <span class="blue">Max = 2.3&#949; (Mean = 1.28&#949;)</span>
195 </p>
196 </td>
197<td>
198 <p>
199 <span class="blue">Max = 2.3&#949; (Mean = 1.29&#949;)</span>
200 </p>
201 </td>
202</tr>
203<tr>
204<td>
205 <p>
206 Bessel I'n: Mathworld Data (Integer Version)
207 </p>
208 </td>
209<td>
210 <p>
211 <span class="blue">Max = 3.61&#949; (Mean = 1.22&#949;)</span>
212 </p>
213 </td>
214<td>
215 <p>
216 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
217 </p>
218 </td>
219<td>
220 <p>
221 <span class="blue">Max = 2.31&#949; (Mean = 1.41&#949;)</span>
222 </p>
223 </td>
224<td>
225 <p>
226 <span class="blue">Max = 701&#949; (Mean = 212&#949;)</span>
227 </p>
228 </td>
229</tr>
230</tbody>
231</table></div>
232</div>
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&#160;6.51.&#160;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">
236<colgroup>
237<col>
238<col>
239<col>
240<col>
241<col>
242</colgroup>
243<thead><tr>
244<th>
245 </th>
246<th>
247 <p>
248 Microsoft Visual C++ version 12.0<br> Win32<br> double
249 </p>
250 </th>
251<th>
252 <p>
253 GNU C++ version 5.1.0<br> linux<br> double
254 </p>
255 </th>
256<th>
257 <p>
258 GNU C++ version 5.1.0<br> linux<br> long double
259 </p>
260 </th>
261<th>
262 <p>
263 Sun compiler version 0x5130<br> Sun Solaris<br> long double
264 </p>
265 </th>
266</tr></thead>
267<tbody>
268<tr>
269<td>
270 <p>
271 Bessel I'0: Mathworld Data
272 </p>
273 </td>
274<td>
275 <p>
276 <span class="blue">Max = 0.885&#949; (Mean = 0.567&#949;)</span>
277 </p>
278 </td>
279<td>
280 <p>
281 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
282 </p>
283 </td>
284<td>
285 <p>
286 <span class="blue">Max = 7.83&#949; (Mean = 2.84&#949;)</span>
287 </p>
288 </td>
289<td>
290 <p>
291 <span class="blue">Max = 6.52&#949; (Mean = 2.26&#949;)</span>
292 </p>
293 </td>
294</tr>
295<tr>
296<td>
297 <p>
298 Bessel I'1: Mathworld Data
299 </p>
300 </td>
301<td>
302 <p>
303 <span class="blue">Max = 1.61&#949; (Mean = 0.786&#949;)</span>
304 </p>
305 </td>
306<td>
307 <p>
308 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
309 </p>
310 </td>
311<td>
312 <p>
313 <span class="blue">Max = 2.3&#949; (Mean = 1.28&#949;)</span>
314 </p>
315 </td>
316<td>
317 <p>
318 <span class="blue">Max = 2.3&#949; (Mean = 1.29&#949;)</span>
319 </p>
320 </td>
321</tr>
322<tr>
323<td>
324 <p>
325 Bessel I'n: Mathworld Data
326 </p>
327 </td>
328<td>
329 <p>
330 <span class="blue">Max = 3.61&#949; (Mean = 1.22&#949;)</span>
331 </p>
332 </td>
333<td>
334 <p>
335 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
336 </p>
337 </td>
338<td>
339 <p>
340 <span class="blue">Max = 2.31&#949; (Mean = 1.41&#949;)</span>
341 </p>
342 </td>
343<td>
344 <p>
345 <span class="blue">Max = 701&#949; (Mean = 212&#949;)</span>
346 </p>
347 </td>
348</tr>
349<tr>
350<td>
351 <p>
352 Bessel I'v: Mathworld Data
353 </p>
354 </td>
355<td>
356 <p>
357 <span class="blue">Max = 3.76e+003&#949; (Mean = 1.19e+003&#949;)</span>
358 </p>
359 </td>
360<td>
361 <p>
362 <span class="blue">Max = 1.62&#949; (Mean = 0.512&#949;)</span>
363 </p>
364 </td>
365<td>
366 <p>
367 <span class="blue">Max = 2.89e+03&#949; (Mean = 914&#949;)</span>
368 </p>
369 </td>
370<td>
371 <p>
372 <span class="blue">Max = 2.89e+03&#949; (Mean = 914&#949;)</span>
373 </p>
374 </td>
375</tr>
376<tr>
377<td>
378 <p>
379 Bessel I'n: Random Data
380 </p>
381 </td>
382<td>
383 <p>
384 <span class="blue">Max = 9.85&#949; (Mean = 1.83&#949;)</span>
385 </p>
386 </td>
387<td>
388 <p>
389 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
390 </p>
391 </td>
392<td>
393 <p>
394 <span class="blue">Max = 6.79&#949; (Mean = 1.13&#949;)</span>
395 </p>
396 </td>
397<td>
398 <p>
399 <span class="blue">Max = 195&#949; (Mean = 37.1&#949;)</span>
400 </p>
401 </td>
402</tr>
403<tr>
404<td>
405 <p>
406 Bessel I'v: Random Data
407 </p>
408 </td>
409<td>
410 <p>
411 <span class="blue">Max = 14&#949; (Mean = 2.46&#949;)</span>
412 </p>
413 </td>
414<td>
415 <p>
416 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
417 </p>
418 </td>
419<td>
420 <p>
421 <span class="blue">Max = 14.1&#949; (Mean = 2.93&#949;)</span>
422 </p>
423 </td>
424<td>
425 <p>
426 <span class="blue">Max = 336&#949; (Mean = 68.5&#949;)</span>
427 </p>
428 </td>
429</tr>
430<tr>
431<td>
432 <p>
433 Bessel I'v: Mathworld Data (large values)
434 </p>
435 </td>
436<td>
437 <p>
438 <span class="blue">Max = 59.5&#949; (Mean = 26.8&#949;)</span>
439 </p>
440 </td>
441<td>
442 <p>
443 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
444 </p>
445 </td>
446<td>
447 <p>
448 <span class="blue">Max = 42.6&#949; (Mean = 20.2&#949;)</span>
449 </p>
450 </td>
451<td>
452 <p>
453 <span class="blue">Max = 43.9&#949; (Mean = 20.7&#949;)</span>
454 </p>
455 </td>
456</tr>
457</tbody>
458</table></div>
459</div>
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&#160;6.52.&#160;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)">
463<colgroup>
464<col>
465<col>
466<col>
467<col>
468<col>
469</colgroup>
470<thead><tr>
471<th>
472 </th>
473<th>
474 <p>
475 Microsoft Visual C++ version 12.0<br> Win32<br> double
476 </p>
477 </th>
478<th>
479 <p>
480 GNU C++ version 5.1.0<br> linux<br> double
481 </p>
482 </th>
483<th>
484 <p>
485 GNU C++ version 5.1.0<br> linux<br> long double
486 </p>
487 </th>
488<th>
489 <p>
490 Sun compiler version 0x5130<br> Sun Solaris<br> long double
491 </p>
492 </th>
493</tr></thead>
494<tbody>
495<tr>
496<td>
497 <p>
498 Bessel J0': Mathworld Data (Integer Version)
499 </p>
500 </td>
501<td>
502 <p>
503 <span class="blue">Max = 6.62&#949; (Mean = 2.55&#949;)</span>
504 </p>
505 </td>
506<td>
507 <p>
508 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
509 </p>
510 </td>
511<td>
512 <p>
513 <span class="blue">Max = 18.9&#949; (Mean = 6.81&#949;)</span>
514 </p>
515 </td>
516<td>
517 <p>
518 <span class="blue">Max = 18.9&#949; (Mean = 6.72&#949;)</span>
519 </p>
520 </td>
521</tr>
522<tr>
523<td>
524 <p>
525 Bessel J0': Mathworld Data (Tricky cases) (Integer Version)
526 </p>
527 </td>
528<td>
529 <p>
530 <span class="blue">Max = 3.67&#949; (Mean = 1.74&#949;)</span>
531 </p>
532 </td>
533<td>
534 <p>
535 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
536 </p>
537 </td>
538<td>
539 <p>
540 <span class="blue">Max = 8.08&#949; (Mean = 4.57&#949;)</span>
541 </p>
542 </td>
543<td>
544 <p>
545 <span class="blue">Max = 7.44&#949; (Mean = 3.31&#949;)</span>
546 </p>
547 </td>
548</tr>
549<tr>
550<td>
551 <p>
552 Bessel J1': Mathworld Data (Integer Version)
553 </p>
554 </td>
555<td>
556 <p>
557 <span class="blue">Max = 0.999&#949; (Mean = 0.627&#949;)</span>
558 </p>
559 </td>
560<td>
561 <p>
562 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
563 </p>
564 </td>
565<td>
566 <p>
567 <span class="blue">Max = 7.9&#949; (Mean = 3.37&#949;)</span>
568 </p>
569 </td>
570<td>
571 <p>
572 <span class="blue">Max = 7.9&#949; (Mean = 3.37&#949;)</span>
573 </p>
574 </td>
575</tr>
576<tr>
577<td>
578 <p>
579 Bessel J1': Mathworld Data (tricky cases) (Integer Version)
580 </p>
581 </td>
582<td>
583 <p>
584 <span class="blue">Max = 288&#949; (Mean = 129&#949;)</span>
585 </p>
586 </td>
587<td>
588 <p>
589 <span class="blue">Max = 287&#949; (Mean = 129&#949;)</span>
590 </p>
591 </td>
592<td>
593 <p>
594 <span class="blue">Max = 5.88e+05&#949; (Mean = 2.63e+05&#949;)</span>
595 </p>
596 </td>
597<td>
598 <p>
599 <span class="blue">Max = 5.88e+05&#949; (Mean = 2.63e+05&#949;)</span>
600 </p>
601 </td>
602</tr>
603<tr>
604<td>
605 <p>
606 Bessel JN': Mathworld Data (Integer Version)
607 </p>
608 </td>
609<td>
610 <p>
611 <span class="blue">Max = 14&#949; (Mean = 6.13&#949;)</span>
612 </p>
613 </td>
614<td>
615 <p>
616 <span class="blue">Max = 0.527&#949; (Mean = 0.128&#949;)</span>
617 </p>
618 </td>
619<td>
620 <p>
621 <span class="blue">Max = 1.29e+03&#949; (Mean = 312&#949;)</span>
622 </p>
623 </td>
624<td>
625 <p>
626 <span class="blue">Max = 1.29e+03&#949; (Mean = 355&#949;)</span>
627 </p>
628 </td>
629</tr>
630</tbody>
631</table></div>
632</div>
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&#160;6.53.&#160;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">
636<colgroup>
637<col>
638<col>
639<col>
640<col>
641<col>
642</colgroup>
643<thead><tr>
644<th>
645 </th>
646<th>
647 <p>
648 Microsoft Visual C++ version 12.0<br> Win32<br> double
649 </p>
650 </th>
651<th>
652 <p>
653 GNU C++ version 5.1.0<br> linux<br> double
654 </p>
655 </th>
656<th>
657 <p>
658 GNU C++ version 5.1.0<br> linux<br> long double
659 </p>
660 </th>
661<th>
662 <p>
663 Sun compiler version 0x5130<br> Sun Solaris<br> long double
664 </p>
665 </th>
666</tr></thead>
667<tbody>
668<tr>
669<td>
670 <p>
671 Bessel J0': Mathworld Data
672 </p>
673 </td>
674<td>
675 <p>
676 <span class="blue">Max = 6.62&#949; (Mean = 2.55&#949;)</span>
677 </p>
678 </td>
679<td>
680 <p>
681 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
682 </p>
683 </td>
684<td>
685 <p>
686 <span class="blue">Max = 18.9&#949; (Mean = 6.81&#949;)</span>
687 </p>
688 </td>
689<td>
690 <p>
691 <span class="blue">Max = 18.9&#949; (Mean = 6.72&#949;)</span>
692 </p>
693 </td>
694</tr>
695<tr>
696<td>
697 <p>
698 Bessel J0': Mathworld Data (Tricky cases)
699 </p>
700 </td>
701<td>
702 <p>
703 <span class="blue">Max = 3.67&#949; (Mean = 1.74&#949;)</span>
704 </p>
705 </td>
706<td>
707 <p>
708 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
709 </p>
710 </td>
711<td>
712 <p>
713 <span class="blue">Max = 8.08&#949; (Mean = 4.57&#949;)</span>
714 </p>
715 </td>
716<td>
717 <p>
718 <span class="blue">Max = 7.44&#949; (Mean = 3.31&#949;)</span>
719 </p>
720 </td>
721</tr>
722<tr>
723<td>
724 <p>
725 Bessel J1': Mathworld Data
726 </p>
727 </td>
728<td>
729 <p>
730 <span class="blue">Max = 0.999&#949; (Mean = 0.627&#949;)</span>
731 </p>
732 </td>
733<td>
734 <p>
735 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
736 </p>
737 </td>
738<td>
739 <p>
740 <span class="blue">Max = 7.9&#949; (Mean = 3.37&#949;)</span>
741 </p>
742 </td>
743<td>
744 <p>
745 <span class="blue">Max = 7.9&#949; (Mean = 3.37&#949;)</span>
746 </p>
747 </td>
748</tr>
749<tr>
750<td>
751 <p>
752 Bessel J1': Mathworld Data (tricky cases)
753 </p>
754 </td>
755<td>
756 <p>
757 <span class="blue">Max = 288&#949; (Mean = 129&#949;)</span>
758 </p>
759 </td>
760<td>
761 <p>
762 <span class="blue">Max = 287&#949; (Mean = 129&#949;)</span>
763 </p>
764 </td>
765<td>
766 <p>
767 <span class="blue">Max = 5.88e+05&#949; (Mean = 2.63e+05&#949;)</span>
768 </p>
769 </td>
770<td>
771 <p>
772 <span class="blue">Max = 5.88e+05&#949; (Mean = 2.63e+05&#949;)</span>
773 </p>
774 </td>
775</tr>
776<tr>
777<td>
778 <p>
779 Bessel JN': Mathworld Data
780 </p>
781 </td>
782<td>
783 <p>
784 <span class="blue">Max = 14&#949; (Mean = 6.13&#949;)</span>
785 </p>
786 </td>
787<td>
788 <p>
789 <span class="blue">Max = 0.527&#949; (Mean = 0.128&#949;)</span>
790 </p>
791 </td>
792<td>
793 <p>
794 <span class="blue">Max = 1.29e+03&#949; (Mean = 312&#949;)</span>
795 </p>
796 </td>
797<td>
798 <p>
799 <span class="blue">Max = 1.29e+03&#949; (Mean = 355&#949;)</span>
800 </p>
801 </td>
802</tr>
803<tr>
804<td>
805 <p>
806 Bessel J': Mathworld Data
807 </p>
808 </td>
809<td>
810 <p>
811 <span class="blue">Max = 23.7&#949; (Mean = 8.01&#949;)</span>
812 </p>
813 </td>
814<td>
815 <p>
816 <span class="blue">Max = 21.5&#949; (Mean = 4.7&#949;)</span>
817 </p>
818 </td>
819<td>
820 <p>
821 <span class="blue">Max = 42.5&#949; (Mean = 9.31&#949;)</span>
822 </p>
823 </td>
824<td>
825 <p>
826 <span class="blue">Max = 42.5&#949; (Mean = 9.32&#949;)</span>
827 </p>
828 </td>
829</tr>
830<tr>
831<td>
832 <p>
833 Bessel J': Mathworld Data (large values)
834 </p>
835 </td>
836<td>
837 <p>
838 <span class="blue">Max = 2.9&#949; (Mean = 1.61&#949;)</span>
839 </p>
840 </td>
841<td>
842 <p>
843 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
844 </p>
845 </td>
846<td>
847 <p>
848 <span class="blue">Max = 989&#949; (Mean = 495&#949;)</span>
849 </p>
850 </td>
851<td>
852 <p>
853 <span class="blue">Max = 989&#949; (Mean = 495&#949;)</span>
854 </p>
855 </td>
856</tr>
857<tr>
858<td>
859 <p>
860 Bessel JN': Random Data
861 </p>
862 </td>
863<td>
864 <p>
865 <span class="blue">Max = 6.34&#949; (Mean = 0.997&#949;)</span>
866 </p>
867 </td>
868<td>
869 <p>
870 <span class="blue">Max = 0.593&#949; (Mean = 0.0396&#949;)</span>
871 </p>
872 </td>
873<td>
874 <p>
875 <span class="blue">Max = 17.3&#949; (Mean = 2.47&#949;)</span>
876 </p>
877 </td>
878<td>
879 <p>
880 <span class="blue">Max = 79.4&#949; (Mean = 16.2&#949;)</span>
881 </p>
882 </td>
883</tr>
884<tr>
885<td>
886 <p>
887 Bessel J': Random Data
888 </p>
889 </td>
890<td>
891 <p>
892 <span class="blue">Max = 176&#949; (Mean = 9.76&#949;)</span>
893 </p>
894 </td>
895<td>
896 <p>
897 <span class="blue">Max = 0.885&#949; (Mean = 0.0457&#949;)</span>
898 </p>
899 </td>
900<td>
901 <p>
902 <span class="blue">Max = 139&#949; (Mean = 6.47&#949;)</span>
903 </p>
904 </td>
905<td>
906 <p>
907 <span class="blue">Max = 279&#949; (Mean = 27.2&#949;)</span>
908 </p>
909 </td>
910</tr>
911<tr>
912<td>
913 <p>
914 Bessel J': Random Data (Tricky large values)
915 </p>
916 </td>
917<td>
918 <p>
919 <span class="blue">Max = 379&#949; (Mean = 45.4&#949;)</span>
920 </p>
921 </td>
922<td>
923 <p>
924 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
925 </p>
926 </td>
927<td>
928 <p>
929 <span class="blue">Max = 474&#949; (Mean = 62.2&#949;)</span>
930 </p>
931 </td>
932<td>
933 <p>
934 <span class="blue">Max = 474&#949; (Mean = 64.5&#949;)</span>
935 </p>
936 </td>
937</tr>
938</tbody>
939</table></div>
940</div>
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&#160;6.54.&#160;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)">
944<colgroup>
945<col>
946<col>
947<col>
948<col>
949<col>
950</colgroup>
951<thead><tr>
952<th>
953 </th>
954<th>
955 <p>
956 Microsoft Visual C++ version 12.0<br> Win32<br> double
957 </p>
958 </th>
959<th>
960 <p>
961 GNU C++ version 5.1.0<br> linux<br> double
962 </p>
963 </th>
964<th>
965 <p>
966 GNU C++ version 5.1.0<br> linux<br> long double
967 </p>
968 </th>
969<th>
970 <p>
971 Sun compiler version 0x5130<br> Sun Solaris<br> long double
972 </p>
973 </th>
974</tr></thead>
975<tbody>
976<tr>
977<td>
978 <p>
979 Bessel K'0: Mathworld Data (Integer Version)
980 </p>
981 </td>
982<td>
983 <p>
984 <span class="blue">Max = 1&#949; (Mean = 0.573&#949;)</span>
985 </p>
986 </td>
987<td>
988 <p>
989 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
990 </p>
991 </td>
992<td>
993 <p>
994 <span class="blue">Max = 1.8&#949; (Mean = 1.02&#949;)</span>
995 </p>
996 </td>
997<td>
998 <p>
999 <span class="blue">Max = 1.8&#949; (Mean = 1.02&#949;)</span>
1000 </p>
1001 </td>
1002</tr>
1003<tr>
1004<td>
1005 <p>
1006 Bessel K'1: Mathworld Data (Integer Version)
1007 </p>
1008 </td>
1009<td>
1010 <p>
1011 <span class="blue">Max = 1.09&#949; (Mean = 0.607&#949;)</span>
1012 </p>
1013 </td>
1014<td>
1015 <p>
1016 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
1017 </p>
1018 </td>
1019<td>
1020 <p>
1021 <span class="blue">Max = 1.96&#949; (Mean = 1&#949;)</span>
1022 </p>
1023 </td>
1024<td>
1025 <p>
1026 <span class="blue">Max = 1.96&#949; (Mean = 1.04&#949;)</span>
1027 </p>
1028 </td>
1029</tr>
1030<tr>
1031<td>
1032 <p>
1033 Bessel K'n: Mathworld Data (Integer Version)
1034 </p>
1035 </td>
1036<td>
1037 <p>
1038 <span class="blue">Max = 4.17&#949; (Mean = 1.74&#949;)</span>
1039 </p>
1040 </td>
1041<td>
1042 <p>
1043 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
1044 </p>
1045 </td>
1046<td>
1047 <p>
1048 <span class="blue">Max = 4.47&#949; (Mean = 2.05&#949;)</span>
1049 </p>
1050 </td>
1051<td>
1052 <p>
1053 <span class="blue">Max = 4.47&#949; (Mean = 1.93&#949;)</span>
1054 </p>
1055 </td>
1056</tr>
1057</tbody>
1058</table></div>
1059</div>
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&#160;6.55.&#160;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">
1063<colgroup>
1064<col>
1065<col>
1066<col>
1067<col>
1068<col>
1069</colgroup>
1070<thead><tr>
1071<th>
1072 </th>
1073<th>
1074 <p>
1075 Microsoft Visual C++ version 12.0<br> Win32<br> double
1076 </p>
1077 </th>
1078<th>
1079 <p>
1080 GNU C++ version 5.1.0<br> linux<br> double
1081 </p>
1082 </th>
1083<th>
1084 <p>
1085 GNU C++ version 5.1.0<br> linux<br> long double
1086 </p>
1087 </th>
1088<th>
1089 <p>
1090 Sun compiler version 0x5130<br> Sun Solaris<br> long double
1091 </p>
1092 </th>
1093</tr></thead>
1094<tbody>
1095<tr>
1096<td>
1097 <p>
1098 Bessel K'0: Mathworld Data
1099 </p>
1100 </td>
1101<td>
1102 <p>
1103 <span class="blue">Max = 1&#949; (Mean = 0.573&#949;)</span>
1104 </p>
1105 </td>
1106<td>
1107 <p>
1108 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
1109 </p>
1110 </td>
1111<td>
1112 <p>
1113 <span class="blue">Max = 1.8&#949; (Mean = 1.02&#949;)</span>
1114 </p>
1115 </td>
1116<td>
1117 <p>
1118 <span class="blue">Max = 1.8&#949; (Mean = 1.02&#949;)</span>
1119 </p>
1120 </td>
1121</tr>
1122<tr>
1123<td>
1124 <p>
1125 Bessel K'1: Mathworld Data
1126 </p>
1127 </td>
1128<td>
1129 <p>
1130 <span class="blue">Max = 1.09&#949; (Mean = 0.607&#949;)</span>
1131 </p>
1132 </td>
1133<td>
1134 <p>
1135 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
1136 </p>
1137 </td>
1138<td>
1139 <p>
1140 <span class="blue">Max = 1.96&#949; (Mean = 1&#949;)</span>
1141 </p>
1142 </td>
1143<td>
1144 <p>
1145 <span class="blue">Max = 1.96&#949; (Mean = 1.04&#949;)</span>
1146 </p>
1147 </td>
1148</tr>
1149<tr>
1150<td>
1151 <p>
1152 Bessel K'n: Mathworld Data
1153 </p>
1154 </td>
1155<td>
1156 <p>
1157 <span class="blue">Max = 4.17&#949; (Mean = 1.74&#949;)</span>
1158 </p>
1159 </td>
1160<td>
1161 <p>
1162 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
1163 </p>
1164 </td>
1165<td>
1166 <p>
1167 <span class="blue">Max = 4.47&#949; (Mean = 2.05&#949;)</span>
1168 </p>
1169 </td>
1170<td>
1171 <p>
1172 <span class="blue">Max = 4.47&#949; (Mean = 1.93&#949;)</span>
1173 </p>
1174 </td>
1175</tr>
1176<tr>
1177<td>
1178 <p>
1179 Bessel K'v: Mathworld Data
1180 </p>
1181 </td>
1182<td>
1183 <p>
1184 <span class="blue">Max = 3.94&#949; (Mean = 1.43&#949;)</span>
1185 </p>
1186 </td>
1187<td>
1188 <p>
1189 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
1190 </p>
1191 </td>
1192<td>
1193 <p>
1194 <span class="blue">Max = 3.94&#949; (Mean = 2.49&#949;)</span>
1195 </p>
1196 </td>
1197<td>
1198 <p>
1199 <span class="blue">Max = 3.94&#949; (Mean = 2.35&#949;)</span>
1200 </p>
1201 </td>
1202</tr>
1203<tr>
1204<td>
1205 <p>
1206 Bessel K'v: Mathworld Data (large values)
1207 </p>
1208 </td>
1209<td>
1210 <p>
1211 <span class="blue">Max = 18.6&#949; (Mean = 12.1&#949;)</span>
1212 </p>
1213 </td>
1214<td>
1215 <p>
1216 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
1217 </p>
1218 </td>
1219<td>
1220 <p>
1221 <span class="blue">Max = 59.2&#949; (Mean = 42.9&#949;)</span>
1222 </p>
1223 </td>
1224<td>
1225 <p>
1226 <span class="blue">Max = 59.8&#949; (Mean = 43.4&#949;)</span>
1227 </p>
1228 </td>
1229</tr>
1230<tr>
1231<td>
1232 <p>
1233 Bessel K'n: Random Data
1234 </p>
1235 </td>
1236<td>
1237 <p>
1238 <span class="blue">Max = 8.18&#949; (Mean = 1.45&#949;)</span>
1239 </p>
1240 </td>
1241<td>
1242 <p>
1243 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
1244 </p>
1245 </td>
1246<td>
1247 <p>
1248 <span class="blue">Max = 4.53&#949; (Mean = 1.12&#949;)</span>
1249 </p>
1250 </td>
1251<td>
1252 <p>
1253 <span class="blue">Max = 4.45&#949; (Mean = 1.23&#949;)</span>
1254 </p>
1255 </td>
1256</tr>
1257<tr>
1258<td>
1259 <p>
1260 Bessel K'v: Random Data
1261 </p>
1262 </td>
1263<td>
1264 <p>
1265 <span class="blue">Max = 8.32&#949; (Mean = 1.65&#949;)</span>
1266 </p>
1267 </td>
1268<td>
1269 <p>
1270 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
1271 </p>
1272 </td>
1273<td>
1274 <p>
1275 <span class="blue">Max = 7.95&#949; (Mean = 1.53&#949;)</span>
1276 </p>
1277 </td>
1278<td>
1279 <p>
1280 <span class="blue">Max = 7.95&#949; (Mean = 1.55&#949;)</span>
1281 </p>
1282 </td>
1283</tr>
1284</tbody>
1285</table></div>
1286</div>
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&#160;6.56.&#160;Error rates for sph_bessel_prime</b></p>
1289<div class="table-contents"><table class="table" summary="Error rates for sph_bessel_prime">
1290<colgroup>
1291<col>
1292<col>
1293<col>
1294<col>
1295<col>
1296</colgroup>
1297<thead><tr>
1298<th>
1299 </th>
1300<th>
1301 <p>
1302 Microsoft Visual C++ version 12.0<br> Win32<br> double
1303 </p>
1304 </th>
1305<th>
1306 <p>
1307 GNU C++ version 5.1.0<br> linux<br> double
1308 </p>
1309 </th>
1310<th>
1311 <p>
1312 GNU C++ version 5.1.0<br> linux<br> long double
1313 </p>
1314 </th>
1315<th>
1316 <p>
1317 Sun compiler version 0x5130<br> Sun Solaris<br> long double
1318 </p>
1319 </th>
1320</tr></thead>
1321<tbody><tr>
1322<td>
1323 <p>
1324 Bessel j': Random Data
1325 </p>
1326 </td>
1327<td>
1328 <p>
1329 <span class="blue">Max = 307&#949; (Mean = 25.2&#949;)</span>
1330 </p>
1331 </td>
1332<td>
1333 <p>
1334 <span class="blue">Max = 0.753&#949; (Mean = 0.0343&#949;)</span>
1335 </p>
1336 </td>
1337<td>
1338 <p>
1339 <span class="blue">Max = 167&#949; (Mean = 12&#949;)</span>
1340 </p>
1341 </td>
1342<td>
1343 <p>
1344 <span class="blue">Max = 167&#949; (Mean = 33.2&#949;)</span>
1345 </p>
1346 </td>
1347</tr></tbody>
1348</table></div>
1349</div>
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&#160;6.57.&#160;Error rates for sph_neumann_prime</b></p>
1352<div class="table-contents"><table class="table" summary="Error rates for sph_neumann_prime">
1353<colgroup>
1354<col>
1355<col>
1356<col>
1357<col>
1358<col>
1359</colgroup>
1360<thead><tr>
1361<th>
1362 </th>
1363<th>
1364 <p>
1365 Microsoft Visual C++ version 12.0<br> Win32<br> double
1366 </p>
1367 </th>
1368<th>
1369 <p>
1370 GNU C++ version 5.1.0<br> linux<br> double
1371 </p>
1372 </th>
1373<th>
1374 <p>
1375 GNU C++ version 5.1.0<br> linux<br> long double
1376 </p>
1377 </th>
1378<th>
1379 <p>
1380 Sun compiler version 0x5130<br> Sun Solaris<br> long double
1381 </p>
1382 </th>
1383</tr></thead>
1384<tbody><tr>
1385<td>
1386 <p>
1387 y': Random Data
1388 </p>
1389 </td>
1390<td>
1391 <p>
1392 <span class="blue">Max = 296&#949; (Mean = 25.6&#949;)</span>
1393 </p>
1394 </td>
1395<td>
1396 <p>
1397 <span class="blue">Max = 0.752&#949; (Mean = 0.062&#949;)</span>
1398 </p>
1399 </td>
1400<td>
1401 <p>
1402 <span class="blue">Max = 158&#949; (Mean = 18.8&#949;)</span>
1403 </p>
1404 </td>
1405<td>
1406 <p>
1407 <span class="blue">Max = 158&#949; (Mean = 20.2&#949;)</span>
1408 </p>
1409 </td>
1410</tr></tbody>
1411</table></div>
1412</div>
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>
1416 </h5>
1417<p>
1418 In the general case, the derivatives are calculated using the relations:
1419 </p>
1420<p>
1421 <span class="inlinemediaobject"><img src="../../../equations/bessel_derivatives1.svg"></span>
1422 </p>
1423<p>
1424 There are also a number of special cases, for large x we have:
1425 </p>
1426<p>
1427 <span class="inlinemediaobject"><img src="../../../equations/bessel_derivatives4.svg"></span>
1428 </p>
1429<p>
1430 And for small x:
1431 </p>
1432<p>
1433 <span class="inlinemediaobject"><img src="../../../equations/bessel_derivatives5.svg"></span>
1434 </p>
1435</div>
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 &#169; 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&#229;de, 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>)
1444 </p>
1445</div></td>
1446</tr></table>
1447<hr>
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="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="../hankel.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
1450</div>
1451</body>
1452</html>
Ceph