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[ceph.git] / ceph / src / boost / libs / math / include / boost / math / constants / calculate_constants.hpp
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FG		 1// Copyright John Maddock 2010, 2012.
2// Copyright Paul A. Bristow 2011, 2012.
3
4// Use, modification and distribution are subject to the
5// Boost Software License, Version 1.0. (See accompanying file
6// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
7
8#ifndef BOOST_MATH_CALCULATE_CONSTANTS_CONSTANTS_INCLUDED
9#define BOOST_MATH_CALCULATE_CONSTANTS_CONSTANTS_INCLUDED
10
11#include <boost/math/special_functions/trunc.hpp>
12
13namespace boost{ namespace math{ namespace constants{ namespace detail{
14
15template <class T>
16template<int N>
17inline T constant_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
18{
19 BOOST_MATH_STD_USING
20
21 return ldexp(acos(T(0)), 1);
22
23 /*
24 // Although this code works well, it's usually more accurate to just call acos
25 // and access the number types own representation of PI which is usually calculated
26 // at slightly higher precision...
27
28 T result;
29 T a = 1;
30 T b;
31 T A(a);
32 T B = 0.5f;
33 T D = 0.25f;
34
35 T lim;
36 lim = boost::math::tools::epsilon<T>();
37
38 unsigned k = 1;
39
40 do
41 {
42 result = A + B;
43 result = ldexp(result, -2);
44 b = sqrt(B);
45 a += b;
46 a = ldexp(a, -1);
47 A = a * a;
48 B = A - result;
49 B = ldexp(B, 1);
50 result = A - B;
51 bool neg = boost::math::sign(result) < 0;
52 if(neg)
53 result = -result;
54 if(result <= lim)
55 break;
56 if(neg)
57 result = -result;
58 result = ldexp(result, k - 1);
59 D -= result;
60 ++k;
61 lim = ldexp(lim, 1);
62 }
63 while(true);
64
65 result = B / D;
66 return result;
67 */
68}
69
70template <class T>
71template<int N>
72inline T constant_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
73{
74 return 2 * pi<T, policies::policy<policies::digits2<N> > >();
75}
76
77template <class T> // 2 / pi
78template<int N>
79inline T constant_two_div_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
80{
81 return 2 / pi<T, policies::policy<policies::digits2<N> > >();
82}
83
84template <class T> // sqrt(2/pi)
85template <int N>
86inline T constant_root_two_div_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
87{
88 BOOST_MATH_STD_USING
89 return sqrt((2 / pi<T, policies::policy<policies::digits2<N> > >()));
90}
91
92template <class T>
93template<int N>
94inline T constant_one_div_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
95{
96 return 1 / two_pi<T, policies::policy<policies::digits2<N> > >();
97}
98
99template <class T>
100template<int N>
101inline T constant_root_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
102{
103 BOOST_MATH_STD_USING
104 return sqrt(pi<T, policies::policy<policies::digits2<N> > >());
105}
106
107template <class T>
108template<int N>
109inline T constant_root_half_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
110{
111 BOOST_MATH_STD_USING
112 return sqrt(pi<T, policies::policy<policies::digits2<N> > >() / 2);
113}
114
115template <class T>
116template<int N>
117inline T constant_root_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
118{
119 BOOST_MATH_STD_USING
120 return sqrt(two_pi<T, policies::policy<policies::digits2<N> > >());
121}
122
123template <class T>
124template<int N>
125inline T constant_log_root_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
126{
127 BOOST_MATH_STD_USING
128 return log(root_two_pi<T, policies::policy<policies::digits2<N> > >());
129}
130
131template <class T>
132template<int N>
133inline T constant_root_ln_four<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
134{
135 BOOST_MATH_STD_USING
136 return sqrt(log(static_cast<T>(4)));
137}
138
139template <class T>
140template<int N>
141inline T constant_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
142{
143 //
144 // Although we can clearly calculate this from first principles, this hooks into
145 // T's own notion of e, which hopefully will more accurate than one calculated to
146 // a few epsilon:
147 //
148 BOOST_MATH_STD_USING
149 return exp(static_cast<T>(1));
150}
151
152template <class T>
153template<int N>
154inline T constant_half<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
155{
156 return static_cast<T>(1) / static_cast<T>(2);
157}
158
159template <class T>
160template<int M>
161inline T constant_euler<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<M>))
162{
163 BOOST_MATH_STD_USING
164 //
165 // This is the method described in:
166 // "Some New Algorithms for High-Precision Computation of Euler's Constant"
167 // Richard P Brent and Edwin M McMillan.
168 // Mathematics of Computation, Volume 34, Number 149, Jan 1980, pages 305-312.
169 // See equation 17 with p = 2.
170 //
171 T n = 3 + (M ? (std::min)(M, tools::digits<T>()) : tools::digits<T>()) / 4;
172 T lim = M ? ldexp(T(1), 1 - (std::min)(M, tools::digits<T>())) : tools::epsilon<T>();
173 T lnn = log(n);
174 T term = 1;
175 T N = -lnn;
176 T D = 1;
177 T Hk = 0;
178 T one = 1;
179
180 for(unsigned k = 1;; ++k)
181 {
182 term *= n * n;
183 term /= k * k;
184 Hk += one / k;
185 N += term * (Hk - lnn);
186 D += term;
187
188 if(term < D * lim)
189 break;
190 }
191 return N / D;
192}
193
194template <class T>
195template<int N>
196inline T constant_euler_sqr<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
197{
198 BOOST_MATH_STD_USING
199 return euler<T, policies::policy<policies::digits2<N> > >()
200 * euler<T, policies::policy<policies::digits2<N> > >();
201}
202
203template <class T>
204template<int N>
205inline T constant_one_div_euler<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
206{
207 BOOST_MATH_STD_USING
208 return static_cast<T>(1)
209 / euler<T, policies::policy<policies::digits2<N> > >();
210}
211
212
213template <class T>
214template<int N>
215inline T constant_root_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
216{
217 BOOST_MATH_STD_USING
218 return sqrt(static_cast<T>(2));
219}
220
221
222template <class T>
223template<int N>
224inline T constant_root_three<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
225{
226 BOOST_MATH_STD_USING
227 return sqrt(static_cast<T>(3));
228}
229
230template <class T>
231template<int N>
232inline T constant_half_root_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
233{
234 BOOST_MATH_STD_USING
235 return sqrt(static_cast<T>(2)) / 2;
236}
237
238template <class T>
239template<int N>
240inline T constant_ln_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
241{
242 //
243 // Although there are good ways to calculate this from scratch, this hooks into
244 // T's own notion of log(2) which will hopefully be accurate to the full precision
245 // of T:
246 //
247 BOOST_MATH_STD_USING
248 return log(static_cast<T>(2));
249}
250
251template <class T>
252template<int N>
253inline T constant_ln_ten<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
254{
255 BOOST_MATH_STD_USING
256 return log(static_cast<T>(10));
257}
258
259template <class T>
260template<int N>
261inline T constant_ln_ln_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
262{
263 BOOST_MATH_STD_USING
264 return log(log(static_cast<T>(2)));
265}
266
267template <class T>
268template<int N>
269inline T constant_third<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
270{
271 BOOST_MATH_STD_USING
272 return static_cast<T>(1) / static_cast<T>(3);
273}
274
275template <class T>
276template<int N>
277inline T constant_twothirds<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
278{
279 BOOST_MATH_STD_USING
280 return static_cast<T>(2) / static_cast<T>(3);
281}
282
283template <class T>
284template<int N>
285inline T constant_two_thirds<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
286{
287 BOOST_MATH_STD_USING
288 return static_cast<T>(2) / static_cast<T>(3);
289}
290
291template <class T>
292template<int N>
293inline T constant_three_quarters<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
294{
295 BOOST_MATH_STD_USING
296 return static_cast<T>(3) / static_cast<T>(4);
297}
298
299template <class T>
300template<int N>
301inline T constant_pi_minus_three<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
302{
303 return pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(3);
304}
305
306template <class T>
307template<int N>
308inline T constant_four_minus_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
309{
310 return static_cast<T>(4) - pi<T, policies::policy<policies::digits2<N> > >();
311}
312
313//template <class T>
314//template<int N>
315//inline T constant_pow23_four_minus_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
316//{
317// BOOST_MATH_STD_USING
318// return pow(four_minus_pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(1.5));
319//}
320
321template <class T>
322template<int N>
323inline T constant_exp_minus_half<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
324{
325 BOOST_MATH_STD_USING
326 return exp(static_cast<T>(-0.5));
327}
328
329// Pi
330template <class T>
331template<int N>
332inline T constant_one_div_root_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
333{
334 return static_cast<T>(1) / root_two<T, policies::policy<policies::digits2<N> > >();
335}
336
337template <class T>
338template<int N>
339inline T constant_one_div_root_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
340{
341 return static_cast<T>(1) / root_pi<T, policies::policy<policies::digits2<N> > >();
342}
343
344template <class T>
345template<int N>
346inline T constant_one_div_root_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
347{
348 return static_cast<T>(1) / root_two_pi<T, policies::policy<policies::digits2<N> > >();
349}
350
351template <class T>
352template<int N>
353inline T constant_root_one_div_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
354{
355 BOOST_MATH_STD_USING
356 return sqrt(static_cast<T>(1) / pi<T, policies::policy<policies::digits2<N> > >());
357}
358
359
360template <class T>
361template<int N>
362inline T constant_four_thirds_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
363{
364 BOOST_MATH_STD_USING
365 return pi<T, policies::policy<policies::digits2<N> > >() * static_cast<T>(4) / static_cast<T>(3);
366}
367
368template <class T>
369template<int N>
370inline T constant_half_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
371{
372 BOOST_MATH_STD_USING
373 return pi<T, policies::policy<policies::digits2<N> > >() / static_cast<T>(2);
374}
375
376
377template <class T>
378template<int N>
379inline T constant_third_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
380{
381 BOOST_MATH_STD_USING
382 return pi<T, policies::policy<policies::digits2<N> > >() / static_cast<T>(3);
383}
384
385template <class T>
386template<int N>
387inline T constant_sixth_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
388{
389 BOOST_MATH_STD_USING
390 return pi<T, policies::policy<policies::digits2<N> > >() / static_cast<T>(6);
391}
392
393template <class T>
394template<int N>
395inline T constant_two_thirds_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
396{
397 BOOST_MATH_STD_USING
398 return pi<T, policies::policy<policies::digits2<N> > >() * static_cast<T>(2) / static_cast<T>(3);
399}
400
401template <class T>
402template<int N>
403inline T constant_three_quarters_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
404{
405 BOOST_MATH_STD_USING
406 return pi<T, policies::policy<policies::digits2<N> > >() * static_cast<T>(3) / static_cast<T>(4);
407}
408
409template <class T>
410template<int N>
411inline T constant_pi_pow_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
412{
413 BOOST_MATH_STD_USING
414 return pow(pi<T, policies::policy<policies::digits2<N> > >(), e<T, policies::policy<policies::digits2<N> > >()); //
415}
416
417template <class T>
418template<int N>
419inline T constant_pi_sqr<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
420{
421 BOOST_MATH_STD_USING
422 return pi<T, policies::policy<policies::digits2<N> > >()
423 * pi<T, policies::policy<policies::digits2<N> > >() ; //
424}
425
426template <class T>
427template<int N>
428inline T constant_pi_sqr_div_six<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
429{
430 BOOST_MATH_STD_USING
431 return pi<T, policies::policy<policies::digits2<N> > >()
432 * pi<T, policies::policy<policies::digits2<N> > >()
433 / static_cast<T>(6); //
434}
435
436
437template <class T>
438template<int N>
439inline T constant_pi_cubed<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
440{
441 BOOST_MATH_STD_USING
442 return pi<T, policies::policy<policies::digits2<N> > >()
443 * pi<T, policies::policy<policies::digits2<N> > >()
444 * pi<T, policies::policy<policies::digits2<N> > >()
445 ; //
446}
447
448template <class T>
449template<int N>
450inline T constant_cbrt_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
451{
452 BOOST_MATH_STD_USING
453 return pow(pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(1)/ static_cast<T>(3));
454}
455
456template <class T>
457template<int N>
458inline T constant_one_div_cbrt_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
459{
460 BOOST_MATH_STD_USING
461 return static_cast<T>(1)
462 / pow(pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(1)/ static_cast<T>(3));
463}
464
465// Euler's e
466
467template <class T>
468template<int N>
469inline T constant_e_pow_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
470{
471 BOOST_MATH_STD_USING
472 return pow(e<T, policies::policy<policies::digits2<N> > >(), pi<T, policies::policy<policies::digits2<N> > >()); //
473}
474
475template <class T>
476template<int N>
477inline T constant_root_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
478{
479 BOOST_MATH_STD_USING
480 return sqrt(e<T, policies::policy<policies::digits2<N> > >());
481}
482
483template <class T>
484template<int N>
485inline T constant_log10_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
486{
487 BOOST_MATH_STD_USING
488 return log10(e<T, policies::policy<policies::digits2<N> > >());
489}
490
491template <class T>
492template<int N>
493inline T constant_one_div_log10_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
494{
495 BOOST_MATH_STD_USING
496 return static_cast<T>(1) /
497 log10(e<T, policies::policy<policies::digits2<N> > >());
498}
499
500// Trigonometric
501
502template <class T>
503template<int N>
504inline T constant_degree<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
505{
506 BOOST_MATH_STD_USING
507 return pi<T, policies::policy<policies::digits2<N> > >()
508 / static_cast<T>(180)
509 ; //
510}
511
512template <class T>
513template<int N>
514inline T constant_radian<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
515{
516 BOOST_MATH_STD_USING
517 return static_cast<T>(180)
518 / pi<T, policies::policy<policies::digits2<N> > >()
519 ; //
520}
521
522template <class T>
523template<int N>
524inline T constant_sin_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
525{
526 BOOST_MATH_STD_USING
527 return sin(static_cast<T>(1)) ; //
528}
529
530template <class T>
531template<int N>
532inline T constant_cos_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
533{
534 BOOST_MATH_STD_USING
535 return cos(static_cast<T>(1)) ; //
536}
537
538template <class T>
539template<int N>
540inline T constant_sinh_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
541{
542 BOOST_MATH_STD_USING
543 return sinh(static_cast<T>(1)) ; //
544}
545
546template <class T>
547template<int N>
548inline T constant_cosh_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
549{
550 BOOST_MATH_STD_USING
551 return cosh(static_cast<T>(1)) ; //
552}
553
554template <class T>
555template<int N>
556inline T constant_phi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
557{
558 BOOST_MATH_STD_USING
559 return (static_cast<T>(1) + sqrt(static_cast<T>(5)) )/static_cast<T>(2) ; //
560}
561
562template <class T>
563template<int N>
564inline T constant_ln_phi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
565{
566 BOOST_MATH_STD_USING
567 //return log(phi<T, policies::policy<policies::digits2<N> > >()); // ???
568 return log((static_cast<T>(1) + sqrt(static_cast<T>(5)) )/static_cast<T>(2) );
569}
570template <class T>
571template<int N>
572inline T constant_one_div_ln_phi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
573{
574 BOOST_MATH_STD_USING
575 return static_cast<T>(1) /
576 log((static_cast<T>(1) + sqrt(static_cast<T>(5)) )/static_cast<T>(2) );
577}
578
579// Zeta
580
581template <class T>
582template<int N>
583inline T constant_zeta_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
584{
585 BOOST_MATH_STD_USING
586
587 return pi<T, policies::policy<policies::digits2<N> > >()
588 * pi<T, policies::policy<policies::digits2<N> > >()
589 /static_cast<T>(6);
590}
591
592template <class T>
593template<int N>
594inline T constant_zeta_three<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
595{
596 // http://mathworld.wolfram.com/AperysConstant.html
597 // http://en.wikipedia.org/wiki/Mathematical_constant
598
599 // http://oeis.org/A002117/constant
600 //T zeta3("1.20205690315959428539973816151144999076"
601 // "4986292340498881792271555341838205786313"
602 // "09018645587360933525814619915");
603
604 //"1.202056903159594285399738161511449990, 76498629234049888179227155534183820578631309018645587360933525814619915" A002117
605 // 1.202056903159594285399738161511449990, 76498629234049888179227155534183820578631309018645587360933525814619915780, +00);
606 //"1.2020569031595942 double
607 // http://www.spaennare.se/SSPROG/ssnum.pdf // section 11, Algorithm for Apery's constant zeta(3).
608 // Programs to Calculate some Mathematical Constants to Large Precision, Document Version 1.50
609
610 // by Stefan Spannare September 19, 2007
611 // zeta(3) = 1/64 * sum
612 BOOST_MATH_STD_USING
613 T n_fact=static_cast<T>(1); // build n! for n = 0.
614 T sum = static_cast<double>(77); // Start with n = 0 case.
615 // for n = 0, (77/1) /64 = 1.203125
616 //double lim = std::numeric_limits<double>::epsilon();
617 T lim = N ? ldexp(T(1), 1 - (std::min)(N, tools::digits<T>())) : tools::epsilon<T>();
618 for(unsigned int n = 1; n < 40; ++n)
619 { // three to five decimal digits per term, so 40 should be plenty for 100 decimal digits.
620 //cout << "n = " << n << endl;
621 n_fact *= n; // n!
622 T n_fact_p10 = n_fact * n_fact * n_fact * n_fact * n_fact * n_fact * n_fact * n_fact * n_fact * n_fact; // (n!)^10
623 T num = ((205 * n * n) + (250 * n) + 77) * n_fact_p10; // 205n^2 + 250n + 77
624 // int nn = (2 * n + 1);
625 // T d = factorial(nn); // inline factorial.
626 T d = 1;
627 for(unsigned int i = 1; i <= (n+n + 1); ++i) // (2n + 1)
628 {
629 d *= i;
630 }
631 T den = d * d * d * d * d; // [(2n+1)!]^5
632 //cout << "den = " << den << endl;
633 T term = num/den;
634 if (n % 2 != 0)
635 { //term *= -1;
636 sum -= term;
637 }
638 else
639 {
640 sum += term;
641 }
642 //cout << "term = " << term << endl;
643 //cout << "sum/64 = " << sum/64 << endl;
644 if(abs(term) < lim)
645 {
646 break;
647 }
648 }
649 return sum / 64;
650}
651
652template <class T>
653template<int N>
654inline T constant_catalan<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
655{ // http://oeis.org/A006752/constant
656 //T c("0.915965594177219015054603514932384110774"
657 //"149374281672134266498119621763019776254769479356512926115106248574");
658
659 // 9.159655941772190150546035149323841107, 74149374281672134266498119621763019776254769479356512926115106248574422619, -01);
660
661 // This is equation (entry) 31 from
662 // http://www-2.cs.cmu.edu/~adamchik/articles/catalan/catalan.htm
663 // See also http://www.mpfr.org/algorithms.pdf
664 BOOST_MATH_STD_USING
665 T k_fact = 1;
666 T tk_fact = 1;
667 T sum = 1;
668 T term;
669 T lim = N ? ldexp(T(1), 1 - (std::min)(N, tools::digits<T>())) : tools::epsilon<T>();
670
671 for(unsigned k = 1;; ++k)
672 {
673 k_fact *= k;
674 tk_fact *= (2 * k) * (2 * k - 1);
675 term = k_fact * k_fact / (tk_fact * (2 * k + 1) * (2 * k + 1));
676 sum += term;
677 if(term < lim)
678 {
679 break;
680 }
681 }
682 return boost::math::constants::pi<T, boost::math::policies::policy<> >()
683 * log(2 + boost::math::constants::root_three<T, boost::math::policies::policy<> >())
684 / 8
685 + 3 * sum / 8;
686}
687
688namespace khinchin_detail{
689
690template <class T>
691T zeta_polynomial_series(T s, T sc, int digits)
692{
693 BOOST_MATH_STD_USING
694 //
695 // This is algorithm 3 from:
696 //
697 // "An Efficient Algorithm for the Riemann Zeta Function", P. Borwein,
698 // Canadian Mathematical Society, Conference Proceedings, 2000.
699 // See: http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf
700 //
701 BOOST_MATH_STD_USING
702 int n = (digits * 19) / 53;
703 T sum = 0;
704 T two_n = ldexp(T(1), n);
705 int ej_sign = 1;
706 for(int j = 0; j < n; ++j)
707 {
708 sum += ej_sign * -two_n / pow(T(j + 1), s);
709 ej_sign = -ej_sign;
710 }
711 T ej_sum = 1;
712 T ej_term = 1;
713 for(int j = n; j <= 2 * n - 1; ++j)
714 {
715 sum += ej_sign * (ej_sum - two_n) / pow(T(j + 1), s);
716 ej_sign = -ej_sign;
717 ej_term *= 2 * n - j;
718 ej_term /= j - n + 1;
719 ej_sum += ej_term;
720 }
721 return -sum / (two_n * (1 - pow(T(2), sc)));
722}
723
724template <class T>
725T khinchin(int digits)
726{
727 BOOST_MATH_STD_USING
728 T sum = 0;
729 T term;
730 T lim = ldexp(T(1), 1-digits);
731 T factor = 0;
732 unsigned last_k = 1;
733 T num = 1;
734 for(unsigned n = 1;; ++n)
735 {
736 for(unsigned k = last_k; k <= 2 * n - 1; ++k)
737 {
738 factor += num / k;
739 num = -num;
740 }
741 last_k = 2 * n;
742 term = (zeta_polynomial_series(T(2 * n), T(1 - T(2 * n)), digits) - 1) * factor / n;
743 sum += term;
744 if(term < lim)
745 break;
746 }
747 return exp(sum / boost::math::constants::ln_two<T, boost::math::policies::policy<> >());
748}
749
750}
751
752template <class T>
753template<int N>
754inline T constant_khinchin<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
755{
756 int n = N ? (std::min)(N, tools::digits<T>()) : tools::digits<T>();
757 return khinchin_detail::khinchin<T>(n);
758}
759
760template <class T>
761template<int N>
762inline T constant_extreme_value_skewness<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
763{ // from e_float constants.cpp
764 // Mathematica: N[12 Sqrt[6] Zeta[3]/Pi^3, 1101]
765 BOOST_MATH_STD_USING
766 T ev(12 * sqrt(static_cast<T>(6)) * zeta_three<T, policies::policy<policies::digits2<N> > >()
767 / pi_cubed<T, policies::policy<policies::digits2<N> > >() );
768
769//T ev(
770//"1.1395470994046486574927930193898461120875997958365518247216557100852480077060706857071875468869385150"
771//"1894272048688553376986765366075828644841024041679714157616857834895702411080704529137366329462558680"
772//"2015498788776135705587959418756809080074611906006528647805347822929577145038743873949415294942796280"
773//"0895597703063466053535550338267721294164578901640163603544404938283861127819804918174973533694090594"
774//"3094963822672055237678432023017824416203652657301470473548274848068762500300316769691474974950757965"
775//"8640779777748741897542093874605477776538884083378029488863880220988107155275203245233994097178778984"
776//"3488995668362387892097897322246698071290011857605809901090220903955815127463328974447572119951192970"
777//"3684453635456559086126406960279692862247058250100678008419431185138019869693206366891639436908462809"
778//"9756051372711251054914491837034685476095423926553367264355374652153595857163724698198860485357368964"
779//"3807049634423621246870868566707915720704996296083373077647528285782964567312903914752617978405994377"
780//"9064157147206717895272199736902453130842229559980076472936976287378945035706933650987259357729800315");
781
782 return ev;
783}
784
785namespace detail{
786//
787// Calculation of the Glaisher constant depends upon calculating the
788// derivative of the zeta function at 2, we can then use the relation:
789// zeta'(2) = 1/6 pi^2 [euler + ln(2pi)-12ln(A)]
790// To get the constant A.
791// See equation 45 at http://mathworld.wolfram.com/RiemannZetaFunction.html.
792//
793// The derivative of the zeta function is computed by direct differentiation
794// of the relation:
795// (1-2^(1-s))zeta(s) = SUM(n=0, INF){ (-n)^n / (n+1)^s }
796// Which gives us 2 slowly converging but alternating sums to compute,
797// for this we use Algorithm 1 from "Convergent Acceleration of Alternating Series",
798// Henri Cohen, Fernando Rodriguez Villegas and Don Zagier, Experimental Mathematics 9:1 (1999).
799// See http://www.math.utexas.edu/users/villegas/publications/conv-accel.pdf
800//
801template <class T>
802T zeta_series_derivative_2(unsigned digits)
803{
804 // Derivative of the series part, evaluated at 2:
805 BOOST_MATH_STD_USING
806 int n = digits * 301 * 13 / 10000;
807 boost::math::itrunc((std::numeric_limits<T>::digits10 + 1) * 1.3);
808 T d = pow(3 + sqrt(T(8)), n);
809 d = (d + 1 / d) / 2;
810 T b = -1;
811 T c = -d;
812 T s = 0;
813 for(int k = 0; k < n; ++k)
814 {
815 T a = -log(T(k+1)) / ((k+1) * (k+1));
816 c = b - c;
817 s = s + c * a;
818 b = (k + n) * (k - n) * b / ((k + T(0.5f)) * (k + 1));
819 }
820 return s / d;
821}
822
823template <class T>
824T zeta_series_2(unsigned digits)
825{
826 // Series part of zeta at 2:
827 BOOST_MATH_STD_USING
828 int n = digits * 301 * 13 / 10000;
829 T d = pow(3 + sqrt(T(8)), n);
830 d = (d + 1 / d) / 2;
831 T b = -1;
832 T c = -d;
833 T s = 0;
834 for(int k = 0; k < n; ++k)
835 {
836 T a = T(1) / ((k + 1) * (k + 1));
837 c = b - c;
838 s = s + c * a;
839 b = (k + n) * (k - n) * b / ((k + T(0.5f)) * (k + 1));
840 }
841 return s / d;
842}
843
844template <class T>
845inline T zeta_series_lead_2()
846{
847 // lead part at 2:
848 return 2;
849}
850
851template <class T>
852inline T zeta_series_derivative_lead_2()
853{
854 // derivative of lead part at 2:
855 return -2 * boost::math::constants::ln_two<T>();
856}
857
858template <class T>
859inline T zeta_derivative_2(unsigned n)
860{
861 // zeta derivative at 2:
862 return zeta_series_derivative_2<T>(n) * zeta_series_lead_2<T>()
863 + zeta_series_derivative_lead_2<T>() * zeta_series_2<T>(n);
864}
865
866} // namespace detail
867
868template <class T>
869template<int N>
870inline T constant_glaisher<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
871{
872
873 BOOST_MATH_STD_USING
874 typedef policies::policy<policies::digits2<N> > forwarding_policy;
875 int n = N ? (std::min)(N, tools::digits<T>()) : tools::digits<T>();
876 T v = detail::zeta_derivative_2<T>(n);
877 v *= 6;
878 v /= boost::math::constants::pi<T, forwarding_policy>() * boost::math::constants::pi<T, forwarding_policy>();
879 v -= boost::math::constants::euler<T, forwarding_policy>();
880 v -= log(2 * boost::math::constants::pi<T, forwarding_policy>());
881 v /= -12;
882 return exp(v);
883
884 /*
885 // from http://mpmath.googlecode.com/svn/data/glaisher.txt
886 // 20,000 digits of the Glaisher-Kinkelin constant A = exp(1/2 - zeta'(-1))
887 // Computed using A = exp((6 (-zeta'(2))/pi^2 + log 2 pi + gamma)/12)
888 // with Euler-Maclaurin summation for zeta'(2).
889 T g(
890 "1.282427129100622636875342568869791727767688927325001192063740021740406308858826"
891 "46112973649195820237439420646120399000748933157791362775280404159072573861727522"
892 "14334327143439787335067915257366856907876561146686449997784962754518174312394652"
893 "76128213808180219264516851546143919901083573730703504903888123418813674978133050"
894 "93770833682222494115874837348064399978830070125567001286994157705432053927585405"
895 "81731588155481762970384743250467775147374600031616023046613296342991558095879293"
896 "36343887288701988953460725233184702489001091776941712153569193674967261270398013"
897 "52652668868978218897401729375840750167472114895288815996668743164513890306962645"
898 "59870469543740253099606800842447417554061490189444139386196089129682173528798629"
899 "88434220366989900606980888785849587494085307347117090132667567503310523405221054"
900 "14176776156308191919997185237047761312315374135304725819814797451761027540834943"
901 "14384965234139453373065832325673954957601692256427736926358821692159870775858274"
902 "69575162841550648585890834128227556209547002918593263079373376942077522290940187");
903
904 return g;
905 */
906}
907
908template <class T>
909template<int N>
910inline T constant_rayleigh_skewness<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
911{ // From e_float
912 // 1100 digits of the Rayleigh distribution skewness
913 // Mathematica: N[2 Sqrt[Pi] (Pi - 3)/((4 - Pi)^(3/2)), 1100]
914
915 BOOST_MATH_STD_USING
916 T rs(2 * root_pi<T, policies::policy<policies::digits2<N> > >()
917 * pi_minus_three<T, policies::policy<policies::digits2<N> > >()
918 / pow(four_minus_pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(3./2))
919 );
920 // 6.31110657818937138191899351544227779844042203134719497658094585692926819617473725459905027032537306794400047264,
921
922 //"0.6311106578189371381918993515442277798440422031347194976580945856929268196174737254599050270325373067"
923 //"9440004726436754739597525250317640394102954301685809920213808351450851396781817932734836994829371322"
924 //"5797376021347531983451654130317032832308462278373358624120822253764532674177325950686466133508511968"
925 //"2389168716630349407238090652663422922072397393006683401992961569208109477307776249225072042971818671"
926 //"4058887072693437217879039875871765635655476241624825389439481561152126886932506682176611183750503553"
927 //"1218982627032068396407180216351425758181396562859085306247387212297187006230007438534686340210168288"
928 //"8956816965453815849613622117088096547521391672977226658826566757207615552041767516828171274858145957"
929 //"6137539156656005855905288420585194082284972984285863898582313048515484073396332610565441264220790791"
930 //"0194897267890422924599776483890102027823328602965235306539844007677157873140562950510028206251529523"
931 //"7428049693650605954398446899724157486062545281504433364675815915402937209673727753199567661561209251"
932 //"4695589950526053470201635372590001578503476490223746511106018091907936826431407434894024396366284848"); ;
933 return rs;
934}
935
936template <class T>
937template<int N>
938inline T constant_rayleigh_kurtosis_excess<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
939{ // - (6 Pi^2 - 24 Pi + 16)/((Pi - 4)^2)
940 // Might provide and calculate this using pi_minus_four.
941 BOOST_MATH_STD_USING
942 return - (((static_cast<T>(6) * pi<T, policies::policy<policies::digits2<N> > >()
943 * pi<T, policies::policy<policies::digits2<N> > >())
944 - (static_cast<T>(24) * pi<T, policies::policy<policies::digits2<N> > >()) + static_cast<T>(16) )
945 /
946 ((pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(4))
947 * (pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(4)))
948 );
949}
950
951template <class T>
952template<int N>
953inline T constant_rayleigh_kurtosis<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
954{ // 3 - (6 Pi^2 - 24 Pi + 16)/((Pi - 4)^2)
955 // Might provide and calculate this using pi_minus_four.
956 BOOST_MATH_STD_USING
957 return static_cast<T>(3) - (((static_cast<T>(6) * pi<T, policies::policy<policies::digits2<N> > >()
958 * pi<T, policies::policy<policies::digits2<N> > >())
959 - (static_cast<T>(24) * pi<T, policies::policy<policies::digits2<N> > >()) + static_cast<T>(16) )
960 /
961 ((pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(4))
962 * (pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(4)))
963 );
964}
965
966}}}} // namespaces
967
968#endif // BOOST_MATH_CALCULATE_CONSTANTS_CONSTANTS_INCLUDED
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