[ceph.git] / ceph / src / boost / libs / math / doc / sf / zeta.qbk

[section:zeta Riemann Zeta Function]

[h4 Synopsis]

``
#include <boost/math/special_functions/zeta.hpp>
``

   namespace boost{ namespace math{
   
   template <class T>
   ``__sf_result`` zeta(T z);
   
   template <class T, class ``__Policy``>
   ``__sf_result`` zeta(T z, const ``__Policy``&);
   
   }} // namespaces
   
The return type of these functions is computed using the __arg_promotion_rules:
the return type is `double` if T is an integer type, and T otherwise.

[optional_policy]

[h4 Description]

   template <class T>
   ``__sf_result`` zeta(T z);
   
   template <class T, class ``__Policy``>
   ``__sf_result`` zeta(T z, const ``__Policy``&);
   
Returns the [@http://mathworld.wolfram.com/RiemannZetaFunction.html zeta function]
of z:

[equation zeta1]

[graph zeta1]

[graph zeta2]

[h4 Accuracy]

The following table shows the peak errors (in units of epsilon) 
found on various platforms with various floating point types, 
along with comparisons to the __gsl and __cephes libraries.
Unless otherwise specified any floating point type that is narrower
than the one shown will have __zero_error.

[table_zeta]

[h4 Testing]

The tests for these functions come in two parts:
basic sanity checks use spot values calculated using
[@http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=Zeta Mathworld's online evaluator],
while accuracy checks use high-precision test values calculated at 1000-bit precision with
[@http://shoup.net/ntl/doc/RR.txt NTL::RR] and this implementation. 
Note that the generic and type-specific
versions of these functions use differing implementations internally, so this
gives us reasonably independent test data.  Using our test data to test other
"known good" implementations also provides an additional sanity check. 

[h4 Implementation]

All versions of these functions first use the usual reflection formulas
to make their arguments positive:

[equation zeta3]

The generic versions of these functions are implemented using the series:

[equation zeta6]

When the significand (mantissa) size is recognised
(currently for 53, 64 and 113-bit reals, plus single-precision 24-bit handled via promotion to double)
then a series of rational approximations [jm_rationals] are used.

For 0 < z < 1 the approximating form is:

[equation zeta4]

For a rational approximation R(1-z) and a constant C.

For 1 < z < 4 the approximating form is:

[equation zeta5]

For a rational approximation R(n-z) and a constant C and integer n.

For z > 4 the approximating form is:

[zeta](z) = 1 + e[super R(z - n)]

For a rational approximation R(z-n) and integer n, note that the accuracy 
required for R(z-n) is not full machine precision, but an absolute error
of: [epsilon]/R(0).  This saves us quite a few digits when dealing with large 
z, especially when [epsilon] is small.

Finally, there are some special cases for integer arguments, there are
closed forms for negative or even integers:

[equation zeta7]

[equation zeta8]

[equation zeta9]

and for positive odd integers we simply cache pre-computed values as these are of great
benefit to some infinite series calculations.

[endsect]
[/ :error_function The Error Functions]

[/ 
  Copyright 2006 John Maddock and Paul A. Bristow.
  Distributed under the Boost Software License, Version 1.0.
  (See accompanying file LICENSE_1_0.txt or copy at
  http://www.boost.org/LICENSE_1_0.txt).
]
Commit	Line	Data
7c673cae FG	1	[section:zeta Riemann Zeta Function]
	2
	3	[h4 Synopsis]
	4
	5	``
	6	#include <boost/math/special_functions/zeta.hpp>
	7	``
	8
	9	namespace boost{ namespace math{
	10
	11	template <class T>
	12	``__sf_result`` zeta(T z);
	13
	14	template <class T, class ``__Policy``>
	15	``__sf_result`` zeta(T z, const ``__Policy``&);
	16
	17	}} // namespaces
	18
	19	The return type of these functions is computed using the __arg_promotion_rules:
	20	the return type is `double` if T is an integer type, and T otherwise.
	21
	22	[optional_policy]
	23
	24	[h4 Description]
	25
	26	template <class T>
	27	``__sf_result`` zeta(T z);
	28
	29	template <class T, class ``__Policy``>
	30	``__sf_result`` zeta(T z, const ``__Policy``&);
	31
	32	Returns the [@http://mathworld.wolfram.com/RiemannZetaFunction.html zeta function]
	33	of z:
	34
	35	[equation zeta1]
	36
	37	[graph zeta1]
	38
	39	[graph zeta2]
	40
	41	[h4 Accuracy]
	42
	43	The following table shows the peak errors (in units of epsilon)
	44	found on various platforms with various floating point types,
	45	along with comparisons to the __gsl and __cephes libraries.
	46	Unless otherwise specified any floating point type that is narrower
	47	than the one shown will have __zero_error.
	48
	49	[table_zeta]
	50
	51	[h4 Testing]
	52
	53	The tests for these functions come in two parts:
	54	basic sanity checks use spot values calculated using
	55	[@http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=Zeta Mathworld's online evaluator],
	56	while accuracy checks use high-precision test values calculated at 1000-bit precision with
	57	[@http://shoup.net/ntl/doc/RR.txt NTL::RR] and this implementation.
	58	Note that the generic and type-specific
	59	versions of these functions use differing implementations internally, so this
	60	gives us reasonably independent test data. Using our test data to test other
	61	"known good" implementations also provides an additional sanity check.
	62
	63	[h4 Implementation]
	64
65	All versions of these functions first use the usual reflection formulas
66	to make their arguments positive:
67
68	[equation zeta3]
69
70	The generic versions of these functions are implemented using the series:
71
72	[equation zeta6]
73
74	When the significand (mantissa) size is recognised
75	(currently for 53, 64 and 113-bit reals, plus single-precision 24-bit handled via promotion to double)
76	then a series of rational approximations [jm_rationals] are used.
77
78	For 0 < z < 1 the approximating form is:
79
80	[equation zeta4]
81
82	For a rational approximation R(1-z) and a constant C.
83
84	For 1 < z < 4 the approximating form is:
85
86	[equation zeta5]
87
88	For a rational approximation R(n-z) and a constant C and integer n.
89
90	For z > 4 the approximating form is:
91
92	[zeta](z) = 1 + e[super R(z - n)]
93
94	For a rational approximation R(z-n) and integer n, note that the accuracy
95	required for R(z-n) is not full machine precision, but an absolute error
96	of: [epsilon]/R(0). This saves us quite a few digits when dealing with large
97	z, especially when [epsilon] is small.
98
99	Finally, there are some special cases for integer arguments, there are
100	closed forms for negative or even integers:
101
102	[equation zeta7]
103
104	[equation zeta8]
105
106	[equation zeta9]
107
108	and for positive odd integers we simply cache pre-computed values as these are of great
109	benefit to some infinite series calculations.
110
111	[endsect]
112	[/ :error_function The Error Functions]
113
114	[/
115	Copyright 2006 John Maddock and Paul A. Bristow.
116	Distributed under the Boost Software License, Version 1.0.
117	(See accompanying file LICENSE_1_0.txt or copy at
118	http://www.boost.org/LICENSE_1_0.txt).
119	]