1 [section:bernoulli_numbers Bernoulli Numbers] 2 3 [@https://en.wikipedia.org/wiki/Bernoulli_number Bernoulli numbers] 4 are a sequence of rational numbers useful for the Taylor series expansion, 5 Euler-Maclaurin formula, and the Riemann zeta function. 6 7 Bernoulli numbers are used in evaluation of some Boost.Math functions, 8 including the __tgamma, __lgamma and polygamma functions. 9 10 [h4 Single Bernoulli number] 11 12 [h4 Synopsis] 13 14 `` 15 #include <boost/math/special_functions/bernoulli.hpp> 16 `` 17 18 namespace boost { namespace math { 19 20 template <class T> 21 T bernoulli_b2n(const int n); // Single Bernoulli number (default policy). 22 23 template <class T, class Policy> 24 T bernoulli_b2n(const int n, const Policy &pol); // User policy for errors etc. 25 26 }} // namespaces 27 28 [h4 Description] 29 30 Both return the (2 * n)[super th] Bernoulli number B[sub 2n]. 31 32 Note that since all odd numbered Bernoulli numbers are zero (apart from B[sub 1] which is [plusminus][frac12]) 33 the interface will only return the even numbered Bernoulli numbers. 34 35 This function uses fast table lookup for low-indexed Bernoulli numbers, while larger values are calculated 36 as needed and then cached. The caching mechanism requires a certain amount of thread safety code, so 37 `unchecked_bernoulli_b2n` may provide a better interface for performance critical code. 38 39 The final __Policy argument is optional and can be used to control the behaviour of the function: 40 how it handles errors, what level of precision to use, etc. 41 42 Refer to __policy_section for more details. 43 44 [h4 Examples] 45 46 [import ../../example/bernoulli_example.cpp] 47 [bernoulli_example_1] 48 49 [bernoulli_output_1] 50 51 [h4 Single (unchecked) Bernoulli number] 52 53 [h4 Synopsis] 54 `` 55 #include <boost/math/special_functions/bernoulli.hpp> 56 57 `` 58 59 template <> 60 struct max_bernoulli_b2n<T>; 61 62 template<class T> 63 inline constexpr T unchecked_bernoulli_b2n(unsigned n); 64 65 `unchecked_bernoulli_b2n` provides access to Bernoulli numbers [*without any checks for overflow or invalid parameters]. 66 It is implemented as a direct (and very fast) table lookup, and while not recommended for general use it can be useful 67 inside inner loops where the ultimate performance is required, and error checking is moved outside the loop. 68 69 The largest value you can pass to `unchecked_bernoulli_b2n<>` is `max_bernoulli_b2n<>::value`: passing values greater than 70 that will result in a buffer overrun error, so it's clearly important to place the error handling in your own code 71 when using this direct interface. 72 73 This function is `constexpr` only if the compiler supports C++14 constexpr functions. 74 75 The value of `boost::math::max_bernoulli_b2n<T>::value` varies by the type T, for types `float`/`double`/`long double` 76 it's the largest value which doesn't overflow the target type: for example, `boost::math::max_bernoulli_b2n<double>::value` is 129. 77 However, for multiprecision types, it's the largest value for which the result can be represented as the ratio of two 64-bit 78 integers, for example `boost::math::max_bernoulli_b2n<boost::multiprecision::cpp_dec_float_50>::value` is just 17. Of course 79 larger indexes can be passed to `bernoulli_b2n<T>(n)`, but then you lose fast table lookup (i.e. values may need to be calculated). 80 81 [bernoulli_example_4] 82 [bernoulli_output_4] 83 84 [h4 Multiple Bernoulli Numbers] 85 86 [h4 Synopsis] 87 88 `` 89 #include <boost/math/special_functions/bernoulli.hpp> 90 `` 91 92 namespace boost { namespace math { 93 94 // Multiple Bernoulli numbers (default policy). 95 template <class T, class OutputIterator> 96 OutputIterator bernoulli_b2n( 97 int start_index, 98 unsigned number_of_bernoullis_b2n, 99 OutputIterator out_it); 100 101 // Multiple Bernoulli numbers (user policy). 102 template <class T, class OutputIterator, class Policy> 103 OutputIterator bernoulli_b2n( 104 int start_index, 105 unsigned number_of_bernoullis_b2n, 106 OutputIterator out_it, 107 const Policy& pol); 108 }} // namespaces 109 110 [h4 Description] 111 112 Two versions of the Bernoulli number function are provided to compute multiple Bernoulli numbers 113 with one call (one with default policy and the other allowing a user-defined policy). 114 115 These return a series of Bernoulli numbers: 116 117 [:B[sub 2*start_index],B[sub 2*(start_index+1)],...,B[sub 2*(start_index+number_of_bernoullis_b2n-1)]] 118 119 [h4 Examples] 120 [bernoulli_example_2] 121 [bernoulli_output_2] 122 [bernoulli_example_3] 123 [bernoulli_output_3] 124 125 The source of this example is at [@../../example/bernoulli_example.cpp bernoulli_example.cpp] 126 127 [h4 Accuracy] 128 129 All the functions usually return values within one ULP (unit in the last place) for the floating-point type. 130 131 [h4 Implementation] 132 133 The implementation details are in [@../../include/boost/math/special_functions/detail/bernoulli_details.hpp bernoulli_details.hpp] 134 and [@../../include/boost/math/special_functions/detail/unchecked_bernoulli.hpp unchecked_bernoulli.hpp]. 135 136 For `i <= max_bernoulli_index<T>::value` this is implemented by simple table lookup from a statically initialized table; 137 for larger values of `i`, this is implemented by the Tangent Numbers algorithm as described in the paper: 138 Fast Computation of Bernoulli, Tangent and Secant Numbers, Richard P. Brent and David Harvey, 139 [@http://arxiv.org/pdf/1108.0286v3.pdf] (2011). 140 141 [@http://mathworld.wolfram.com/TangentNumber.html Tangent (or Zag) numbers] 142 (an even alternating permutation number) are defined 143 and their generating function is also given therein. 144 145 The relation of Tangent numbers with Bernoulli numbers ['B[sub i]] 146 is given by Brent and Harvey's equation 14: 147 148 [equation tangent_numbers] 149 150 Their relation with Bernoulli numbers ['B[sub i]] are defined by 151 152 if i > 0 and i is even then 153 [equation bernoulli_numbers] [br] 154 elseif i == 0 then ['B[sub i]] = 1 [br] 155 elseif i == 1 then ['B[sub i]] = -1/2 [br] 156 elseif i < 0 or i is odd then ['B[sub i]] = 0 157 158 Note that computed values are stored in a fixed-size table, access is thread safe via atomic operations (i.e. lock 159 free programming), this imparts a much lower overhead on access to cached values than might otherwise be expected - 160 typically for multiprecision types the cost of thread synchronisation is negligible, while for built in types 161 this code is not normally executed anyway. For very large arguments which cannot be reasonably computed or 162 stored in our cache, an asymptotic expansion [@http://www.luschny.de/math/primes/bernincl.html due to Luschny] is used: 163 164 [equation bernoulli_numbers2] 165 166 [endsect] [/section:bernoulli_numbers Bernoulli Numbers] 167 168 [section:tangent_numbers Tangent Numbers] 169 170 [@http://en.wikipedia.org/wiki/Tangent_numbers Tangent numbers], 171 also called a zag function. See also 172 [@http://mathworld.wolfram.com/TangentNumber.html Tangent number]. 173 174 From the number, An, of alternating permutations of the set {1, ..., n}, 175 the numbers A2n+1 with odd indices are called tangent numbers or zag numbers. 176 The first few values are 1, 2, 16, 272, 7936, 353792, 22368256, 1903757312 ... 177 (sequence [@http://oeis.org/A000182 A000182 in OEIS]). 178 They are called tangent numbers because they appear as 179 numerators in the Maclaurin series of tan x. 180 181 Tangent numbers are used in the computation of Bernoulli numbers, 182 but are also made available here. 183 184 [h4 Synopsis] 185 `` 186 #include <boost/math/special_functions/detail/bernoulli.hpp> 187 `` 188 189 template <class T> 190 T tangent_t2n(const int i); // Single tangent number (default policy). 191 192 template <class T, class Policy> 193 T tangent_t2n(const int i, const Policy &pol); // Single tangent number (user policy). 194 195 // Multiple tangent numbers (default policy). 196 template <class T, class OutputIterator> 197 OutputIterator tangent_t2n(const int start_index, 198 const unsigned number_of_tangent_t2n, 199 OutputIterator out_it); 200 201 // Multiple tangent numbers (user policy). 202 template <class T, class OutputIterator, class Policy> 203 OutputIterator tangent_t2n(const int start_index, 204 const unsigned number_of_tangent_t2n, 205 OutputIterator out_it, 206 const Policy& pol); 207 208 [h4 Examples] 209 210 [tangent_example_1] 211 212 The output is: 213 [tangent_output_1] 214 215 The source of this example is at [../../example/bernoulli_example.cpp bernoulli_example.cpp] 216 217 [endsect] [/section:tangent_numbers Tangent Numbers] 218 219 [/ 220 Copyright 2013, 2014 Nikhar Agrawal, Christopher Kormanyos, John Maddock, Paul A. Bristow. 221 Distributed under the Boost Software License, Version 1.0. 222 (See accompanying file LICENSE_1_0.txt or copy at 223 http://www.boost.org/LICENSE_1_0.txt). 224 ] 225