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Bernoulli Numbers

Bernoulli numbers are a sequence of rational numbers useful for the Taylor series expansion, Euler-Maclaurin formula, and the Riemann zeta function.

Bernoulli numbers are used in evaluation of some Boost.Math functions, including the tgamma, lgamma and polygamma functions.

Single Bernoulli number
Synopsis
#include <boost/math/special_functions/bernoulli.hpp>
namespace boost { namespace math {

template <class T>
T bernoulli_b2n(const int n);  // Single Bernoulli number (default policy).

template <class T, class Policy>
T bernoulli_b2n(const int n, const Policy &pol); // User policy for errors etc.

}} // namespaces
Description

Both return the (2 * n)th Bernoulli number B2n.

Note that since all odd numbered Bernoulli numbers are zero (apart from B1 which is -½) the interface will only return the even numbered Bernoulli numbers.

This function uses fast table lookup for low-indexed Bernoulli numbers, while larger values are calculated as needed and then cached. The caching mechanism requires a certain amount of thread safety code, so unchecked_bernoulli_b2n may provide a better interface for performance critical code.

The final Policy argument is optional and can be used to control the behaviour of the function: how it handles errors, what level of precision to use, etc.

Refer to Policies for more details.

Examples

A simple example computes the value of B4 where the return type is double, note that the argument to bernoulli_b2n is 2 not 4 since it computes B2N.

try
{ // It is always wise to use try'n'catch blocks around Boost.Math functions
  // so that any informative error messages can be displayed in the catch block.
std::cout
  << std::setprecision(std::numeric_limits<double>::digits10)
  << boost::math::bernoulli_b2n<double>(2) << std::endl;

So B4 == -1/30 == -0.0333333333333333

If we use Boost.Multiprecision and its 50 decimal digit floating-point type cpp_dec_float_50, we can calculate the value of much larger numbers like B200 and also obtain much higher precision.

std::cout
  << std::setprecision(std::numeric_limits<boost::multiprecision::cpp_dec_float_50>::digits10)
  << boost::math::bernoulli_b2n<boost::multiprecision::cpp_dec_float_50>(100) << std::endl;
-3.6470772645191354362138308865549944904868234686191e+215
Single (unchecked) Bernoulli number
Synopsis
#include <boost/math/special_functions/bernoulli.hpp>
template <>
struct max_bernoulli_b2n<T>;

template<class T>
inline T unchecked_bernoulli_b2n(unsigned n);

unchecked_bernoulli_b2n provides access to Bernoulli numbers without any checks for overflow or invalid parameters. It is implemented as a direct (and very fast) table lookup, and while not recommended for general use it can be useful inside inner loops where the ultimate performance is required, and error checking is moved outside the loop.

The largest value you can pass to unchecked_bernoulli_b2n<> is max_bernoulli_b2n<>::value: passing values greater than that will result in a buffer overrun error, so it's clearly important to place the error handling in your own code when using this direct interface.

The value of boost::math::max_bernoulli_b2n<T>::value varies by the type T, for types float/double/long double it's the largest value which doesn't overflow the target type: for example, boost::math::max_bernoulli_b2n<double>::value is 129. However, for multiprecision types, it's the largest value for which the result can be represented as the ratio of two 64-bit integers, for example boost::math::max_bernoulli_b2n<boost::multiprecision::cpp_dec_float_50>::value is just 17. Of course 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).

/*For example:
*/
   std::cout << "boost::math::max_bernoulli_b2n<float>::value = "  << boost::math::max_bernoulli_b2n<float>::value << std::endl;
   std::cout << "Maximum Bernoulli number using float is " << boost::math::bernoulli_b2n<float>( boost::math::max_bernoulli_b2n<float>::value) << std::endl;
   std::cout << "boost::math::max_bernoulli_b2n<double>::value = "  << boost::math::max_bernoulli_b2n<double>::value << std::endl;
   std::cout << "Maximum Bernoulli number using double is " << boost::math::bernoulli_b2n<double>( boost::math::max_bernoulli_b2n<double>::value) << std::endl;
boost::math::max_bernoulli_b2n<float>::value = 32
Maximum Bernoulli number using float is -2.0938e+038
boost::math::max_bernoulli_b2n<double>::value = 129
Maximum Bernoulli number using double is 1.33528e+306
Multiple Bernoulli Numbers
Synopsis
#include <boost/math/special_functions/bernoulli.hpp>
namespace boost { namespace math {

// Multiple Bernoulli numbers (default policy).
template <class T, class OutputIterator>
OutputIterator bernoulli_b2n(
  int start_index,
  unsigned number_of_bernoullis_b2n,
  OutputIterator out_it);

// Multiple Bernoulli numbers (user policy).
template <class T, class OutputIterator, class Policy>
OutputIterator bernoulli_b2n(
  int start_index,
  unsigned number_of_bernoullis_b2n,
  OutputIterator out_it,
  const Policy& pol);
}} // namespaces
Description

Two versions of the Bernoulli number function are provided to compute multiple Bernoulli numbers with one call (one with default policy and the other allowing a user-defined policy).

These return a series of Bernoulli numbers:

B2*start_index,B2*(start_index+1),...,B2*(start_index+number_of_bernoullis_b2n-1)

Examples

We can compute and save all the float-precision Bernoulli numbers from one call.

std::vector<float> bn; // Space for 32-bit `float` precision Bernoulli numbers.

// Start with Bernoulli number 0.
boost::math::bernoulli_b2n<float>(0, 32, std::back_inserter(bn)); // Fill vector with even Bernoulli numbers.

for(size_t i = 0; i < bn.size(); i++)
{ // Show vector of even Bernoulli numbers, showing all significant decimal digits.
    std::cout << std::setprecision(std::numeric_limits<float>::digits10)
        << i*2 << ' '
        << bn[i]
        << std::endl;
}
0 1
2 0.166667
4 -0.0333333
6 0.0238095
8 -0.0333333
10 0.0757576
12 -0.253114
14 1.16667
16 -7.09216
18 54.9712
20 -529.124
22 6192.12
24 -86580.3
26 1.42552e+006
28 -2.72982e+007
30 6.01581e+008
32 -1.51163e+010
34 4.29615e+011
36 -1.37117e+013
38 4.88332e+014
40 -1.92966e+016
42 8.41693e+017
44 -4.03381e+019
46 2.11507e+021
48 -1.20866e+023
50 7.50087e+024
52 -5.03878e+026
54 3.65288e+028
56 -2.84988e+030
58 2.38654e+032
60 -2.14e+034
62 2.0501e+036

Of course, for any floating-point type, there is a maximum Bernoulli number that can be computed before it overflows the exponent. By default policy, if we try to compute too high a Bernoulli number, an exception will be thrown.

try
{
  std::cout
  << std::setprecision(std::numeric_limits<float>::digits10)
  << "Bernoulli number " << 33 * 2 <<std::endl;

  std::cout << boost::math::bernoulli_b2n<float>(33) << std::endl;
}
catch (std::exception ex)
{
  std::cout << "Thrown Exception caught: " << ex.what() << std::endl;
}

and we will get a helpful error message (provided try'n'catch blocks are used).

Bernoulli number 66
Thrown Exception caught: Error in function boost::math::bernoulli_b2n<float>(n):
Overflow evaluating function at 33

The source of this example is at bernoulli_example.cpp

Accuracy

All the functions usually return values within one ULP (unit in the last place) for the floating-point type.

Implementation

The implementation details are in bernoulli_details.hpp and unchecked_bernoulli.hpp.

For i <= max_bernoulli_index<T>::value this is implemented by simple table lookup from a statically initialized table; for larger values of i, this is implemented by the Tangent Numbers algorithm as described in the paper: Fast Computation of Bernoulli, Tangent and Secant Numbers, Richard P. Brent and David Harvey, http://arxiv.org/pdf/1108.0286v3.pdf (2011).

Tangent (or Zag) numbers (an even alternating permutation number) are defined and their generating function is also given therein.

The relation of Tangent numbers with Bernoulli numbers Bi is given by Brent and Harvey's equation 14:

  

Their relation with Bernoulli numbers Bi are defined by

if i > 0 and i is even then   
elseif i == 0 then Bi = 1
elseif i == 1 then Bi = -1/2
elseif i < 0 or i is odd then Bi = 0

Note that computed values are stored in a fixed-size table, access is thread safe via atomic operations (i.e. lock free programming), this imparts a much lower overhead on access to cached values than might otherwise be expected - typically for multiprecision types the cost of thread synchronisation is negligible, while for built in types this code is not normally executed anyway. For very large arguments which cannot be reasonably computed or stored in our cache, an asymptotic expansion due to Luschny is used:


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