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224 lines
8.3 KiB
Plaintext
224 lines
8.3 KiB
Plaintext
[section:bernoulli_numbers Bernoulli Numbers]
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[@https://en.wikipedia.org/wiki/Bernoulli_number Bernoulli numbers]
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are a sequence of rational numbers useful for the Taylor series expansion,
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Euler-Maclaurin formula, and the Riemann zeta function.
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Bernoulli numbers are used in evaluation of some Boost.Math functions,
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including the __tgamma, __lgamma and polygamma functions.
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[h4 Single Bernoulli number]
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[h4 Synopsis]
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``
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#include <boost/math/special_functions/bernoulli.hpp>
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``
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namespace boost { namespace math {
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template <class T>
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T bernoulli_b2n(const int n); // Single Bernoulli number (default policy).
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template <class T, class Policy>
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T bernoulli_b2n(const int n, const Policy &pol); // User policy for errors etc.
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}} // namespaces
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[h4 Description]
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Both return the (2 * n)[super th] Bernoulli number B[sub 2n].
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Note that since all odd numbered Bernoulli numbers are zero (apart from B[sub 1] which is [plusminus][frac12])
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the interface will only return the even numbered Bernoulli numbers.
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This function uses fast table lookup for low-indexed Bernoulli numbers, while larger values are calculated
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as needed and then cached. The caching mechanism requires a certain amount of thread safety code, so
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`unchecked_bernoulli_b2n` may provide a better interface for performance critical code.
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The final __Policy argument is optional and can be used to control the behaviour of the function:
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how it handles errors, what level of precision to use, etc.
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Refer to __policy_section for more details.
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[h4 Examples]
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[import ../../example/bernoulli_example.cpp]
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[bernoulli_example_1]
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[bernoulli_output_1]
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[h4 Single (unchecked) Bernoulli number]
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[h4 Synopsis]
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``
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#include <boost/math/special_functions/bernoulli.hpp>
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``
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template <>
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struct max_bernoulli_b2n<T>;
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template<class T>
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inline T unchecked_bernoulli_b2n(unsigned n);
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`unchecked_bernoulli_b2n` provides access to Bernoulli numbers [*without any checks for overflow or invalid parameters].
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It is implemented as a direct (and very fast) table lookup, and while not recommended for general use it can be useful
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inside inner loops where the ultimate performance is required, and error checking is moved outside the loop.
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The largest value you can pass to `unchecked_bernoulli_b2n<>` is `max_bernoulli_b2n<>::value`: passing values greater than
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that will result in a buffer overrun error, so it's clearly important to place the error handling in your own code
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when using this direct interface.
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The value of `boost::math::max_bernoulli_b2n<T>::value` varies by the type T, for types `float`/`double`/`long double`
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it's the largest value which doesn't overflow the target type: for example, `boost::math::max_bernoulli_b2n<double>::value` is 129.
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However, for multiprecision types, it's the largest value for which the result can be represented as the ratio of two 64-bit
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integers, for example `boost::math::max_bernoulli_b2n<boost::multiprecision::cpp_dec_float_50>::value` is just 17. Of course
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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).
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[bernoulli_example_4]
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[bernoulli_output_4]
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[h4 Multiple Bernoulli Numbers]
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[h4 Synopsis]
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``
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#include <boost/math/special_functions/bernoulli.hpp>
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``
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namespace boost { namespace math {
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// Multiple Bernoulli numbers (default policy).
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template <class T, class OutputIterator>
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OutputIterator bernoulli_b2n(
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int start_index,
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unsigned number_of_bernoullis_b2n,
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OutputIterator out_it);
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// Multiple Bernoulli numbers (user policy).
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template <class T, class OutputIterator, class Policy>
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OutputIterator bernoulli_b2n(
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int start_index,
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unsigned number_of_bernoullis_b2n,
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OutputIterator out_it,
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const Policy& pol);
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}} // namespaces
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[h4 Description]
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Two versions of the Bernoulli number function are provided to compute multiple Bernoulli numbers
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with one call (one with default policy and the other allowing a user-defined policy).
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These return a series of Bernoulli numbers:
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[:B[sub 2*start_index],B[sub 2*(start_index+1)],...,B[sub 2*(start_index+number_of_bernoullis_b2n-1)]]
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[h4 Examples]
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[bernoulli_example_2]
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[bernoulli_output_2]
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[bernoulli_example_3]
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[bernoulli_output_3]
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The source of this example is at [@../../example/bernoulli_example.cpp bernoulli_example.cpp]
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[h4 Accuracy]
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All the functions usually return values within one ULP (unit in the last place) for the floating-point type.
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[h4 Implementation]
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The implementation details are in [@../../include/boost/math/special_functions/detail/bernoulli_details.hpp bernoulli_details.hpp]
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and [@../../include/boost/math/special_functions/detail/unchecked_bernoulli.hpp unchecked_bernoulli.hpp].
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For `i <= max_bernoulli_index<T>::value` this is implemented by simple table lookup from a statically initialized table;
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for larger values of `i`, this is implemented by the Tangent Numbers algorithm as described in the paper:
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Fast Computation of Bernoulli, Tangent and Secant Numbers, Richard P. Brent and David Harvey,
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[@http://arxiv.org/pdf/1108.0286v3.pdf] (2011).
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[@http://mathworld.wolfram.com/TangentNumber.html Tangent (or Zag) numbers]
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(an even alternating permutation number) are defined
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and their generating function is also given therein.
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The relation of Tangent numbers with Bernoulli numbers ['B[sub i]]
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is given by Brent and Harvey's equation 14:
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__spaces[equation tangent_numbers]
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Their relation with Bernoulli numbers ['B[sub i]] are defined by
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if i > 0 and i is even then
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__spaces[equation bernoulli_numbers] [br]
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elseif i == 0 then ['B[sub i]] = 1 [br]
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elseif i == 1 then ['B[sub i]] = -1/2 [br]
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elseif i < 0 or i is odd then ['B[sub i]] = 0
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Note that computed values are stored in a fixed-size table, access is thread safe via atomic operations (i.e. lock
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free programming), this imparts a much lower overhead on access to cached values than might otherwise be expected -
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typically for multiprecision types the cost of thread synchronisation is negligible, while for built in types
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this code is not normally executed anyway. For very large arguments which cannot be reasonably computed or
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stored in our cache, an asymptotic expansion [@http://www.luschny.de/math/primes/bernincl.html due to Luschny] is used:
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[equation bernoulli_numbers2]
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[endsect] [/section:bernoulli_numbers Bernoulli Numbers]
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[section:tangent_numbers Tangent Numbers]
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[@http://en.wikipedia.org/wiki/Tangent_numbers Tangent numbers],
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also called a zag function. See also
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[@http://mathworld.wolfram.com/TangentNumber.html Tangent number].
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From the number, An, of alternating permutations of the set {1, ..., n},
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the numbers A2n+1 with odd indices are called tangent numbers or zag numbers.
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The first few values are 1, 2, 16, 272, 7936, 353792, 22368256, 1903757312 ...
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(sequence [@http://oeis.org/A000182 A000182 in OEIS]).
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They are called tangent numbers because they appear as
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numerators in the Maclaurin series of tan x.
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Tangent numbers are used in the computation of Bernoulli numbers,
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but are also made available here.
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[h4 Synopsis]
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``
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#include <boost/math/special_functions/detail/bernoulli.hpp>
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``
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template <class T>
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T tangent_t2n(const int i); // Single tangent number (default policy).
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template <class T, class Policy>
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T tangent_t2n(const int i, const Policy &pol); // Single tangent number (user policy).
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// Multiple tangent numbers (default policy).
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template <class T, class OutputIterator>
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OutputIterator tangent_t2n(const int start_index,
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const unsigned number_of_tangent_t2n,
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OutputIterator out_it);
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// Multiple tangent numbers (user policy).
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template <class T, class OutputIterator, class Policy>
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OutputIterator tangent_t2n(const int start_index,
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const unsigned number_of_tangent_t2n,
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OutputIterator out_it,
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const Policy& pol);
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[h4 Examples]
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[tangent_example_1]
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The output is:
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[tangent_output_1]
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The source of this example is at [../../example/bernoulli_example.cpp bernoulli_example.cpp]
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[endsect] [/section:tangent_numbers Tangent Numbers]
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[/
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Copyright 2013, 2014 Nikhar Agrawal, Christopher Kormanyos, John Maddock, Paul A. Bristow.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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