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[section:factorials Factorials and Binomial Coefficients]
[section:sf_factorial Factorial]
[h4 Synopsis]
``
#include <boost/math/special_functions/factorials.hpp>
``
namespace boost{ namespace math{
template <class T>
T factorial(unsigned i);
template <class T, class ``__Policy``>
T factorial(unsigned i, const ``__Policy``&);
template <class T>
T unchecked_factorial(unsigned i);
template <class T>
struct max_factorial;
}} // namespaces
[h4 Description]
template <class T>
T factorial(unsigned i);
template <class T, class ``__Policy``>
T factorial(unsigned i, const ``__Policy``&);
Returns [^i!].
[optional_policy]
For [^i <= max_factorial<T>::value] this is implemented by table lookup,
for larger values of [^i], this function is implemented in terms of __tgamma.
If [^i] is so large that the result can not be represented in type T, then
calls __overflow_error.
template <class T>
T unchecked_factorial(unsigned i);
Returns [^i!].
Internally this function performs table lookup of the result. Further it performs
no range checking on the value of i: it is up to the caller to ensure
that [^i <= max_factorial<T>::value]. This function is intended to be used
inside inner loops that require fast table lookup of factorials, but requires
care to ensure that argument [^i] never grows too large.
template <class T>
struct max_factorial
{
static const unsigned value = X;
};
This traits class defines the largest value that can be passed to
[^unchecked_factorial]. The member `value` can be used where integral
constant expressions are required: for example to define the size of
further tables that depend on the factorials.
[h4 Accuracy]
For arguments smaller than `max_factorial<T>::value`
the result should be
correctly rounded. For larger arguments the accuracy will be the same
as for __tgamma.
[h4 Testing]
Basic sanity checks and spot values to verify the data tables:
the main tests for the __tgamma function handle those cases already.
[h4 Implementation]
The factorial function is table driven for small arguments, and is
implemented in terms of __tgamma for larger arguments.
[endsect]
[section:sf_double_factorial Double Factorial]
``
#include <boost/math/special_functions/factorials.hpp>
``
namespace boost{ namespace math{
template <class T>
T double_factorial(unsigned i);
template <class T, class ``__Policy``>
T double_factorial(unsigned i, const ``__Policy``&);
}} // namespaces
Returns [^i!!].
[optional_policy]
May return the result of __overflow_error if the result is too large
to represent in type T. The implementation is designed to be optimised
for small /i/ where table lookup of i! is possible.
[h4 Accuracy]
The implementation uses a trivial adaptation of
the factorial function, so error rates should be no more than a couple
of epsilon higher.
[h4 Testing]
The spot tests for the double factorial use data
generated by functions.wolfram.com.
[h4 Implementation]
The double factorial is implemented in terms of the factorial and gamma
functions using the relations:
(2n)!! = 2[super n ] * n!
(2n+1)!! = (2n+1)! / (2[super n ] n!)
and
(2n-1)!! = [Gamma]((2n+1)/2) * 2[super n ] / sqrt(pi)
[endsect]
[section:sf_rising_factorial Rising Factorial]
``
#include <boost/math/special_functions/factorials.hpp>
``
namespace boost{ namespace math{
template <class T>
``__sf_result`` rising_factorial(T x, int i);
template <class T, class ``__Policy``>
``__sf_result`` rising_factorial(T x, int i, const ``__Policy``&);
}} // namespaces
Returns the rising factorial of /x/ and /i/:
rising_factorial(x, i) = [Gamma](x + i) / [Gamma](x);
or
rising_factorial(x, i) = x(x+1)(x+2)(x+3)...(x+i)
Note that both /x/ and /i/ can be negative as well as positive.
[optional_policy]
May return the result of __overflow_error if the result is too large
to represent in type T.
The return type of these functions is computed using the __arg_pomotion_rules:
the type of the result is `double` if T is an integer type, otherwise the type
of the result is T.
[h4 Accuracy]
The accuracy will be the same as
the __tgamma_delta_ratio function.
[h4 Testing]
The spot tests for the rising factorials use data generated
by functions.wolfram.com.
[h4 Implementation]
Rising and falling factorials are implemented as ratios of gamma functions
using __tgamma_delta_ratio. Optimisations for
small integer arguments are handled internally by that function.
[endsect]
[section:sf_falling_factorial Falling Factorial]
``
#include <boost/math/special_functions/factorials.hpp>
``
namespace boost{ namespace math{
template <class T>
``__sf_result`` falling_factorial(T x, unsigned i);
template <class T, class ``__Policy``>
``__sf_result`` falling_factorial(T x, unsigned i, const ``__Policy``&);
}} // namespaces
Returns the falling factorial of /x/ and /i/:
falling_factorial(x, i) = x(x-1)(x-2)(x-3)...(x-i+1)
Note that this function is only defined for positive /i/, hence the
`unsigned` second argument. Argument /x/ can be either positive or
negative however.
[optional_policy]
May return the result of __overflow_error if the result is too large
to represent in type T.
The return type of these functions is computed using the __arg_pomotion_rules:
the type of the result is `double` if T is an integer type, otherwise the type
of the result is T.
[h4 Accuracy]
The accuracy will be the same as
the __tgamma_delta_ratio function.
[h4 Testing]
The spot tests for the falling factorials use data generated by
functions.wolfram.com.
[h4 Implementation]
Rising and falling factorials are implemented as ratios of gamma functions
using __tgamma_delta_ratio. Optimisations for
small integer arguments are handled internally by that function.
[endsect]
[section:sf_binomial Binomial Coefficients]
``
#include <boost/math/special_functions/binomial.hpp>
``
namespace boost{ namespace math{
template <class T>
T binomial_coefficient(unsigned n, unsigned k);
template <class T, class ``__Policy``>
T binomial_coefficient(unsigned n, unsigned k, const ``__Policy``&);
}} // namespaces
Returns the binomial coefficient: [sub n]C[sub k].
Requires k <= n.
[optional_policy]
May return the result of __overflow_error if the result is too large
to represent in type T.
[h4 Accuracy]
The accuracy will be the same as for the
factorials for small arguments (i.e. no more than one or two epsilon),
and the __beta function for larger arguments.
[h4 Testing]
The spot tests for the binomial coefficients use data
generated by functions.wolfram.com.
[h4 Implementation]
Binomial coefficients are calculated using table lookup of factorials
where possible using:
[sub n]C[sub k] = n! / (k!(n-k)!)
Otherwise it is implemented in terms of the beta function using the relations:
[sub n]C[sub k] = 1 / (k * __beta(k, n-k+1))
and
[sub n]C[sub k] = 1 / ((n-k) * __beta(k+1, n-k))
[endsect]
[endsect][/section:factorials Factorials]
[/
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).
]
|
