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#### Factorial

##### 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

##### Description

| Important
The functions described below are templates where the template argument T CANNOT be deduced from the arguments passed to the function. Therefore if you write something like: `boost::math::factorial(2);` You will get a (perhaps perplexing) compiler error, ususally indicating that there is no such function to be found. Instead you need to specify the return type explicity and write: `boost::math::factorial<double>(2);` So that the return type is known. Furthermore, the template argument must be a real-valued type such as `float` or `double` and not an integer type - that would overflow far too easily for quite small values of parameter `i`! The source code `static_assert` and comment just after the will be:
    BOOST_STATIC_ASSERT(!boost::is_integral<T>::value);
    // factorial<unsigned int>(n) is not implemented
    // because it would overflow integral type T for too small n
    // to be useful. Use instead a floating-point type,
    // and convert to an unsigned type if essential, for example:
    // unsigned int nfac = static_cast<unsigned int>(factorial<double>(n));
    // See factorial documentation for more detail.

    template <class T>
    T factorial(unsigned i);

    template <class T, class Policy>
    T factorial(unsigned i, const Policy&);

Returns `i!`.

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 the policy documentation for more details.

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.

##### 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.

##### Testing

Basic sanity checks and spot values to verify the data tables: the main tests for the tgamma function handle those cases already.

##### Implementation

The factorial function is table driven for small arguments, and is implemented in terms of tgamma for larger arguments.