void hyp2F1(T& result, const T& a, const T& b, const T& c, const T& x)
{
// Compute the series representation of hyperg_2f1 taken from
// Abramowitz and Stegun 15.1.1.
// There are no checks on input range or parameter boundaries.
typedef typename boost::multiprecision::detail::canonical<boost::uint32_t, T>::type ui_type;
T x_pow_n_div_n_fact(x);
T pochham_a (a);
T pochham_b (b);
T pochham_c (c);
T ap (a);
T bp (b);
T cp (c);
eval_multiply(result, pochham_a, pochham_b);
eval_divide(result, pochham_c);
eval_multiply(result, x_pow_n_div_n_fact);
eval_add(result, ui_type(1));
T lim;
eval_ldexp(lim, result, 1 - boost::multiprecision::detail::digits2<number<T, et_on> >::value);
if(eval_get_sign(lim) < 0)
lim.negate();
ui_type n;
T term;
static const unsigned series_limit =
boost::multiprecision::detail::digits2<number<T, et_on> >::value < 100
? 100 : boost::multiprecision::detail::digits2<number<T, et_on> >::value;
// Series expansion of hyperg_2f1(a, b; c; x).
for(n = 2; n < series_limit; ++n)
{
eval_multiply(x_pow_n_div_n_fact, x);
eval_divide(x_pow_n_div_n_fact, n);
eval_increment(ap);
eval_multiply(pochham_a, ap);
eval_increment(bp);
eval_multiply(pochham_b, bp);
eval_increment(cp);
eval_multiply(pochham_c, cp);
eval_multiply(term, pochham_a, pochham_b);
eval_divide(term, pochham_c);
eval_multiply(term, x_pow_n_div_n_fact);
eval_add(result, term);
if(eval_get_sign(term) < 0)
term.negate();
if(lim.compare(term) >= 0)
break;
}
if(n > series_limit)
BOOST_THROW_EXCEPTION(std::runtime_error("H2F1 failed to converge."));
}
void hyp0F1(T& result, const T& b, const T& x)
{
typedef typename boost::multiprecision::detail::canonical<boost::int32_t, T>::type si_type;
typedef typename boost::multiprecision::detail::canonical<boost::uint32_t, T>::type ui_type;
typedef typename T::exponent_type exp_type;
typedef typename boost::multiprecision::detail::canonical<exp_type, T>::type canonical_exp_type;
typedef typename mpl::front<typename T::float_types>::type fp_type;
// Compute the series representation of Hypergeometric0F1 taken from
// http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric0F1/06/01/01/
// There are no checks on input range or parameter boundaries.
T x_pow_n_div_n_fact(x);
T pochham_b (b);
T bp (b);
eval_divide(result, x_pow_n_div_n_fact, pochham_b);
eval_add(result, ui_type(1));
si_type n;
T tol;
tol = ui_type(1);
eval_ldexp(tol, tol, 1 - boost::multiprecision::detail::digits2<number<T, et_on> >::value);
eval_multiply(tol, result);
if(eval_get_sign(tol) < 0)
tol.negate();
T term;
static const unsigned series_limit =
boost::multiprecision::detail::digits2<number<T, et_on> >::value < 100
? 100 : boost::multiprecision::detail::digits2<number<T, et_on> >::value;
// Series expansion of hyperg_0f1(; b; x).
for(n = 2; n < series_limit; ++n)
{
eval_multiply(x_pow_n_div_n_fact, x);
eval_divide(x_pow_n_div_n_fact, n);
eval_increment(bp);
eval_multiply(pochham_b, bp);
eval_divide(term, x_pow_n_div_n_fact, pochham_b);
eval_add(result, term);
bool neg_term = eval_get_sign(term) < 0;
if(neg_term)
term.negate();
if(term.compare(tol) <= 0)
break;
}
if(n >= series_limit)
BOOST_THROW_EXCEPTION(std::runtime_error("H0F1 Failed to Converge"));
}
void hyp1F0(T& H1F0, const T& a, const T& x)
{
// Compute the series representation of Hypergeometric1F0 taken from
// http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric1F0/06/01/01/
// and also see the corresponding section for the power function (i.e. x^a).
// There are no checks on input range or parameter boundaries.
typedef typename boost::multiprecision::detail::canonical<int, T>::type si_type;
typedef typename boost::multiprecision::detail::canonical<unsigned, T>::type ui_type;
typedef typename T::exponent_type exp_type;
typedef typename boost::multiprecision::detail::canonical<exp_type, T>::type canonical_exp_type;
typedef typename mpl::front<typename T::float_types>::type fp_type;
BOOST_ASSERT(&H1F0 != &x);
BOOST_ASSERT(&H1F0 != &a);
T x_pow_n_div_n_fact(x);
T pochham_a (a);
T ap (a);
eval_multiply(H1F0, pochham_a, x_pow_n_div_n_fact);
eval_add(H1F0, si_type(1));
T lim;
eval_ldexp(lim, H1F0, 1 - boost::multiprecision::detail::digits2<number<T, et_on> >::value);
if(eval_get_sign(lim) < 0)
lim.negate();
si_type n;
T term, part;
static const unsigned series_limit =
boost::multiprecision::detail::digits2<number<T, et_on> >::value < 100
? 100 : boost::multiprecision::detail::digits2<number<T, et_on> >::value;
// Series expansion of hyperg_1f0(a; ; x).
for(n = 2; n < series_limit; n++)
{
eval_multiply(x_pow_n_div_n_fact, x);
eval_divide(x_pow_n_div_n_fact, n);
eval_increment(ap);
eval_multiply(pochham_a, ap);
eval_multiply(term, pochham_a, x_pow_n_div_n_fact);
eval_add(H1F0, term);
if(eval_get_sign(term) < 0)
term.negate();
if(lim.compare(term) >= 0)
break;
}
if(n >= series_limit)
BOOST_THROW_EXCEPTION(std::runtime_error("H1F0 failed to converge"));
}