// (C) Copyright John Maddock 2006. | |
// Use, modification and distribution are subject to 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) | |
#ifndef BOOST_MATH_SPECIAL_FUNCTIONS_IGAMMA_INVERSE_HPP | |
#define BOOST_MATH_SPECIAL_FUNCTIONS_IGAMMA_INVERSE_HPP | |
#ifdef _MSC_VER | |
#pragma once | |
#endif | |
#include <boost/math/tools/tuple.hpp> | |
#include <boost/math/special_functions/gamma.hpp> | |
#include <boost/math/special_functions/sign.hpp> | |
#include <boost/math/tools/roots.hpp> | |
#include <boost/math/policies/error_handling.hpp> | |
namespace boost{ namespace math{ | |
namespace detail{ | |
template <class T> | |
T find_inverse_s(T p, T q) | |
{ | |
// | |
// Computation of the Incomplete Gamma Function Ratios and their Inverse | |
// ARMIDO R. DIDONATO and ALFRED H. MORRIS, JR. | |
// ACM Transactions on Mathematical Software, Vol. 12, No. 4, | |
// December 1986, Pages 377-393. | |
// | |
// See equation 32. | |
// | |
BOOST_MATH_STD_USING | |
T t; | |
if(p < 0.5) | |
{ | |
t = sqrt(-2 * log(p)); | |
} | |
else | |
{ | |
t = sqrt(-2 * log(q)); | |
} | |
static const double a[4] = { 3.31125922108741, 11.6616720288968, 4.28342155967104, 0.213623493715853 }; | |
static const double b[5] = { 1, 6.61053765625462, 6.40691597760039, 1.27364489782223, 0.3611708101884203e-1 }; | |
T s = t - tools::evaluate_polynomial(a, t) / tools::evaluate_polynomial(b, t); | |
if(p < 0.5) | |
s = -s; | |
return s; | |
} | |
template <class T> | |
T didonato_SN(T a, T x, unsigned N, T tolerance = 0) | |
{ | |
// | |
// Computation of the Incomplete Gamma Function Ratios and their Inverse | |
// ARMIDO R. DIDONATO and ALFRED H. MORRIS, JR. | |
// ACM Transactions on Mathematical Software, Vol. 12, No. 4, | |
// December 1986, Pages 377-393. | |
// | |
// See equation 34. | |
// | |
T sum = 1; | |
if(N >= 1) | |
{ | |
T partial = x / (a + 1); | |
sum += partial; | |
for(unsigned i = 2; i <= N; ++i) | |
{ | |
partial *= x / (a + i); | |
sum += partial; | |
if(partial < tolerance) | |
break; | |
} | |
} | |
return sum; | |
} | |
template <class T, class Policy> | |
inline T didonato_FN(T p, T a, T x, unsigned N, T tolerance, const Policy& pol) | |
{ | |
// | |
// Computation of the Incomplete Gamma Function Ratios and their Inverse | |
// ARMIDO R. DIDONATO and ALFRED H. MORRIS, JR. | |
// ACM Transactions on Mathematical Software, Vol. 12, No. 4, | |
// December 1986, Pages 377-393. | |
// | |
// See equation 34. | |
// | |
BOOST_MATH_STD_USING | |
T u = log(p) + boost::math::lgamma(a + 1, pol); | |
return exp((u + x - log(didonato_SN(a, x, N, tolerance))) / a); | |
} | |
template <class T, class Policy> | |
T find_inverse_gamma(T a, T p, T q, const Policy& pol, bool* p_has_10_digits) | |
{ | |
// | |
// In order to understand what's going on here, you will | |
// need to refer to: | |
// | |
// Computation of the Incomplete Gamma Function Ratios and their Inverse | |
// ARMIDO R. DIDONATO and ALFRED H. MORRIS, JR. | |
// ACM Transactions on Mathematical Software, Vol. 12, No. 4, | |
// December 1986, Pages 377-393. | |
// | |
BOOST_MATH_STD_USING | |
T result; | |
*p_has_10_digits = false; | |
if(a == 1) | |
{ | |
result = -log(q); | |
BOOST_MATH_INSTRUMENT_VARIABLE(result); | |
} | |
else if(a < 1) | |
{ | |
T g = boost::math::tgamma(a, pol); | |
T b = q * g; | |
BOOST_MATH_INSTRUMENT_VARIABLE(g); | |
BOOST_MATH_INSTRUMENT_VARIABLE(b); | |
if((b > 0.6) || ((b >= 0.45) && (a >= 0.3))) | |
{ | |
// DiDonato & Morris Eq 21: | |
// | |
// There is a slight variation from DiDonato and Morris here: | |
// the first form given here is unstable when p is close to 1, | |
// making it impossible to compute the inverse of Q(a,x) for small | |
// q. Fortunately the second form works perfectly well in this case. | |
// | |
T u; | |
if((b * q > 1e-8) && (q > 1e-5)) | |
{ | |
u = pow(p * g * a, 1 / a); | |
BOOST_MATH_INSTRUMENT_VARIABLE(u); | |
} | |
else | |
{ | |
u = exp((-q / a) - constants::euler<T>()); | |
BOOST_MATH_INSTRUMENT_VARIABLE(u); | |
} | |
result = u / (1 - (u / (a + 1))); | |
BOOST_MATH_INSTRUMENT_VARIABLE(result); | |
} | |
else if((a < 0.3) && (b >= 0.35)) | |
{ | |
// DiDonato & Morris Eq 22: | |
T t = exp(-constants::euler<T>() - b); | |
T u = t * exp(t); | |
result = t * exp(u); | |
BOOST_MATH_INSTRUMENT_VARIABLE(result); | |
} | |
else if((b > 0.15) || (a >= 0.3)) | |
{ | |
// DiDonato & Morris Eq 23: | |
T y = -log(b); | |
T u = y - (1 - a) * log(y); | |
result = y - (1 - a) * log(u) - log(1 + (1 - a) / (1 + u)); | |
BOOST_MATH_INSTRUMENT_VARIABLE(result); | |
} | |
else if (b > 0.1) | |
{ | |
// DiDonato & Morris Eq 24: | |
T y = -log(b); | |
T u = y - (1 - a) * log(y); | |
result = y - (1 - a) * log(u) - log((u * u + 2 * (3 - a) * u + (2 - a) * (3 - a)) / (u * u + (5 - a) * u + 2)); | |
BOOST_MATH_INSTRUMENT_VARIABLE(result); | |
} | |
else | |
{ | |
// DiDonato & Morris Eq 25: | |
T y = -log(b); | |
T c1 = (a - 1) * log(y); | |
T c1_2 = c1 * c1; | |
T c1_3 = c1_2 * c1; | |
T c1_4 = c1_2 * c1_2; | |
T a_2 = a * a; | |
T a_3 = a_2 * a; | |
T c2 = (a - 1) * (1 + c1); | |
T c3 = (a - 1) * (-(c1_2 / 2) + (a - 2) * c1 + (3 * a - 5) / 2); | |
T c4 = (a - 1) * ((c1_3 / 3) - (3 * a - 5) * c1_2 / 2 + (a_2 - 6 * a + 7) * c1 + (11 * a_2 - 46 * a + 47) / 6); | |
T c5 = (a - 1) * (-(c1_4 / 4) | |
+ (11 * a - 17) * c1_3 / 6 | |
+ (-3 * a_2 + 13 * a -13) * c1_2 | |
+ (2 * a_3 - 25 * a_2 + 72 * a - 61) * c1 / 2 | |
+ (25 * a_3 - 195 * a_2 + 477 * a - 379) / 12); | |
T y_2 = y * y; | |
T y_3 = y_2 * y; | |
T y_4 = y_2 * y_2; | |
result = y + c1 + (c2 / y) + (c3 / y_2) + (c4 / y_3) + (c5 / y_4); | |
BOOST_MATH_INSTRUMENT_VARIABLE(result); | |
if(b < 1e-28f) | |
*p_has_10_digits = true; | |
} | |
} | |
else | |
{ | |
// DiDonato and Morris Eq 31: | |
T s = find_inverse_s(p, q); | |
BOOST_MATH_INSTRUMENT_VARIABLE(s); | |
T s_2 = s * s; | |
T s_3 = s_2 * s; | |
T s_4 = s_2 * s_2; | |
T s_5 = s_4 * s; | |
T ra = sqrt(a); | |
BOOST_MATH_INSTRUMENT_VARIABLE(ra); | |
T w = a + s * ra + (s * s -1) / 3; | |
w += (s_3 - 7 * s) / (36 * ra); | |
w -= (3 * s_4 + 7 * s_2 - 16) / (810 * a); | |
w += (9 * s_5 + 256 * s_3 - 433 * s) / (38880 * a * ra); | |
BOOST_MATH_INSTRUMENT_VARIABLE(w); | |
if((a >= 500) && (fabs(1 - w / a) < 1e-6)) | |
{ | |
result = w; | |
*p_has_10_digits = true; | |
BOOST_MATH_INSTRUMENT_VARIABLE(result); | |
} | |
else if (p > 0.5) | |
{ | |
if(w < 3 * a) | |
{ | |
result = w; | |
BOOST_MATH_INSTRUMENT_VARIABLE(result); | |
} | |
else | |
{ | |
T D = (std::max)(T(2), T(a * (a - 1))); | |
T lg = boost::math::lgamma(a, pol); | |
T lb = log(q) + lg; | |
if(lb < -D * 2.3) | |
{ | |
// DiDonato and Morris Eq 25: | |
T y = -lb; | |
T c1 = (a - 1) * log(y); | |
T c1_2 = c1 * c1; | |
T c1_3 = c1_2 * c1; | |
T c1_4 = c1_2 * c1_2; | |
T a_2 = a * a; | |
T a_3 = a_2 * a; | |
T c2 = (a - 1) * (1 + c1); | |
T c3 = (a - 1) * (-(c1_2 / 2) + (a - 2) * c1 + (3 * a - 5) / 2); | |
T c4 = (a - 1) * ((c1_3 / 3) - (3 * a - 5) * c1_2 / 2 + (a_2 - 6 * a + 7) * c1 + (11 * a_2 - 46 * a + 47) / 6); | |
T c5 = (a - 1) * (-(c1_4 / 4) | |
+ (11 * a - 17) * c1_3 / 6 | |
+ (-3 * a_2 + 13 * a -13) * c1_2 | |
+ (2 * a_3 - 25 * a_2 + 72 * a - 61) * c1 / 2 | |
+ (25 * a_3 - 195 * a_2 + 477 * a - 379) / 12); | |
T y_2 = y * y; | |
T y_3 = y_2 * y; | |
T y_4 = y_2 * y_2; | |
result = y + c1 + (c2 / y) + (c3 / y_2) + (c4 / y_3) + (c5 / y_4); | |
BOOST_MATH_INSTRUMENT_VARIABLE(result); | |
} | |
else | |
{ | |
// DiDonato and Morris Eq 33: | |
T u = -lb + (a - 1) * log(w) - log(1 + (1 - a) / (1 + w)); | |
result = -lb + (a - 1) * log(u) - log(1 + (1 - a) / (1 + u)); | |
BOOST_MATH_INSTRUMENT_VARIABLE(result); | |
} | |
} | |
} | |
else | |
{ | |
T z = w; | |
T ap1 = a + 1; | |
T ap2 = a + 2; | |
if(w < 0.15f * ap1) | |
{ | |
// DiDonato and Morris Eq 35: | |
T v = log(p) + boost::math::lgamma(ap1, pol); | |
T s = 1; | |
z = exp((v + w) / a); | |
s = boost::math::log1p(z / ap1 * (1 + z / ap2)); | |
z = exp((v + z - s) / a); | |
s = boost::math::log1p(z / ap1 * (1 + z / ap2)); | |
z = exp((v + z - s) / a); | |
s = boost::math::log1p(z / ap1 * (1 + z / ap2 * (1 + z / (a + 3)))); | |
z = exp((v + z - s) / a); | |
BOOST_MATH_INSTRUMENT_VARIABLE(z); | |
} | |
if((z <= 0.01 * ap1) || (z > 0.7 * ap1)) | |
{ | |
result = z; | |
if(z <= 0.002 * ap1) | |
*p_has_10_digits = true; | |
BOOST_MATH_INSTRUMENT_VARIABLE(result); | |
} | |
else | |
{ | |
// DiDonato and Morris Eq 36: | |
T ls = log(didonato_SN(a, z, 100, T(1e-4))); | |
T v = log(p) + boost::math::lgamma(ap1, pol); | |
z = exp((v + z - ls) / a); | |
result = z * (1 - (a * log(z) - z - v + ls) / (a - z)); | |
BOOST_MATH_INSTRUMENT_VARIABLE(result); | |
} | |
} | |
} | |
return result; | |
} | |
template <class T, class Policy> | |
struct gamma_p_inverse_func | |
{ | |
gamma_p_inverse_func(T a_, T p_, bool inv) : a(a_), p(p_), invert(inv) | |
{ | |
// | |
// If p is too near 1 then P(x) - p suffers from cancellation | |
// errors causing our root-finding algorithms to "thrash", better | |
// to invert in this case and calculate Q(x) - (1-p) instead. | |
// | |
// Of course if p is *very* close to 1, then the answer we get will | |
// be inaccurate anyway (because there's not enough information in p) | |
// but at least we will converge on the (inaccurate) answer quickly. | |
// | |
if(p > 0.9) | |
{ | |
p = 1 - p; | |
invert = !invert; | |
} | |
} | |
boost::math::tuple<T, T, T> operator()(const T& x)const | |
{ | |
BOOST_FPU_EXCEPTION_GUARD | |
// | |
// Calculate P(x) - p and the first two derivates, or if the invert | |
// flag is set, then Q(x) - q and it's derivatives. | |
// | |
typedef typename policies::evaluation<T, Policy>::type value_type; | |
typedef typename lanczos::lanczos<T, Policy>::type evaluation_type; | |
typedef typename policies::normalise< | |
Policy, | |
policies::promote_float<false>, | |
policies::promote_double<false>, | |
policies::discrete_quantile<>, | |
policies::assert_undefined<> >::type forwarding_policy; | |
BOOST_MATH_STD_USING // For ADL of std functions. | |
T f, f1; | |
value_type ft; | |
f = static_cast<T>(boost::math::detail::gamma_incomplete_imp( | |
static_cast<value_type>(a), | |
static_cast<value_type>(x), | |
true, invert, | |
forwarding_policy(), &ft)); | |
f1 = static_cast<T>(ft); | |
T f2; | |
T div = (a - x - 1) / x; | |
f2 = f1; | |
if((fabs(div) > 1) && (tools::max_value<T>() / fabs(div) < f2)) | |
{ | |
// overflow: | |
f2 = -tools::max_value<T>() / 2; | |
} | |
else | |
{ | |
f2 *= div; | |
} | |
if(invert) | |
{ | |
f1 = -f1; | |
f2 = -f2; | |
} | |
return boost::math::make_tuple(f - p, f1, f2); | |
} | |
private: | |
T a, p; | |
bool invert; | |
}; | |
template <class T, class Policy> | |
T gamma_p_inv_imp(T a, T p, const Policy& pol) | |
{ | |
BOOST_MATH_STD_USING // ADL of std functions. | |
static const char* function = "boost::math::gamma_p_inv<%1%>(%1%, %1%)"; | |
BOOST_MATH_INSTRUMENT_VARIABLE(a); | |
BOOST_MATH_INSTRUMENT_VARIABLE(p); | |
if(a <= 0) | |
policies::raise_domain_error<T>(function, "Argument a in the incomplete gamma function inverse must be >= 0 (got a=%1%).", a, pol); | |
if((p < 0) || (p > 1)) | |
policies::raise_domain_error<T>(function, "Probabilty must be in the range [0,1] in the incomplete gamma function inverse (got p=%1%).", p, pol); | |
if(p == 1) | |
return tools::max_value<T>(); | |
if(p == 0) | |
return 0; | |
bool has_10_digits; | |
T guess = detail::find_inverse_gamma<T>(a, p, 1 - p, pol, &has_10_digits); | |
if((policies::digits<T, Policy>() <= 36) && has_10_digits) | |
return guess; | |
T lower = tools::min_value<T>(); | |
if(guess <= lower) | |
guess = tools::min_value<T>(); | |
BOOST_MATH_INSTRUMENT_VARIABLE(guess); | |
// | |
// Work out how many digits to converge to, normally this is | |
// 2/3 of the digits in T, but if the first derivative is very | |
// large convergence is slow, so we'll bump it up to full | |
// precision to prevent premature termination of the root-finding routine. | |
// | |
unsigned digits = policies::digits<T, Policy>(); | |
if(digits < 30) | |
{ | |
digits *= 2; | |
digits /= 3; | |
} | |
else | |
{ | |
digits /= 2; | |
digits -= 1; | |
} | |
if((a < 0.125) && (fabs(gamma_p_derivative(a, guess, pol)) > 1 / sqrt(tools::epsilon<T>()))) | |
digits = policies::digits<T, Policy>() - 2; | |
// | |
// Go ahead and iterate: | |
// | |
boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>(); | |
guess = tools::halley_iterate( | |
detail::gamma_p_inverse_func<T, Policy>(a, p, false), | |
guess, | |
lower, | |
tools::max_value<T>(), | |
digits, | |
max_iter); | |
policies::check_root_iterations(function, max_iter, pol); | |
BOOST_MATH_INSTRUMENT_VARIABLE(guess); | |
if(guess == lower) | |
guess = policies::raise_underflow_error<T>(function, "Expected result known to be non-zero, but is smaller than the smallest available number.", pol); | |
return guess; | |
} | |
template <class T, class Policy> | |
T gamma_q_inv_imp(T a, T q, const Policy& pol) | |
{ | |
BOOST_MATH_STD_USING // ADL of std functions. | |
static const char* function = "boost::math::gamma_q_inv<%1%>(%1%, %1%)"; | |
if(a <= 0) | |
policies::raise_domain_error<T>(function, "Argument a in the incomplete gamma function inverse must be >= 0 (got a=%1%).", a, pol); | |
if((q < 0) || (q > 1)) | |
policies::raise_domain_error<T>(function, "Probabilty must be in the range [0,1] in the incomplete gamma function inverse (got q=%1%).", q, pol); | |
if(q == 0) | |
return tools::max_value<T>(); | |
if(q == 1) | |
return 0; | |
bool has_10_digits; | |
T guess = detail::find_inverse_gamma<T>(a, 1 - q, q, pol, &has_10_digits); | |
if((policies::digits<T, Policy>() <= 36) && has_10_digits) | |
return guess; | |
T lower = tools::min_value<T>(); | |
if(guess <= lower) | |
guess = tools::min_value<T>(); | |
// | |
// Work out how many digits to converge to, normally this is | |
// 2/3 of the digits in T, but if the first derivative is very | |
// large convergence is slow, so we'll bump it up to full | |
// precision to prevent premature termination of the root-finding routine. | |
// | |
unsigned digits = policies::digits<T, Policy>(); | |
if(digits < 30) | |
{ | |
digits *= 2; | |
digits /= 3; | |
} | |
else | |
{ | |
digits /= 2; | |
digits -= 1; | |
} | |
if((a < 0.125) && (fabs(gamma_p_derivative(a, guess, pol)) > 1 / sqrt(tools::epsilon<T>()))) | |
digits = policies::digits<T, Policy>(); | |
// | |
// Go ahead and iterate: | |
// | |
boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>(); | |
guess = tools::halley_iterate( | |
detail::gamma_p_inverse_func<T, Policy>(a, q, true), | |
guess, | |
lower, | |
tools::max_value<T>(), | |
digits, | |
max_iter); | |
policies::check_root_iterations(function, max_iter, pol); | |
if(guess == lower) | |
guess = policies::raise_underflow_error<T>(function, "Expected result known to be non-zero, but is smaller than the smallest available number.", pol); | |
return guess; | |
} | |
} // namespace detail | |
template <class T1, class T2, class Policy> | |
inline typename tools::promote_args<T1, T2>::type | |
gamma_p_inv(T1 a, T2 p, const Policy& pol) | |
{ | |
typedef typename tools::promote_args<T1, T2>::type result_type; | |
return detail::gamma_p_inv_imp( | |
static_cast<result_type>(a), | |
static_cast<result_type>(p), pol); | |
} | |
template <class T1, class T2, class Policy> | |
inline typename tools::promote_args<T1, T2>::type | |
gamma_q_inv(T1 a, T2 p, const Policy& pol) | |
{ | |
typedef typename tools::promote_args<T1, T2>::type result_type; | |
return detail::gamma_q_inv_imp( | |
static_cast<result_type>(a), | |
static_cast<result_type>(p), pol); | |
} | |
template <class T1, class T2> | |
inline typename tools::promote_args<T1, T2>::type | |
gamma_p_inv(T1 a, T2 p) | |
{ | |
return gamma_p_inv(a, p, policies::policy<>()); | |
} | |
template <class T1, class T2> | |
inline typename tools::promote_args<T1, T2>::type | |
gamma_q_inv(T1 a, T2 p) | |
{ | |
return gamma_q_inv(a, p, policies::policy<>()); | |
} | |
} // namespace math | |
} // namespace boost | |
#endif // BOOST_MATH_SPECIAL_FUNCTIONS_IGAMMA_INVERSE_HPP | |