# Boost C++ Libraries

...one of the most highly regarded and expertly designed C++ library projects in the world.

### boost/math/special_functions/detail/igamma_inverse.hpp

```//  (C) Copyright John Maddock 2006.
//  Use, modification and distribution are subject to the
//  Boost Software License, Version 1.0. (See accompanying file

#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);
z = exp((v + w) / a);
s = boost::math::log1p(z / ap1 * (1 + z / ap2), pol);
z = exp((v + z - s) / a);
s = boost::math::log1p(z / ap1 * (1 + z / ap2), pol);
z = exp((v + z - s) / a);
s = boost::math::log1p(z / ap1 * (1 + z / ap2 * (1 + z / (a + 3))), pol);
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(static_cast<T>(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)
return 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))
return policies::raise_domain_error<T>(function, "Probability must be in the range [0,1] in the incomplete gamma function inverse (got p=%1%).", p, pol);
if(p == 1)
return policies::raise_overflow_error<T>(function, 0, Policy());
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;
//
//
std::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<T>(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)
return 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))
return policies::raise_domain_error<T>(function, "Probability must be in the range [0,1] in the incomplete gamma function inverse (got q=%1%).", q, pol);
if(q == 0)
return policies::raise_overflow_error<T>(function, 0, Policy());
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>();
//
//
std::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<T>(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

```