summaryrefslogtreecommitdiff
path: root/boost/math/special_functions/detail/igamma_inverse.hpp
diff options
context:
space:
mode:
Diffstat (limited to 'boost/math/special_functions/detail/igamma_inverse.hpp')
-rw-r--r--boost/math/special_functions/detail/igamma_inverse.hpp551
1 files changed, 551 insertions, 0 deletions
diff --git a/boost/math/special_functions/detail/igamma_inverse.hpp b/boost/math/special_functions/detail/igamma_inverse.hpp
new file mode 100644
index 0000000000..53875ff83e
--- /dev/null
+++ b/boost/math/special_functions/detail/igamma_inverse.hpp
@@ -0,0 +1,551 @@
+// (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);
+ 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<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)
+ 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<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
+
+
+
generated by cgit v1.2.3 (git 2.25.1) at 2024-06-01 13:08:16 +0000