552 lines
18 KiB
Plaintext
552 lines
18 KiB
Plaintext
// (C) Copyright John Maddock 2006.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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#ifndef BOOST_MATH_SPECIAL_FUNCTIONS_IGAMMA_INVERSE_HPP
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#define BOOST_MATH_SPECIAL_FUNCTIONS_IGAMMA_INVERSE_HPP
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#ifdef _MSC_VER
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#pragma once
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#endif
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#include <boost/math/tools/tuple.hpp>
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#include <boost/math/special_functions/gamma.hpp>
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#include <boost/math/special_functions/sign.hpp>
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#include <boost/math/tools/roots.hpp>
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#include <boost/math/policies/error_handling.hpp>
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namespace boost{ namespace math{
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namespace detail{
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template <class T>
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T find_inverse_s(T p, T q)
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{
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//
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// Computation of the Incomplete Gamma Function Ratios and their Inverse
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// ARMIDO R. DIDONATO and ALFRED H. MORRIS, JR.
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// ACM Transactions on Mathematical Software, Vol. 12, No. 4,
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// December 1986, Pages 377-393.
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//
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// See equation 32.
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//
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BOOST_MATH_STD_USING
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T t;
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if(p < 0.5)
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{
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t = sqrt(-2 * log(p));
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}
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else
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{
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t = sqrt(-2 * log(q));
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}
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static const double a[4] = { 3.31125922108741, 11.6616720288968, 4.28342155967104, 0.213623493715853 };
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static const double b[5] = { 1, 6.61053765625462, 6.40691597760039, 1.27364489782223, 0.3611708101884203e-1 };
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T s = t - tools::evaluate_polynomial(a, t) / tools::evaluate_polynomial(b, t);
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if(p < 0.5)
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s = -s;
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return s;
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}
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template <class T>
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T didonato_SN(T a, T x, unsigned N, T tolerance = 0)
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{
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//
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// Computation of the Incomplete Gamma Function Ratios and their Inverse
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// ARMIDO R. DIDONATO and ALFRED H. MORRIS, JR.
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// ACM Transactions on Mathematical Software, Vol. 12, No. 4,
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// December 1986, Pages 377-393.
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//
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// See equation 34.
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//
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T sum = 1;
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if(N >= 1)
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{
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T partial = x / (a + 1);
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sum += partial;
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for(unsigned i = 2; i <= N; ++i)
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{
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partial *= x / (a + i);
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sum += partial;
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if(partial < tolerance)
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break;
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}
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}
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return sum;
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}
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template <class T, class Policy>
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inline T didonato_FN(T p, T a, T x, unsigned N, T tolerance, const Policy& pol)
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{
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//
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// Computation of the Incomplete Gamma Function Ratios and their Inverse
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// ARMIDO R. DIDONATO and ALFRED H. MORRIS, JR.
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// ACM Transactions on Mathematical Software, Vol. 12, No. 4,
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// December 1986, Pages 377-393.
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//
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// See equation 34.
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//
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BOOST_MATH_STD_USING
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T u = log(p) + boost::math::lgamma(a + 1, pol);
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return exp((u + x - log(didonato_SN(a, x, N, tolerance))) / a);
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}
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template <class T, class Policy>
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T find_inverse_gamma(T a, T p, T q, const Policy& pol, bool* p_has_10_digits)
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{
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//
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// In order to understand what's going on here, you will
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// need to refer to:
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//
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// Computation of the Incomplete Gamma Function Ratios and their Inverse
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// ARMIDO R. DIDONATO and ALFRED H. MORRIS, JR.
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// ACM Transactions on Mathematical Software, Vol. 12, No. 4,
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// December 1986, Pages 377-393.
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//
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BOOST_MATH_STD_USING
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T result;
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*p_has_10_digits = false;
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if(a == 1)
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{
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result = -log(q);
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BOOST_MATH_INSTRUMENT_VARIABLE(result);
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}
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else if(a < 1)
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{
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T g = boost::math::tgamma(a, pol);
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T b = q * g;
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BOOST_MATH_INSTRUMENT_VARIABLE(g);
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BOOST_MATH_INSTRUMENT_VARIABLE(b);
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if((b > 0.6) || ((b >= 0.45) && (a >= 0.3)))
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{
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// DiDonato & Morris Eq 21:
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//
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// There is a slight variation from DiDonato and Morris here:
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// the first form given here is unstable when p is close to 1,
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// making it impossible to compute the inverse of Q(a,x) for small
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// q. Fortunately the second form works perfectly well in this case.
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//
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T u;
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if((b * q > 1e-8) && (q > 1e-5))
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{
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u = pow(p * g * a, 1 / a);
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BOOST_MATH_INSTRUMENT_VARIABLE(u);
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}
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else
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{
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u = exp((-q / a) - constants::euler<T>());
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BOOST_MATH_INSTRUMENT_VARIABLE(u);
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}
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result = u / (1 - (u / (a + 1)));
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BOOST_MATH_INSTRUMENT_VARIABLE(result);
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}
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else if((a < 0.3) && (b >= 0.35))
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{
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// DiDonato & Morris Eq 22:
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T t = exp(-constants::euler<T>() - b);
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T u = t * exp(t);
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result = t * exp(u);
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BOOST_MATH_INSTRUMENT_VARIABLE(result);
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}
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else if((b > 0.15) || (a >= 0.3))
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{
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// DiDonato & Morris Eq 23:
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T y = -log(b);
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T u = y - (1 - a) * log(y);
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result = y - (1 - a) * log(u) - log(1 + (1 - a) / (1 + u));
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BOOST_MATH_INSTRUMENT_VARIABLE(result);
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}
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else if (b > 0.1)
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{
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// DiDonato & Morris Eq 24:
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T y = -log(b);
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T u = y - (1 - a) * log(y);
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result = y - (1 - a) * log(u) - log((u * u + 2 * (3 - a) * u + (2 - a) * (3 - a)) / (u * u + (5 - a) * u + 2));
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BOOST_MATH_INSTRUMENT_VARIABLE(result);
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}
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else
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{
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// DiDonato & Morris Eq 25:
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T y = -log(b);
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T c1 = (a - 1) * log(y);
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T c1_2 = c1 * c1;
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T c1_3 = c1_2 * c1;
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T c1_4 = c1_2 * c1_2;
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T a_2 = a * a;
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T a_3 = a_2 * a;
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T c2 = (a - 1) * (1 + c1);
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T c3 = (a - 1) * (-(c1_2 / 2) + (a - 2) * c1 + (3 * a - 5) / 2);
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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);
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T c5 = (a - 1) * (-(c1_4 / 4)
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+ (11 * a - 17) * c1_3 / 6
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+ (-3 * a_2 + 13 * a -13) * c1_2
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+ (2 * a_3 - 25 * a_2 + 72 * a - 61) * c1 / 2
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+ (25 * a_3 - 195 * a_2 + 477 * a - 379) / 12);
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T y_2 = y * y;
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T y_3 = y_2 * y;
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T y_4 = y_2 * y_2;
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result = y + c1 + (c2 / y) + (c3 / y_2) + (c4 / y_3) + (c5 / y_4);
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BOOST_MATH_INSTRUMENT_VARIABLE(result);
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if(b < 1e-28f)
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*p_has_10_digits = true;
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}
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}
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else
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{
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// DiDonato and Morris Eq 31:
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T s = find_inverse_s(p, q);
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BOOST_MATH_INSTRUMENT_VARIABLE(s);
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T s_2 = s * s;
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T s_3 = s_2 * s;
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T s_4 = s_2 * s_2;
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T s_5 = s_4 * s;
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T ra = sqrt(a);
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BOOST_MATH_INSTRUMENT_VARIABLE(ra);
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T w = a + s * ra + (s * s -1) / 3;
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w += (s_3 - 7 * s) / (36 * ra);
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w -= (3 * s_4 + 7 * s_2 - 16) / (810 * a);
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w += (9 * s_5 + 256 * s_3 - 433 * s) / (38880 * a * ra);
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BOOST_MATH_INSTRUMENT_VARIABLE(w);
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if((a >= 500) && (fabs(1 - w / a) < 1e-6))
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{
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result = w;
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*p_has_10_digits = true;
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BOOST_MATH_INSTRUMENT_VARIABLE(result);
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}
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else if (p > 0.5)
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{
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if(w < 3 * a)
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{
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result = w;
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BOOST_MATH_INSTRUMENT_VARIABLE(result);
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}
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else
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{
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T D = (std::max)(T(2), T(a * (a - 1)));
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T lg = boost::math::lgamma(a, pol);
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T lb = log(q) + lg;
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if(lb < -D * 2.3)
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{
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// DiDonato and Morris Eq 25:
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T y = -lb;
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T c1 = (a - 1) * log(y);
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T c1_2 = c1 * c1;
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T c1_3 = c1_2 * c1;
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T c1_4 = c1_2 * c1_2;
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T a_2 = a * a;
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T a_3 = a_2 * a;
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T c2 = (a - 1) * (1 + c1);
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T c3 = (a - 1) * (-(c1_2 / 2) + (a - 2) * c1 + (3 * a - 5) / 2);
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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);
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T c5 = (a - 1) * (-(c1_4 / 4)
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+ (11 * a - 17) * c1_3 / 6
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+ (-3 * a_2 + 13 * a -13) * c1_2
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+ (2 * a_3 - 25 * a_2 + 72 * a - 61) * c1 / 2
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+ (25 * a_3 - 195 * a_2 + 477 * a - 379) / 12);
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T y_2 = y * y;
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T y_3 = y_2 * y;
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T y_4 = y_2 * y_2;
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result = y + c1 + (c2 / y) + (c3 / y_2) + (c4 / y_3) + (c5 / y_4);
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BOOST_MATH_INSTRUMENT_VARIABLE(result);
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}
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else
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{
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// DiDonato and Morris Eq 33:
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T u = -lb + (a - 1) * log(w) - log(1 + (1 - a) / (1 + w));
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result = -lb + (a - 1) * log(u) - log(1 + (1 - a) / (1 + u));
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BOOST_MATH_INSTRUMENT_VARIABLE(result);
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}
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}
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}
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else
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{
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T z = w;
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T ap1 = a + 1;
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T ap2 = a + 2;
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if(w < 0.15f * ap1)
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{
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// DiDonato and Morris Eq 35:
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T v = log(p) + boost::math::lgamma(ap1, pol);
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z = exp((v + w) / a);
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s = boost::math::log1p(z / ap1 * (1 + z / ap2), pol);
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z = exp((v + z - s) / a);
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s = boost::math::log1p(z / ap1 * (1 + z / ap2), pol);
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z = exp((v + z - s) / a);
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s = boost::math::log1p(z / ap1 * (1 + z / ap2 * (1 + z / (a + 3))), pol);
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z = exp((v + z - s) / a);
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BOOST_MATH_INSTRUMENT_VARIABLE(z);
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}
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if((z <= 0.01 * ap1) || (z > 0.7 * ap1))
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{
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result = z;
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if(z <= 0.002 * ap1)
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*p_has_10_digits = true;
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BOOST_MATH_INSTRUMENT_VARIABLE(result);
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}
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else
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{
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// DiDonato and Morris Eq 36:
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T ls = log(didonato_SN(a, z, 100, T(1e-4)));
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T v = log(p) + boost::math::lgamma(ap1, pol);
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z = exp((v + z - ls) / a);
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result = z * (1 - (a * log(z) - z - v + ls) / (a - z));
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BOOST_MATH_INSTRUMENT_VARIABLE(result);
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}
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}
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}
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return result;
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}
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template <class T, class Policy>
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struct gamma_p_inverse_func
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{
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gamma_p_inverse_func(T a_, T p_, bool inv) : a(a_), p(p_), invert(inv)
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{
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//
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// If p is too near 1 then P(x) - p suffers from cancellation
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// errors causing our root-finding algorithms to "thrash", better
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// to invert in this case and calculate Q(x) - (1-p) instead.
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//
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// Of course if p is *very* close to 1, then the answer we get will
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// be inaccurate anyway (because there's not enough information in p)
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// but at least we will converge on the (inaccurate) answer quickly.
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//
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if(p > 0.9)
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{
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p = 1 - p;
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invert = !invert;
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}
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}
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boost::math::tuple<T, T, T> operator()(const T& x)const
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{
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BOOST_FPU_EXCEPTION_GUARD
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//
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// Calculate P(x) - p and the first two derivates, or if the invert
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// flag is set, then Q(x) - q and it's derivatives.
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//
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typedef typename policies::evaluation<T, Policy>::type value_type;
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// typedef typename lanczos::lanczos<T, Policy>::type evaluation_type;
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typedef typename policies::normalise<
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Policy,
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policies::promote_float<false>,
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policies::promote_double<false>,
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policies::discrete_quantile<>,
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policies::assert_undefined<> >::type forwarding_policy;
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BOOST_MATH_STD_USING // For ADL of std functions.
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T f, f1;
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value_type ft;
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f = static_cast<T>(boost::math::detail::gamma_incomplete_imp(
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static_cast<value_type>(a),
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static_cast<value_type>(x),
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true, invert,
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forwarding_policy(), &ft));
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f1 = static_cast<T>(ft);
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T f2;
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T div = (a - x - 1) / x;
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f2 = f1;
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if((fabs(div) > 1) && (tools::max_value<T>() / fabs(div) < f2))
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{
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// overflow:
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f2 = -tools::max_value<T>() / 2;
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}
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else
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{
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f2 *= div;
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}
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if(invert)
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{
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f1 = -f1;
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f2 = -f2;
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}
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return boost::math::make_tuple(static_cast<T>(f - p), f1, f2);
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}
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private:
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T a, p;
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bool invert;
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};
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template <class T, class Policy>
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T gamma_p_inv_imp(T a, T p, const Policy& pol)
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{
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BOOST_MATH_STD_USING // ADL of std functions.
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static const char* function = "boost::math::gamma_p_inv<%1%>(%1%, %1%)";
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BOOST_MATH_INSTRUMENT_VARIABLE(a);
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BOOST_MATH_INSTRUMENT_VARIABLE(p);
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if(a <= 0)
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return policies::raise_domain_error<T>(function, "Argument a in the incomplete gamma function inverse must be >= 0 (got a=%1%).", a, pol);
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if((p < 0) || (p > 1))
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return 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);
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if(p == 1)
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return policies::raise_overflow_error<T>(function, 0, Policy());
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if(p == 0)
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return 0;
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bool has_10_digits;
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T guess = detail::find_inverse_gamma<T>(a, p, 1 - p, pol, &has_10_digits);
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if((policies::digits<T, Policy>() <= 36) && has_10_digits)
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return guess;
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T lower = tools::min_value<T>();
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if(guess <= lower)
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guess = tools::min_value<T>();
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BOOST_MATH_INSTRUMENT_VARIABLE(guess);
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//
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// Work out how many digits to converge to, normally this is
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// 2/3 of the digits in T, but if the first derivative is very
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// large convergence is slow, so we'll bump it up to full
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// precision to prevent premature termination of the root-finding routine.
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//
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unsigned digits = policies::digits<T, Policy>();
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if(digits < 30)
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{
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digits *= 2;
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digits /= 3;
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}
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else
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{
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digits /= 2;
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digits -= 1;
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}
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if((a < 0.125) && (fabs(gamma_p_derivative(a, guess, pol)) > 1 / sqrt(tools::epsilon<T>())))
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digits = policies::digits<T, Policy>() - 2;
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//
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// Go ahead and iterate:
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//
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boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>();
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guess = tools::halley_iterate(
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detail::gamma_p_inverse_func<T, Policy>(a, p, false),
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guess,
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lower,
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tools::max_value<T>(),
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digits,
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max_iter);
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policies::check_root_iterations<T>(function, max_iter, pol);
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BOOST_MATH_INSTRUMENT_VARIABLE(guess);
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if(guess == lower)
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guess = policies::raise_underflow_error<T>(function, "Expected result known to be non-zero, but is smaller than the smallest available number.", pol);
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return guess;
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}
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template <class T, class Policy>
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T gamma_q_inv_imp(T a, T q, const Policy& pol)
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{
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BOOST_MATH_STD_USING // ADL of std functions.
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static const char* function = "boost::math::gamma_q_inv<%1%>(%1%, %1%)";
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if(a <= 0)
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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, "Probabilty 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>();
|
|
//
|
|
// 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
|
|
|
|
|
|
|