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1// boost\math\special_functions\negative_binomial.hpp2 3// Copyright Paul A. Bristow 2007.4// Copyright John Maddock 2007.5 6// Use, modification and distribution are subject to the7// Boost Software License, Version 1.0.8// (See accompanying file LICENSE_1_0.txt9// or copy at http://www.boost.org/LICENSE_1_0.txt)10 11// http://en.wikipedia.org/wiki/negative_binomial_distribution12// http://mathworld.wolfram.com/NegativeBinomialDistribution.html13// http://documents.wolfram.com/teachersedition/Teacher/Statistics/DiscreteDistributions.html14 15// The negative binomial distribution NegativeBinomialDistribution[n, p]16// is the distribution of the number (k) of failures that occur in a sequence of trials before17// r successes have occurred, where the probability of success in each trial is p.18 19// In a sequence of Bernoulli trials or events20// (independent, yes or no, succeed or fail) with success_fraction probability p,21// negative_binomial is the probability that k or fewer failures22// precede the r th trial's success.23// random variable k is the number of failures (NOT the probability).24 25// Negative_binomial distribution is a discrete probability distribution.26// But note that the negative binomial distribution27// (like others including the binomial, Poisson & Bernoulli)28// is strictly defined as a discrete function: only integral values of k are envisaged.29// However because of the method of calculation using a continuous gamma function,30// it is convenient to treat it as if a continuous function,31// and permit non-integral values of k.32 33// However, by default the policy is to use discrete_quantile_policy.34 35// To enforce the strict mathematical model, users should use conversion36// on k outside this function to ensure that k is integral.37 38// MATHCAD cumulative negative binomial pnbinom(k, n, p)39 40// Implementation note: much greater speed, and perhaps greater accuracy,41// might be achieved for extreme values by using a normal approximation.42// This is NOT been tested or implemented.43 44#ifndef BOOST_MATH_SPECIAL_NEGATIVE_BINOMIAL_HPP45#define BOOST_MATH_SPECIAL_NEGATIVE_BINOMIAL_HPP46 47#include <boost/math/tools/config.hpp>48#include <boost/math/tools/tuple.hpp>49#include <boost/math/tools/numeric_limits.hpp>50#include <boost/math/tools/cstdint.hpp>51#include <boost/math/distributions/fwd.hpp>52#include <boost/math/special_functions/beta.hpp> // for ibeta(a, b, x) == Ix(a, b).53#include <boost/math/distributions/complement.hpp> // complement.54#include <boost/math/distributions/detail/common_error_handling.hpp> // error checks domain_error & logic_error.55#include <boost/math/special_functions/fpclassify.hpp> // isnan.56#include <boost/math/tools/roots.hpp> // for root finding.57#include <boost/math/distributions/detail/inv_discrete_quantile.hpp>58#include <boost/math/policies/error_handling.hpp>59 60#if defined (BOOST_MSVC)61# pragma warning(push)62// This believed not now necessary, so commented out.63//# pragma warning(disable: 4702) // unreachable code.64// in domain_error_imp in error_handling.65#endif66 67namespace boost68{69 namespace math70 {71 namespace negative_binomial_detail72 {73 // Common error checking routines for negative binomial distribution functions:74 template <class RealType, class Policy>75 BOOST_MATH_GPU_ENABLED inline bool check_successes(const char* function, const RealType& r, RealType* result, const Policy& pol)76 {77 if( !(boost::math::isfinite)(r) || (r <= 0) )78 {79 *result = policies::raise_domain_error<RealType>(80 function,81 "Number of successes argument is %1%, but must be > 0 !", r, pol);82 return false;83 }84 return true;85 }86 template <class RealType, class Policy>87 BOOST_MATH_GPU_ENABLED inline bool check_success_fraction(const char* function, const RealType& p, RealType* result, const Policy& pol)88 {89 if( !(boost::math::isfinite)(p) || (p < 0) || (p > 1) )90 {91 *result = policies::raise_domain_error<RealType>(92 function,93 "Success fraction argument is %1%, but must be >= 0 and <= 1 !", p, pol);94 return false;95 }96 return true;97 }98 template <class RealType, class Policy>99 BOOST_MATH_GPU_ENABLED inline bool check_dist(const char* function, const RealType& r, const RealType& p, RealType* result, const Policy& pol)100 {101 return check_success_fraction(function, p, result, pol)102 && check_successes(function, r, result, pol);103 }104 template <class RealType, class Policy>105 BOOST_MATH_GPU_ENABLED inline bool check_dist_and_k(const char* function, const RealType& r, const RealType& p, RealType k, RealType* result, const Policy& pol)106 {107 if(check_dist(function, r, p, result, pol) == false)108 {109 return false;110 }111 if( !(boost::math::isfinite)(k) || (k < 0) )112 { // Check k failures.113 *result = policies::raise_domain_error<RealType>(114 function,115 "Number of failures argument is %1%, but must be >= 0 !", k, pol);116 return false;117 }118 return true;119 } // Check_dist_and_k120 121 template <class RealType, class Policy>122 BOOST_MATH_GPU_ENABLED inline bool check_dist_and_prob(const char* function, const RealType& r, RealType p, RealType prob, RealType* result, const Policy& pol)123 {124 if((check_dist(function, r, p, result, pol) && detail::check_probability(function, prob, result, pol)) == false)125 {126 return false;127 }128 return true;129 } // check_dist_and_prob130 } // namespace negative_binomial_detail131 132 template <class RealType = double, class Policy = policies::policy<> >133 class negative_binomial_distribution134 {135 public:136 typedef RealType value_type;137 typedef Policy policy_type;138 139 BOOST_MATH_GPU_ENABLED negative_binomial_distribution(RealType r, RealType p) : m_r(r), m_p(p)140 { // Constructor.141 RealType result;142 negative_binomial_detail::check_dist(143 "negative_binomial_distribution<%1%>::negative_binomial_distribution",144 m_r, // Check successes r > 0.145 m_p, // Check success_fraction 0 <= p <= 1.146 &result, Policy());147 } // negative_binomial_distribution constructor.148 149 // Private data getter class member functions.150 BOOST_MATH_GPU_ENABLED RealType success_fraction() const151 { // Probability of success as fraction in range 0 to 1.152 return m_p;153 }154 BOOST_MATH_GPU_ENABLED RealType successes() const155 { // Total number of successes r.156 return m_r;157 }158 159 BOOST_MATH_GPU_ENABLED static RealType find_lower_bound_on_p(160 RealType trials,161 RealType successes,162 RealType alpha) // alpha 0.05 equivalent to 95% for one-sided test.163 {164 constexpr auto function = "boost::math::negative_binomial<%1%>::find_lower_bound_on_p";165 RealType result = 0; // of error checks.166 RealType failures = trials - successes;167 if(false == detail::check_probability(function, alpha, &result, Policy())168 && negative_binomial_detail::check_dist_and_k(169 function, successes, RealType(0), failures, &result, Policy()))170 {171 return result;172 }173 // Use complement ibeta_inv function for lower bound.174 // This is adapted from the corresponding binomial formula175 // here: http://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm176 // This is a Clopper-Pearson interval, and may be overly conservative,177 // see also "A Simple Improved Inferential Method for Some178 // Discrete Distributions" Yong CAI and K. KRISHNAMOORTHY179 // http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf180 //181 return ibeta_inv(successes, failures + 1, alpha, static_cast<RealType*>(nullptr), Policy());182 } // find_lower_bound_on_p183 184 BOOST_MATH_GPU_ENABLED static RealType find_upper_bound_on_p(185 RealType trials,186 RealType successes,187 RealType alpha) // alpha 0.05 equivalent to 95% for one-sided test.188 {189 constexpr auto function = "boost::math::negative_binomial<%1%>::find_upper_bound_on_p";190 RealType result = 0; // of error checks.191 RealType failures = trials - successes;192 if(false == negative_binomial_detail::check_dist_and_k(193 function, successes, RealType(0), failures, &result, Policy())194 && detail::check_probability(function, alpha, &result, Policy()))195 {196 return result;197 }198 if(failures == 0)199 return 1;200 // Use complement ibetac_inv function for upper bound.201 // Note adjusted failures value: *not* failures+1 as usual.202 // This is adapted from the corresponding binomial formula203 // here: http://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm204 // This is a Clopper-Pearson interval, and may be overly conservative,205 // see also "A Simple Improved Inferential Method for Some206 // Discrete Distributions" Yong CAI and K. KRISHNAMOORTHY207 // http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf208 //209 return ibetac_inv(successes, failures, alpha, static_cast<RealType*>(nullptr), Policy());210 } // find_upper_bound_on_p211 212 // Estimate number of trials :213 // "How many trials do I need to be P% sure of seeing k or fewer failures?"214 215 BOOST_MATH_GPU_ENABLED static RealType find_minimum_number_of_trials(216 RealType k, // number of failures (k >= 0).217 RealType p, // success fraction 0 <= p <= 1.218 RealType alpha) // risk level threshold 0 <= alpha <= 1.219 {220 constexpr auto function = "boost::math::negative_binomial<%1%>::find_minimum_number_of_trials";221 // Error checks:222 RealType result = 0;223 if(false == negative_binomial_detail::check_dist_and_k(224 function, RealType(1), p, k, &result, Policy())225 && detail::check_probability(function, alpha, &result, Policy()))226 { return result; }227 228 result = ibeta_inva(k + 1, p, alpha, Policy()); // returns n - k229 return result + k;230 } // RealType find_number_of_failures231 232 BOOST_MATH_GPU_ENABLED static RealType find_maximum_number_of_trials(233 RealType k, // number of failures (k >= 0).234 RealType p, // success fraction 0 <= p <= 1.235 RealType alpha) // risk level threshold 0 <= alpha <= 1.236 {237 constexpr auto function = "boost::math::negative_binomial<%1%>::find_maximum_number_of_trials";238 // Error checks:239 RealType result = 0;240 if(false == negative_binomial_detail::check_dist_and_k(241 function, RealType(1), p, k, &result, Policy())242 && detail::check_probability(function, alpha, &result, Policy()))243 { return result; }244 245 result = ibetac_inva(k + 1, p, alpha, Policy()); // returns n - k246 return result + k;247 } // RealType find_number_of_trials complemented248 249 private:250 RealType m_r; // successes.251 RealType m_p; // success_fraction252 }; // template <class RealType, class Policy> class negative_binomial_distribution253 254 typedef negative_binomial_distribution<double> negative_binomial; // Reserved name of type double.255 256 #ifdef __cpp_deduction_guides257 template <class RealType>258 negative_binomial_distribution(RealType,RealType)->negative_binomial_distribution<typename boost::math::tools::promote_args<RealType>::type>;259 #endif260 261 template <class RealType, class Policy>262 BOOST_MATH_GPU_ENABLED inline const boost::math::pair<RealType, RealType> range(const negative_binomial_distribution<RealType, Policy>& /* dist */)263 { // Range of permissible values for random variable k.264 using boost::math::tools::max_value;265 return boost::math::pair<RealType, RealType>(static_cast<RealType>(0), max_value<RealType>()); // max_integer?266 }267 268 template <class RealType, class Policy>269 BOOST_MATH_GPU_ENABLED inline const boost::math::pair<RealType, RealType> support(const negative_binomial_distribution<RealType, Policy>& /* dist */)270 { // Range of supported values for random variable k.271 // This is range where cdf rises from 0 to 1, and outside it, the pdf is zero.272 using boost::math::tools::max_value;273 return boost::math::pair<RealType, RealType>(static_cast<RealType>(0), max_value<RealType>()); // max_integer?274 }275 276 template <class RealType, class Policy>277 BOOST_MATH_GPU_ENABLED inline RealType mean(const negative_binomial_distribution<RealType, Policy>& dist)278 { // Mean of Negative Binomial distribution = r(1-p)/p.279 return dist.successes() * (1 - dist.success_fraction() ) / dist.success_fraction();280 } // mean281 282 //template <class RealType, class Policy>283 //inline RealType median(const negative_binomial_distribution<RealType, Policy>& dist)284 //{ // Median of negative_binomial_distribution is not defined.285 // return policies::raise_domain_error<RealType>(BOOST_CURRENT_FUNCTION, "Median is not implemented, result is %1%!", std::numeric_limits<RealType>::quiet_NaN());286 //} // median287 // Now implemented via quantile(half) in derived accessors.288 289 template <class RealType, class Policy>290 BOOST_MATH_GPU_ENABLED inline RealType mode(const negative_binomial_distribution<RealType, Policy>& dist)291 { // Mode of Negative Binomial distribution = floor[(r-1) * (1 - p)/p]292 BOOST_MATH_STD_USING // ADL of std functions.293 return floor((dist.successes() -1) * (1 - dist.success_fraction()) / dist.success_fraction());294 } // mode295 296 template <class RealType, class Policy>297 BOOST_MATH_GPU_ENABLED inline RealType skewness(const negative_binomial_distribution<RealType, Policy>& dist)298 { // skewness of Negative Binomial distribution = 2-p / (sqrt(r(1-p))299 BOOST_MATH_STD_USING // ADL of std functions.300 RealType p = dist.success_fraction();301 RealType r = dist.successes();302 303 return (2 - p) /304 sqrt(r * (1 - p));305 } // skewness306 307 template <class RealType, class Policy>308 BOOST_MATH_GPU_ENABLED inline RealType kurtosis(const negative_binomial_distribution<RealType, Policy>& dist)309 { // kurtosis of Negative Binomial distribution310 // http://en.wikipedia.org/wiki/Negative_binomial is kurtosis_excess so add 3311 RealType p = dist.success_fraction();312 RealType r = dist.successes();313 return 3 + (6 / r) + ((p * p) / (r * (1 - p)));314 } // kurtosis315 316 template <class RealType, class Policy>317 BOOST_MATH_GPU_ENABLED inline RealType kurtosis_excess(const negative_binomial_distribution<RealType, Policy>& dist)318 { // kurtosis excess of Negative Binomial distribution319 // http://mathworld.wolfram.com/Kurtosis.html table of kurtosis_excess320 RealType p = dist.success_fraction();321 RealType r = dist.successes();322 return (6 - p * (6-p)) / (r * (1-p));323 } // kurtosis_excess324 325 template <class RealType, class Policy>326 BOOST_MATH_GPU_ENABLED inline RealType variance(const negative_binomial_distribution<RealType, Policy>& dist)327 { // Variance of Binomial distribution = r (1-p) / p^2.328 return dist.successes() * (1 - dist.success_fraction())329 / (dist.success_fraction() * dist.success_fraction());330 } // variance331 332 // RealType standard_deviation(const negative_binomial_distribution<RealType, Policy>& dist)333 // standard_deviation provided by derived accessors.334 // RealType hazard(const negative_binomial_distribution<RealType, Policy>& dist)335 // hazard of Negative Binomial distribution provided by derived accessors.336 // RealType chf(const negative_binomial_distribution<RealType, Policy>& dist)337 // chf of Negative Binomial distribution provided by derived accessors.338 339 template <class RealType, class Policy>340 BOOST_MATH_GPU_ENABLED inline RealType pdf(const negative_binomial_distribution<RealType, Policy>& dist, const RealType& k)341 { // Probability Density/Mass Function.342 BOOST_FPU_EXCEPTION_GUARD343 344 constexpr auto function = "boost::math::pdf(const negative_binomial_distribution<%1%>&, %1%)";345 346 RealType r = dist.successes();347 RealType p = dist.success_fraction();348 RealType result = 0;349 if(false == negative_binomial_detail::check_dist_and_k(350 function,351 r,352 dist.success_fraction(),353 k,354 &result, Policy()))355 {356 return result;357 }358 359 result = (p/(r + k)) * ibeta_derivative(r, static_cast<RealType>(k+1), p, Policy());360 // Equivalent to:361 // return exp(lgamma(r + k) - lgamma(r) - lgamma(k+1)) * pow(p, r) * pow((1-p), k);362 return result;363 } // negative_binomial_pdf364 365 template <class RealType, class Policy>366 BOOST_MATH_GPU_ENABLED inline RealType cdf(const negative_binomial_distribution<RealType, Policy>& dist, const RealType& k)367 { // Cumulative Distribution Function of Negative Binomial.368 constexpr auto function = "boost::math::cdf(const negative_binomial_distribution<%1%>&, %1%)";369 using boost::math::ibeta; // Regularized incomplete beta function.370 // k argument may be integral, signed, or unsigned, or floating point.371 // If necessary, it has already been promoted from an integral type.372 RealType p = dist.success_fraction();373 RealType r = dist.successes();374 // Error check:375 RealType result = 0;376 if(false == negative_binomial_detail::check_dist_and_k(377 function,378 r,379 dist.success_fraction(),380 k,381 &result, Policy()))382 {383 return result;384 }385 386 RealType probability = ibeta(r, static_cast<RealType>(k+1), p, Policy());387 // Ip(r, k+1) = ibeta(r, k+1, p)388 return probability;389 } // cdf Cumulative Distribution Function Negative Binomial.390 391 template <class RealType, class Policy>392 BOOST_MATH_GPU_ENABLED inline RealType cdf(const complemented2_type<negative_binomial_distribution<RealType, Policy>, RealType>& c)393 { // Complemented Cumulative Distribution Function Negative Binomial.394 395 constexpr auto function = "boost::math::cdf(const negative_binomial_distribution<%1%>&, %1%)";396 using boost::math::ibetac; // Regularized incomplete beta function complement.397 // k argument may be integral, signed, or unsigned, or floating point.398 // If necessary, it has already been promoted from an integral type.399 RealType const& k = c.param;400 negative_binomial_distribution<RealType, Policy> const& dist = c.dist;401 RealType p = dist.success_fraction();402 RealType r = dist.successes();403 // Error check:404 RealType result = 0;405 if(false == negative_binomial_detail::check_dist_and_k(406 function,407 r,408 p,409 k,410 &result, Policy()))411 {412 return result;413 }414 // Calculate cdf negative binomial using the incomplete beta function.415 // Use of ibeta here prevents cancellation errors in calculating416 // 1-p if p is very small, perhaps smaller than machine epsilon.417 // Ip(k+1, r) = ibetac(r, k+1, p)418 // constrain_probability here?419 RealType probability = ibetac(r, static_cast<RealType>(k+1), p, Policy());420 // Numerical errors might cause probability to be slightly outside the range < 0 or > 1.421 // This might cause trouble downstream, so warn, possibly throw exception, but constrain to the limits.422 return probability;423 } // cdf Cumulative Distribution Function Negative Binomial.424 425 template <class RealType, class Policy>426 BOOST_MATH_GPU_ENABLED inline RealType quantile(const negative_binomial_distribution<RealType, Policy>& dist, const RealType& P)427 { // Quantile, percentile/100 or Percent Point Negative Binomial function.428 // Return the number of expected failures k for a given probability p.429 430 // Inverse cumulative Distribution Function or Quantile (percentile / 100) of negative_binomial Probability.431 // MAthCAD pnbinom return smallest k such that negative_binomial(k, n, p) >= probability.432 // k argument may be integral, signed, or unsigned, or floating point.433 // BUT Cephes/CodeCogs says: finds argument p (0 to 1) such that cdf(k, n, p) = y434 constexpr auto function = "boost::math::quantile(const negative_binomial_distribution<%1%>&, %1%)";435 BOOST_MATH_STD_USING // ADL of std functions.436 437 RealType p = dist.success_fraction();438 RealType r = dist.successes();439 // Check dist and P.440 RealType result = 0;441 if(false == negative_binomial_detail::check_dist_and_prob442 (function, r, p, P, &result, Policy()))443 {444 return result;445 }446 447 // Special cases.448 if (P == 1)449 { // Would need +infinity failures for total confidence.450 result = policies::raise_overflow_error<RealType>(451 function,452 "Probability argument is 1, which implies infinite failures !", Policy());453 return result;454 // usually means return +std::numeric_limits<RealType>::infinity();455 // unless #define BOOST_MATH_THROW_ON_OVERFLOW_ERROR456 }457 if (P == 0)458 { // No failures are expected if P = 0.459 return 0; // Total trials will be just dist.successes.460 }461 if (P <= pow(dist.success_fraction(), dist.successes()))462 { // p <= pdf(dist, 0) == cdf(dist, 0)463 return 0;464 }465 if(p == 0)466 { // Would need +infinity failures for total confidence.467 result = policies::raise_overflow_error<RealType>(468 function,469 "Success fraction is 0, which implies infinite failures !", Policy());470 return result;471 // usually means return +std::numeric_limits<RealType>::infinity();472 // unless #define BOOST_MATH_THROW_ON_OVERFLOW_ERROR473 }474 /*475 // Calculate quantile of negative_binomial using the inverse incomplete beta function.476 using boost::math::ibeta_invb;477 return ibeta_invb(r, p, P, Policy()) - 1; //478 */479 RealType guess = 0;480 RealType factor = 5;481 if(r * r * r * P * p > 0.005)482 guess = detail::inverse_negative_binomial_cornish_fisher(r, p, RealType(1-p), P, RealType(1-P), Policy());483 484 if(guess < 10)485 {486 //487 // Cornish-Fisher Negative binomial approximation not accurate in this area:488 //489 guess = BOOST_MATH_GPU_SAFE_MIN(RealType(r * 2), RealType(10));490 }491 else492 factor = (1-P < sqrt(tools::epsilon<RealType>())) ? 2 : (guess < 20 ? 1.2f : 1.1f);493 BOOST_MATH_INSTRUMENT_CODE("guess = " << guess);494 //495 // Max iterations permitted:496 //497 boost::math::uintmax_t max_iter = policies::get_max_root_iterations<Policy>();498 typedef typename Policy::discrete_quantile_type discrete_type;499 return detail::inverse_discrete_quantile(500 dist,501 P,502 false,503 guess,504 factor,505 RealType(1),506 discrete_type(),507 max_iter);508 } // RealType quantile(const negative_binomial_distribution dist, p)509 510 template <class RealType, class Policy>511 BOOST_MATH_GPU_ENABLED inline RealType quantile(const complemented2_type<negative_binomial_distribution<RealType, Policy>, RealType>& c)512 { // Quantile or Percent Point Binomial function.513 // Return the number of expected failures k for a given514 // complement of the probability Q = 1 - P.515 constexpr auto function = "boost::math::quantile(const negative_binomial_distribution<%1%>&, %1%)";516 BOOST_MATH_STD_USING517 518 // Error checks:519 RealType Q = c.param;520 const negative_binomial_distribution<RealType, Policy>& dist = c.dist;521 RealType p = dist.success_fraction();522 RealType r = dist.successes();523 RealType result = 0;524 if(false == negative_binomial_detail::check_dist_and_prob(525 function,526 r,527 p,528 Q,529 &result, Policy()))530 {531 return result;532 }533 534 // Special cases:535 //536 if(Q == 1)537 { // There may actually be no answer to this question,538 // since the probability of zero failures may be non-zero,539 return 0; // but zero is the best we can do:540 }541 if(Q == 0)542 { // Probability 1 - Q == 1 so infinite failures to achieve certainty.543 // Would need +infinity failures for total confidence.544 result = policies::raise_overflow_error<RealType>(545 function,546 "Probability argument complement is 0, which implies infinite failures !", Policy());547 return result;548 // usually means return +std::numeric_limits<RealType>::infinity();549 // unless #define BOOST_MATH_THROW_ON_OVERFLOW_ERROR550 }551 if (-Q <= boost::math::powm1(dist.success_fraction(), dist.successes(), Policy()))552 { // q <= cdf(complement(dist, 0)) == pdf(dist, 0)553 return 0; //554 }555 if(p == 0)556 { // Success fraction is 0 so infinite failures to achieve certainty.557 // Would need +infinity failures for total confidence.558 result = policies::raise_overflow_error<RealType>(559 function,560 "Success fraction is 0, which implies infinite failures !", Policy());561 return result;562 // usually means return +std::numeric_limits<RealType>::infinity();563 // unless #define BOOST_MATH_THROW_ON_OVERFLOW_ERROR564 }565 //return ibetac_invb(r, p, Q, Policy()) -1;566 RealType guess = 0;567 RealType factor = 5;568 if(r * r * r * (1-Q) * p > 0.005)569 guess = detail::inverse_negative_binomial_cornish_fisher(r, p, RealType(1-p), RealType(1-Q), Q, Policy());570 571 if(guess < 10)572 {573 //574 // Cornish-Fisher Negative binomial approximation not accurate in this area:575 //576 guess = BOOST_MATH_GPU_SAFE_MIN(RealType(r * 2), RealType(10));577 }578 else579 factor = (Q < sqrt(tools::epsilon<RealType>())) ? 2 : (guess < 20 ? 1.2f : 1.1f);580 BOOST_MATH_INSTRUMENT_CODE("guess = " << guess);581 //582 // Max iterations permitted:583 //584 boost::math::uintmax_t max_iter = policies::get_max_root_iterations<Policy>();585 typedef typename Policy::discrete_quantile_type discrete_type;586 return detail::inverse_discrete_quantile(587 dist,588 Q,589 true,590 guess,591 factor,592 RealType(1),593 discrete_type(),594 max_iter);595 } // quantile complement596 597 } // namespace math598} // namespace boost599 600// This include must be at the end, *after* the accessors601// for this distribution have been defined, in order to602// keep compilers that support two-phase lookup happy.603#include <boost/math/distributions/detail/derived_accessors.hpp>604 605#if defined (BOOST_MSVC)606# pragma warning(pop)607#endif608 609#endif // BOOST_MATH_SPECIAL_NEGATIVE_BINOMIAL_HPP610