Stan Math Library  2.20.0
reverse mode automatic differentiation
grad_reg_inc_gamma.hpp
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1 #ifndef STAN_MATH_PRIM_SCAL_FUN_GRAD_REG_INC_GAMMA_HPP
2 #define STAN_MATH_PRIM_SCAL_FUN_GRAD_REG_INC_GAMMA_HPP
3 
13 #include <cmath>
14 #include <limits>
15 
16 namespace stan {
17 namespace math {
18 
48 template <typename T1, typename T2>
49 typename return_type<T1, T2>::type grad_reg_inc_gamma(T1 a, T2 z, T1 g, T1 dig,
50  double precision = 1e-6,
51  int max_steps = 1e5) {
52  using std::exp;
53  using std::fabs;
54  using std::log;
55  typedef typename return_type<T1, T2>::type TP;
56 
57  if (is_any_nan(a, z, g, dig))
58  return std::numeric_limits<TP>::quiet_NaN();
59 
60  T2 l = log(z);
61  if (z >= a && z >= 8) {
62  // asymptotic expansion http://dlmf.nist.gov/8.11#E2
63  TP S = 0;
64  T1 a_minus_one_minus_k = a - 1;
65  T1 fac = a_minus_one_minus_k; // falling_factorial(a-1, k)
66  T1 dfac = 1; // d/da[falling_factorial(a-1, k)]
67  T2 zpow = z; // z ** k
68  TP delta = dfac / zpow;
69 
70  for (int k = 1; k < 10; ++k) {
71  a_minus_one_minus_k -= 1;
72 
73  S += delta;
74 
75  zpow *= z;
76  dfac = a_minus_one_minus_k * dfac + fac;
77  fac *= a_minus_one_minus_k;
78  delta = dfac / zpow;
79 
80  if (is_inf(delta))
81  domain_error("grad_reg_inc_gamma", "is not converging", "", "");
82  }
83 
84  return gamma_q(a, z) * (l - dig) + exp(-z + (a - 1) * l) * S / g;
85  } else {
86  // gradient of series expansion http://dlmf.nist.gov/8.7#E3
87  TP S = 0;
88  TP log_s = 0.0;
89  double s_sign = 1.0;
90  T2 log_z = log(z);
91  TP log_delta = log_s - multiply_log(2, a);
92  for (int k = 1; k <= max_steps; ++k) {
93  S += s_sign >= 0.0 ? exp(log_delta) : -exp(log_delta);
94  log_s += log_z - log(k);
95  s_sign = -s_sign;
96  log_delta = log_s - multiply_log(2, k + a);
97  if (is_inf(log_delta))
98  domain_error("grad_reg_inc_gamma", "is not converging", "", "");
99  if (log_delta <= log(precision))
100  return gamma_p(a, z) * (dig - l) + exp(a * l) * S / g;
101  }
102  domain_error("grad_reg_inc_gamma", "k (internal counter)", max_steps,
103  "exceeded ",
104  " iterations, gamma function gradient did not converge.");
105  return INFTY;
106  }
107 }
108 
109 } // namespace math
110 } // namespace stan
111 #endif
fvar< T > fabs(const fvar< T > &x)
Definition: fabs.hpp:15
fvar< T > log(const fvar< T > &x)
Definition: log.hpp:12
bool is_any_nan(const T &x)
Returns true if the input is NaN and false otherwise.
Definition: is_any_nan.hpp:21
return_type< T1, T2 >::type grad_reg_inc_gamma(T1 a, T2 z, T1 g, T1 dig, double precision=1e-6, int max_steps=1e5)
Gradient of the regularized incomplete gamma functions igamma(a, z)
boost::math::tools::promote_args< double, typename scalar_type< T >::type, typename return_type< Types_pack... >::type >::type type
Definition: return_type.hpp:36
fvar< T > exp(const fvar< T > &x)
Definition: exp.hpp:11
void domain_error(const char *function, const char *name, const T &y, const char *msg1, const char *msg2)
Throw a domain error with a consistently formatted message.
int is_inf(const fvar< T > &x)
Returns 1 if the input&#39;s value is infinite and 0 otherwise.
Definition: is_inf.hpp:20
fvar< T > gamma_p(const fvar< T > &x1, const fvar< T > &x2)
Definition: gamma_p.hpp:15
double e()
Return the base of the natural logarithm.
Definition: constants.hpp:87
fvar< T > multiply_log(const fvar< T > &x1, const fvar< T > &x2)
const double INFTY
Positive infinity.
Definition: constants.hpp:48
fvar< T > gamma_q(const fvar< T > &x1, const fvar< T > &x2)
Definition: gamma_q.hpp:13

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