Stan Math Library  2.15.0
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 
10 #include <cmath>
11 #include <limits>
12 
13 namespace stan {
14  namespace math {
15 
44  template<typename T>
45  T grad_reg_inc_gamma(T a, T z, T g, T dig, double precision = 1e-6,
46  int max_steps = 1e5) {
47  using std::domain_error;
48  using std::exp;
49  using std::fabs;
50  using std::log;
51 
52  if (is_nan(a)) return std::numeric_limits<T>::quiet_NaN();
53  if (is_nan(z)) return std::numeric_limits<T>::quiet_NaN();
54  if (is_nan(g)) return std::numeric_limits<T>::quiet_NaN();
55  if (is_nan(dig)) return std::numeric_limits<T>::quiet_NaN();
56 
57  T l = log(z);
58  if (z >= a && z >= 8) {
59  // asymptotic expansion http://dlmf.nist.gov/8.11#E2
60  T S = 0;
61  T a_minus_one_minus_k = a - 1;
62  T fac = a_minus_one_minus_k; // falling_factorial(a-1, k)
63  T dfac = 1; // d/da[falling_factorial(a-1, k)]
64  T zpow = z; // z ** k
65  T delta = dfac / zpow;
66 
67  for (int k = 1; k < 10; ++k) {
68  a_minus_one_minus_k -= 1;
69 
70  S += delta;
71 
72  zpow *= z;
73  dfac = a_minus_one_minus_k * dfac + fac;
74  fac *= a_minus_one_minus_k;
75  delta = dfac / zpow;
76 
77  if (is_inf(delta))
78  stan::math::domain_error("grad_reg_inc_gamma",
79  "is not converging", "", "");
80  }
81 
82  return gamma_q(a, z) * (l - dig) + exp(-z + (a - 1) * l) * S / g;
83  } else {
84  // gradient of series expansion http://dlmf.nist.gov/8.7#E3
85 
86  T S = 0;
87  T log_s = 0.0;
88  double s_sign = 1.0;
89  T log_z = log(z);
90  T log_delta = log_s - 2 * log(a);
91  for (int k = 1; k <= max_steps; ++k) {
92  S += s_sign >= 0.0 ? exp(log_delta) : -exp(log_delta);
93  log_s += log_z - log(k);
94  s_sign = -s_sign;
95  log_delta = log_s - 2 * log(k + a);
96  if (is_inf(log_delta))
97  stan::math::domain_error("grad_reg_inc_gamma",
98  "is not converging", "", "");
99  if (log_delta <= log(precision))
100  return gamma_p(a, z) * ( dig - l ) + exp( a * l ) * S / g;
101  }
102  stan::math::domain_error("grad_reg_inc_gamma",
103  "k (internal counter)",
104  max_steps, "exceeded ",
105  " iterations, gamma function gradient did not converge.");
106  return std::numeric_limits<T>::infinity();
107  }
108  }
109 
110  }
111 }
112 #endif
fvar< T > fabs(const fvar< T > &x)
Definition: fabs.hpp:15
fvar< T > log(const fvar< T > &x)
Definition: log.hpp:14
fvar< T > exp(const fvar< T > &x)
Definition: exp.hpp:10
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:21
fvar< T > gamma_p(const fvar< T > &x1, const fvar< T > &x2)
Definition: gamma_p.hpp:14
T grad_reg_inc_gamma(T a, T z, T g, T dig, double precision=1e-6, int max_steps=1e5)
Gradient of the regularized incomplete gamma functions igamma(a, z)
double e()
Return the base of the natural logarithm.
Definition: constants.hpp:94
int is_nan(const fvar< T > &x)
Returns 1 if the input&#39;s value is NaN and 0 otherwise.
Definition: is_nan.hpp:21
fvar< T > gamma_q(const fvar< T > &x1, const fvar< T > &x2)
Definition: gamma_q.hpp:14

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