Stan Math Library  2.12.0
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 
9 #include <cmath>
10 
11 namespace stan {
12  namespace math {
13 
39  template<typename T>
40  T grad_reg_inc_gamma(T a, T z, T g, T dig, double precision = 1e-6) {
41  using std::domain_error;
42  using std::exp;
43  using std::fabs;
44  using std::log;
45 
46  T l = log(z);
47  if (z >= a && z >= 8) {
48  // asymptotic expansion http://dlmf.nist.gov/8.11#E2
49  T S = 0;
50  T a_minus_one_minus_k = a - 1;
51  T fac = a_minus_one_minus_k; // falling_factorial(a-1, k)
52  T dfac = 1; // d/da[falling_factorial(a-1, k)]
53  T zpow = z; // z ** k
54  T delta = dfac / zpow;
55 
56  for (int k = 1; k < 10; ++k) {
57  a_minus_one_minus_k -= 1;
58 
59  S += delta;
60 
61  zpow *= z;
62  dfac = a_minus_one_minus_k * dfac + fac;
63  fac *= a_minus_one_minus_k;
64  delta = dfac / zpow;
65 
66  if (is_inf(delta))
67  stan::math::domain_error("grad_reg_inc_gamma",
68  "is not converging", "", "");
69  }
70 
71  return gamma_q(a, z) * (l - dig) + exp(-z + (a - 1) * l) * S / g;
72  } else {
73  // gradient of series expansion http://dlmf.nist.gov/8.7#E3
74 
75  T S = 0;
76  T s = 1;
77  int k = 0;
78  T delta = s / square(a);
79  while (fabs(delta) > precision) {
80  S += delta;
81  ++k;
82  s *= - z / k;
83  delta = s / square(k + a);
84  if (is_inf(delta))
85  stan::math::domain_error("grad_reg_inc_gamma",
86  "is not converging", "", "");
87  }
88  return gamma_p(a, z) * ( dig - l ) + exp( a * l ) * S / g;
89  }
90  }
91 
92  }
93 }
94 #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 > square(const fvar< T > &x)
Definition: square.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'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
double e()
Return the base of the natural logarithm.
Definition: constants.hpp:94
T grad_reg_inc_gamma(T a, T z, T g, T dig, double precision=1e-6)
Gradient of the regularized incomplete gamma functions igamma(a, z)
fvar< T > gamma_q(const fvar< T > &x1, const fvar< T > &x2)
Definition: gamma_q.hpp:14

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