Skip to contents

Gamma Functions

Source: R/gamma_functions.R
gamma_functions.Rd

Functions to compute the gamma function, its logarithm, digamma, trigamma, polygamma, and various incomplete gamma functions.

Gamma Function Gamma(z):

  • tgamma(z): Computes \(\Gamma(z)\), the true gamma function

  • tgamma1pm1(z): Computes \(\Gamma(1+z) - 1\) with enhanced numerical stability for very small z values, avoiding precision loss

  • lgamma_boost(z): Returns \(\log(\Gamma(z))\), the logarithm of the gamma function

Derivative Functions:

  • digamma_boost(z): The digamma function, the first derivative of the logarithm of the Gamma function \(\psi(z) = \frac{d}{dz}\log\Gamma(z) = \frac{\Gamma'(z)}{\Gamma(z)}\),

  • trigamma_boost(z): The trigamma function, the second derivative of the logarithm of the Gamma function \(\psi_1(z) = \frac{d^2}{dz^2}\log\Gamma(z)\)

  • polygamma(n, z): The nth derivative of the digamma function (n-th order polygamma) \(\psi^{(m)}(z) = \frac{d^m}{dz^m}\psi(z)\)

Ratios:

  • tgamma_ratio(a, b): Computes \(\Gamma(a)/\Gamma(b)\)

  • tgamma_delta_ratio(a, delta): Computes \(\Gamma(a)/\Gamma(a + \delta)\)

Incomplete Gamma Functions:

These functions require a > 0 and z >= 0.

  • Normalised (Regularised) Functions (return values in \([0, 1]\)):

    • gamma_p(a, z): Normalised lower incomplete gamma \(P(a, z) = \frac{\gamma(a, z)}{\Gamma(z)}\)

    • gamma_q(a, z): Normalised upper incomplete gamma \(Q(a, z) = \frac{\Gamma(a, z)}{\Gamma(a)}\)

  • Non-normalised Functions (return values in \([0, Gamma(a)]\)):

    • tgamma_lower(a, z): Full lower incomplete gamma function \(\gamma(a, z) = \int_{0}^z t^{a-1}e^{-t}dt\)

    • tgamma_upper(a, z): Full upper incomplete gamma function \(\Gamma(a, z) = \int_{z}^\infty t^{a-1}e^{-t}dt\)

Inverse Functions:

  • gamma_p_inv(a, p): Returns \(z\) such that \(p = P(a, z)\)

  • gamma_q_inv(a, q): Returns \(z\) such that \(q = Q(a, z)\)

  • gamma_p_inva(z, p): Returns \(a\) such that \(p = P(a, z)\)

  • gamma_q_inva(z, q): Returns \(a\) such that \(q = Q(a, z)\)

Derivative:

  • gamma_p_derivative(a, z): Computes the derivative of the normalised lower incomplete gamma function with respect to z: \(\frac{\partial}{\partial z}P(a, z) = \frac{e^{-z}z^{a-1}}{\Gamma(a)}\)

Usage

tgamma(z)

tgamma1pm1(z)

lgamma_boost(z)

digamma_boost(z)

trigamma_boost(z)

polygamma(n, z)

tgamma_ratio(a, b)

tgamma_delta_ratio(a, delta)

gamma_p(a, z)

gamma_q(a, z)

tgamma_lower(a, z)

tgamma_upper(a, z)

gamma_q_inv(a, q)

gamma_p_inv(a, p)

gamma_q_inva(z, q)

gamma_p_inva(z, p)

gamma_p_derivative(a, z)

Arguments

z

Input numeric value for the gamma function

n

Order of the polygamma function (non-negative integer)

a

Argument for the incomplete gamma functions (must be positive)

b

Denominator argument for the ratio of gamma functions

delta

Increment for the ratio of gamma functions

q

Probability value for the incomplete gamma functions (0 <= q <= 1)

p

Probability value for the incomplete gamma functions (0 <= p <= 1)

Value

A single numeric value with the computed gamma function, logarithm, digamma, trigamma, polygamma, or incomplete gamma functions.

See also

Boost Documentation for more details on the mathematical background.

Examples

if (FALSE) { # \dontrun{
# Gamma function for z = 5
tgamma(5)
# Gamma function for (1 + z) - 1, where z = 5
tgamma1pm1(5)
# Logarithm of the gamma function for z = 5
lgamma_boost(5)
# Digamma function for z = 5
digamma_boost(5)
# Trigamma function for z = 5
trigamma_boost(5)
# Polygamma function of order 1 for z = 5
polygamma(1, 5)
# Ratio of gamma functions for a = 5, b = 3
tgamma_ratio(5, 3)
# Ratio of gamma functions with delta for a = 5, delta = 2
tgamma_delta_ratio(5, 2)
# Normalised lower incomplete gamma function P(a, z) for a = 5, z = 2
gamma_p(5, 2)
# Normalised upper incomplete gamma function Q(a, z) for a = 5, z = 2
gamma_q(5, 2)
# Full lower incomplete gamma function for a = 5, z = 2
tgamma_lower(5, 2)
# Full upper incomplete gamma function for a = 5, z = 2
tgamma_upper(5, 2)
# Inverse of the normalised upper incomplete gamma function for a = 5, q = 0.5
gamma_q_inv(5, 0.5)
# Inverse of the normalised lower incomplete gamma function for a = 5, p = 0.5
gamma_p_inv(5, 0.5)
# Inverse of the normalised upper incomplete gamma function with respect to a for z = 2, q = 0.5
gamma_q_inva(2, 0.5)
# Inverse of the normalised lower incomplete gamma function with respect to a for z = 2, p = 0.5
gamma_p_inva(2, 0.5)
# Derivative of the normalised lower incomplete gamma function for a = 5, z = 2
gamma_p_derivative(5, 2)
} # }