gammaincc

→Tensor

gammaincc(input: Tensor, other: Tensor)

source edit

Regularized upper incomplete gamma function $Q(a, x)$ .

Q(a, x) = \frac{1}{\Gamma(a)} \int_x^\infty t^{a-1} e^{-t}\, dt = 1 - P(a, x)

Evaluated directly rather than as 1 - gammainc, so a small upper tail keeps its relative precision.

Parameters

inputTensor

Shape

a \geq 0

otherTensor

Lower limit

x \geq 0

; broadcast against input.

Returns

Tensor

$Q(a, x) \in [0, 1]$ , in the promoted dtype of the two.

Raises

NotImplementedError

From backward, when input requires a gradient: only other is differentiable, with

\partial Q / \partial x = -x^{a-1} e^{-x} / \Gamma(a)

Notes

The branches and edge values mirror gammainc's: $Q(a, 0) = 1$ , $Q(a, \infty) = 0$ , $Q(0, x) = 0$ for $x > 0$ , $Q(\infty, x) = 1$ for a finite x. For a tiny shape the reference framework evaluates $\ln\Gamma(1 + a)$ after rounding $1 + a$ to 1 and loses the Euler-Mascheroni term: at $a = 10^{-30}$ its value is 4.4% above the one mpmath computes at $x = 10^{-6}$ , 103% above at $x = 0.5$ and 286% above at $x = 1.05$ . Here that term is kept, and the value agrees with mpmath's to float64 rounding.

Examples

>>> import lucid
>>> from lucid.special import gammaincc
>>> gammaincc(lucid.tensor([1.0, 2.0]), lucid.tensor([1.0, 10.0]))
tensor([0.3679, 0.0004994])

Used by 1

lucid.special