gammainc

→Tensor

gammainc(input: Tensor, other: Tensor)

source edit

Regularized lower incomplete gamma function $P(a, x)$ .

P(a, x) = \frac{1}{\Gamma(a)} \int_0^x t^{a-1} e^{-t}\, dt

The CDF of a Gamma(a, 1) variable at x — what lucid.distributions.Gamma.cdf evaluates.

Parameters

inputTensor

Shape

a \geq 0

otherTensor

Upper limit

x \geq 0

; broadcast against input.

Returns

Tensor

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

Raises

NotImplementedError

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

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

Notes

Evaluated by the algorithm of Cephes' igam as SciPy refines it: the power series below $x \approx a$ , 1 - Q by the continued fraction above, and Temme's uniform asymptotic expansion where a is large and x near it ( $20 < a < 200$ with $|x - a|/a < 0.3$ , or $a > 200$ with $|x - a|/a < 4.5/\sqrt{a}$ ). The edges follow the reference framework: $P(a, 0) = 0$ , $P(a, \infty) = 1$ , $P(0, x) = 1$ for $x > 0$ , $P(\infty, x) = 0$ for a finite x; a negative argument, $P(0, 0)$ , $P(\infty, \infty)$ and NaN answer NaN.

Examples

>>> import lucid
>>> from lucid.special import gammainc
>>> gammainc(lucid.tensor([1.0, 2.0, 3.0]), lucid.tensor([1.0, 2.0, 3.0]))
tensor([0.6321, 0.594, 0.5768])

Used by 1

lucid.special