i1e

→Tensor

i1e(x: Tensor)

source

Exponentially scaled modified Bessel function of order 1.

Computes $I_1(x)\, e^{-|x|}$ , the bounded counterpart of i1. Because $I_1$ itself grows exponentially with $|x|$ , this scaled form is the safer building block for log-likelihood expressions (Rice / Skellam distributions and similar) where evaluating I_1 directly would overflow.

Parameters

xTensor

Input tensor; any floating-point dtype.

Returns

Tensor

$I_1(x)\, e^{-|x|}$ element-wise, same shape and dtype as x.

Notes

Definition:

I_1(x)\, e^{-|x|}.

The function is odd in x, equals 0 at x = 0, and approaches $1/\sqrt{2\pi x}$ as $x \to \infty$ . Together with i0e this forms the standard scaled-Bessel pair for stable evaluation of rotational-symmetric likelihoods.

Examples

>>> import lucid
>>> from lucid.special import i1e
>>> i1e(lucid.tensor([0.0, 1.0, 5.0]))
Tensor([0.0000, 0.2079, 0.1640])