polygamma

→Tensor

polygamma(n: int, x: Tensor)

source

Polygamma function $\psi^{(n)}(x)$ .

The n-th derivative of the digamma function. Polygamma functions appear in the derivatives of log-partition functions of the Gamma / Beta / Dirichlet families and are needed for evaluating Fisher information matrices for these distributions analytically.

Parameters

nint

Non-negative integer order. Only

n \in \{0, 1, 2, 3\}

are supported; n = 0 recovers lucid.digamma, n = 1 is the trigamma function. Higher orders raise NotImplementedError.

xTensor

Real argument; any floating-point dtype.

Returns

Tensor

$\psi^{(n)}(x)$ element-wise, same shape and dtype as x.

Notes

Definition (for n >= 1):

\psi^{(n)}(x) = \frac{d^{n+1}}{dx^{n+1}} \log \Gamma(x) = (-1)^{n+1}\, n!\, \sum_{k=0}^\infty \frac{1}{(x + k)^{n+1}}.

Implementation: shift x upward by $K = 6$ using the recurrence

\psi^{(n)}(x) = \psi^{(n)}(x + 1) + \frac{(-1)^{n+1} n!}{x^{n+1}},

accumulate the per-step corrections, then evaluate the Bernoulli asymptotic series at the shifted argument where it is well-conditioned. Accuracy is roughly seven decimal digits across the positive-real regime.

Raises ValueError for n < 0 and NotImplementedError for n >= 4.

Examples

>>> import lucid
>>> from lucid.special import polygamma
>>> polygamma(1, lucid.tensor([1.0, 2.0, 5.0]))
Tensor([1.6449, 0.6449, 0.2213])