naginterfaces.library.specfun.polygamma_​deriv

naginterfaces.library.specfun.polygamma_deriv(x, n, m)

polygamma_deriv returns a sequence of values of scaled derivatives of the psi function $\psi \left(x\right)$ (also known as the digamma function).

For full information please refer to the NAG Library document for s14ad

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s14adf.html

Parameters

xfloat

The argument $x$ of the function.

nint

The index of the first member $n$ of the sequence of functions.

mint

The number of members $m$ required in the sequence $w(\text{k},x)$, for $\text{k}=n,\dots ,n+m-1$.

Returns

ansfloat, ndarray, shape $\left(m\right)$

The first $m$ elements of $ans$ contain the required values $w(\text{k},x)$, for $\text{k}=n,\dots ,n+m-1$.

Raises

NagValueError

(errno $1$)

On entry, $x=⟨value⟩$.

Constraint: $x>0.0$.

(errno $2$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

(errno $3$)

On entry, $m=⟨value⟩$.

Constraint: $m\ge 1$.

(errno $4$)

Computation abandoned due to the likelihood of underflow.

(errno $5$)

Computation abandoned due to the likelihood of overflow.

(errno $6$)

There is not enough internal workspace to continue computation. $m$ is probably too large.

Notes

polygamma_deriv computes $m$ values of the function

$$w(k,x)=\frac{{(-1)}^{k+1}{\psi }^{\left(k\right)}\left(x\right)}{k!}\text{,}$$

for $x>0$, $k=n$, $n+1,\dots ,n+m-1$, where $\psi$ is the psi function

$$\psi \left(x\right)=\frac{d}{dx}ln\left(\Gamma \right)\left(x\right)=\frac{{\Gamma }^{\prime }\left(x\right)}{\Gamma \left(x\right)}\text{,}$$

and ${\psi }^{\left(k\right)}$ denotes the $k$th derivative of $\psi$.

The function is derived from the function PSIFN in Amos (1983). The basic method of evaluation of $w(k,x)$ is the asymptotic series

$$w(k,x)\sim ϵ(k,x)+\frac{1}{2x^{k+1}}+\frac{1}{x^k}\sum _{j=1}^{\infty }{B}_{2j}\frac{(2j+k-1)!}{\left(2j\right)!k!x^{2j}}$$

for large $x$ greater than a machine-dependent value $x_{min}$, followed by backward recurrence using

$$w(k,x)=w(k,x+1)+x^{-k-1}$$

for smaller values of $x$, where $ϵ(k,x)=-ln\left(x\right)$ when $k=0$, $ϵ(k,x)=\frac{1}{k{x}^k}$ when $k>0$, and $B_{2j}$, $j=1,2,\dots$, are the Bernoulli numbers.

When $k$ is large, the above procedure may be inefficient, and the expansion

$$w(k,x)=\sum _{j=1}^{\infty }\frac{1}{{(x+j)}^{k+1}}\text{,}$$

which converges rapidly for large $k$, is used instead.

References

NIST Digital Library of Mathematical Functions

Amos, D E, 1983, Algorithm 610: A portable FORTRAN subroutine for derivatives of the psi function, ACM Trans. Math. Software (9), 494–502