naginterfaces.library.specfun.psi_​deriv_​real

naginterfaces.library.specfun.psi_deriv_real(x, k)

psi_deriv_real returns the value of the $k$th derivative of the psi function $\psi \left(x\right)$ for real $x$ and $k=0,1,\dots ,6$.

For full information please refer to the NAG Library document for s14ae

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/s/s14aef.html

Parameters

xfloat

The argument $x$ of the function.

kint

The function ${\psi }^{\left(k\right)}\left(x\right)$ to be evaluated.

Returns

pkxfloat

The value of the $k$th derivative of the psi function $\psi \left(x\right)$ for real $x$ and $k=0,1,\dots ,6$.

Raises

NagValueError

(errno $1$)

On entry, $x$ is ‘too close’ to a non-positive integer: $x=⟨value⟩$ and $nint\left(x\right)=⟨value⟩$.

(errno $1$)

On entry, $k=⟨value⟩$.

Constraint: $k\le 6$.

(errno $1$)

On entry, $k=⟨value⟩$.

Constraint: $k\ge 0$.

(errno $2$)

Evaluation abandoned due to likelihood of underflow.

(errno $3$)

Evaluation abandoned due to likelihood of overflow.

Notes

psi_deriv_real evaluates an approximation to the $k$th derivative of the psi function $\psi \left(x\right)$ given by

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

where $x$ is real with $x\ne 0,-1,-2,\dots$ and $k=0,1,\dots ,6$. For negative noninteger values of $x$, the recurrence relationship

$${\psi }^{\left(k\right)}(x+1)={\psi }^{\left(k\right)}\left(x\right)+\frac{d^k}{d{x}^k}\left(\frac{1}{x}\right)$$

is used. The value of $\frac{{(-1)}^{k+1}{\psi }^{\left(k\right)}\left(x\right)}{k!}$ is obtained by a call to polygamma_deriv(), which is based on the function PSIFN in Amos (1983).

Note that ${\psi }^{\left(k\right)}\left(x\right)$ is also known as the polygamma function. Specifically, ${\psi }^{\left(0\right)}\left(x\right)$ is often referred to as the digamma function and ${\psi }^{\left(1\right)}\left(x\right)$ as the trigamma function in the literature. Further details can be found in Abramowitz and Stegun (1972).

References

Abramowitz, M and Stegun, I A, 1972, Handbook of Mathematical Functions, (3rd Edition), Dover Publications

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