NAG Library Manual, Mark 27.3
Interfaces:  FL   CL   CPP   AD 
NAG CL Interface Introduction
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S (Specfun) Chapter Introduction
s14af:  FL   CL   CPP   AD 

NAG CL Interface
s14afc (psi_​deriv_​complex)

NAG Library Manual, Mark 27.3
Interfaces:  FL   CL   CPP   AD 
NAG CL Interface Introduction
S (Specfun) Chapter Contents
S (Specfun) Chapter Introduction
s14af:  FL   CL   CPP   AD 

▸▿ Contents

© The Numerical Algorithms Group Ltd. 2021
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1 Purpose

s14afc returns the value of the kth derivative of the psi function ψ(z) for complex z and k=0,1,…,4.

2 Specification

#include <nag.h>
Complex  s14afc (Complex z, Integer k, NagError *fail)
The function may be called by the names: s14afc, nag_specfun_psi_deriv_complex or nag_complex_polygamma.

3 Description

s14afc evaluates an approximation to the kth derivative of the psi function ψ(z) given by
ψ (k) (z)=dkdzk ψ(z)=dkdzk (ddz loge⁡Γ(z)) ,  
where z=x+iy is complex provided y≠0 and k=0,1,…,4. If y=0, z is real and thus ψ (k) (z) is singular when z=0,-1,-2,….
Note that ψ (k) (z) is also known as the polygamma function. Specifically, ψ (0) (z) is often referred to as the digamma function and ψ (1) (z) as the trigamma function in the literature. Further details can be found in Abramowitz and Stegun (1972).
s14afc is based on a modification of the method proposed by Kölbig (1972).
To obtain the value of ψ (k) (z) when z is real, s14aec can be used.

4 References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Kölbig K S (1972) Programs for computing the logarithm of the gamma function, and the digamma function, for complex arguments Comp. Phys. Comm. 4 221–226

5 Arguments

1: z – Complex Input
On entry: the argument z of the function.
Constraint: z.re must not be ‘too close’ (see Section 6) to a non-positive integer when z.im=0.0.
2: k – Integer Input
On entry: the function ψ(k)(z) to be evaluated.
Constraint: 0≤k≤4.
3: fail – NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_COMPLEX
On entry, z.re is ‘too close’ to a non-positive integer when z.im=0.0: z.re=⟨value⟩, nint(z.re)=⟨value⟩.
NE_INT
On entry, k=⟨value⟩.
Constraint: k≤4.
On entry, k=⟨value⟩.
Constraint: k≥0.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_OVERFLOW_LIKELY
Evaluation abandoned due to likelihood of overflow.

7 Accuracy

Empirical tests have shown that the maximum relative error is a loss of approximately two decimal places of precision.

8 Parallelism and Performance

s14afc is not threaded in any implementation.

9 Further Comments

None.

10 Example

This example evaluates the psi (trigamma) function ψ (1) (z) at z=-1.5+2.5i, and prints the results.

10.1 Program Text

Program Text (s14afce.c)

10.2 Program Data

Program Data (s14afce.d)

10.3 Program Results

Program Results (s14afce.r)
NAG Library Manual, Mark 27.3
Interfaces:  FL   CL   CPP   AD 
NAG CL Interface Introduction
S (Specfun) Chapter Contents
S (Specfun) Chapter Introduction
s14af:  FL   CL   CPP   AD 
© The Numerical Algorithms Group Ltd. 2021