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s14ae:  FL   CL   CPP   AD 

NAG FL Interface
s14aef (psi_​deriv_​real)

NAG Library Manual, Mark 27.3
Interfaces:  FL   CL   CPP   AD 
NAG FL Interface Introduction
S (Specfun) Chapter Contents
S (Specfun) Chapter Introduction
s14ae:  FL   CL   CPP   AD 

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© The Numerical Algorithms Group Ltd. 2021
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1 Purpose

s14aef returns the value of the kth derivative of the psi function ψ(x) for real x and k=0,1,…,6, via the function name.

2 Specification

Fortran Interface
Function s14aef ( x, k, ifail)
Real (Kind=nag_wp) :: s14aef
Integer, Intent (In) :: k
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x
C Header Interface
#include <nag.h>
double  s14aef_ (const double *x, const Integer *k, Integer *ifail)
The routine may be called by the names s14aef or nagf_specfun_psi_deriv_real.

3 Description

s14aef evaluates an approximation to the kth derivative of the psi function ψ(x) given by
ψ (k) (x)=dkdxk ψ(x)=dkdxk (ddx loge⁡Γ(x)) ,  
where x is real with x≠0,-1,-2,… and k=0,1,…,6. For negative noninteger values of x, the recurrence relationship
ψ (k) (x+1)=ψ (k) (x)+dkdxk (1x)  
is used. The value of (-1)k+1ψ (k) (x) k! is obtained by a call to s14adf, which is based on the routine PSIFN in Amos (1983).
Note that ψ (k) (x) is also known as the polygamma function. Specifically, ψ (0) (x) is often referred to as the digamma function and ψ (1) (x) as the trigamma function in the literature. Further details can be found in Abramowitz and Stegun (1972).

4 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

5 Arguments

1: x – Real (Kind=nag_wp) Input
On entry: the argument x of the function.
Constraint: x must not be ‘too close’ (see Section 6) to a non-positive integer.
2: k – Integer Input
On entry: the function ψ(k)(x) to be evaluated.
Constraint: 0≤k≤6.
3: ifail – Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
ifail=1
On entry, k=⟨value⟩.
Constraint: k≤6.
On entry, k=⟨value⟩.
Constraint: k≥0.
On entry, x is ‘too close’ to a non-positive integer: x=⟨value⟩ and nint(x)=⟨value⟩.
ifail=2
Evaluation abandoned due to likelihood of underflow.
ifail=3
Evaluation abandoned due to likelihood of overflow.
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
ifail=-999
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

All constants in s14adf are given to approximately 18 digits of precision. If t denotes the number of digits of precision in the floating-point arithmetic being used, then clearly the maximum number in the results obtained is limited by p=min(t,18). Empirical tests by Amos (1983) have shown that the maximum relative error is a loss of approximately two decimal places of precision. Further tests with the function -ψ (0) (x) have shown somewhat improved accuracy, except at points near the positive zero of ψ (0) (x) at x=1.46…, where only absolute accuracy can be obtained.

8 Parallelism and Performance

s14aef is not threaded in any implementation.

9 Further Comments

None.

10 Example

This example evaluates ψ (2) (x) at x=2.5, and prints the results.

10.1 Program Text

Program Text (s14aefe.f90)

10.2 Program Data

Program Data (s14aefe.d)

10.3 Program Results

Program Results (s14aefe.r)
NAG Library Manual, Mark 27.3
Interfaces:  FL   CL   CPP   AD 
NAG FL Interface Introduction
S (Specfun) Chapter Contents
S (Specfun) Chapter Introduction
s14ae:  FL   CL   CPP   AD 
© The Numerical Algorithms Group Ltd. 2021