NAG Library Routine Document

s14aef (psi_deriv_real)


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


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 <nagmk26.h>
double  s14aef_ (const double *x, const Integer *k, Integer *ifail)


s14aef evaluates an approximation to the kth derivative of the psi function ψx given by
ψ k x=dkdxk ψx=dkdxk ddx logeΓx ,  
where x is real with x0,-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 -1k+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).


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


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 – IntegerInput
On entry: the function ψkx to be evaluated.
Constraint: 0k6.
3:     ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1 or 1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1 or 1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this argument, the recommended value is 0. 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).

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:
On entry, k=value.
Constraint: k6.
On entry, k=value.
Constraint: k0.
On entry, x is ‘too close’ to a non-positive integer: x=value and nintx=value.
Evaluation abandoned due to likelihood of underflow.
Evaluation abandoned due to likelihood of overflow.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.


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=mint,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.

Parallelism and Performance

s14aef is not threaded in any implementation.

Further Comments



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

Program Text

Program Text (s14aefe.f90)

Program Data

Program Data (s14aefe.d)

Program Results

Program Results (s14aefe.r)