NAG Library Routine Document
S14AFF returns the value of the th derivative of the psi function for complex and , via the function name.
|COMPLEX (KIND=nag_wp) S14AFF
S14AFF evaluates an approximation to the
th derivative of the psi function
is complex provided
is real and thus
is singular when
is also known as the polygamma
is often referred to as the digamma
as the trigamma
function in the literature. Further details can be found in Abramowitz and Stegun (1972)
S14AFF is based on a modification of the method proposed by Kölbig (1972)
To obtain the value of
is real, S14AEF
can be used.
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
- 1: Z – COMPLEX (KIND=nag_wp)Input
On entry: the argument of the function.
must not be ‘too close’ (see Section 6
) to a non-positive integer when
- 2: K – INTEGERInput
On entry: the function to be evaluated.
- 3: IFAIL – INTEGERInput/Output
must be set to
. If you are unfamiliar with this parameter you should refer to Section 3.3
in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Errors or warnings detected by the routine:
|or|| is ‘too close’ to a non-positive integer when . That is, .|
The evaluation has been abandoned due to the likelihood of overflow. The result is returned as zero.
Empirical tests have shown that the maximum relative error is a loss of approximately two decimal places of precision.
This example evaluates the psi (trigamma) function at , and prints the results.
9.1 Program Text
Program Text (s14affe.f90)
9.2 Program Data
Program Data (s14affe.d)
9.3 Program Results
Program Results (s14affe.r)