The routine may be called by the names s14aff or nagf_specfun_psi_deriv_complex.
s14aff evaluates an approximation to the th derivative of the psi function given by
where is complex provided and . If , is real and thus is singular when .
Note that is also known as the polygamma function. Specifically, is often referred to as the digamma function and 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 when 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: – 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: – IntegerInput
On entry: the function to be evaluated.
3: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, .
On entry, .
On entry, is ‘too close’ to a non-positive integer when : , .
Evaluation abandoned due to likelihood of overflow.
An unexpected error has been triggered by this routine. Please
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
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.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
Empirical tests have shown that the maximum relative error is a loss of approximately two decimal places of precision.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
s14aff is not threaded in any implementation.
This example evaluates the psi (trigamma) function at , and prints the results.