The routine may be called by the names s14acf or nagf_specfun_polygamma.
s14acf returns a value of the function . The psi function is computed without the logarithmic term so that when is large, sums or differences of psi functions may be computed without unnecessary loss of precision, by analytically combining the logarithmic terms. For example, the difference has an asymptotic behaviour for large given by .
Computing directly would amount to subtracting two large numbers which are close to and to produce a small number close to , resulting in a loss of significant digits. However, using this routine to compute , we can compute , and the dominant logarithmic term may be computed accurately from its power series when is large. Thus we avoid the unnecessary loss of precision.
The routine is derived from the routine PSIFN in Amos (1983).
Amos D E (1983) Algorithm 610: A portable FORTRAN subroutine for derivatives of the psi function ACM Trans. Math. Software9 494–502
1: – Real (Kind=nag_wp)Input
On entry: the argument of the function.
2: – 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, .
Computation halted due to likelihood of underflow. x may be too large. .
Computation halted due to likelihood of overflow. x may be too small. .
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.
All constants in s14acf are given to approximately digits of precision. Calling the number of digits of precision in the floating-point arithmetic being used , then clearly the maximum number of correct digits in the results obtained is limited by .
With the above proviso, results returned by this routine should be accurate almost to full precision, except at points close to the zero of , , where only absolute rather than relative accuracy can be obtained.
8Parallelism and Performance
s14acf is not threaded in any implementation.
The example program reads values of the argument from a file, evaluates the function at each value of and prints the results.