The function may be called by the names: s14acc, nag_specfun_polygamma or nag_polygamma_fun.
s14acc 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 function 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 function is derived from the function 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: – doubleInput
On entry: the argument of the function.
2: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL 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 CL Interface for further information.
Computation halted due to likelihood of overflow. x may be too small. .
On entry, .
Computation halted due to likelihood of underflow. x may be too large. .
All constants in s14acc 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 function 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
s14acc 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.