NAG FL Interface
s14acf (polygamma)

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1 Purpose

s14acf returns a value of the function ψ(x)-lnx, where ψ is the psi function ψ(x)= ddx lnΓ(x)= Γ(x) Γ(x) .

2 Specification

Fortran Interface
Function s14acf ( x, ifail)
Real (Kind=nag_wp) :: s14acf
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x
C Header Interface
#include <nag.h>
double  s14acf_ (const double *x, Integer *ifail)
The routine may be called by the names s14acf or nagf_specfun_polygamma.

3 Description

s14acf returns a value of the function ψ(x)-lnx. The psi function is computed without the logarithmic term so that when x 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 d=ψ (x+12)-ψ(x) has an asymptotic behaviour for large x given by dln(x+12)-lnx+O( 1x2)ln(1+ 12x ) 12x .
Computing d directly would amount to subtracting two large numbers which are close to ln(x+12) and lnx to produce a small number close to 12x , resulting in a loss of significant digits. However, using this routine to compute f(x)=ψ(x)-lnx, we can compute d=f (x+12)-f(x)+ln(1+12x ) , and the dominant logarithmic term may be computed accurately from its power series when x is large. Thus we avoid the unnecessary loss of precision.
The routine is derived from the routine PSIFN in Amos (1983).

4 References

NIST Digital Library of Mathematical Functions
Amos D E (1983) Algorithm 610: A portable FORTRAN subroutine for derivatives of the psi function ACM Trans. Math. Software 9 494–502

5 Arguments

1: x Real (Kind=nag_wp) Input
On entry: the argument x of the function.
Constraint: x>0.0.
2: ifail Integer Input/Output
On entry: ifail must be set to 0, -1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of -1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value -1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. 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).

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:
ifail=1
On entry, x=value.
Constraint: x>0.0.
ifail=2
Computation halted due to likelihood of underflow. x may be too large. x=value.
ifail=3
Computation halted due to likelihood of overflow. x may be too small. x=value.
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
ifail=-399
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.
ifail=-999
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

All constants in s14acf are given to approximately 18 digits of precision. Calling the number of digits of precision in the floating-point arithmetic being used t, then clearly the maximum number of correct digits in the results obtained is limited by p=min(t,18).
With the above proviso, results returned by this routine should be accurate almost to full precision, except at points close to the zero of ψ(x), x1.461632, where only absolute rather than relative accuracy can be obtained.

8 Parallelism and Performance

s14acf is not threaded in any implementation.

9 Further Comments

None.

10 Example

The example program reads values of the argument x from a file, evaluates the function at each value of x and prints the results.

10.1 Program Text

Program Text (s14acfe.f90)

10.2 Program Data

Program Data (s14acfe.d)

10.3 Program Results

Program Results (s14acfe.r)
GnuplotProduced by GNUPLOT 4.6 patchlevel 3 −7 −6 −5 −4 −3 −2 −1 0 0 1 2 3 4 5 6 7 8 ψ(x)ln x x Example Program Returns a Value of the Function ψ(x)ln x gnuplot_plot_1