NAG Library Routine Document

s14ahf (gamma_log_scaled_real)


    1  Purpose
    7  Accuracy


s14ahf returns the value of lnGx, the scaled logarithm of the gamma function Γx, via the function name.


Fortran Interface
Function s14ahf ( x, ifail)
Real (Kind=nag_wp):: s14ahf
Integer, Intent (Inout):: ifail
Real (Kind=nag_wp), Intent (In):: x
C Header Interface
#include nagmk26.h
double  s14ahf_ (const double *x, Integer *ifail)


s14ahf calculates an approximate value for lnGx, where Gx=Γx+1/xex. This is a variant of the lnΓx function (see also s14abf), which avoids rounding problems for very large arguments by computing lnΓx with the Stirling approximation factored out.
For 0<x<15, lnGx=lnΓx+1-xlnx+x;
and for 15x, lnGx=12 lnx+ln2π+1xR1/x2, where R is a suitable Remez approximation.
For x0.0, the value lnGx is undefined; s14ahf returns zero and exits with ifail=1.


Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications


1:     x – Real (Kind=nag_wp)Input
On entry: the argument x of the function.
Constraint: x>0.0.
2:     ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1​ or ​1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this argument, the recommended value is 0. 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).

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:
On entry, x0.0. On soft failure, the function value returned is zero.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.


s14ahf has been designed to produce full relative accuracy for all input arguments. Empirical results obtained by comparing with multiprecision software confirm this.

Parallelism and Performance

s14ahf is not threaded in any implementation.

Further Comments



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

Program Text

Program Text (s14ahfe.f90)

Program Data

Program Data (s14ahfe.d)

Program Results

Program Results (s14ahfe.r)

GnuplotProduced by GNUPLOT 4.6 patchlevel 3 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 1 2 3 4 5 6 7 8 lnG(x) x Example Program Returns the Value of lnG(x), the Scaled Logarithm of the Gamma Function Γ(x) gnuplot_plot_1
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017