NAG CL Interface
s14abc (gamma_​log_​real)

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1 Purpose

s14abc returns the value of the logarithm of the gamma function, lnΓ(x).

2 Specification

#include <nag.h>
double  s14abc (double x, NagError *fail)
The function may be called by the names: s14abc, nag_specfun_gamma_log_real or nag_log_gamma.

3 Description

s14abc calculates an approximate value for lnΓ(x). It is based on rational Chebyshev expansions.
Denote by Rn,mi(x)=Pni(x)/Qmi(x) a ratio of polynomials of degree n in the numerator and m in the denominator. Then:
For each expansion, the specific values of n and m are selected to be minimal such that the maximum relative error in the expansion is of the order 10-d, where d is the maximum number of decimal digits that can be accurately represented for the particular implementation (see X02BEC).
Let ε denote machine precision and let xhuge denote the largest positive model number (see X02ALC). For x<0.0 the value lnΓ(x) is not defined; s14abc returns zero and exits with fail.code= NE_REAL_ARG_LE. It also exits with fail.code= NE_REAL_ARG_LE when x=0.0, and in this case the value xhuge is returned. For x in the interval (0.0,ε], the function lnΓ(x)=-ln(x) to machine accuracy.
Now denote by xbig the largest allowable argument for lnΓ(x) on the machine. For (xbig)1/4<xxbig the Rn,m4(1/x2) term in Equation (1) is negligible. For x>xbig there is a danger of setting overflow, and so s14abc exits with fail.code= NE_REAL_ARG_GT and returns xhuge. The value of xbig is given in the Users' Note for your implementation.

4 References

NIST Digital Library of Mathematical Functions
Cody W J and Hillstrom K E (1967) Chebyshev approximations for the natural logarithm of the gamma function Math.Comp. 21 198–203

5 Arguments

1: x double Input
On entry: the argument x of the function.
Constraint: x>0.0.
2: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error 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.
On entry, x=value and the constant xbig=value.
Constraint: xxbig.
On entry, x=value.
Constraint: x>0.0.

7 Accuracy

Let δ and ε be the relative errors in the argument and result respectively, and E be the absolute error in the result.
If δ is somewhat larger than machine precision, then
E |x×Ψ(x)| δ   and   ε | x×Ψ(x) lnΓ (x) | δ  
where Ψ(x) is the digamma function Γ(x) Γ(x) . Figure 1 and Figure 2 show the behaviour of these error amplification factors.
Figure 1
Figure 1
Figure 2
Figure 2
These show that relative error can be controlled, since except near x=1 or 2 relative error is attenuated by the function or at least is not greatly amplified.
For large x, ε(1+ 1lnx ) δ and for small x, ε 1lnx δ.
The function lnΓ(x) has zeros at x=1 and 2 and hence relative accuracy is not maintainable near those points. However, absolute accuracy can still be provided near those zeros as is shown above.
If however, δ is of the order of machine precision, then rounding errors in the function's internal arithmetic may result in errors which are slightly larger than those predicted by the equalities. It should be noted that even in areas where strong attenuation of errors is predicted the relative precision is bounded by the effective machine precision.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s14abc is not threaded in any implementation.

9 Further Comments


10 Example

This example 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 (s14abce.c)

10.2 Program Data

Program Data (s14abce.d)

10.3 Program Results

Program Results (s14abce.r)
GnuplotProduced by GNUPLOT 5.4 patchlevel 6 0 2 4 6 8 0 1 2 3 4 5 6 7 8 lnΓ(x) x "s14abfe.r" Example Program Returned Values for the Logarithm of the Gamma Function, lnΓ(x)