NAG Library Manual, Mark 28.3
Interfaces:  FL   CL   CPP   AD 
NAG CL Interface Introduction
S (Specfun) Chapter Contents
S (Specfun) Chapter Introduction
s14ah:  FL   CL   CPP   AD 

NAG CL Interface
s14ahc (gamma_​log_​scaled_​real)

NAG Library Manual, Mark 28.3
Interfaces:  FL   CL   CPP   AD 
NAG CL Interface Introduction
S (Specfun) Chapter Contents
S (Specfun) Chapter Introduction
s14ah:  FL   CL   CPP   AD 

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© The Numerical Algorithms Group Ltd. 2022
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1 Purpose

s14ahc returns the value of ln⁡G(x), the scaled logarithm of the gamma function Γ(x).

2 Specification

#include <nag.h>
double  s14ahc (double x, NagError *fail)
The function may be called by the names: s14ahc, nag_specfun_gamma_log_scaled_real or nag_scaled_log_gamma.

3 Description

s14ahc calculates an approximate value for ln⁡G(x), where G(x)=Γ(x+1)/(xe)x. This is a variant of the ln⁡Γ(x) function (see also s14abc), which avoids rounding problems for very large arguments by computing ln⁡Γ(x) with the Stirling approximation factored out.
For 0<x<15, ln⁡G(x)=ln⁡Γ(x+1)-x⁢ln⁡x+x;
and for 15≤x, ln⁡G(x)=12 ln⁡x+ln(2π)+1xR(1/x2), where R is a suitable Remez approximation.
For x≤0.0, the value ln⁡G(x) is undefined; s14ahc returns zero and exits with fail.code= NE_REAL_ARG_LE.

4 References

NIST Digital Library of Mathematical Functions

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

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_INTERNAL_ERROR
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.
NE_NO_LICENCE
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.
NE_REAL_ARG_LE
On entry, x=⟨value⟩.
Constraint: x>0.0.

7 Accuracy

s14ahc has been designed to produce full relative accuracy for all input arguments. Empirical results obtained by comparing with multiprecision software confirm this.

8 Parallelism and Performance

s14ahc is not threaded in any implementation.

9 Further Comments

None.

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 (s14ahce.c)

10.2 Program Data

Program Data (s14ahce.d)

10.3 Program Results

Program Results (s14ahce.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
NAG Library Manual, Mark 28.3
Interfaces:  FL   CL   CPP   AD 
NAG CL Interface Introduction
S (Specfun) Chapter Contents
S (Specfun) Chapter Introduction
s14ah:  FL   CL   CPP   AD 
© The Numerical Algorithms Group Ltd. 2022