nag_gamma (s14aac) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_gamma (s14aac)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_gamma (s14aac) returns the value of the gamma function Γx.

2  Specification

#include <nag.h>
#include <nags.h>
double  nag_gamma (double x, NagError *fail)

3  Description

nag_gamma (s14aac) evaluates an approximation to the gamma function Γx. The function is based on the Chebyshev expansion:
Γ1+u=r=0arTrt,   where ​ 0u<1,t=2u-1,
and uses the property Γ1+x=xΓx. If x=N+1+u where N is integral and 0u<1 then it follows that:
There are four possible failures for this function:
(i) if x is too large, there is a danger of overflow since Γx could become too large to be represented in the machine;
(ii) if x is too large and negative, there is a danger of underflow;
(iii) if x is equal to a negative integer, Γx would overflow since it has poles at such points;
(iv) if x is too near zero, there is again the danger of overflow on some machines. For small x, Γx1/x , and on some machines there exists a range of nonzero but small values of x for which 1/x is larger than the greatest representable value.

4  References

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

5  Arguments

1:     xdoubleInput
On entry: the argument x of the function.
Constraint: x must not be zero or a negative integer.
2:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

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.
NE_REAL_ARG_GT
On entry, x=value.
Constraint: xvalue.
The argument is too large, the function returns the approximate value of Γ x  at the nearest valid argument.
NE_REAL_ARG_LT
On entry, x=value. The function returns zero.
Constraint: xvalue.
The argument is too large and negative, the function returns zero.
NE_REAL_ARG_NEG_INT
On entry, x=value.
Constraint: x must not be a negative integer.
The argument is a negative integer, at which values Γ x  is infinite. The function returns a large positive value.
NE_REAL_ARG_TOO_SMALL
On entry, x=value.
Constraint: xvalue.
The argument is too close to zero, the function returns the approximate value of Γ x  at the nearest valid argument.

7  Accuracy

Let δ and ε be the relative errors in the argument and the result respectively. If δ is somewhat larger than the machine precision (i.e., is due to data errors etc.), then ε and δ are approximately related by:
εxΨxδ
(provided ε is also greater than the representation error). Here Ψx is the digamma function Γx Γx . Figure 1 shows the behaviour of the error amplification factor xΨx.
If δ is of the same order as machine precision, then rounding errors could make ε slightly larger than the above relation predicts.
There is clearly a severe, but unavoidable, loss of accuracy for arguments close to the poles of Γx at negative integers. However relative accuracy is preserved near the pole at x=0 right up to the point of failure arising from the danger of overflow.
Also accuracy will necessarily be lost as x becomes large since in this region
εδxlnx.
However since Γx increases rapidly with x, the function must fail due to the danger of overflow before this loss of accuracy is too great. (For example, for x=20, the amplification factor 60.)
Figure 1
Figure 1

8  Parallelism and Performance

Not applicable.

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 (s14aace.c)

10.2  Program Data

Program Data (s14aace.d)

10.3  Program Results

Program Results (s14aace.r)

Produced by GNUPLOT 4.4 patchlevel 0 -4 -2 0 2 4 -4 -3 -2 -1 0 1 2 3 4 G(x) x Example Program Returned Values for the Gamma Function G(x)

nag_gamma (s14aac) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014