NAG CL Interface
s14aac (gamma)

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1 Purpose

s14aac returns the value of the gamma function Γ(x).

2 Specification

#include <nag.h>
double  s14aac (double x, NagError *fail)
The function may be called by the names: s14aac, nag_specfun_gamma or nag_gamma.

3 Description

s14aac evaluates an approximation to the gamma function Γ(x). The function is based on the Chebyshev expansion:
Γ(1+u) = r=0 ar Tr (t)  
where 0u<1,t = 2u-1, and uses the property Γ(1+x) = xΓ(x) . If x=N+1+u where N is integral and 0u<1 then it follows that:
for N>0, Γ(x)=(x-1)(x-2)(x-N)Γ(1+u),
for N=0, Γ(x)=Γ(1+u),
for N<0, Γ(x) = Γ(1+u) x(x+1)(x+2)(x-N-1) .
There are four possible failures for this function:
  1. (i)if x is too large, there is a danger of overflow since Γ(x) could become too large to be represented in the machine;
  2. (ii)if x is too large and negative, there is a danger of underflow;
  3. (iii)if x is equal to a negative integer, Γ(x) would overflow since it has poles at such points;
  4. (iv)if x is too near zero, there is again the danger of overflow on some machines. For small x, Γ(x)1/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

NIST Digital Library of Mathematical Functions

5 Arguments

1: x double Input
On entry: the argument x of the function.
Constraint: x must not be zero or a negative integer.
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.
Constraint: xvalue.
The argument is too large, the function returns the approximate value of Γ (x) at the nearest valid argument.
On entry, x=value. The function returns zero.
Constraint: xvalue.
The argument is too large and negative, the function returns zero.
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.
On entry, x=value.
Constraint: |x|value.
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:
(provided ε is also greater than the representation error). Here Ψ(x) is the digamma function Γ(x) Γ(x) . Figure 1 shows the behaviour of the error amplification factor |xΨ(x)|.
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 Γ(x) 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
However, since Γ(x) 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

Background information to multithreading can be found in the Multithreading documentation.
s14aac 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 (s14aace.c)

10.2 Program Data

Program Data (s14aace.d)

10.3 Program Results

Program Results (s14aace.r)
GnuplotProduced by GNUPLOT 5.4 patchlevel 6 −4 −2 0 2 4 −4 −3 −2 −1 0 1 2 3 4 Γ(x) x "s14aafe.r" Example Program Returned Values for the Gamma Function Γ(x)