NAG CL Interface
s14anc (gamma_​vector)

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1 Purpose

s14anc returns an array of values of the gamma function Γ(x).

2 Specification

#include <nag.h>
void  s14anc (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)
The function may be called by the names: s14anc, nag_specfun_gamma_vector or nag_gamma_vector.

3 Description

s14anc evaluates an approximation to the gamma function Γ(x) for an array of arguments xi, for i=1,2,,n. 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: n Integer Input
On entry: n, the number of points.
Constraint: n0.
2: x[n] const double Input
On entry: the argument xi of the function, for i=1,2,,n.
Constraint: x[i-1]0-, for i=1,2,,n.
3: f[n] double Output
On exit: Γ(xi), the function values.
4: ivalid[n] Integer Output
On exit: ivalid[i-1] contains the error code for xi, for i=1,2,,n.
ivalid[i-1]=0
No error.
ivalid[i-1]=1
xi is too large and positive. f[i-1] contains the approximate value of Γ(xi) at the nearest valid argument. The threshold value is the same as for fail.code= NE_REAL_ARG_GT in s14aac , as defined in the Users' Note for your implementation.
ivalid[i-1]=2
xi is too large and negative. f[i-1] contains zero. The threshold value is the same as for fail.code= NE_REAL_ARG_LT in s14aac , as defined in the Users' Note for your implementation.
ivalid[i-1]=3
xi is too close to zero. f[i-1] contains the approximate value of Γ(xi) at the nearest valid argument. The threshold value is the same as for fail.code= NE_REAL_ARG_LT in s14aac , as defined in the Users' Note for your implementation.
ivalid[i-1]=4
xi is a negative integer, at which values Γ(xi) are infinite. f[i-1] contains a large positive value.
5: 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_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
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.
NW_IVALID
On entry, at least one value of x was invalid.
Check ivalid for more information.

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:
ε|xΨ(x)|δ  
(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
εδxlnx.  
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

s14anc is not threaded in any implementation.

9 Further Comments

None.

10 Example

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

10.1 Program Text

Program Text (s14ance.c)

10.2 Program Data

Program Data (s14ance.d)

10.3 Program Results

Program Results (s14ance.r)