NAG FL Interface
s14anf (gamma_​vector)

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1 Purpose

s14anf returns an array of values of the gamma function Γ(x).

2 Specification

Fortran Interface
Subroutine s14anf ( n, x, f, ivalid, ifail)
Integer, Intent (In) :: n
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: ivalid(n)
Real (Kind=nag_wp), Intent (In) :: x(n)
Real (Kind=nag_wp), Intent (Out) :: f(n)
C Header Interface
#include <nag.h>
void  s14anf_ (const Integer *n, const double x[], double f[], Integer ivalid[], Integer *ifail)
The routine may be called by the names s14anf or nagf_specfun_gamma_vector.

3 Description

s14anf evaluates an approximation to the gamma function Γ(x) for an array of arguments xi, for i=1,2,,n. The routine 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 routine:
  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) Real (Kind=nag_wp) array Input
On entry: the argument xi of the function, for i=1,2,,n.
Constraint: x(i)0-, for i=1,2,,n.
3: f(n) Real (Kind=nag_wp) array Output
On exit: Γ(xi), the function values.
4: ivalid(n) Integer array Output
On exit: ivalid(i) contains the error code for xi, for i=1,2,,n.
ivalid(i)=0
No error.
ivalid(i)=1
xi is too large and positive. f(i) contains the approximate value of Γ(xi) at the nearest valid argument. The threshold value is the same as for ifail=1 in s14aaf , as defined in the the Users' Note for your implementation.
ivalid(i)=2
xi is too large and negative. f(i) contains zero. The threshold value is the same as for ifail=2 in s14aaf , as defined in the the Users' Note for your implementation.
ivalid(i)=3
xi is too close to zero. f(i) contains the approximate value of Γ(xi) at the nearest valid argument. The threshold value is the same as for ifail=2 in s14aaf , as defined in the the Users' Note for your implementation.
ivalid(i)=4
xi is a negative integer, at which values Γ(xi) are infinite. f(i) contains a large positive value.
5: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
ifail=1
On entry, at least one value of x was invalid.
Check ivalid for more information.
ifail=2
On entry, n=value.
Constraint: n0.
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
ifail=-999
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further 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 routine 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

s14anf 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 (s14anfe.f90)

10.2 Program Data

Program Data (s14anfe.d)

10.3 Program Results

Program Results (s14anfe.r)