s14aa returns the value of the gamma function Γx.


public static double s14aa(
	double x,
	out int ifail
Visual Basic
Public Shared Function s14aa ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double
Visual C++
static double s14aa(
	double x, 
	[OutAttribute] int% ifail
static member s14aa : 
        x : float * 
        ifail : int byref -> float 


Type: System..::..Double
On entry: the argument x of the function.
Constraint: x must not be zero or a negative integer.
Type: System..::..Int32%
On exit: ifail=0 unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Return Value

s14aa returns the value of the gamma function Γx.


s14aa evaluates an approximation to the gamma function Γx. The method 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:
  • for N>0, Γx=x-1x-2x-NΓ1+u,
  • for N=0, Γx=Γ1+u,
  • for N<0, Γx=Γ1+uxx+1x+2x-N-1.
There are four possible failures for this method:
(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.


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

Error Indicators and Warnings

Errors or warnings detected by the method:
The argument is too large. On failure the method returns the approximate value of Γx at the nearest valid argument.
The argument is too large and negative. On failure the method returns zero.
The argument is too close to zero. On failure the method returns the approximate value of Γx at the nearest valid argument.
The argument is a negative integer, at which value Γx is infinite. On failure the method returns a large positive value.
An error occured, see message report.


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 Ψx is the digamma function ΓxΓx. 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
However since Γx increases rapidly with x, the method 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

Parallelism and Performance


Further Comments



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

Example program (C#): s14aae.cs

Example program data: s14aae.d

Example program results: s14aae.r

See Also