NAG Library Function Document
nag_gamma (s14aac) returns the value of the Gamma function .
||nag_gamma (double x,
nag_gamma (s14aac) evaluates
The function is based on a Chebyshev expansion for
, and uses the property
is integral and
then it follows that:
- for .
There are four possible failures for this function:
||if is too large, there is a danger of overflow since could become too large to be represented in the machine;
||if is too large and negative, there is a danger of underflow;
||if is equal to a negative integer, would overflow since it has poles at such points;
||if is too near zero, there is again the danger of overflow on some machines.
For small , , and on some machines there exists a range of nonzero but small values of for which is larger than the greatest representable value.
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
x – doubleInput
On entry: the argument of the function.
must not be zero or a negative integer.
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
On entry, .
The argument is too large, the function returns the approximate value of at the nearest valid argument.
On entry, x
must not be less than
The argument is too large and negative, the function returns zero.
On entry, x
must not be effectively a negative integer:
The argument is a negative integer, at which values
is infinite. The function returns a large positive value.
On entry, x
must be greater than
The argument is too close to zero, the function returns the approximate value of
at the nearest valid argument.
Let and be the relative errors in the argument and the result respectively. If is somewhat larger than the machine precision (i.e., is due to data errors etc.), then and are approximately related by (provided is also greater than the representation error). Here is the digamma function .
If is of the same order as machine precision, then rounding errors could make slightly larger than the above relation predicts.
There is clearly a severe, but unavoidable, loss of accuracy for arguments close to the poles of at negative integers. However, relative accuracy is preserved near the pole at right up to the point of failure arising from the danger of setting overflow.
Also accuracy will necessarily be lost as becomes large since in this region . However, since increases rapidly with , the function must fail due to the danger of setting overflow before this loss of accuracy is too great. For example, for , the amplification factor .
The following program reads values of the argument from a file, evaluates the function at each value of and prints the results.
9.1 Program Text
Program Text (s14aace.c)
9.2 Program Data
Program Data (s14aace.d)
9.3 Program Results
Program Results (s14aace.r)