for $N > 0$ ,	$Γ (x) = (x - 1) (x - 2) \dots (x - N) Γ (1 + u)$ ,
for $N = 0$ ,	$Γ (x) = Γ (1 + u)$ ,
for $N < 0$ ,	$Γ (x) = \frac{Γ (1 + u)}{x (x + 1) (x + 2) \dots (x - N - 1)}$ .

There are four possible failures for this routine:

(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;
(ii)if $x$ is too large and negative, there is a danger of underflow;
(iii)if $x$ is equal to a negative integer, $Γ (x)$ 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 $x$ , $Γ (x) ≃ 1 / x$ , 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: $x$ – Real (Kind=nag_wp) Input: On entry: the argument $x$ of the function.

Constraint: $x$ must not be zero or a negative integer.
2: $ifail$ – 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

−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, $x = ⟨ value ⟩$ .
Constraint: $x \leq ⟨ value ⟩$ .
The argument is too large, the function returns the approximate value of $Γ (x)$ at the nearest valid argument.

$ifail = 2$: On entry, $x = ⟨ value ⟩$ . The function returns zero.
Constraint: $x \geq ⟨ value ⟩$ .
The argument is too large and negative, the function returns zero.

$ifail = 3$: On entry, $x = ⟨ value ⟩$ .
Constraint: $| x | \geq ⟨ value ⟩$ .
The argument is too close to zero, the function returns the approximate value of $Γ (x)$ at the nearest valid argument.

$ifail = 4$: 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.

$ifail = - 99$: 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 ≃ | x Ψ (x) | δ

(provided

ε

is also greater than the representation error). Here

Ψ (x)

is the digamma function

\frac{Γ^{'} (x)}{Γ (x)}

. Figure 1 shows the behaviour of the error amplification factor

| x Ψ (x) |

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

ε ≃ δ x \ln x .

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

8 Parallelism and Performance

s14aaf is not threaded in any implementation.

9 Further Comments

None.

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.

s14aa: FL CL CPP AD

NAG FL Interfaces14aaf (gamma)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
s14aaf (gamma)