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 function:

(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: $n$ – Integer Input

On entry:

n

, the number of points.

Constraint:

n \geq 0

2: $x [n]$ – const double Input

On entry: the argument

x_{i}

of the function, for

i = 1, 2, \dots, n

Constraint:

x [i - 1] \notin ℤ_{0}^{-}

, for

i = 1, 2, \dots, n

3: $f [n]$ – double Output

On exit:

Γ (x_{i})

, the function values.

4: $ivalid [n]$ – Integer Output

On exit:

ivalid [i - 1]

contains the error code for

x_{i}

, for

i = 1, 2, \dots, n

$ivalid [i - 1] = 0$: No error.
$ivalid [i - 1] = 1$: $x_{i}$ is too large and positive. $f [i - 1]$ contains the approximate value of $Γ (x_{i})$ 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.
$ivalid [i - 1] = 2$: $x_{i}$ 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.
$ivalid [i - 1] = 3$: $x_{i}$ is too close to zero. $f [i - 1]$ contains the approximate value of $Γ (x_{i})$ 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.
$ivalid [i - 1] = 4$: $x_{i}$ is a negative integer, at which values $Γ (x_{i})$ 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: $n \geq 0$ .
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 ≃ | 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 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

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

x_{i}

and prints the results.

s14an: FL CL CPP AD

NAG CL Interfaces14anc (gamma_​vector)

▸▿ 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 CL Interface
s14anc (gamma_vector)