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	s14apf_ (const Integer n, const double x[], double f[], Integer ivalid[], Integer ifail)

The routine may be called by the names s14apf or nagf_specfun_gamma_log_real_vector.

3 Description

s14apf calculates an approximate value for

\ln Γ (x)

for an array of arguments

x_{i}

, for

i = 1, 2, \dots, n

. It is based on rational Chebyshev expansions.

Denote by

R_{n, m}^{i} (x) = P_{n}^{i} (x) / Q_{m}^{i} (x)

a ratio of polynomials of degree

n

in the numerator and

m

in the denominator. Then:

for $0 < x \leq 1 / 2$ ,

$\ln Γ (x) \approx - \ln (x) + x R_{n, m}^{1} (x + 1);$
for $1 / 2 < x \leq 3 / 2$ ,

$\ln Γ (x) \approx (x - 1) R_{n, m}^{1} (x);$
for $3 / 2 < x \leq 4$ ,

$\ln Γ (x) \approx (x - 2) R_{n, m}^{2} (x);$
for $4 < x \leq 12$ ,

$\ln Γ (x) \approx R_{n, m}^{3} (x);$
and for $12 < x$ ,

$\ln Γ (x) \approx (x - \frac{1}{2}) \ln (x) - x + \ln (\sqrt{2 π}) + \frac{1}{x} R_{n, m}^{4} (1 / x^{2}) .$ (1)

For each expansion, the specific values of

n

and

m

are selected to be minimal such that the maximum relative error in the expansion is of the order

10^{- d}

, where

d

is the maximum number of decimal digits that can be accurately represented for the particular implementation (see x02bef).

Let

ε

denote machine precision and let

x_{huge}

denote the largest positive model number (see x02alf). For

x < 0.0

the value

\ln Γ (x)

is not defined; s14apf returns zero and exits with

ifail = 1

. It also exits with

ifail = 1

when

x = 0.0

, and in this case the value

x_{huge}

is returned. For

x

in the interval

(0.0, ε]

, the function

\ln Γ (x) = - \ln (x)

to machine accuracy.

Now denote by

x_{big}

the largest allowable argument for

\ln Γ (x)

on the machine. For

{(x_{big})}^{1 / 4} < x \leq x_{big}

the

R_{n, m}^{4} (1 / x^{2})

term in Equation (1) is negligible. For

x > x_{big}

there is a danger of setting overflow, and so s14apf exits with

ifail = 2

and returns

x_{huge}

. The value of

x_{big}

is given in the Users' Note for your implementation.

4 References

NIST Digital Library of Mathematical Functions

Cody W J and Hillstrom K E (1967) Chebyshev approximations for the natural logarithm of the gamma function Math.Comp. 21 198–203

5 Arguments

1: $n$ – Integer Input

On entry:

n

, the number of points.

Constraint:

n \geq 0

2: $x (n)$ – Real (Kind=nag_wp) array Input

On entry: the argument

x_{i}

of the function, for

i = 1, 2, \dots, n

Constraint:

x (i) > 0

, for

i = 1, 2, \dots, n

3: $f (n)$ – Real (Kind=nag_wp) array Output

On exit:

\ln Γ (x_{i})

, the function values.

4: $ivalid (n)$ – Integer array Output

On exit:

ivalid (i)

contains the error code for

x_{i}

, for

i = 1, 2, \dots, n

$ivalid (i) = 0$: No error.
$ivalid (i) = 1$: $x_{i} \leq 0$ .
$ivalid (i) = 2$: $x_{i}$ is too large and positive. The threshold value is the same as for $ifail = 2$ in s14abf , as defined in the Users' Note for your implementation.

5: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

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

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, at least one value of x was invalid.
Check ivalid for more information.

$ifail = 2$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

$ifail = - 99$: 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 result respectively, and

E

be the absolute error in the result.

δ

is somewhat larger than machine precision, then

E ≃ | x \times Ψ (x) | δ and ε ≃ | \frac{x \times Ψ (x)}{\ln Γ (x)} | δ

where

Ψ (x)

is the digamma function

\frac{Γ^{'} (x)}{Γ (x)}

. Figure 1 and Figure 2 show the behaviour of these error amplification factors.

Figure 1

Figure 2

These show that relative error can be controlled, since except near

x = 1

2

relative error is attenuated by the function or at least is not greatly amplified.

For large

x

ε ≃ (1 + \frac{1}{\ln x}) δ

and for small

x

ε ≃ \frac{1}{\ln x} δ

The function

\ln Γ (x)

has zeros at

x = 1

and

2

and hence relative accuracy is not maintainable near those points. However, absolute accuracy can still be provided near those zeros as is shown above.

If however,

δ

is of the order of machine precision, then rounding errors in the routine's internal arithmetic may result in errors which are slightly larger than those predicted by the equalities. It should be noted that even in areas where strong attenuation of errors is predicted the relative precision is bounded by the effective machine precision.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

s14apf 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.

s14ap: FL CL CPP AD PY MB

NAG FL Interfaces14apf (gamma_​log_​real_​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 FL Interface
s14apf (gamma_log_real_vector)