Integer, Intent (In)	::	n
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	ivalid(n)
Real (Kind=nag_wp), Intent (In)	::	a(n), b(n)
Real (Kind=nag_wp), Intent (Out)	::	f(n)

C Header Interface

#include <nag.h>

void	s14cpf_ (const Integer n, const double a[], const double b[], double f[], Integer ivalid[], Integer ifail)

The routine may be called by the names s14cpf or nagf_specfun_beta_log_real_vector.

3 Description

s14cpf calculates values for

\ln B (a, b)

, for arrays of arguments

a_{i}

and

b_{i}

, for

i = 1, 2, \dots, n

, where

B

is the beta function given by

B (a, b) = \int_{0}^{1} t^{a - 1} {(1 - t)}^{b - 1} dt

or equivalently

B (a, b) = \frac{Γ (a) Γ (b)}{Γ (a + b)}

and

Γ (x)

is the gamma function. Note that the beta function is symmetric, so that

B (a, b) = B (b, a)

In order to efficiently obtain accurate results several methods are used depending on the parameters

a

and

b

Let

a_{0} = \min (a, b)

and

b_{0} = \max (a, b)

. Then:

for

a_{0} \geq 8

\begin{matrix} \ln B & = & 0.5 \ln (2 π) - 0.5 \ln (b_{0}) + Δ (a_{0}) + Δ (b_{0}) - Δ (a_{0} + b_{0}) - u - v; \end{matrix}

where

$Δ (a_{0}) = \ln Γ (a_{0}) - (a_{0} - 0.5) \ln a_{0} + a_{0} - 0.5 \ln (2 π)$ ,
$u = - (a_{0} - 0.5) \ln [\frac{a_{0}}{a_{0} + b_{0}}]$ and
$v = b_{0} \ln (1 + \frac{a_{0}}{b_{0}})$ ;

for

a_{0} < 1

for $b_{0} \geq 8$ ,
$\ln B = \ln Γ (a_{0}) + \ln \frac{Γ (b_{0})}{Γ (a_{0} + b_{0})};$
for $b_{0} < 8$ ,
$\ln B = \ln Γ (a_{0}) + \ln Γ (b_{0}) - \ln Γ (a_{0} + b_{0});$

for

2 < a_{0} < 8

a_{0}

is reduced to the interval

[1, 2]

B (a, b) = \frac{a_{0} - 1}{a_{0} + b_{0} - 1} B (a_{0} - 1, b_{0})

;

for

1 \leq a_{0} \leq 2

for $b_{0} \geq 8$ ,
$\ln B = \ln Γ (a_{0}) + \ln \frac{Γ (b_{0})}{Γ (a_{0} + b_{0})};$
for $2 < b_{0} < 8$ , $b_{0}$ is reduced to the interval $[1, 2]$ ;
for $b_{0} \leq 2$ ,
$\ln B = \ln Γ (a_{0}) + \ln Γ (b_{0}) - \ln Γ (a_{0} + b_{0}) .$

s14cpf is derived from BETALN in DiDonato and Morris (1992).

4 References

DiDonato A R and Morris A H (1992) Algorithm 708: Significant digit computation of the incomplete beta function ratios ACM Trans. Math. Software 18 360–373

5 Arguments

1: $n$ – Integer Input

On entry:

n

, the number of points.

Constraint:

n \geq 0

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

On entry: the argument

a_{i}

of the function, for

i = 1, 2, \dots, n

Constraint:

a (i) > 0

, for

i = 1, 2, \dots, n

3: $b (n)$ – Real (Kind=nag_wp) array Input

On entry: the argument

b_{i}

of the function, for

i = 1, 2, \dots, n

Constraint:

b (i) > 0

, for

i = 1, 2, \dots, n

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

On exit:

\ln B (a_{i}, b_{i})

, the function values.

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

On exit:

ivalid (i)

contains the error code for

a_{i}

and

b_{i}

, for

i = 1, 2, \dots, n

$ivalid (i) = 0$: No error.
$ivalid (i) = 1$: $a_{i} or b_{i} \leq 0$ .

6: $ifail$ – 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 a or b 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

s14cpf should produce full relative accuracy for all input arguments.

8 Parallelism and Performance

s14cpf is not threaded in any implementation.

9 Further Comments

None.

10 Example

This example reads values of a and b from a file, evaluates the function at each value of

a_{i}

and

b_{i}

and prints the results.

s14cp: FL CL CPP AD

NAG FL Interfaces14cpf (beta_​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
s14cpf (beta_log_real_vector)