NAG Library Routine Document

S14CBF

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

S14CBF returns the value of the logarithm of the beta function,

\ln B (a, b)

, via the routine name.

2 Specification

FUNCTION S14CBF ( A, B, IFAIL)

REAL (KIND=nag_wp) S14CBF

INTEGER IFAIL

REAL (KIND=nag_wp) A, B

3 Description

S14CBF calculates values for

\ln B (a, b)

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 $b_{0} \leq 2$ ,
$\ln B = \ln Γ (a_{0}) + \ln Γ (b_{0}) - \ln Γ (a_{0} + b_{0}) .$

S14CBF 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 Parameters

1: A – REAL (KIND=nag_wp)Input: On entry: the argument $a$ of the function.
Constraint: $A > 0.0$ .
2: B – REAL (KIND=nag_wp)Input: On entry: the argument $b$ of the function.
Constraint: $B > 0.0$ .
3: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$ . 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, $A = ⟨value⟩$ .
Constraint: $A > 0.0$ .

On entry, $B = ⟨value⟩$ .
Constraint: $B > 0.0$ .

7 Accuracy

S14CBF should produce full relative accuracy for all input arguments.

8 Further Comments

None.

9 Example

This example reads values of the arguments

a

and

b

from a file, evaluates the function and prints the results.

NAG Library Routine DocumentS14CBF