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NAG Library Function Document

nag_log_beta (s14cbc)

▸▿ 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

1 Purpose

nag_log_beta (s14cbc) returns the value of the logarithm of the beta function,

\ln B (a, b)

, via the routine name.

2 Specification

#include <nag.h>

#include <nags.h>

double

nag_log_beta (double a, double b, NagError *fail)

3 Description

nag_log_beta (s14cbc) 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 $b_{0} \leq 2$ ,
$\ln B = \ln Γ (a_{0}) + \ln Γ (b_{0}) - \ln Γ (a_{0} + b_{0}) .$

nag_log_beta (s14cbc) 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: $a$ – doubleInput: On entry: the argument $a$ of the function.

Constraint: $a > 0.0$ .
2: $b$ – doubleInput: On entry: the argument $b$ of the function.

Constraint: $b > 0.0$ .
3: $fail$ – NagError *Input/Output: The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL: On entry, $a = 〈value〉$ .
Constraint: $a > 0.0$ .

On entry, $b = 〈value〉$ .
Constraint: $b > 0.0$ .

7 Accuracy

nag_log_beta (s14cbc) should produce full relative accuracy for all input arguments.

8 Parallelism and Performance

nag_log_beta (s14cbc) is not threaded in any implementation.

9 Further Comments

None.

10 Example

This example reads values of the arguments

a

and

b

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

NAG Library Function Documentnag_log_beta (s14cbc)