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 parameter $a$ of the function.
Constraint: $a > 0.0$ .
2: b – doubleInput: On entry: the parameter $b$ of the function.
Constraint: $b > 0.0$ .
3: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

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.
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

Not applicable.

9 Further Comments

None.

10 Example

This example reads values of the parameters

a

and

b

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

NAG Library Function Documentnag_log_beta (s14cbc)