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_specfun_beta_log_real (s14cb) is derived from BETALN in DiDonato and Morris (1992).

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

Parameters

Compulsory Input Parameters

1: $a$ – double scalar: The argument $a$ of the function.

Constraint: $a > 0.0$ .
2: $b$ – double scalar: The argument $b$ of the function.

Constraint: $b > 0.0$ .

Optional Input Parameters

None.

Output Parameters

1: $result$ – double scalar: The result of the function.
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: Constraint: $a > 0.0$ .

Constraint: $b > 0.0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

nag_specfun_beta_log_real (s14cb) should produce full relative accuracy for all input arguments.

Further Comments

None.

Example

This example reads values of the arguments

a

and

b

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

Open in the MATLAB editor: s14cb_example

function s14cb_example


fprintf('s14cb example results\n\n');

a = [0.2; 0.4; 0.6; 0.8; 1.0; 1.0; 1.0; 2.0; 3.0; 4.0; ...
     5.0; 6.0; 6.0; 6.0; 6.0; 6.0; 7.0];
b = [1.0; 1.0; 1.0; 1.0; 0.2; 0.4; 1.0; 2.0; 3.0; 4.0; ...
     5.0; 2.0; 3.0; 4.0; 5.0; 6.0; 7.0];
lb = zeros(numel(a), 1);
for i = 1:numel(a)
  [lb(i), ifail] = s14cb(a(i), b(i));
end
fprintf('\n  a    b        ln(beta(a,b))\n');
fprintf('%5.2f%5.2f%17.4e\n', vertcat(a', b', lb'));

s14cb example results


  a    b        ln(beta(a,b))
 0.20 1.00       1.6094e+00
 0.40 1.00       9.1629e-01
 0.60 1.00       5.1083e-01
 0.80 1.00       2.2314e-01
 1.00 0.20       1.6094e+00
 1.00 0.40       9.1629e-01
 1.00 1.00       0.0000e+00
 2.00 2.00      -1.7918e+00
 3.00 3.00      -3.4012e+00
 4.00 4.00      -4.9416e+00
 5.00 5.00      -6.4457e+00
 6.00 2.00      -3.7377e+00
 6.00 3.00      -5.1240e+00
 6.00 4.00      -6.2226e+00
 6.00 5.00      -7.1389e+00
 6.00 6.00      -7.9273e+00
 7.00 7.00      -9.3937e+00

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox