$0 \leq x \leq 1$ ,
$a \geq 0$ and $b \geq 0$ ,
and the beta function $B (a, b)$ is defined as $B (a, b) = \int_{0}^{1} t^{a - 1} {(1 - t)}^{b - 1} d t = \frac{Γ (a) Γ (b)}{Γ (a + b)}$ where $Γ (y)$ is the gamma function.

Several methods are used to evaluate the functions depending on the arguments

a

b

and

x

. The methods include Wise's asymptotic expansion (see Wise (1950)) when

a > b

, continued fraction derived by DiDonato and Morris (1992) when

a

b > 1

, and power series when

b \leq 1

b \times x \leq 0.7

. When both

a

and

b

are large, specifically

a

b \geq 15

, the DiDonato and Morris (1992) asymptotic expansion is employed for greater efficiency.

Once either

I_{x} (a, b)

I_{y} (b, a)

is computed, the other is obtained by subtraction from

1

. In order to avoid loss of relative precision in this subtraction, the smaller of

I_{x} (a, b)

and

I_{y} (b, a)

is computed first.

nag_specfun_beta_incomplete (s14cc) is derived from BRATIO 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

Wise M E (1950) The incomplete beta function as a contour integral and a quickly converging series for its inverse Biometrika 37 208–218

Parameters

Compulsory Input Parameters

1: $a$ – double scalar

The argument

a

of the function.

Constraint:

a \geq 0.0

2: $b$ – double scalar

The argument

b

of the function.

Constraints:

$b \geq 0.0$ ;
either $b \neq 0.0$ or $a \neq 0.0$ .

3: $x$ – double scalar

x

, upper limit of integration.

Constraints:

$0.0 \leq x \leq 1.0$ ;
either $x \neq 0.0$ or $a \neq 0.0$ ;
either $1 - x \neq 0.0$ or $b \neq 0.0$ .

Optional Input Parameters

None.

Output Parameters

1: $w$ – double scalar: The value of the incomplete beta function $I_{x} (a, b)$ evaluated from zero to $x$ .
2: $w1$ – double scalar: The value of the complement of the incomplete beta function $1 - I_{x} (a, b)$ , i.e., the incomplete beta function evaluated from $x$ to one.
3: $ifail$ – 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: $a \geq 0.0$ .

Constraint: $b \geq 0.0$ .

$ifail = 2$: On entry, a and b were zero.
Constraint: a or b must be nonzero.

$ifail = 3$: Constraint: $0.0 \leq x \leq 1.0$ .

$ifail = 4$: On entry, x and a were zero.
Constraint: x or a must be nonzero.

$ifail = 5$: On entry, $1.0 - x$ and b were zero.
Constraint: $1.0 - x$ or b must be nonzero.

$ifail = - 99$: 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_incomplete (s14cc) is designed to maintain relative accuracy for all arguments. For very tiny results (of the order of machine precision or less) some relative accuracy may be lost – loss of three or four decimal places has been observed in experiments. For other arguments full relative accuracy may be expected.

Further Comments

None.

Example

This example reads values of the arguments

a

and

b

from a file, evaluates the function and its complement for

10

different values of

x

and prints the results.

Open in the MATLAB editor: s14cc_example

function s14cc_example


fprintf('s14cc example results\n\n');

a = 5.3;
b = 10.1;
fprintf('\n   a     b     x          Ix(a,b)          1-Ix(a,b)\n');
for x = 0.01:0.01:0.1
  [w, w1, ifail] = s14cc(a, b, x);
  fprintf('%6.2f%6.2f%6.2f%17.4e%17.4e\n', a, b, x, w, w1);
end

s14cc example results


   a     b     x          Ix(a,b)          1-Ix(a,b)
  5.30 10.10  0.01       6.4755e-08       1.0000e+00
  5.30 10.10  0.02       2.3613e-06       1.0000e+00
  5.30 10.10  0.03       1.8734e-05       9.9998e-01
  5.30 10.10  0.04       7.9575e-05       9.9992e-01
  5.30 10.10  0.05       2.3997e-04       9.9976e-01
  5.30 10.10  0.06       5.8255e-04       9.9942e-01
  5.30 10.10  0.07       1.2174e-03       9.9878e-01
  5.30 10.10  0.08       2.2797e-03       9.9772e-01
  5.30 10.10  0.09       3.9249e-03       9.9608e-01
  5.30 10.10  0.10       6.3236e-03       9.9368e-01

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

Chapter Contents

Chapter Introduction

NAG Toolbox