$0 \leq x \leq 1$ ,
$a \geq 0$ 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.

s14cqc is derived from BRATIO 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

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

5 Arguments

1: $n$ – Integer Input

On entry:

n

, the number of points.

Constraint:

n \geq 0

2: $a [n]$ – const double Input

On entry: the argument

a_{i}

of the function, for

i = 1, 2, \dots, n

Constraint:

a [i - 1] \geq 0.0

, for

i = 1, 2, \dots, n

3: $b [n]$ – const double Input

On entry: the argument

b_{i}

of the function, for

i = 1, 2, \dots, n

Constraints:

$b [i - 1] \geq 0.0$ , for $i = 1, 2, \dots, n$ ;
$b [i - 1] \neq 0.0$ or $a [i - 1] \neq 0.0$ , for $i = 1, 2, \dots, n$ .

4: $x [n]$ – const double Input

On entry:

x_{i}

, the upper limit of integration, for

i = 1, 2, \dots, n

Constraints:

$x [i - 1] \geq 0.0$ , for $i = 1, 2, \dots, n$ ;
$x [i - 1] \neq 0.0$ or $a [i - 1] \neq 0.0$ , for $i = 1, 2, \dots, n$ ;
$1 - x [i - 1] \neq 0.0$ or $b [i - 1] \neq 0.0$ , for $i = 1, 2, \dots, n$ .

5: $w [n]$ – double Output

On exit: the values of the incomplete beta function

I_{x_{i}} (a_{i}, b_{i})

evaluated from zero to

x_{i}

6: $w1 [n]$ – double Output

On exit: the values of the complement of the incomplete beta function

1 - I_{x_{i}} (a_{i}, b_{i})

, i.e., the incomplete beta function evaluated from

x_{i}

to one.

7: $ivalid [n]$ – Integer Output

On exit:

ivalid [i - 1]

contains the error code for the

i

th evaluation, for

i = 1, 2, \dots, n

$ivalid [i - 1] = 0$: No error.
$ivalid [i - 1] = 1$: $a_{i} or b_{i} < 0$ .
$ivalid [i - 1] = 2$: Both $a_{i} and b_{i} = 0$ .
$ivalid [i - 1] = 3$: $x_{i} \notin [0, 1]$ .
$ivalid [i - 1] = 4$: Both $x_{i} and a_{i} = 0$ .
$ivalid [i - 1] = 5$: Both $1 - x_{i} and b_{i} = 0$ .

8: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NW_IVALID: On entry, at least one argument had an invalid value.
Check ivalid for more information.

7 Accuracy

s14cqc 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.

8 Parallelism and Performance

s14cqc is not threaded in any implementation.

9 Further Comments

None.

10 Example

This example reads

10

values for each vector argument

a

b

and

x

from a file. It then evaluates the function and its complement for each set of values.

s14cq: FL CL CPP AD

NAG CL Interfaces14cqc (beta_​incomplete_​vector)

▸▿ 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

NAG CL Interface
s14cqc (beta_incomplete_vector)