$0 \leq x \leq 1$ ,
$a \geq 0$ 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_incomplete_beta (s14ccc) 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: $a$ – doubleInput

On entry: the argument

a

of the function.

Constraint:

a \geq 0.0

2: $b$ – doubleInput

On entry: the argument

b

of the function.

Constraints:

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

3: $x$ – doubleInput

On entry:

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 $b \neq 0.0$ .

4: $w$ – double *Output

On exit: the value of the incomplete beta function

I_{x} (a, b)

evaluated from zero to

x

5: $w1$ – double *Output

On exit: 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.

6: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 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_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
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 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: $a \geq 0.0$ .

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

On entry, $x = 〈value〉$ .
Constraint: $0.0 \leq x \leq 1.0$ .
NE_REAL_2: On entry, $1.0 - x$ and b were zero.
Constraint: $1.0 - x$ or b must be nonzero.

On entry, a and b were zero.
Constraint: a or b must be nonzero.

On entry, x and a were zero.
Constraint: x or a must be nonzero.

7

Accuracy

nag_incomplete_beta (s14ccc) 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

nag_incomplete_beta (s14ccc) 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 its complement for

10

different values of

x

and prints the results.

NAG Library Function Document

nag_incomplete_beta (s14ccc)

▸▿ 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

2

Specification

3

Description