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

s14ccf 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: $a$ – Real (Kind=nag_wp) Input

On entry: the argument

a

of the function.

Constraint:

a \geq 0.0

2: $b$ – Real (Kind=nag_wp) Input

On entry: the argument

b

of the function.

Constraints:

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

3: $x$ – Real (Kind=nag_wp) Input

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: $w$ – Real (Kind=nag_wp) Output

On exit: the value of the incomplete beta function

I_{x} (a, b)

evaluated from zero to

x

5: $w1$ – Real (Kind=nag_wp) 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: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $a = ⟨ value ⟩$ .
Constraint: $a \geq 0.0$ .

On entry, $b = ⟨ value ⟩$ .
Constraint: $b \geq 0.0$ .

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

$ifail = 3$: On entry, $x = ⟨ value ⟩$ .
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.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

s14ccf 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

s14ccf 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.

s14cc: FL CL CPP AD

NAG FL Interfaces14ccf (beta_​incomplete)

▸▿ 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 FL Interface
s14ccf (beta_incomplete)