Integer, Intent (In)	::	n
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	ivalid(n)
Real (Kind=nag_wp), Intent (In)	::	a(n), b(n), x(n)
Real (Kind=nag_wp), Intent (Out)	::	w(n), w1(n)

C Header Interface

#include <nag.h>

void	s14cqf_ (const Integer n, const double a[], const double b[], const double x[], double w[], double w1[], Integer ivalid[], Integer ifail)

The routine may be called by the names s14cqf or nagf_specfun_beta_incomplete_vector.

3 Description

s14cqf evaluates the regularized incomplete beta function

I_{x} (a, b)

and its complement

1 - I_{x} (a, b)

in the normalized form, for arrays of arguments

x_{i}

a_{i}

and

b_{i}

, for

i = 1, 2, \dots, n

. The incomplete beta function and its complement are given by

\begin{matrix} I_{x} (a, b) & = & \frac{1}{B (a, b)} \int_{0}^{x} t^{a - 1} {(1 - t)}^{b - 1} d t \\ 1 - I_{x} (a, b) & = & I_{y} (b, a), where ​ y = 1 - x, \end{matrix}

with

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

s14cqf 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: $a (n)$ – Real (Kind=nag_wp) array Input

On entry: the argument

a_{i}

of the function, for

i = 1, 2, \dots, n

Constraint:

a (i) \geq 0.0

, for

i = 1, 2, \dots, n

3: $b (n)$ – Real (Kind=nag_wp) array Input

On entry: the argument

b_{i}

of the function, for

i = 1, 2, \dots, n

Constraints:

$b (i) \geq 0.0$ , for $i = 1, 2, \dots, n$ ;
$b (i) \neq 0.0$ or $a (i) \neq 0.0$ , for $i = 1, 2, \dots, n$ .

4: $x (n)$ – Real (Kind=nag_wp) array Input

On entry:

x_{i}

, the upper limit of integration, for

i = 1, 2, \dots, n

Constraints:

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

5: $w (n)$ – Real (Kind=nag_wp) array Output

On exit: the values of the incomplete beta function

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

evaluated from zero to

x_{i}

6: $w1 (n)$ – Real (Kind=nag_wp) array 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 array Output

On exit:

ivalid (i)

contains the error code for the

i

th evaluation, for

i = 1, 2, \dots, n

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

8: $ifail$ – 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, at least one argument had an invalid value.
Check ivalid for more information.

$ifail = 2$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

$ifail = - 99$: 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

s14cqf 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

Background information to multithreading can be found in the Multithreading documentation.

s14cqf 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 PY MB

NAG FL Interfaces14cqf (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 FL Interface
s14cqf (beta_incomplete_vector)