NAG Library Function Document

nag_incomplete_beta (s14ccc)

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

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 parameter

a

of the function.

Constraint:

a \geq 0.0

2: b – doubleInput

On entry: the parameter

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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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.
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

Not applicable.

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 Documentnag_incomplete_beta (s14ccc)

+− Contents

1 Purpose

2 Specification

3 Description