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NAG Library Manual

# NAG Library Function Documentnag_incomplete_beta (s14ccc)

## 1  Purpose

nag_incomplete_beta (s14ccc) computes values for the incomplete beta function ${I}_{x}\left(a,b\right)$ and its complement $1-{I}_{x}\left(a,b\right)$.

## 2  Specification

 #include #include
 void nag_incomplete_beta (double a, double b, double x, double *w, double *w1, NagError *fail)

## 3  Description

nag_incomplete_beta (s14ccc) evaluates the incomplete beta function and its complement in the normalized form
 $Ixa,b = 1 Ba,b ∫ 0 x ta-1 1-t b-1 dt 1–Ix a,b = Iy b,a , where ​ y=1-x ,$
with
• $0\le x\le 1$,
• $a\ge 0$ and $b\ge 0$,
• and the beta function $B\left(a,b\right)$ is defined as $B\left(a,b\right)=\underset{0}{\overset{1}{\int }}{t}^{a-1}{\left(1-t\right)}^{b-1}dt=\frac{\Gamma \left(a\right)\Gamma \left(b\right)}{\Gamma \left(a+b\right)}$ where $\Gamma \left(y\right)$ 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 $b\le 1$ or $b×x\le 0.7$. When both $a$ and $b$ are large, specifically $a$, $b\ge 15$, the DiDonato and Morris (1992) asymptotic expansion is employed for greater efficiency.
Once either ${I}_{x}\left(a,b\right)$ or ${I}_{y}\left(b,a\right)$ is computed, the other is obtained by subtraction from $1$. In order to avoid loss of relative precision in this subtraction, the smaller of ${I}_{x}\left(a,b\right)$ and ${I}_{y}\left(b,a\right)$ 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:    $\mathbf{a}$doubleInput
On entry: the parameter $a$ of the function.
Constraint: ${\mathbf{a}}\ge 0.0$.
2:    $\mathbf{b}$doubleInput
On entry: the parameter $b$ of the function.
Constraints:
• ${\mathbf{b}}\ge 0.0$;
• either ${\mathbf{b}}\ne 0.0$ or ${\mathbf{a}}\ne 0.0$.
3:    $\mathbf{x}$doubleInput
On entry: $x$, upper limit of integration.
Constraints:
• $0.0\le {\mathbf{x}}\le 1.0$;
• either ${\mathbf{x}}\ne 0.0$ or ${\mathbf{a}}\ne 0.0$;
• either $1-{\mathbf{x}}\ne 0.0$ or ${\mathbf{b}}\ne 0.0$.
4:    $\mathbf{w}$double *Output
On exit: the value of the incomplete beta function ${I}_{x}\left(a,b\right)$ evaluated from zero to $x$.
5:    $\mathbf{w1}$double *Output
On exit: the value of the complement of the incomplete beta function $1-{I}_{x}\left(a,b\right)$, i.e., the incomplete beta function evaluated from $x$ to one.
6:    $\mathbf{fail}$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.
See Section 3.2.1.2 in the Essential Introduction for further information.
On entry, argument $〈\mathit{\text{value}}〉$ 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 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
NE_REAL
On entry, ${\mathbf{a}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{a}}\ge 0.0$.
On entry, ${\mathbf{b}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{b}}\ge 0.0$.
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: $0.0\le {\mathbf{x}}\le 1.0$.
NE_REAL_2
On entry, $1.0-{\mathbf{x}}$ and b were zero.
Constraint: $1.0-{\mathbf{x}}$ 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.

Not applicable.

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.

### 10.1  Program Text

Program Text (s14ccce.c)

### 10.2  Program Data

Program Data (s14ccce.d)

### 10.3  Program Results

Program Results (s14ccce.r)