naginterfaces.library.specfun.beta_​incomplete

naginterfaces.library.specfun.beta_incomplete(a, b, x)

beta_incomplete computes values for the regularized incomplete beta function $I_x(a,b)$ and its complement $1-I_x(a,b)$.

For full information please refer to the NAG Library document for s14cc

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/s/s14ccf.html

Parameters

afloat

The argument $a$ of the function.

bfloat

The argument $b$ of the function.

xfloat

$x$, upper limit of integration.

Returns

wfloat

The value of the incomplete beta function $I_x(a,b)$ evaluated from zero to $x$.

w1float

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.

Raises

NagValueError

(errno $1$)

On entry, $b=⟨value⟩$.

Constraint: $b\ge 0.0$.

(errno $1$)

On entry, $a=⟨value⟩$.

Constraint: $a\ge 0.0$.

(errno $2$)

On entry, $a$ and $b$ were zero.

Constraint: $a$ or $b$ must be nonzero.

(errno $3$)

On entry, $x=⟨value⟩$.

Constraint: $0.0\le x\le 1.0$.

(errno $4$)

On entry, $x$ and $a$ were zero.

Constraint: $x$ or $a$ must be nonzero.

(errno $5$)

On entry, $1.0-x$ and $b$ were zero.

Constraint: $1.0-x$ or $b$ must be nonzero.

Notes

beta_incomplete evaluates the regularized incomplete beta function and its complement in the normalized form

$$\begin{array}{c}\begin{array}{ccc}{I}_x(a,b)& =& \frac{1}{B(a,b)}{\int }_0^x{t}^{a-1}{(1-t)}^{b-1}dt\\ 1-I_x(a,b)& =& I_y(b,a)\text{, where}y=1-x\text{,}\end{array}\end{array}$$

with

$0\le x\le 1$,

$a\ge 0$ and $b\ge 0$,

and the beta function $B(a,b)$ is defined as $B(a,b)={\int }_0^1{t}^{a-1}{(1-t)}^{b-1}dt=\frac{\Gamma \left(a\right)\Gamma \left(b\right)}{\Gamma (a+b)}$ where $\Gamma \left(y\right)$ 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 $b\le 1$ or $b\times x\le 0.7$. 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 $I_x(a,b)$ or $I_y(b,a)$ 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.

beta_incomplete is derived from BRATIO in DiDonato and Morris (1992).

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