naginterfaces.library.specfun.beta_​incomplete_​vector

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

beta_incomplete_vector computes an array of 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 s14cq

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/s/s14cqf.html

Parameters

afloat, array-like, shape $\left(n\right)$

The argument $a_{\text{i}}$ of the function, for $\text{i}=1,2,\dots ,n$.

bfloat, array-like, shape $\left(n\right)$

The argument $b_{\text{i}}$ of the function, for $\text{i}=1,2,\dots ,n$.

xfloat, array-like, shape $\left(n\right)$

$x_{\text{i}}$, the upper limit of integration, for $\text{i}=1,2,\dots ,n$.

Returns

wfloat, ndarray, shape $\left(n\right)$

The values of the incomplete beta function $I_{x_i}(a_i,b_i)$ evaluated from zero to $x_i$.

w1float, ndarray, shape $\left(n\right)$

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.

ivalidint, ndarray, shape $\left(n\right)$

$ivalid[\text{i}-1]$ contains the error code for the $\text{i}$th evaluation, for $\text{i}=1,2,\dots ,n$.

$ivalid[i-1]=0$

No error.

$ivalid[i-1]=1$

$a_i\text{or}{b}_i<0$.

$ivalid[i-1]=2$

Both $a_i\text{and}{b}_i=0$.

$ivalid[i-1]=3$

$x_i\notin [0,1]$.

$ivalid[i-1]=4$

Both $x_i\text{and}{a}_i=0$.

$ivalid[i-1]=5$

Both $1-x_i\text{and}{b}_i=0$.

Raises

NagValueError

(errno $2$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

Warns

NagAlgorithmicWarning

(errno $1$)

On entry, at least one argument had an invalid value.

Check $ivalid$ for more information.

Notes

beta_incomplete_vector 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 $\text{i}=1,2,\dots ,n$. The incomplete beta function and its complement are given by

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

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_vector 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