naginterfaces.library.specfun.beta_​log_​real_​vector

naginterfaces.library.specfun.beta_log_real_vector(a, b)

beta_log_real_vector returns an array of values of the logarithm of the beta function, $ln\left(B\right)(a,b)$.

For full information please refer to the NAG Library document for s14cp

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/s/s14cpf.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 $\text{i}=1,2,\dots ,n$.

Returns

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

$ln\left(B\right)(a_i,b_i)$, the function values.

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

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

$ivalid[i-1]=0$

No error.

$ivalid[i-1]=1$

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

Raises

NagValueError

(errno $2$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

Warns

NagAlgorithmicWarning

(errno $1$)

On entry, at least one value of $a$ or $b$ was invalid.

Check $ivalid$ for more information.

Notes

beta_log_real_vector calculates values for $ln\left(B\right)(a,b)$, for arrays of arguments $a_{\text{i}}$ and $b_{\text{i}}$, for $\text{i}=1,2,\dots ,n$, where $B$ is the beta function given by

$$B(a,b)={\int }_0^1{t}^{a-1}{(1-t)}^{b-1}\text{dt}$$

or equivalently

$$B(a,b)=\frac{\Gamma \left(a\right)\Gamma \left(b\right)}{\Gamma (a+b)}$$

and $\Gamma \left(x\right)$ is the gamma function. Note that the beta function is symmetric, so that $B(a,b)=B(b,a)$.

In order to efficiently obtain accurate results several methods are used depending on the parameters $a$ and $b$.

Let $a_0=min(a,b)$ and $b_0=max(a,b)$. Then:

for $a_0\ge 8$,

$$\begin{array}{ccc}ln\left(B\right)& =& 0.5ln\left(\left(2\pi \right)-0.5\right)ln\left(b_0\right)+\Delta \left(a_0\right)+\Delta \left(b_0\right)-\Delta (a_0+b_0)-u-v\text{;}\end{array}$$

where

$\Delta \left(a_0\right)=ln\left(\Gamma \left(a_0\right)\right)-(a_0-0.5)ln\left(a_0\right)+a_0-0.5ln\left(2\pi \right)$,

$u=-(a_0-0.5)ln\left[\frac{a_0}{a_0+b_0}\right]$ and

$v=b_{0}ln(1+\frac{a_0}{b_0})$;

for $a_0<1$,

for $b_0\ge 8$,

$$ln\left(B\right)=ln\left(\Gamma \right)\left(a_0\right)+ln\left(\frac{\Gamma \left(b_0\right)}{\Gamma (a_0+b_0)}\right)\text{;}$$

for $b_0<8$,

$$ln\left(B\right)=ln\left(\Gamma \right)\left(a_0\right)+ln\left(\Gamma \right)\left(b_0\right)-ln\left(\Gamma \right)(a_0+b_0)\text{;}$$

for $2<a_0<8$, $a_0$ is reduced to the interval $[1,2]$ by $B(a,b)=\frac{a_0-1}{a_0+b_0-1}B(a_0-1,b_0)$;

for $1\le a_0\le 2$,

for $b_0\ge 8$,

$$ln\left(B\right)=ln\left(\Gamma \right)\left(a_0\right)+ln\left(\frac{\Gamma \left(b_0\right)}{\Gamma (a_0+b_0)}\right)\text{;}$$

for $2<b_0<8$, $b_0$ is reduced to the interval $[1,2]$;

for $b_0\le 2$,

$$ln\left(B\right)=ln\left(\Gamma \right)\left(a_0\right)+ln\left(\Gamma \right)\left(b_0\right)-ln\left(\Gamma \right)(a_0+b_0)\text{.}$$

beta_log_real_vector is derived from BETALN 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