naginterfaces.library.specfun.beta_log_real¶

naginterfaces.library.specfun.beta_log_real(a, b)[source]¶

beta_log_real returns the value of the logarithm of the beta function, $l n (B) (a, b)$ .

For full information please refer to the NAG Library document for s14cb

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s14cbf.html

Parameters

afloat: The argument $a$ of the function.
bfloat: The argument $b$ of the function.

Returns

lbabfloat: The value of the logarithm of the beta function, $l n (B) (a, b)$ .

Raises

NagValueError

(errno $1$ )

On entry, $b = ⟨ v a l u e ⟩$ .

Constraint: $b > 0.0$ .

(errno $1$ )

On entry, $a = ⟨ v a l u e ⟩$ .

Constraint: $a > 0.0$ .

Notes

beta_log_real calculates values for $l n (B) (a, b)$ where $B$ is the beta function given by

B (a, b) = \int_{0}^{1} t^{a - 1} {(1 - t)}^{b - 1} dt

or equivalently

B (a, b) = \frac{Γ (a) Γ (b)}{Γ (a + b)}

and $Γ (x)$ 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 $a_{0} = m i n (a, b)$ and $b_{0} = m a x (a, b)$ . Then:

for $a_{0} \geq 8$ ,

\begin{matrix} l n (B) & = & 0.5 l n ((2 π) - 0.5) l n (b_{0}) + Δ (a_{0}) + Δ (b_{0}) - Δ (a_{0} + b_{0}) - u - v; \end{matrix}

where

$Δ (a_{0}) = l n (Γ (a_{0})) - (a_{0} - 0.5) l n (a_{0}) + a_{0} - 0.5 l n (2 π)$ ,

$u = - (a_{0} - 0.5) l n [\frac{a_{0}}{a_{0} + b_{0}}]$ and

$v = b_{0} l n (1 + \frac{a_{0}}{b_{0}})$ .

for $a_{0} < 1$ ,

for $b_{0} \geq 8$ ,

$l n (B) = l n (Γ) (a_{0}) + l n (\frac{Γ (b_{0})}{Γ (a_{0} + b_{0})});$

for $b_{0} < 8$ ,

$l n (B) = l n (Γ) (a_{0}) + l n (Γ) (b_{0}) - l n (Γ) (a_{0} + b_{0});$

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 $1 \leq a_{0} \leq 2$ ,

for $b_{0} \geq 8$ ,

$l n (B) = l n (Γ) (a_{0}) + l n (\frac{Γ (b_{0})}{Γ (a_{0} + b_{0})});$

for $2 < b_{0} < 8$ , $b_{0}$ is reduced to the interval $[1, 2]$ ;

for $b_{0} \leq 2$ ,

$l n (B) = l n (Γ) (a_{0}) + l n (Γ) (b_{0}) - l n (Γ) (a_{0} + b_{0}) .$

beta_log_real 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

NAG and Python

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naginterfaces.library.specfun.beta_log_real¶

naginterfaces.library.specfun.beta_​log_​real¶

naginterfaces.library.specfun.beta_log_real¶