# NAG CL Interfaces14cbc (beta_​log_​real)

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## 1Purpose

s14cbc returns the value of the logarithm of the beta function, $\mathrm{ln}B\left(a,b\right)$.

## 2Specification

 #include
 double s14cbc (double a, double b, NagError *fail)
The function may be called by the names: s14cbc, nag_specfun_beta_log_real or nag_log_beta.

## 3Description

s14cbc calculates values for $\mathrm{ln}B\left(a,b\right)$ where $B$ is the beta function given by
 $B(a,b) = ∫ 0 1 ta-1 (1-t) b-1 dt$
or equivalently
 $B(a,b) = Γ(a) Γ(b) Γ(a+b)$
and $\Gamma \left(x\right)$ is the gamma function. Note that the beta function is symmetric, so that $B\left(a,b\right)=B\left(b,a\right)$.
In order to efficiently obtain accurate results several methods are used depending on the parameters $a$ and $b$.
Let ${a}_{0}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(a,b\right)$ and ${b}_{0}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(a,b\right)$. Then:
for ${a}_{0}\ge 8$,
 $ln⁡B = 0.5 ln⁡ (2π) -0.5 ln(b0) + Δ(a0) + Δ (b0) - Δ (a0+b0) - u - v ;$
where
• $\Delta \left({a}_{0}\right)=\mathrm{ln}\Gamma \left({a}_{0}\right)-\left({a}_{0}-0.5\right)\mathrm{ln}{a}_{0}+{a}_{0}-0.5\mathrm{ln}\left(2\pi \right)$,
• $u=-\left({a}_{0}-0.5\right)\mathrm{ln}\left[\frac{{a}_{0}}{{a}_{0}+{b}_{0}}\right]$  and
• $v={b}_{0}\mathrm{ln}\left(1+\frac{{a}_{0}}{{b}_{0}}\right)$.
for ${a}_{0}<1$,
• for ${b}_{0}\ge 8$,
 $ln⁡B = ln⁡Γ (a0) + ln⁡ Γ (b0) Γ (a0+b0) ;$
• for ${b}_{0}<8$,
 $ln⁡B = ln⁡Γ (a0) + ln⁡Γ (b0) - ln⁡Γ (a0+b0) ;$
for $2<{a}_{0}<8$, ${a}_{0}$ is reduced to the interval $\left[1,2\right]$ by $B\left(a,b\right)=\frac{{a}_{0}-1}{{a}_{0}+{b}_{0}-1}B\left({a}_{0}-1,{b}_{0}\right)$;
for $1\le {a}_{0}\le 2$,
• for ${b}_{0}\ge 8$,
 $ln⁡B = ln⁡Γ (a0) + ln⁡ Γ (b0) Γ (a0+b0) ;$
• for $2<{b}_{0}<8$, ${b}_{0}$ is reduced to the interval $\left[1,2\right]$;
• for ${b}_{0}\le 2$,
 $ln⁡B = ln⁡Γ (a0) + ln⁡Γ (b0) - ln⁡Γ (a0+b0) .$
s14cbc is derived from BETALN in DiDonato and Morris (1992).

## 4References

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

## 5Arguments

1: $\mathbf{a}$double Input
On entry: the argument $a$ of the function.
Constraint: ${\mathbf{a}}>0.0$.
2: $\mathbf{b}$double Input
On entry: the argument $b$ of the function.
Constraint: ${\mathbf{b}}>0.0$.
3: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL
On entry, ${\mathbf{a}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{a}}>0.0$.
On entry, ${\mathbf{b}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{b}}>0.0$.

## 7Accuracy

s14cbc should produce full relative accuracy for all input arguments.

## 8Parallelism and Performance

s14cbc is not threaded in any implementation.

None.

## 10Example

This example reads values of the arguments $a$ and $b$ from a file, evaluates the function and prints the results.

### 10.1Program Text

Program Text (s14cbce.c)

### 10.2Program Data

Program Data (s14cbce.d)

### 10.3Program Results

Program Results (s14cbce.r)