s Chapter Contents
s Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_log_beta (s14cbc)

## 1  Purpose

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

## 2  Specification

 #include #include
 double nag_log_beta (double a, double b, NagError *fail)

## 3  Description

nag_log_beta (s14cbc) calculates values for $\mathrm{ln}B\left(a,b\right)$ where $B$ is the beta function given by
 $Ba,b = ∫ 0 1 ta-1 1-t b-1 dt$
or equivalently
 $Ba,b = Γa Γb Γa+b$
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 ${a}_{0}\ge 8$,
 $ln⁡B = 0.5 ln⁡ 2π -0.5 lnb0 + Δa0 + Δ b0 - Δ a0+b0 - u - v ;$
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 ${b}_{0}\le 2$,
 $ln⁡B = ln⁡Γ a0 + ln⁡Γ b0 - ln⁡Γ a0 + b0 .$
nag_log_beta (s14cbc) is derived from BETALN in DiDonato and Morris (1992).

## 4  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

## 5  Arguments

1:    $\mathbf{a}$doubleInput
On entry: the argument $a$ of the function.
Constraint: ${\mathbf{a}}>0.0$.
2:    $\mathbf{b}$doubleInput
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 2.7 in How to Use the NAG Library and its Documentation).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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: ${\mathbf{b}}>0.0$.

## 7  Accuracy

nag_log_beta (s14cbc) should produce full relative accuracy for all input arguments.

## 8  Parallelism and Performance

nag_log_beta (s14cbc) is not threaded in any implementation.

None.

## 10  Example

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

### 10.1  Program Text

Program Text (s14cbce.c)

### 10.2  Program Data

Program Data (s14cbce.d)

### 10.3  Program Results

Program Results (s14cbce.r)