# NAG Library Routine Document

## 1Purpose

s14cbf returns the value of the logarithm of the beta function, $\mathrm{ln}B\left(a,b\right)$, via the routine name.

## 2Specification

Fortran Interface
 Function s14cbf ( a, b,
 Real (Kind=nag_wp) :: s14cbf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: a, b
#include nagmk26.h
 double s14cbf_ (const double *a, const double *b, Integer *ifail)

## 3Description

s14cbf 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 ${b}_{0}\le 2$,
 $ln⁡B = ln⁡Γ a0 + ln⁡Γ b0 - ln⁡Γ a0 + b0 .$
s14cbf 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:     $\mathbf{a}$ – Real (Kind=nag_wp)Input
On entry: the argument $a$ of the function.
Constraint: ${\mathbf{a}}>0.0$.
2:     $\mathbf{b}$ – Real (Kind=nag_wp)Input
On entry: the argument $b$ of the function.
Constraint: ${\mathbf{b}}>0.0$.
3:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{a}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{a}}>0.0$.
On entry, ${\mathbf{b}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{b}}>0.0$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

s14cbf should produce full relative accuracy for all input arguments.

## 8Parallelism and Performance

s14cbf 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 (s14cbfe.f90)

### 10.2Program Data

Program Data (s14cbfe.d)

### 10.3Program Results

Program Results (s14cbfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017