# NAG FL Interfaces14cpf (beta_​log_​real_​vector)

## 1Purpose

s14cpf returns an array of values of the logarithm of the beta function, $\mathrm{ln}B\left(a,b\right)$.

## 2Specification

Fortran Interface
 Subroutine s14cpf ( n, a, b, f,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: ivalid(n) Real (Kind=nag_wp), Intent (In) :: a(n), b(n) Real (Kind=nag_wp), Intent (Out) :: f(n)
#include <nag.h>
 void s14cpf_ (const Integer *n, const double a[], const double b[], double f[], Integer ivalid[], Integer *ifail)
The routine may be called by the names s14cpf or nagf_specfun_beta_log_real_vector.

## 3Description

s14cpf calculates values for $\mathrm{ln}B\left(a,b\right)$, for arrays of arguments ${a}_{\mathit{i}}$ and ${b}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, 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 ${b}_{0}\le 2$,
 $ln⁡B = ln⁡Γ a0 + ln⁡Γ b0 - ln⁡Γ a0 + b0 .$
s14cpf 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: $\mathbf{n}$Integer Input
On entry: $n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{a}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
On entry: the argument ${a}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: ${\mathbf{a}}\left(\mathit{i}\right)>0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3: $\mathbf{b}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
On entry: the argument ${b}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: ${\mathbf{b}}\left(\mathit{i}\right)>0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
4: $\mathbf{f}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: $\mathrm{ln}B\left({a}_{i},{b}_{i}\right)$, the function values.
5: $\mathbf{ivalid}\left({\mathbf{n}}\right)$Integer array Output
On exit: ${\mathbf{ivalid}}\left(\mathit{i}\right)$ contains the error code for ${a}_{\mathit{i}}$ and ${b}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
${\mathbf{ivalid}}\left(i\right)=0$
No error.
${\mathbf{ivalid}}\left(i\right)=1$
${a}_{i}\text{​ or ​}{b}_{i}\le 0$.
6: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value 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 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, at least one value of a or b was invalid.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

s14cpf should produce full relative accuracy for all input arguments.

## 8Parallelism and Performance

s14cpf is not threaded in any implementation.

None.

## 10Example

This example reads values of a and b from a file, evaluates the function at each value of ${a}_{i}$ and ${b}_{i}$ and prints the results.

### 10.1Program Text

Program Text (s14cpfe.f90)

### 10.2Program Data

Program Data (s14cpfe.d)

### 10.3Program Results

Program Results (s14cpfe.r)