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

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## 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
 $B(a,b) = ∫ 0 1 ta-1 (1-t) b-1 dt$
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 ${b}_{0}\le 2$,
 $ln⁡B = ln⁡Γ (a0) + ln⁡Γ (b0) - ln⁡Γ (a0+b0) .$
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 $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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, 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)