# NAG CL Interfaces14cpc (beta_​log_​real_​vector)

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## 1Purpose

s14cpc returns an array of values of the logarithm of the beta function, $\mathrm{ln}B\left(a,b\right)$.

## 2Specification

 #include
 void s14cpc (Integer n, const double a[], const double b[], double f[], Integer ivalid[], NagError *fail)
The function may be called by the names: s14cpc, nag_specfun_beta_log_real_vector or nag_log_beta_vector.

## 3Description

s14cpc 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 ${b}_{0}\le 2$,
 $ln⁡B = ln⁡Γ (a0) + ln⁡Γ (b0) - ln⁡Γ (a0+b0) .$
s14cpc 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: $\mathbf{a}\left[{\mathbf{n}}\right]$const double Input
On entry: the argument ${a}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: ${\mathbf{a}}\left[\mathit{i}-1\right]>0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3: $\mathbf{b}\left[{\mathbf{n}}\right]$const double Input
On entry: the argument ${b}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: ${\mathbf{b}}\left[\mathit{i}-1\right]>0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
4: $\mathbf{f}\left[{\mathbf{n}}\right]$double Output
On exit: $\mathrm{ln}B\left({a}_{i},{b}_{i}\right)$, the function values.
5: $\mathbf{ivalid}\left[{\mathbf{n}}\right]$Integer Output
On exit: ${\mathbf{ivalid}}\left[\mathit{i}-1\right]$ contains the error code for ${a}_{\mathit{i}}$ and ${b}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
${\mathbf{ivalid}}\left[i-1\right]=0$
No error.
${\mathbf{ivalid}}\left[i-1\right]=1$
${a}_{i}\text{​ or ​}{b}_{i}\le 0$.
6: $\mathbf{fail}$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.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
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.
NW_IVALID
On entry, at least one value of a or b was invalid.

## 7Accuracy

s14cpc should produce full relative accuracy for all input arguments.

## 8Parallelism and Performance

s14cpc 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 (s14cpce.c)

### 10.2Program Data

Program Data (s14cpce.d)

### 10.3Program Results

Program Results (s14cpce.r)