NAG FL Interfaces14cbf (beta_​log_​real)

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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 <nag.h>
 double s14cbf_ (const double *a, const double *b, Integer *ifail)
The routine may be called by the names s14cbf or nagf_specfun_beta_log_real.

3Description

s14cbf calculates values for $\mathrm{ln}B\left(a,b\right)$ 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 ${b}_{0}\le 2$,
 $ln⁡B = ln⁡Γ (a0) + ln⁡Γ (b0) - ln⁡Γ (a0+b0) .$
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: $\mathbf{ifail}$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, ${\mathbf{a}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{a}}>0.0$.
On entry, ${\mathbf{b}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{b}}>0.0$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
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

s14cbf should produce full relative accuracy for all input arguments.

8Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
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)