NAG FL Interface
s14cpf (beta_​log_​real_​vector)

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1 Purpose

s14cpf returns an array of values of the logarithm of the beta function, lnB(a,b).

2 Specification

Fortran Interface
Subroutine s14cpf ( n, a, b, f, ivalid, ifail)
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)
C Header Interface
#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.

3 Description

s14cpf calculates values for lnB(a,b), for arrays of arguments ai and bi, for i=1,2,,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 Γ(x) is the gamma function. Note that the beta function is symmetric, so that B(a,b)=B(b,a).
In order to efficiently obtain accurate results several methods are used depending on the parameters a and b.
Let a0=min(a,b) and b0=max(a,b). Then:
for a08,
lnB = 0.5 ln (2π) -0.5 ln(b0) + Δ(a0) + Δ (b0) - Δ (a0+b0) - u - v ;  
where for a0<1, for 2<a0<8, a0 is reduced to the interval [1,2] by B(a,b)=a0-1a0+b0-1 B(a0-1,b0);
for 1a02,
s14cpf is derived from BETALN in DiDonato and Morris (1992).

4 References

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

5 Arguments

1: n Integer Input
On entry: n, the number of points.
Constraint: n0.
2: a(n) Real (Kind=nag_wp) array Input
On entry: the argument ai of the function, for i=1,2,,n.
Constraint: a(i)>0, for i=1,2,,n.
3: b(n) Real (Kind=nag_wp) array Input
On entry: the argument bi of the function, for i=1,2,,n.
Constraint: b(i)>0, for i=1,2,,n.
4: f(n) Real (Kind=nag_wp) array Output
On exit: lnB(ai,bi), the function values.
5: ivalid(n) Integer array Output
On exit: ivalid(i) contains the error code for ai and bi, for i=1,2,,n.
No error.
ai​ or ​bi0.
6: 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 -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry 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:
On entry, at least one value of a or b was invalid.
Check ivalid for more information.
On entry, n=value.
Constraint: n0.
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.
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.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

s14cpf should produce full relative accuracy for all input arguments.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s14cpf is not threaded in any implementation.

9 Further Comments


10 Example

This example reads values of a and b from a file, evaluates the function at each value of ai and bi and prints the results.

10.1 Program Text

Program Text (s14cpfe.f90)

10.2 Program Data

Program Data (s14cpfe.d)

10.3 Program Results

Program Results (s14cpfe.r)