nag_incomplete_beta (s14ccc) (PDF version)
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NAG Library Manual

NAG Library Function Document

nag_incomplete_beta (s14ccc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_incomplete_beta (s14ccc) computes values for the incomplete beta function Ixa,b and its complement 1-Ixa,b.

2  Specification

#include <nag.h>
#include <nags.h>
void  nag_incomplete_beta (double a, double b, double x, double *w, double *w1, NagError *fail)

3  Description

nag_incomplete_beta (s14ccc) evaluates the incomplete beta function and its complement in the normalized form
Ixa,b = 1 Ba,b 0 x ta-1 1-t b-1 dt 1Ix a,b = Iy b,a , where ​ y=1-x ,
Several methods are used to evaluate the functions depending on the parameters a, b and x. The methods include Wise's asymptotic expansion (see Wise (1950)) when a>b, continued fraction derived by DiDonato and Morris (1992) when a, b>1, and power series when b1 or b×x0.7. When both a and b are large, specifically a, b15, the DiDonato and Morris (1992) asymptotic expansion is employed for greater efficiency.
Once either Ixa,b or Iyb,a is computed, the other is obtained by subtraction from 1. In order to avoid loss of relative precision in this subtraction, the smaller of Ixa,b and Iyb,a is computed first.
nag_incomplete_beta (s14ccc) is derived from BRATIO 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
Wise M E (1950) The incomplete beta function as a contour integral and a quickly converging series for its inverse Biometrika 37 208–218

5  Arguments

1:     adoubleInput
On entry: the parameter a of the function.
Constraint: a0.0.
2:     bdoubleInput
On entry: the parameter b of the function.
  • b0.0;
  • either b0.0 or a0.0.
3:     xdoubleInput
On entry: x, upper limit of integration.
  • 0.0x1.0;
  • either x0.0 or a0.0;
  • either 1-x0.0 or b0.0.
4:     wdouble *Output
On exit: the value of the incomplete beta function Ixa,b evaluated from zero to x.
5:     w1double *Output
On exit: the value of the complement of the incomplete beta function 1-Ixa,b, i.e., the incomplete beta function evaluated from x to one.
6:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
On entry, argument value had an illegal value.
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.
On entry, a=value.
Constraint: a0.0.
On entry, b=value.
Constraint: b0.0.
On entry, x=value.
Constraint: 0.0x1.0.
On entry, 1.0-x and b were zero.
Constraint: 1.0-x or b must be nonzero.
On entry, a and b were zero.
Constraint: a or b must be nonzero.
On entry, x and a were zero.
Constraint: x or a must be nonzero.

7  Accuracy

nag_incomplete_beta (s14ccc) is designed to maintain relative accuracy for all arguments. For very tiny results (of the order of machine precision or less) some relative accuracy may be lost – loss of three or four decimal places has been observed in experiments. For other arguments full relative accuracy may be expected.

8  Parallelism and Performance

Not applicable.

9  Further Comments


10  Example

This example reads values of the arguments a and b from a file, evaluates the function and its complement for 10 different values of x and prints the results.

10.1  Program Text

Program Text (s14ccce.c)

10.2  Program Data

Program Data (s14ccce.d)

10.3  Program Results

Program Results (s14ccce.r)

nag_incomplete_beta (s14ccc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014