incbet

Incomplete beta integral.

SYNOPSIS:

double a, b, x, y, incbet();
y = incbet(a, b, x);

DESCRIPTION:
Returns the incomplete beta integral of the arguments, evaluated from zero to x. The function is defined as:
                  x
     -            -
    | (a+b)      | | a-1     b-1
-----------    |   t   (1-t)   dt.
   -     -     | |
| (a) | (b)   -
                 0

The domain of definition is 0 <= x <= 1. In this implementation a and b are restricted to positive values. The integral from x to 1 can be obtained by the symmetry relation:
    1 - incbet(a, b, x) = incbet(b, a, 1-x).
The integral is evaluated by a continued fraction expansion or, when b*x is small, by a power series.

ACCURACY:

Tested at uniformly distributed random points (a,b,x) with a and b in "domain" and x between 0 and 1.

                                        Relative error
arithmetic   domain     # trials      peak         rms
    IEEE      0,5         10000       6.9e-15     4.5e-16
    IEEE      0,85       250000       2.2e-13     1.7e-14
    IEEE      0,1000      30000       5.3e-12     6.3e-13
    IEEE      0,10000    250000       9.3e-11     7.1e-12
    IEEE      0,100000    10000       8.7e-10     4.8e-11

Outputs smaller than the IEEE gradual underflow threshold were excluded from these statistics.

ERROR MESSAGES:

   message         condition      value returned
   domain          x<0, x>1          0.0
   underflow                         0.0