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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