Incomplete beta function¶
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incbet(a, b, x)¶ Returns incomplete beta integral of the arguments, evaluated from zero to x. The function is defined as
Parameters:
See also incbi().
Description¶
\[\frac{\Gamma(a+b)}{\Gamma(a)+\Gamma(b)} \int_0^xt^{a-1}(1-t)^{b-1} dt\]
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 may 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 |
|---|---|---|
| incbet domain | x < 0, x > 1 | 0.0 |
| incbet underflow | 0.0 |
Reference: http://www.netlib.org/cephes/doubldoc.html#incbet