betai

betai (a, b, x [, accuracy ])

The betai function returns the regularized incomplete beta function I_x (a, b ),

$$\displaystyle I_{x}(a, b)=\frac{B(x ; a, b)}{B(a, b)} .$$

Here

$$\displaystyle B(x ; a, b)=\int_{0}^{x} t^{a-1}(1-t)^{b-1} d t .$$

where a, b > 0, and 0 <= x <= 1.

Optionally, accuracy can be used to specify the desired fractional accuracy.

Details

The accuracy parameter specifies the fractional accuracy that you desire. That is, if you set accuracy to 1e-7, that means that you wish that the absolute value of (f_actual-f_returned)/f_actual be less than 10^-7.

Larger values of accuracy (poorer accuracy) result in evaluation of fewer terms of a series, which means the function executes somewhat faster.

A single-precision level of accuracy is about 3x10^-7, double-precision about 2x10^-16. The betai function will return full double-precision accuracy for small values of a and b. Achievable accuracy declines as a and b increase:

a	b	x	betai	Accuracy Achievable
1	1.5	0.5	0.646447	full double precision (2x10-16)
8	10	0.5	0.685470	6x10-16
20	21	0.5	0.562685	2x10-15
20	21	0.1	1.87186e-10	5x10-15