Efficient algorithms for the inversion of the cumulative central beta   distribution
  Accurate and efficient algorithms for the inversion of the cumulative central beta distribution are described. The algorithms are based on the combination of a fourth-order fixed point method with good non-local convergence properties (the Schwarzian-Newton method), asymptotic inversion methods and sharp bounds in the tails of the distribution function. 
A. Gil, J. Segura, N. M. Temme
math.NA cs.NA math.CA