  The Zernike radial polynomials are a system of orthogonal polynomials over
the unit interval with weight x. They are used as basis functions in optics to
expand fields over the cross section of circular pupils. To calculate the roots
of Zernike polynomials, we optimize the generic iterative numerical Newton's
Method that iterates on zeros of functions with third order convergence. The
technique is based on rewriting the polynomials as Gauss Hypergeometric
Functions, reduction of second order derivatives to first order derivatives,
and evaluation of some ratios of derivatives by terminating continued
fractions.
  A PARI program and a short table of zeros complete up to polynomials of 40th
order are included.
