Third Order Newton's Method for Zernike Polynomial Zeros

Details Third Order Newton's Method for Zernike Polynomial Zeros Third Order Newton's Method for Zernike Polynomial Zeros

2007-05-09

arxiv.org/abs/0705.1329v2

Mathematics

Numerical Analysis

Improved eqs (1),(3) and (48). More references. Expanded Table in Appendix

Scientific paper

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 20th order are included.

Affiliated with

Also associated with

No associations

Rating

Third Order Newton's Method for Zernike Polynomial Zeros does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Third Order Newton's Method for Zernike Polynomial Zeros, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Third Order Newton's Method for Zernike Polynomial Zeros will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-646242