Third Order Newton's Method for Zernike Polynomial Zeros
Richard J. Mathar
TL;DR
The work addresses efficient computation of zeros of Zernike radial polynomials R_n^m, essential for optical pupil expansions, by leveraging Halley’s third-order Newton method and reformulating the problem through Gauss hypergeometric functions. By deriving f''/f' from f/f' via structure relations and using terminating continued fractions for F, the method computes key derivative ratios without direct polynomial evaluation, enabling fast, stable root finding. Key contributions include explicit derivative-ratio machinery, recurrence-based lifting/lowering of n, robust initial guesses (including a shooting-based bootstrap), and a PARI implementation with root tables up to order 40. The approach yields practical gains in accuracy and speed for optical wavefront analyses and quadrature-related tasks in circular geometries, with explicit numerical artifacts documented through the provided tables and code.
Abstract
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.