Core Concepts
A third-order Newton's method is implemented to efficiently compute the roots of Zernike polynomials by leveraging their representation as terminating Gaussian hypergeometric functions.
Abstract
The paper presents an efficient numerical method for finding the roots of Zernike polynomials, which are widely used in optics to expand fields over the cross-section of circular pupils. The key insights are:
Zernike polynomials can be represented as terminating Gaussian hypergeometric functions, which enables the use of recurrence relations and continued fractions to compute the ratios of derivatives required for the third-order Newton's method.
Derivatives of Zernike polynomials are expressed in terms of derivatives of the hypergeometric functions, allowing efficient computation without directly evaluating the polynomials.
A shooting method is proposed to generate accurate initial guesses for the roots by leveraging information from previously computed roots.
The implementation includes a PARI program and a table of roots up to 40th order polynomials, which can be used as a reference.
The paper demonstrates how the representation of Zernike polynomials as hypergeometric functions, along with the use of advanced numerical techniques, can lead to an efficient root-finding algorithm with third-order convergence. This work contributes to the efficient processing and analysis of content related to computational complexity and numerical methods.