Householder Transformation Method Implementation for Zernike Polynomial Wavefront Fitting

This article presents a comprehensive MATLAB implementation of the Householder transformation method, specifically designed for calculating Zernike polynomial wavefront fitting coefficients. The program employs QR decomposition via Householder reflections to effectively avoid numerical instabilities associated with ill-conditioned equations, ensuring more accurate coefficient calculations and superior fitting results. The algorithm works by constructing orthogonal matrices through successive Householder transformations, which triangularize the design matrix while preserving numerical stability. Key MATLAB functions utilized include householder vector computation, orthogonal transformation application, and back-substitution for solving the triangular system. Additionally, we explore fundamental concepts of Zernike polynomials, their critical applications in optical system analysis, wavefront aberration characterization, and relevant mathematical foundations. The implementation features automatic condition number checking and includes optimization for handling large-scale fitting problems commonly encountered in optical engineering applications. This resource aims to provide practical understanding of Zernike polynomial applications and enhance proficiency in numerical methods for wavefront analysis.