Zernike多项式零点的三阶牛顿法
Third Order Newton's Method for Zernike Polynomial Zeros
AI总结：
本文提出一种三阶收敛的牛顿法变体，通过高斯超几何函数和连分数技术高效计算Zernike多项式零点，并提供PARI程序及40阶以内的零点表。
摘要：
Zernike径向多项式是在单位区间上以权重x正交的多项式系统。在光学中，它们被用作基函数，以在圆形光瞳的横截面上展开场。为了计算Zernike多项式的根，我们优化了通用的迭代数值牛顿法，该方法以三阶收敛速度对函数的零点进行迭代。该技术基于将多项式重写为高斯超几何函数，将二阶导数降为一阶导数，并通过终止连分数计算某些导数之比。文中包含一个PARI程序和一个直至40阶多项式的零点简表。
英文摘要：
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.
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