Third-order iterative methods for zeros of solutions of second-order linear ODEs

Abstract

This thesis presents three iterative methods for computing the zeros of solutions to second order linear ordinary differential equations (ODEs). All the iterative methods proposed in this thesis exhibit third-order convergence. Sufficient conditions ensuring the theoretical monotone (nonlocal) convergence of the proposed iterative methods have been established, representing one of the major contributions of this work. Algorithms based on these iterative methods are developed to solve benchmark problems such as the inversion of cumulative central and noncentral distribution functions, the computation of zeros of highly oscillatory special functions such as Bessel and cylinder functions, and the computation of nodes and weights for Gauss–Hermite and Gauss–Legendre quadratures. Numerical studies of the proposed algorithms support the theoretical claims. To demonstrate the efficiency of the proposed algorithms, a numerical comparative study with existing methods in the literature is also included.