On the Numerical Calculation of the Roots of Special Functions Satisfying Second Order Ordinary Differential Equations

DOI10.1137/16M1057139zbMath1355.65037arXiv1512.08357MaRDI QIDQ2954482

James Bremer

Publication date: 13 January 2017

Published in: SIAM Journal on Scientific Computing (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1512.08357

zbMATH Keywords

rootsnumerical experimentsfast algorithmsspecial functionssecond-order ordinary differential equationsGaussian quadrature rules

Mathematics Subject Classification ID

Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) (33C45) Computation of special functions and constants, construction of tables (65D20) Approximate quadratures (41A55) Numerical quadrature and cubature formulas (65D32) Numerical approximation and evaluation of special functions (33F05)

Related Items (7)

An algorithm for the rapid numerical evaluation of Bessel functions of real orders and arguments ⋮ Quasi-analytical root-finding for non-polynomial functions ⋮ Phase function methods for second order linear ordinary differential equations with turning points ⋮ High-order asymptotic expansions of Gaussian quadrature rules with classical and generalized weight functions ⋮ Noniterative Computation of Gauss--Jacobi Quadrature ⋮ Asymptotic-numerical approximations for highly oscillatory second-order differential equations by the phase function method ⋮ Fast, reliable and unrestricted iterative computation of Gauss-Hermite and Gauss-Laguerre quadratures

Uses Software

• DLMF
• FastGaussQuadrature

Cites Work

• Improved estimates for nonoscillatory phase functions
• On the existence of nonoscillatory phase functions for second order ordinary differential equations in the high-frequency regime
• The ``phase function method to solve second-order asymptotically polynomial differential equations
• Spectral deferred correction methods for ordinary differential equations
• On the numerical solution of second order ordinary differential equations in the high-frequency regime
• On the asymptotics of Bessel functions in the Fresnel regime
• On the zeros of the hypergeometric function
• Fast computation of Gauss quadrature nodes and weights on the whole real line
• Iteration-Free Computation of Gauss--Legendre Quadrature Nodes and Weights
• $\mathcal{O}(1)$ Computation of Legendre Polynomials and Gauss--Legendre Nodes and Weights for Parallel Computing
• Reliable Computation of the Zeros of Solutions of Second Order Linear ODEs Using a Fourth Order Method
• Bessel Functions for Large Arguments
• Spectral Integration and Two-Point Boundary Value Problems
• A Fast Algorithm for the Calculation of the Roots of Special Functions
• A First Course in the Numerical Analysis of Differential Equations
• Bound on the Extreme Zeros of Orthogonal Polynomials
• On Computing the Points and Weights for Gauss--Legendre Quadrature
• Fast and Accurate Computation of Gauss--Legendre and Gauss--Jacobi Quadrature Nodes and Weights
• Calculation of Gauss Quadrature Rules
• An asymptotic expansion for a ratio of products of gamma functions
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