A Fast Algorithm for the Calculation of the Roots of Special Functions

Abstract

We describe a procedure for the determination of the roots of functions satisfying second-order ordinary differential equations, including the classical special functions. The scheme is based on a combination of the Prufer transform with the classical Taylor series method for the solution of ordinary differential equations, and requires 0(1) operations for the determination of each root. When the functions in question are classical orthogonal polynomials (Legendre, Hermite, etc.), the techniques presented here also provide tools for the evaluation of the weights for the associated Gaussian quadratures. The performance of the scheme for several classical special functions (prolate spheroidal wave functions, Bessel functions, and Legendre, Hermite, and Laguerre polynomials) is illustrated with numerical examples.

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Document Details

Document Type
Technical Report
Publication Date
Aug 23, 2006
Accession Number
ADA635869

Entities

People

  • Andreas Glaser
  • Vladimir Rokhlin, Jr.
  • Xiangtao Liu

Organizations

  • Yale University

Tags

DTIC Thesaurus Topics

  • Abstracts
  • Algorithms
  • Bessel Functions
  • Computer Science
  • Department Of Defense
  • Differential Equations
  • Equations
  • Gaussian Quadrature
  • Information Operations
  • Mathematics
  • Partial Differential Equations
  • Polynomials
  • Wave Functions

Fields of Study

  • Mathematics

Readers

  • Calculus or Mathematical Analysis