An algorithm for the rapid numerical evaluation of Bessel functions of real orders and arguments

@article{Bremer2019AnAF,
  title={An algorithm for the rapid numerical evaluation of Bessel functions of real orders and arguments},
  author={James Bremer},
  journal={Advances in Computational Mathematics},
  year={2019},
  volume={45},
  pages={173-211}
}
  • J. Bremer
  • Published 22 May 2017
  • Mathematics
  • Advances in Computational Mathematics
We describe a method for the rapid numerical evaluation of the Bessel functions of the first and second kinds of nonnegative real orders and positive arguments. Our algorithm makes use of the well-known observation that although the Bessel functions themselves are expensive to represent via piecewise polynomial expansions, the logarithms of certain solutions of Bessel’s equation are not. We exploit this observation by numerically precomputing the logarithms of carefully chosen Bessel functions… 
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References

SHOWING 1-10 OF 30 REFERENCES
On the Numerical Calculation of the Roots of Special Functions Satisfying Second Order Ordinary Differential Equations
  • J. Bremer
  • Computer Science, Mathematics
    SIAM J. Sci. Comput.
  • 2017
We describe a method for calculating the roots of special functions satisfying second order linear ordinary differential equations. It exploits the recent observation that the solutions of a large
On the Evaluation of Bessel Functions
Abstract In the present paper we describe an algorithm for the evaluation of Bessel functions J ν ( x ), Y ν ( x ) and H ( j ) ν ( x ) ( j = 1, 2) of arbitrary positive orders and arguments at a
Numerical methods for special functions
TLDR
This book provides an up-to-date overview of methods for computing special functions and discusses when to use them in standard parameter domains, as well as in large and complex domains.
Improved estimates for nonoscillatory phase functions
Recently, it was observed that solutions of a large class of highly oscillatory second order linear ordinary differential equations can be approximated using nonoscillatory phase functions. In
An algorithm for the rapid evaluation of special function transforms
A Fast Butterfly Algorithm for the Computation of Fourier Integral Operators
TLDR
This paper introduces a novel algorithm running in O(N^2 log N) time, i.e., with near-optimal computational complexity, and whose overall structure follows that of the butterfly algorithm.
A First Course in the Numerical Analysis of Differential Equations: Stiff equations
Cambridge University Press. Paperback. Book Condition: New. Paperback. 480 pages. Numerical analysis presents different faces to the world. For mathematicians it is a bona fide mathematical theory
Accuracy and stability of numerical algorithms, Second Edition
TLDR
This second edition gives a thorough, up-to-date treatment of the behavior of numerical algorithms in finite precision arithmetic and combines algorithmic derivations, perturbation theory, and rounding error analysis.
Fast Computation of Fourier Integral Operators
TLDR
A new numerical algorithm which requires O(N^{2.5} \log N) operations and as low as $O(\sqrt{N})$ in storage space (the constants in front of these estimates are small).
...
...