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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2024 MCQ for Mathematics of Computation is 1.78.

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New backward recurrences for Bessel functions
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by Henry C. Thacher PDF
Math. Comp. 33 (1979), 744-764 Request permission

Abstract:

The recurrences for the coefficients of appropriate power series may be used with the Miller algorithm to evaluate ${J_v}(x)\;(|x|\;{\text {small}})$, ${e^x}{K_v}(x)\;(\operatorname {Re} x > 0,\;|x|\;{\text {large}})$, and the modulus and phase of $H_v^{(1)}(x)\;(\operatorname {Re} x > 0,\;|x|\;{\text {large}})$. The first converges slightly faster than the power series or the classical recurrence, but requires more arithmetic; the last three give both better ultimate precision and faster convergence than the corresponding asymptotic series. The analysis also leads to a formal continued fraction for ${K_{v + 1}}(x)/{K_v}(x)$ the convergence of which increases with $|x|$. The procedures were tested numerically both for integer and fractional values of v, and for real and complex x.
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Additional Information
  • © Copyright 1979 American Mathematical Society
  • Journal: Math. Comp. 33 (1979), 744-764
  • MSC: Primary 65D20; Secondary 33A40
  • DOI: https://doi.org/10.1090/S0025-5718-1979-0521289-1
  • MathSciNet review: 521289