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A recursive algorithm for an efficient and accurate computation of incomplete Bessel functions

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Abstract

In a previous work, we developed an algorithm for the computation of incomplete Bessel functions, which pose as a numerical challenge, based on the \(G_{n}^{(1)}\) transformation and Slevinsky-Safouhi formula for differentiation. In the present contribution, we improve this existing algorithm for incomplete Bessel functions by developing a recurrence relation for the numerator sequence and the denominator sequence whose ratio forms the sequence of approximations. By finding this recurrence relation, we reduce the complexity from \({\mathcal O}(n^{4})\) to \({\mathcal O}(n)\). We plot relative error showing that the algorithm is capable of extremely high accuracy for incomplete Bessel functions.

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Funding

HS acknowledges the financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) — Grant RGPIN-2016-04317. RMS acknowledges the financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) — Grant RGPIN-2017-05514.

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Authors and Affiliations

  1. Department of Mathematics, University of Manitoba, 186 Dysart Rd, Winnipeg, R3T 2M8, Manotoba, Canada

    Richard M. Slevinsky

  2. Campus Saint-Jean, University of Alberta, 8406, 91 Street, Edmonton, T6C 4G9, Alberta, Canada

    Hassan Safouhi

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  1. Richard M. Slevinsky
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  2. Hassan Safouhi
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Correspondence to Hassan Safouhi.

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Slevinsky, R.M., Safouhi, H. A recursive algorithm for an efficient and accurate computation of incomplete Bessel functions. Numer Algor 92, 973–983 (2023). https://doi.org/10.1007/s11075-022-01438-0

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Keywords

Mathematical subject classification (2010)

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