Taylor Numbers for the Fast Multipole Method

We present a novel linear algebra formulation of the math behind the FastMultipole Method (FMM). For this, we introduce the Taylor numbers. They are polynomials governed by differential algebra but used like integers and floating points. They help simplify the FMM math and potentially any scientific programming project.

Content

Author and article information

Journal

Title: ScienceOpen Preprints

Publisher: ScienceOpen

Publication date (Electronic preprint): 28 August 2022

Affiliations

[1 ] none

Author notes

[* ]Email: tiliavirga@ 123456gmail.com .

Author information

Ulf Lindqvist https://orcid.org/0000-0003-4545-3436

Article

DOI: 10.14293/S2199-1006.1.SOR-.PPRXR9A.v1

SO-VID: 5de21447-a15b-41da-a414-e8bd039a88f6

License:

This work has been published open access under Creative Commons Attribution License CC BY 4.0 , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Conditions, terms of use and publishing policy can be found at www.scienceopen.com .

History

Date received : 28 August 2022

Data availability: All data generated or analysed during this study are included in this published article (and its supplementary information files).

ScienceOpen disciplines: Data structures & Algorithms,Mathematical & Computational physics,Bioinformatics & Computational biology

Keywords: Fast Multipole Method (FMM),Differential Algebra (DA),Polynomial rings,Automatic differentiation,N-body problems,Molecular Dynamics (MD)

References

Berz M. Differential algebraic description of beam dynamics to very high orders. 1988. Office of Scientific and Technical Information (OSTI). [Cross Ref]
Bauch H.. Atanassova, L.; Herzberger, .I., Computer Arithmetic and Enclosure Methods. Amsterdam etc., North-Holland 1992. X, 504 pp., Dfl. 260.00. ISBN 0-444-89834-4. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik. Vol. 74(7)1994. Wiley. [Cross Ref]
Borsellino A., Poggio T.. Convolution and correlation algebras. Kybernetik. Vol. 13(2):113–122. 1973. Springer Science and Business Media LLC. [Cross Ref]
The VNR Concise Encyclopedia of Mathematics. 1990. Springer Netherlands. [Cross Ref]
Greengard Leslie F.. The Rapid Evaluation of Potential Fields in Particle Systems. 1988. The MIT Press. [Cross Ref]
Kalman Dan. Doubly Recursive Multivariate Automatic Differentiation. Mathematics Magazine. Vol. 75(3):187–202. 2002. Informa UK Limited. [Cross Ref]
Goodstein R. L.. Infinite sequences and series. By Konrad Knopp, Translated by F. Bagemihl. Pp.186. $1.75. 1956. (Dover Publications, New York). The Mathematical Gazette. Vol. 41(337)1957. Cambridge University Press (CUP). [Cross Ref]
The synthetic theorySynthetic Differential Geometry. p. 1–96. 2006. Cambridge University Press. [Cross Ref]
Efficient recurrence relations for univariate and multivariate Taylor series coefficientsConference Publications. 2013. AIMS Press. [Cross Ref]
Pennestrì E., Stefanelli R.. Linear algebra and numerical algorithms using dual numbers. Multibody System Dynamics. Vol. 18(3):323–344. 2007. Springer Science and Business Media LLC. [Cross Ref]
Rotman Joseph. Commutative rings IAdvanced Modern Algebra. p. 81–171. 2010. American Mathematical Society. [Cross Ref]
Shanker B., Huang H.. Accelerated Cartesian expansions – A fast method for computing of potentials of the form R−ν for all real ν. Journal of Computational Physics. Vol. 226(1):732–753. 2007. Elsevier BV. [Cross Ref]
Zhang He, Berz Martin. The fast multipole method in the differential algebra framework. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment. Vol. 645(1):338–344. 2011. Elsevier BV. [Cross Ref]

Comments

Comment on this article

[1] Berz M. Differential algebraic description of beam dynamics to very high orders. 1988. Office of Scientific and Technical Information (OSTI). [Cross Ref]

[2] Bauch H.. Atanassova, L.; Herzberger, .I., Computer Arithmetic and Enclosure Methods. Amsterdam etc., North-Holland 1992. X, 504 pp., Dfl. 260.00. ISBN 0-444-89834-4. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik. Vol. 74(7)1994. Wiley. [Cross Ref]

[3] Borsellino A., Poggio T.. Convolution and correlation algebras. Kybernetik. Vol. 13(2):113–122. 1973. Springer Science and Business Media LLC. [Cross Ref]

[4] The VNR Concise Encyclopedia of Mathematics. 1990. Springer Netherlands. [Cross Ref]

[5] Greengard Leslie F.. The Rapid Evaluation of Potential Fields in Particle Systems. 1988. The MIT Press. [Cross Ref]

[6] Kalman Dan. Doubly Recursive Multivariate Automatic Differentiation. Mathematics Magazine. Vol. 75(3):187–202. 2002. Informa UK Limited. [Cross Ref]

[7] Goodstein R. L.. Infinite sequences and series. By Konrad Knopp, Translated by F. Bagemihl. Pp.186. $1.75. 1956. (Dover Publications, New York). The Mathematical Gazette. Vol. 41(337)1957. Cambridge University Press (CUP). [Cross Ref]

[8] The synthetic theorySynthetic Differential Geometry. p. 1–96. 2006. Cambridge University Press. [Cross Ref]

[9] Efficient recurrence relations for univariate and multivariate Taylor series coefficientsConference Publications. 2013. AIMS Press. [Cross Ref]

[10] Pennestrì E., Stefanelli R.. Linear algebra and numerical algorithms using dual numbers. Multibody System Dynamics. Vol. 18(3):323–344. 2007. Springer Science and Business Media LLC. [Cross Ref]

[11] Rotman Joseph. Commutative rings IAdvanced Modern Algebra. p. 81–171. 2010. American Mathematical Society. [Cross Ref]

[12] Shanker B., Huang H.. Accelerated Cartesian expansions – A fast method for computing of potentials of the form R−ν for all real ν. Journal of Computational Physics. Vol. 226(1):732–753. 2007. Elsevier BV. [Cross Ref]

[13] Zhang He, Berz Martin. The fast multipole method in the differential algebra framework. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment. Vol. 645(1):338–344. 2011. Elsevier BV. [Cross Ref]