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Fast algorithms for Jacobi expansions via nonoscillatory phase functions

IMA JOURNAL OF NUMERICAL ANALYSIS 2020年第3期40卷 2019-2051页

作者： Bremer, James Yang, Haizhao Univ Calif Davis Dept Math Davis CA 95616 USA Natl Univ Singapore Dept Math Singapore 119076 Singapore

We describe a suite of fast algorithms for evaluating Jacobi polynomials, applying the corresponding discrete Sturm-Liouville eigentransforms and calculating Gauss-Jacobi quadrature rules. Our approach, which applies in the case in which both of the parameters alpha and beta in Jacobi's differential equation are of magnitude less than 1/2, is based on the well-known fact that in this regime Jacobi's differential equation admits a nonoscillatory phase function that can be loosely approximated via an affine function over much of its domain. We illustrate this with several numerical experiments, the source code for which is publicly available.

关键词： fast algorithms fast special function transforms butterfly algorithms special functions nonoscillatory phase functions asymptotic methods

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An algorithm for the numerical evaluation of the associated Legendre functions that runs in time independent of degree and order

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JOURNAL OF COMPUTATIONAL PHYSICS 2018年 360卷 15-38页

作者： Bremer, James Univ Calif Davis Dept Math Davis CA 95616 USA

We describe a method for the numerical evaluation of normalized versions of the associated Legendre functions P-v(-mu) and Q(v)(-mu) of degrees 0 <= v <= 1,000,000 and orders -v <= mu <= v for arguments in the interval (-1, 1). Our algorithm, which runs in time independent of v and mu, is based on the fact that while the associated Legendre functions themselves are extremely expensive to represent via polynomial expansions, the logarithms of certain solutions of the differential equation defining them are not. We exploit this by numerically precomputing the logarithms of carefully chosen solutions of the associated Legendre differential equation and representing them via piecewise trivariate Chebyshev expansions. These precomputed expansions, which allow for the rapid evaluation of the associated Legendre functions over a large swath of parameter domain mentioned above, are supplemented with asymptotic and series expansions in order to cover it entirely. The results of numerical experiments demonstrating the efficacy of our approach are presented, and our code for evaluating the associated Legendre functions is publicly available. (c) 2018 Elsevier Inc. All rights reserved.

关键词： Associated Legendre functions Fast algorithms butterfly algorithms Special functions Nonoscillatory phase functions Asymptotic methods

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On MLMDA/butterfly Compressibility of Inverse Integral Operators

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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS 2013年第1期12卷 31-34页

作者： Guo, Han Hu, Jun Michielssen, Eric Univ Elect Sci & Technol China Dept Microwave Engn Chengdu 611731 Peoples R China Univ Michigan Dept Elect Engn & Comp Sci Radiat Lab Ann Arbor MI 48109 USA

The multilevel matrix decomposition algorithm (MLMDA) is shown to permit effective compression of inverse integral operators pertinent to the analysis of scattering from electrically large structures. Observed compression ratios exceed those realized by low-rank (LR) compression methods, leading to substantial memory savings and a faster application of the inverse operator, and suggesting a new application for schemes traditionally used for compressing forward integral operators.

关键词： butterfly algorithms direct solver integral equations multilevel matrix decomposition

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