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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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On the nonoscillatory phase function for Legendre's differential equation

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JOURNAL OF COMPUTATIONAL PHYSICS 2017年 350卷 326-342页

作者： Bremer, James Rokhlin, Vladimir Univ Calif Davis Dept Math Davis CA 95616 USA Yale Univ Dept Comp Sci New Haven CT 06520 USA

We express a certain complex-valued solution of Legendre's differential equation as the product of an oscillatory exponential function and an integral involving only nonoscillatory elementary functions. By calculating the logarithmic derivative of this solution, we show that Legendre's differential equation admits a nonoscillatory phase function. Moreover, we derive from our expression an asymptotic expansion useful for evaluating Legendre functions of the first and second kinds of large orders, as well as the derivative of the nonoscillatory phase function. Our asymptotic expansion is not as efficient as the well-known uniform asymptotic expansion of Olver;however, unlike Olver's expansion, it coefficients can be easily obtained. Numerical experiments demonstrating the properties of our asymptotic expansion are presented. (C) 2017 Elsevier Inc. All rights reserved.

关键词： Special functions Fast algorithms nonoscillatory phase functions

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

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2019年第1期45卷 173-211页

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

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 and representing them with piecewise bivariate Chebyshev expansions. Our scheme is able to evaluate Bessel functions of orders between 0 and 1,000,000,000 at essentially any positive real argument. In that regime, it is competitive with existing methods for the rapid evaluation of Bessel functions and has at least three advantages over them. First, our approach is quite general and can be readily applied to many other special functions which satisfy second order ordinary differential equations. Second, by calculating the logarithms of the Bessel functions rather than the Bessel functions themselves, we avoid many issues which arise from numerical overflow and underflow. Third, in the oscillatory regime, our algorithm calculates the values of a nonoscillatory phase function for Bessel's differential equation and its derivative. These quantities are useful for computing the zeros of Bessel functions, as well as for rapidly applying the Fourier-Bessel transform. The results of extensive numerical experiments demonstrating the efficacy of our algorithm are presented. A Fortran package which includes our code for evaluating the Bessel functions is publicly available.

关键词： Special functions Fast algorithms nonoscillatory phase functions

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