Using Parity to Accelerate the Computation of the Zeros of Truncated Legendre and Gegenbauer Polynomial Series and Gaussian Quadrature
Year: 2020
Author: John P. Boyd
East Asian Journal on Applied Mathematics, Vol. 10 (2020), Iss. 2 : pp. 217–242
Abstract
Any function $u(x)$ can be decomposed into its parts that are symmetric and antisymmetric with respect to the origin. The zeros, maxima and minima of a truncated spectral series of degree $N$ can always be computed as the eigenvalues of the sparse $N$-dimensional companion matrix whose elements are trivial functions of the coefficients of the spectral series. Here, we show that the matrix dimension can be halved if the series has definite parity. A series of Legendre and Gegenbauer polynomials has even parity if only even degree coefficients are nonzero and odd parity if the sum includes odd degrees only. We give the elements of the parity-exploiting companion matrices explicitly. We also give the coefficients of parity-exploiting recurrences for computing the orthogonal polynomials of even degree only or odd degree only without the wasteful computation of all polynomials of the opposite parity. For an $N$-point Gaussian quadrature, the quadrature points are the eigenvalues of a symmetric tridiagonal matrix of dimension $N$ ("Jacobi matrix"). We give the explicit elements of symmetric tridiagonal matrices of dimension $N$/2 that do the same job.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/eajam.140819.271119
East Asian Journal on Applied Mathematics, Vol. 10 (2020), Iss. 2 : pp. 217–242
Published online: 2020-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 26
Keywords: Gaussian quadrature Legendre polynomials Gegenbauer polynomials parity root-finding.
Author Details
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Using parity to accelerate Hermite function computations: Zeros of truncated Hermite series, Gaussian quadrature and Clenshaw summation
Boyd, John P.
Mathematics and Computers in Simulation, Vol. 207 (2023), Iss. P.521
https://doi.org/10.1016/j.matcom.2022.12.006 [Citations: 1]