“…The authors of [14], for instance, take it to be 1000 `n `t1.1γu in their implementation. The experiments of [15], however, suggest that the necessary dimension grows as O `n `?nγ ˘. It is difficult to find a simple formula which suffices in all cases of interest.…”
Section: Numerical Experimentsmentioning
confidence: 99%
“…The Osipov-Xiao-Rokhlin method [20,14], which is the standard approach to the numerical calculation of Ps n pz; γq and the corresponding Sturm-Liouville eigenvalue χ n pγq, operates by representing a solution of (3) as a finite Legendre expansion. While the dependence of its running time on the parameters γ and n is not fully understood, the numerical experiments of [15] suggest that it grows as O `n `?nγ ˘, at least for large values of n and γ.…”
In addition to being the eigenfunctions of the restricted Fourier operator, the angular spheroidal wave functions of the first kind of order zero and nonnegative integer characteristic exponents are the solutions of a singular self-adjoint Sturm-Liouville problem. The running time of the standard algorithm for the numerical evaluation of their Sturm-Liouville eigenvalues grows with both bandlimit and characteristic exponent. Here, we describe a new approach whose running time is bounded independent of these parameters. Although the Sturm-Liouville eigenvalues are of little interest themselves, our algorithm is a component of a fast scheme for the numerical evaluation of the prolate spheroidal wave functions developed by one of the authors. We illustrate the performance of our method with numerical experiments.
“…The authors of [14], for instance, take it to be 1000 `n `t1.1γu in their implementation. The experiments of [15], however, suggest that the necessary dimension grows as O `n `?nγ ˘. It is difficult to find a simple formula which suffices in all cases of interest.…”
Section: Numerical Experimentsmentioning
confidence: 99%
“…The Osipov-Xiao-Rokhlin method [20,14], which is the standard approach to the numerical calculation of Ps n pz; γq and the corresponding Sturm-Liouville eigenvalue χ n pγq, operates by representing a solution of (3) as a finite Legendre expansion. While the dependence of its running time on the parameters γ and n is not fully understood, the numerical experiments of [15] suggest that it grows as O `n `?nγ ˘, at least for large values of n and γ.…”
In addition to being the eigenfunctions of the restricted Fourier operator, the angular spheroidal wave functions of the first kind of order zero and nonnegative integer characteristic exponents are the solutions of a singular self-adjoint Sturm-Liouville problem. The running time of the standard algorithm for the numerical evaluation of their Sturm-Liouville eigenvalues grows with both bandlimit and characteristic exponent. Here, we describe a new approach whose running time is bounded independent of these parameters. Although the Sturm-Liouville eigenvalues are of little interest themselves, our algorithm is a component of a fast scheme for the numerical evaluation of the prolate spheroidal wave functions developed by one of the authors. We illustrate the performance of our method with numerical experiments.
“…Boyd [8] provided the algorithms and Matlab codes for computing the PSWFs, eigenvalues and their zeros etc.. The recent work [73] had deeper insights into the truncation of the infinite eigensystem where the Legendre-Galerkin method was applied to solve the Sturm-Liouville problem (2.21). One can refer to [66] for some delicate algorithms to enhance the codes in [8] for certain range of the parameters.…”
Section: Evaluation Of Pswfs and The Associated Eigenvaluesmentioning
confidence: 99%
“…Schmutzhard et al [73] provided insightful observations on the choice of M for a prescribed accuracy. We next present some stable formulas for computing the eigenvalues {λ n (c)} of the integral operator (see [92]), where we note that the magnitude of λ n (c) with c > 0 is exponentially small for large n. Taking x = 0 in (2.13), and using (3.1) and the property…”
Section: Evaluation Of Pswfs and The Associated Eigenvaluesmentioning
confidence: 99%
“…(i) Study of their analytic and asymptotic properties, numerical evaluations, prolate quadrature, interpolation and related issues (see, e.g., [4,6,8,25,37,40,43,60,66,70,73,91,[99][100][101]);…”
This paper is devoted to a review of the prolate spheroidal wave functions (PSWFs) and their variants from the viewpoint of spectral/spectral-element approximations using such functions as basis functions. We demonstrate the pros and cons over their polynomial counterparts, and put the emphasis on the construction of essential building blocks for efficient spectral algorithms.
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