Weideman, J.A.C. and Trefethen, L.N. (1988) The Eigenvalues of Second-Order Spectral Differentiation Matrices. SIAM Journal on Numerical Analysis, 25, 1279-1298. http://dx.doi.org/10.1137/0725072

has been cited by the following article:

• TITLE: The Asymptotic Eigenvalues of First-Order Spectral Differentiation Matrices

  AUTHORS: Jue Wang, Fabian Waleffe

  KEYWORDS: Asymptotic Analysis, Spectral Approximations, Jacobi Polynomials, Collocation, Eigenvalues

  JOURNAL NAME: Journal of Applied Mathematics and Physics, Vol.2 No.5, April 25, 2014

  ABSTRACT: We complete and extend the asymptotic analysis of the spectrum of Jacobi Tau approximations that were first considered by Dubiner. The asymptotic formulas for Jacobi polynomials PN(α ,β ) ,α ,β > -1 are derived and confirmed by numerical approximations. More accurate results for the slowest decaying mode are obtained. We explain where the large negative eigenvalues come from. Furthermore, we show that a large negative eigenvalue of order N2 appears for -1 1 . The eigenvalues for Legendre polynomials are directly related to the roots of the spherical Bessel and Hankel functions that are involved in solving Helmholtz equation inspherical coordinates.