On spectral properties of some Chebyshev-type methods. Dimension vs. structure

Abstract

The aim of the present paper is to analyze the non-normality of the matrices (finite dimensional operators) which result when some Chebyshev-type methods are used in order to solve second order differential two-point boundary value problem.

We consider in turn the classical Chebyshev-tau method as well as two variants of the Chebyshev-Galerkin method. As measure of non-normality we use the non-normality ratio introduced in a previous paper. The competition between the dimension of matrices (the order of approximation) and their structure (the numerical method itself) with respect to normality is the core of our study.

It is observed that for quasi normal matrices, i.e., non-normality ratio close to 0, exhibiting pure real spectrum, this measure remains the unique indicator of non-normality. In such cases the pseudospectrum tells nothing about non-normality.

Authors

C.I. Gheorghiu
Tiberiu Popoviciu Institute of Numerical Analysis

Keywords

Non-normal matrices; scalar measure; Chebyshev spectral approximation; two point boundary value problem.

References

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Cite this paper as

C.I. Gheorghiu, On spectral properties of Chebyshev-type methods. Dimensions vs. structure, Studia Univ. Babeş-Bolyai Math., (2005) 61-66.

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Babes-Bolyai University

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0252-1938

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2065-961x

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References

[1] Gheorghiu, C.I., A new Scalar Measure on Non-normalityApplication to the Characterization of some Spectral Methods,  under evaluation at SIAM J. Sci. Comput,
[2] Gheorghiu, C.I., Pop, I.S., On the Chebyshev-tau approximation for some Singularly Perturbed Two-Point Boundary Value Problems,  Rev. d’Analyse Numerique et de Theorie de l’Approximation, 24(1995), 125-129.
[3] Gottlieb, D., Orszag, S.A., Numerical Analysis of Spectral Methods: Theory and Applications, SIAM Philadelphia, 1977.
[4] Pop, I. S., Numerical approximation of differential equations with spectral methods (in Romanian), Technical report “Babes-Bolyai” University Cluj-Napoca, Faculty of Mathematics and Informatics, Cluj-Napoca, 1995.
[5] Pop, I.S. A stabilized approach for the Chebyshev-tau method, Studia Univ. “Babes-Bolyai”, Mathematica, 42(1997), 67-79.
[6] Pop, I.S., Gheorghiu, C.I., Chebyshev-Galerkin Methode for Eigenvalue Problems,Proceedings of ICAOR, Cluj-Napoca, 1996, vol. II, 217-220.
[7] Shen, J., Efficient spectral-Galerkin method II, Direct solvers of second and fourth equations using Chebyshev polynomials, SIAM J. Sci. Stat. Comput., 16(1995), 74-87.
[8] Trefethen, L.N., Computation of Pseudospectra, Acta Numerica, 1999, 247-295.

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