Commenced in January 2007
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The Ratios between the Spectral Norm, the Numerical Radius and the Spectral Radius
Authors: Kui Du
Abstract:
Recently, Uhlig [Numer. Algorithms, 52(3):335-353, 2009] proposed open questions about the ratios between the spectral norm, the numerical radius and the spectral radius of a square matrix. In this note, we provide some observations to answer these questions.
Keywords: Spectral norm, Numerical radius, Spectral radius, Ratios
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1077307
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[1] B. Beckermann, S. A. Goreinov, and E. E. Tyrtyshnikov. Some Remarks on the Elman Estimate for GMRES. SIAM J. Matrix Anal. Appl., 27(3):772-778, 2006.
[2] L. Caston, M. Savova, I. Spitkovsky, and N. Zobin. On eigenvalues and boundary curvature of the numerical range. Linear Algebra Appl., 322(1-3):129-140, 2001.
[3] M. Eiermann and O. Ernst. Geometric aspects of the theory of Krylov subspace methods. Acta Numerica, 10:251-312, 2003.
[4] S. C. Eisenstat, H. C. Elman, and M. H. Schultz. Variational iterative methods for nonsymmetric systems of linear equations. SIAM J. Numer. Anal., 20(2):345-357, 1983.
[5] M. Goldberg, E. Tadmor, and G. Zwas. The numerical radius and spectral matrices. Linear Multilinear Algebra, 2:317-326, 1975.
[6] A. Greenbaum. Iterative Methods for Solving Linear Systems. SIAM, Philadelphia, 1997.
[7] A. Greenbaum. Generalizations of the field of values useful in the study of polynomial functions of a matrix. Linear Algebra Appl., 347(1- 3):233-249, 2002.
[8] L. Hogben. Handbook of Linear Algebra. Chapman & Hall, 2007.
[9] R. Horn and C. Johnson. Topics in Matrix analysis. Cambridge University Press, 1991.
[10] I. C. F. Ipsen. A note on the field of values of non-normal matrices. Tech. Rep. CRSC-TR98-26, Department of Mathematics, North Carolina State University, 1998.
[11] O. Nevanlinna. Convergence of iterations for linear equations. Birkh¨auser, 1993.
[12] Y. Saad and M. H. Schultz. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput., 7(3):856-869, 1986.
[13] F. Uhlig. Geometric computation of the numerical radius of a matrix. Numer. Algorithms, 52(3):335-353, 2009.