On Best Approximations of Polynomials in Matrices in the Matrix 2-Norm

Authors: J. Liesen, Petr Tichý
Publication year: 2009
Publication type: Článek v odborném periodiku
Periodical: SIAM Journal on Matrix Analysis and Applications
Projects: Theory of Krylov subspace methods and its relationship to other mathematical disciplines
Bibliographic citation: LIESEN, J., TICHÝ, Petr. On Best Approximations of Polynomials in Matrices in the Matrix 2-Norm. SIAM Journal on Matrix Analysis and Applications. 2009, 31(2), 853-863. ISSN 0895-4798. E-ISSN 1095-7162
Dostupné z: doi:10.1137/080728299
DOI: 10.1137/080728299
ID: 330308

Abstract

We show that certain matrix approximation problems in the matrix 2-norm have uniquely defined solutions, despite the lack of strict convexity of the matrix 2-norm. The problems we consider are generalizations of the ideal Arnoldi and ideal GMRES approximation problems introduced by Greenbaum and Trefethen [SIAM J. Sci. Comput., 15 (1994), pp. 359-368]. We also discuss general characterizations of best approximation in the matrix 2-norm and provide an example showing that a known sufficient condition for uniqueness in these characterizations is not necessary.