The Chebyshev Polynomials of a Matrix
- 1 January 1998
- journal article
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Matrix Analysis and Applications
- Vol. 20 (2) , 400-419
- https://doi.org/10.1137/s0895479896303739
Abstract
A Chebyshev polynomial of a square matrix A is a monic polynomial p of specified degree that minimizes |p (A)|2. The study of such polynomials is motivated by the analysis of Krylov subspace iterations in numerical linear algebra. An algorithm is presented for computing these polynomials based on reduction to a semidefinite program which is then solved by a primal-dual interior point method. Examples of Chebyshev polynomials of matrices are presented, and it is noted that if A is far from normal, the lemniscates of these polynomials tend to approximate pseudospectra of A.Keywords
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