Evaluating non-analytic functions of matrices

Nir Sharon; Yoel Shkolnisky

doi:10.1016/j.jmaa.2018.02.029

Evaluating non-analytic functions of matrices

Nir Sharon^*, Yoel Shkolnisky

^*Corresponding author for this work

School of Mathematical Sciences

Princeton University

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

The paper revisits the classical problem of evaluating f(A) for a real function f and a matrix A with real spectrum. The evaluation is based on expanding f in Chebyshev polynomials, and the focus of the paper is to study the convergence rates of these expansions. In particular, we derive bounds on the convergence rates which reveal the relation between the smoothness of f and the diagonalizability of the matrix A. We present several numerical examples to illustrate our analysis.

Original language	English
Pages (from-to)	613-636
Number of pages	24
Journal	Journal of Mathematical Analysis and Applications
Volume	462
Issue number	1
DOIs	https://doi.org/10.1016/j.jmaa.2018.02.029
State	Published - 1 Jun 2018

Keywords

Chebyshev polynomials
Convergence rates
Jordan blocks
Matrix Chebyshev expansion
Matrix functions

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This output contributes to the following UN Sustainable Development Goals (SDGs)

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10.1016/j.jmaa.2018.02.029

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AB - The paper revisits the classical problem of evaluating f(A) for a real function f and a matrix A with real spectrum. The evaluation is based on expanding f in Chebyshev polynomials, and the focus of the paper is to study the convergence rates of these expansions. In particular, we derive bounds on the convergence rates which reveal the relation between the smoothness of f and the diagonalizability of the matrix A. We present several numerical examples to illustrate our analysis.

KW - Chebyshev polynomials

KW - Convergence rates

KW - Jordan blocks

KW - Matrix Chebyshev expansion

KW - Matrix functions

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Evaluating non-analytic functions of matrices

Abstract

Keywords

UN SDGs

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