Evaluating Non-Analytic Functions of Matrices
For numerical linear algebra researchers, it provides theoretical convergence guarantees for Chebyshev-based matrix function evaluation, but the results are incremental as they extend existing analysis to non-diagonalizable matrices.
The paper analyzes convergence rates of Chebyshev polynomial expansions for evaluating matrix functions f(A) with real spectrum, deriving bounds that relate function smoothness to matrix diagonalizability, supported by numerical examples.
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.