Optimal Polynomial Approximation to Rational Matrix Functions Using the Arnoldi Algorithm

TL;DR

This paper develops a method to find the best polynomial approximation to rational matrix functions applied to a matrix-vector product using the Arnoldi algorithm, improving efficiency in large-scale computations.

Contribution

It introduces a procedure to compute optimal Krylov subspace approximations for rational functions of matrices, including error analysis and additional Arnoldi steps for improved accuracy.

Findings

The method achieves optimal approximation in the D(A)*D(A)-norm.

Eigenvalues alone do not predict convergence for nonnormal matrices.

Additional Arnoldi steps improve approximation quality.

Abstract

Given an $n$ by $n$ matrix $A$ and an $n$-vector $b$, along with a rational function $R(z) := D(z )^{-1} N(z)$, we show how to find the optimal approximation to $R(A) b$ from the Krylov space, $\mbox{span}( b, Ab, \ldots , A^{k-1} b)$, using the basis vectors produced by the Arnoldi algorithm. To find this optimal approximation requires running $\max \{ \mbox{deg} (D) , \mbox{deg} (N) \} - 1$ extra Arnoldi steps and solving a $k + \max \{ \mbox{deg} (D) , \mbox{deg} (N) \}$ by $k$ least squares problem. Here {\em optimal} is taken to mean optimal in the $D(A )^{*} D(A)$-norm. Similar to the case for linear systems, we show that eigenvalues alone cannot provide information about the convergence behavior of this algorithm and we discuss other possible error bounds for highly nonnormal matrices.