Inexact Arnoldi residual estimates and decay properties for functions of non-Hermitian matrices

Abstract

This paper derives a priori residual-type bounds for the Arnoldi approximation of a matrix function together with a strategy for setting the iteration accuracies in the inexact Arnoldi approximation of matrix functions. Such results are based on the decay behavior of the entries of functions of banded matrices. Specifically, a priori decay bounds for the entries of functions of banded non-Hermitian matrices will be exploited, using Faber polynomial approximation. Numerical experiments illustrate the quality of the results.

Appendix. Technical proofs

1.1 Proof of corollary 2.4

Let $\rho = \sqrt{a^2 - b^2}$ be the distance between the foci and the center of the ellipse (i.e., the boundary of E), and let $R = (a + b)/\rho$. Then a conformal map for E is

$$\begin{aligned} \phi (w) = \frac{w - c - \sqrt{(w-c)^2 - \rho ^2 }}{\rho R}, \end{aligned}$$

(A.1)

and its inverse is

$$\begin{aligned} \psi (z) = \frac{\rho }{2}\left( Rz + \frac{1}{Rz} \right) + c \,; \end{aligned}$$

(A.2)

see, e.g., [41, chapter II, Example 3]. Notice that

$$\begin{aligned} \max _{|z| = \tau } |e^{\psi (z)}| = \max _{|z| = \tau } e^{\mathfrak {R}(\psi (z))} = e^{\frac{\rho }{2}\left( R\tau + \frac{1}{R\tau } \right) + c_1}. \end{aligned}$$

Hence by Theorem 2.3 we get

$$\begin{aligned} \left| \left( e^{A} \right) _{k,\ell } \right| \le 2 \frac{\tau }{\tau - 1} e^{c_1} e^{\frac{\rho }{2}\left( R\tau + \frac{1}{R\tau } \right) } \left( \frac{1}{\tau } \right) ^\xi . \end{aligned}$$

The optimal value of $\tau > 1$ that minimizes $e^{\frac{\rho }{2}\left( R\tau + \frac{1}{R\tau } \right) } \left( \frac{1}{\tau } \right) ^\xi$ is

$$\begin{aligned} \tau = \frac{\xi + \sqrt{\xi ^2 + \rho ^2 }}{\rho R}. \end{aligned}$$

Moreover, the condition $\tau >1$ is satisfied if and only if $\xi > \frac{\rho }{2}\left( R - \frac{1}{R} \right) = b$. Finally, noticing that

$$\begin{aligned} \psi \left( \frac{\xi + \sqrt{\xi ^2 + \rho ^2 }}{\rho R}\right) - c_1 = \frac{1}{2}\left( \xi + \sqrt{\xi ^2 + \rho ^2 } + \frac{\rho ^2}{\xi + \sqrt{\xi ^2 + \rho ^2 }} \right) = \xi q(\xi ), \end{aligned}$$

and collecting $\xi$ the proof is completed. $\square$

1.2 Proof of Corollary 2.5

The function $f(z) = \exp (- \sqrt{z})$ is analytic on $\mathbb {C} \setminus (-\infty ,0)$. Since we consider the principal square root, then $\mathfrak {R}(\sqrt{z})\ge 0$, and

$$\begin{aligned} |\exp (- \sqrt{z})| = \exp (-\mathfrak {R}(\sqrt{z})) \le 1. \end{aligned}$$

Hence, by Theorem 2.3 we can determine $\tau$ for which

$$\begin{aligned} \left| \left( e^{-\sqrt{A}} \right) _{k,\ell } \right| \le 2 \frac{\tau }{\tau - 1} \left( \frac{1}{\tau } \right) ^\xi . \end{aligned}$$

For every $\varepsilon > 0$ close enough to zero, we set the parameter

$$\begin{aligned} \tau _\varepsilon = |\phi (\varepsilon )| = \left| \frac{c - \varepsilon + \sqrt{(c-\varepsilon )^2 -\rho ^2}}{\rho R} \right| , \end{aligned}$$

with $\phi (w)$ as in (A.1) and $\psi (z)$ its inverse (A.2). Then the ellipse $\{ \psi (z), \, |z| = \tau _\varepsilon \}$ is contained in $\mathbb {C}{\setminus }(-\infty ,0]$. Letting $\varepsilon \rightarrow 0$ concludes the proof. $\square$