On the exponential of semi-infinite quasi-Toeplitz matrices

Abstract

Let $a(z)=\sum _{i\in {\mathbb {Z}}}a_iz^i$ be a complex valued function defined for $|z|=1$, such that $\sum _{i\in {\mathbb {Z}}}|a_i|<\infty$; define $T(a)=(t_{i,j})_{i,j\in {\mathbb {Z}}^+}, t_{i,j}=a_{j-i}$ for $i,j\in {\mathbb {Z}}^+$, the semi-infinite Toeplitz matrix associated with the symbol a(z); let $E=(e_{i,j})_{i,j\in {\mathbb {Z}}^+}$ be a compact operator in $\ell ^p$, with $1\le p\le \infty .$ A semi-infinite matrix of the kind $A=T(a)+E$ is said quasi-Toeplitz (QT). The problem of the computation of $\exp (A)$ or $\exp (A)v$, with A quasi-Toeplitz and v a vector, arises in many applications. We prove that the exponential of a QT-matrix A is QT, that is, $\exp (A) = T(\exp (a))+F$ where F is a compact operator in $\ell ^p$. This property allows the design of an algorithm for computing $\exp (A)$ and $\exp (A)v$ up to any precision. The case of families of $n\times n$ matrices obtained by truncating infinite QT-matrices to finite size is also considered. Numerical experiments show the effectiveness of this approach.