On the exponential of semi-infinite quasi-Toeplitz matrices

Abstract

Let a(z) = i∈Z a i z i be a complex valued function defined for |z| = 1, such that i∈Z |ia i | < ∞, and let E = (e i,j ) i,j∈Z + be such thatToeplitz matrix associated with the symbol a(z), that is, t i,j = a j−i for i, j ∈ Z + . We analyze theoretical and computational properties of the exponential of A. More specifically, it is shown that exp(A) = T (exp(a)) + F where F = (f i,j ) i,j∈Z + is such that i,j∈Z + |f i,j | is finite, i.e., exp(A) is a semi-infinite quasi-Toeplitz matrix as well, and an effective algorithm for its computation is given. These results can be extended from the function exp(z) to any function f (z) satisfying mild conditions, and can be applied to finite quasi-Toeplitz matrices.

Quasi-Toeplitz matrix arithmetic: a MATLAB toolbox

“…= u −1 e u we arrive at b − a −1 . = a(e l + e u + ae ul ), which proves (13). Equations (14) are an immediate consequence of the definitions of e l and e u .…”

“…In [2,7,8,13], the class QT of semi-infinite Quasi-Toeplitz (QT) matrices has been introduced. This set is formed by matrices of the kind A = T (a) + E where, in general, a(z) = i∈Z a i z i is a Laurent series such that a W = +∞ i=−∞ |a i | is finite, and E is a compact correction.…”

“…In this paper, by continuing the work started in [2,7,8,13], we analyze the representation of QT matrices by means of a finite number of parameters, in a sort of analogy with the finite floating point representation of real numbers. Moreover, we investigate some computational issues related to the definition and the implementation of a matrix arithmetic in this class.…”

“…This norm is different from the one used in [2,7,8,13]: it is slightly more general, and still makes the set QT a Banach algebra. It can be shown that any value of α 1+ √ 5 2 would make this set a Banach algebra.…”

“…We refer to such matrices as Quasi-Toeplitz matrices, in short QT matrices. Their computational properties have been investigated in [2,7,8,13].…”