Numerische Mathematik

, Volume 141, Issue 2, pp 319–351 | Cite as

On the exponential of semi-infinite quasi-Toeplitz matrices

  • Dario A. BiniEmail author
  • Beatrice Meini


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.

Mathematics Subject Classification

65F60 15A16 15B05 47B35 



The authors wish to thank Robert Luce for providing the software for computing the matrix exponential of a finite Toeplitz matrix based on the displacement rank and the anonymous referees who provided useful suggestions and remarks which helped to improve the presentation of the paper.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly

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