Semi-infinite quasi-Toeplitz matrices with applications to QBD stochastic processes

Dario A. Bini, Stefano Massei, Beatrice Meini

Research output: Contribution to journalArticleAcademicpeer-review

11 Citations (Scopus)


Denote by W1 the set of complex valued functions of the form a(z) = Σi=-∞ +∞ aizi such that i=-∞ +∞ |iai| < ∞. We call QT-matrix a quasi-Toeplitz matrix A, associated with a symbol a(z) ∈ W1, of the form A = T(a) + E, where T(a) = (ti,j)i,j∈ℤ + is the semi-infinite Toeplitz matrix such that ti,j = aj-i, for i, j ∈ ℤ+, and E = (ei,j)i,j∈ℤ + is a semi-infinite matrix such thatΣi,j=1 +∞ |ei,j| is finite. We prove that the class of QT-matrices is a Banach algebra with a suitable sub-multiplicative matrix norm. We introduce a finite representation of QT-matrices together with algorithms which implement elementary matrix operations. An application to solving quadratic matrix equations of the kind AX2 +BX +C = 0, encountered in the solution of Quasi-Birth and Death (QBD) stochastic processes with a denumerable set of phases, is presented where A,B,C are QT-matrices.

Original languageEnglish
Pages (from-to)2811-2830
Number of pages20
JournalMathematics of Computation
Issue number314
Publication statusPublished - 2018
Externally publishedYes

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© 2017 American Mathematical Society.

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