Quasi-Toeplitz matrix arithmetic: a MATLAB toolbox

Бини, Дарио Андреа; Массеи, Стефано; Robol, Leonardo

doi:10.1007/s11075-018-0571-6

Cited by 24 publications

(32 citation statements)

References 36 publications

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“…Using an induction argument on k, we show part 1 of the theorem i.e., A (9) and Corollary 8, we deduce that A…”

Section: Alm Meanmentioning

confidence: 82%

“…, A p ∈ QT . We rely on the CQT-Toolbox [9] for computations in the QT algebra but in order to compute geometric means, we need to compute some fundamental functions of QT matrices, namely, the p-th root and in particular the square root, that we will discuss in the following. In this section, without loss of generality, we assume that the matrix A = T (a) + E A ∈ QT is such that 0 ≤ a(z) ≤ 1.…”

Section: Computational Issuesmentioning

confidence: 99%

“…This property holds even in the case where there is an overlapping of the supports in the two corners of the corrections, or if the bandwidth r takes large values. In this situation, the implementation of the arithmetic is still possible [9], but its not efficient from the computational point of view.…”

Section: Geometric Means Of Finite Qt Matricesmentioning

confidence: 99%

“…The algorithms of Section 4 have been implemented in MATLAB by following the lines of the corresponding implementations for finite positive definite matrices of the Matrix Means Toolbox (http://bezout.dm.unipi.it/software/ mmtoolbox/). They rely on the package CQT-Toolbox of [9] for the storage and arithmetic of QT matrices. The software can be provided by the authors upon request.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Then we deal with the problem of computing G. We introduce and analyze algorithms for computing the Toeplitz part T (g) and the correction part K G in a very efficient way, both in the case of infinite and of finite Toeplitz matrices. A set of numerical experiments shows that the geometric means of infinite quasi-Toeplitz matrices can be easily and accurately computed by relying on simple MATLAB implementations, based on the CQT-Toolbox of [9].…”

Section: Introductionmentioning

confidence: 99%

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On functions of quasi-Toeplitz matrices

Бини¹,

Массеи²,

Meini³

2017

Sb. Math.

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Abstract. Let a(z) = i∈Z a i z i be a complex valued continuous function, defined for |z| = 1, such that +∞ i=−∞ |ia i | < ∞. Consider the semi-infinite Toeplitz matrix T (a) = (t i,j ) i,j∈Z + associated with the symbol a(z) such that t i,j = a j−i . A quasi-Toeplitz matrix associated with the continuous symbol a(z) is a matrix of the form A = T (a) + E where E = (e i,j ), i,j∈Z + |e i,j | < ∞, and is called a CQT-matrix. Given a function f (x) and a CQT matrix M , we provide conditions under which f (M ) is well defined and is a CQT matrix. Moreover, we introduce a parametrization of CQT matrices and algorithms for the computation of f (M ). We treat the case where f (x) is assigned in terms of power series and the case where f (x) is defined in terms of a Cauchy integral. This analysis is applied also to finite matrices which can be written as the sum of a Toeplitz matrix and of a low rank correction.Key words. Matrix functions, Toeplitz matrices, Infinite matrices 1. Introduction. Functions of finite matrices have received a lot of interest in the literature both for their theoretical properties and for the many applications they have in real world problems. We refer the interested reader to the book [16] for more details. Among the available different definitions of matrix function, which are equivalent under mild hypotheses, many rely on the Jordan canonical form of the matrix argument or on its Schur form. As a consequence, they are not directly generalizable to the setting of infinite matrices. However, there are two definitions which apparently seem to be more suited for extending the concept of function to infinite matrices. They rely on the Laurent series expansion, and on the integral representation through the Dunford Cauchy formula [16].In the set of matrices with infinite size there is a class which is a cornerstone for numerical linear algebra. It is the class of Toeplitz matrices T (a) = (t i,j ) associated with a function a(z) = i∈Z a i z i , called symbol, defined by t i,j = a j−i . Toeplitz matrices are widely analyzed in the literature, from the pioneering papers by O. Toeplitz, to the seminal monograph by Grenander and Szegő [14], until to the more recent and wide contributions given by several international research groups, including, but not limited to, the books In some problems, typically encountered in the analysis of stochastic processes, like bi-dimensional random walks in the quarter-plane, one has to deal with matrices of the kind A = T (a) + E, where T (a) is a semi-infinite Toeplitz matrix, while E = (e i,j ) is a non-Toeplitz correction such that i,j∈Z + |e i,j | < ∞. We call this class of matrices Quasi-Toeplitz matrices, in short QT matrices, and denote them with QT . We consider the subset of QT matrices whose symbol a(z) is differentiable and both a(z) and a(z) ′ belong to the Wiener class. We call the latter as the set of CQT matrices and we denote it with CQT .From the computational point of view, one has to solve matrix equations where the coefficients are CQT matrices, o...

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“…Using an induction argument on k, we show part 1 of the theorem i.e., A (9) and Corollary 8, we deduce that A…”

Section: Alm Meanmentioning

confidence: 82%

Section: Computational Issuesmentioning

confidence: 99%

Section: Geometric Means Of Finite Qt Matricesmentioning

confidence: 99%

Section: Numerical Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On functions of quasi-Toeplitz matrices

Бини¹,

Массеи²,

Meini³

2017

Sb. Math.

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Algorithms for Approximating Means of Semi-infinite Quasi-Toeplitz Matrices

Бини

Iannazzo

Meng

2021

Lecture Notes in Computer Science

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On the exponential of semi-infinite quasi-Toeplitz matrices

Бини

Meini

2018

Numer. Math.

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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.

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Quasi-Toeplitz matrix arithmetic: a MATLAB toolbox

Cited by 24 publications

References 36 publications

On functions of quasi-Toeplitz matrices

On functions of quasi-Toeplitz matrices

Algorithms for Approximating Means of Semi-infinite Quasi-Toeplitz Matrices

On the exponential of semi-infinite quasi-Toeplitz matrices

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