General Solution of the Poisson Equation for Quasi-Birth-and-Death Processes

Бини, Дарио Андреа; Dendievel, Sarah; Latouche, Guy; Meini, Beatrice

doi:10.1137/16m1065045

Cited by 11 publications

(9 citation statements)

References 17 publications

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“…Assume we are given a canonical factorization of A(z −1 ), it is a natural question to figure out if also A(z −1 ) admits a canonical factorization, and if it is related to the one of A(z −1 ). This issue is motivated by applications in the solution of the Poisson problem in stochastic models, where both the canonical factorizations of A(z) and A(z −1 ) are needed for the existence of the solution and for providing its explicit expression [1].…”

Section: Brauer's Theorem and Canonical Factorizationsmentioning

confidence: 99%

Generalization of the Brauer Theorem to Matrix Polynomials and Matrix Laurent Series

Бини

Meini

2017

Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics

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Given a square matrix A, Brauer's theorem [Duke Math. J. 19 (1952), [75][76][77][78][79][80][81][82][83][84][85][86][87][88][89][90][91] shows how to modify one single eigenvalue of A via a rank-one perturbation, without changing any of the remaining eigenvalues. We reformulate Brauer's theorem in functional form and provide extensions to matrix polynomials and to matrix Laurent series A(z) together with generalizations to shifting a set of eigenvalues. We provide conditions under which the modified function A(z) has a canonical factorization A(z) = U (z) L(z −1 ) and we provide explicit expressions of the factors U (z) and L(z). Similar conditions and expressions are given for the factorization of A(z −1 ). Some applications are discussed. * Work supported by Gruppo Nazionale di Calcolo Scientifico (GNCS) of INdAM, and by the PRA project "Mathematical models and computational methods for complex networks" funded by the University of Pisa. inverse eigenvalue problem [10], [21], to accelerating the convergence of iterative methods for solving matrix equations in particular quadratic equations [20], equations related to QBD Markov chains [17], [14], to M/G/1-type Markov chains [5], [3], or for algebraic Riccati equations in [15], [16], [19], and [2].In this paper we revisit Brauer's theorem in functional form and generalize it to a non-degenerate n × n matrix function A(z) analytic in the open annulusHere, non-degenerate means that det A(z) is not identically zero. Throughout the paper we assume that all the matrix functions are nondegenerate.Let λ ∈ A r1,r2 be an eigenvalue of A(z), i.e., such that det A(λ) = 0 and u ∈ C n , u = 0, be a corresponding eigenvector, i.e., such that A(λ)u = 0. We prove that for any v ∈ C n such that v * u = 1 and for any µ ∈ C, the functionwhere v * denotes the transpose conjugate of v. Moreover, A(z) has all the eigenvalues of A(z), except for λ which, if µ ∈ A r1,r2 , is replaced by µ, otherwise is removed. We provide explicit expressions which relate the coefficients A i of A(z) = i∈Z z i A i to those of A(z).This transformation, called right shift, is obtained by using a right eigenvector u of A(z). A similar version, called left shift, is given by using a left eigenvector v of A(z). Similarly, a combination of the left and the right shift, leads to a transformation called double shift, which enables one to shift a pair of eigenvalues by operating to the right and to the left of A(z).This result, restricted to a matrix Laurent polynomial A(z) = k −h z i A i , provides a shifted function which is still a matrix Laurent polynomial A(z) = k i=−h z i A i . In the particular case where A(z) = zI − A, we find that A(z) = zI − A is such that A is the matrix given in the classical theorem of Brauer [9]. Therefore, these results provide an extension of Brauer's Theorem 1.1 to matrix Laurent series and to matrix polynomials.A further generalization is obtained by performing the right, left, or double shift simultaneously to a packet of eigenvalues once we are given an invariant (right or le...

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Section: Brauer's Theorem and Canonical Factorizationsmentioning

confidence: 99%

Generalization of the Brauer Theorem to Matrix Polynomials and Matrix Laurent Series

Бини

Meini

2017

Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics

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“…In particular, the fact that some important space-time covariance matrices have the Toeplitz-block Toeplitz form stimulated the study of the inverted multi-dimensional Toeplitz matrices. Among various other interesting recent works on the Toeplitz matrices, convolution operators and their applications, we mention [1,4,5,9,12,15,21,31,38,39,60].…”

Section: Introductionmentioning

confidence: 99%

On the inversion of the block double-structured and of the triple-structured Toeplitz matrices and on the corresponding reflection coefficients

Roitberg¹,

Sakhnovich²

2020

Preprint

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The results on the inversion of convolution operators as well as Toeplitz (and block Toeplitz) matrices in the 1-D (one-dimensional) case are classical and have numerous applications. Last year, we considered the 2-D case of Toeplitz-block Toeplitz (TBT) matrices, described a minimal information, which is necessary to recover the inverse matrices, and gave a complete characterisation of the inverse matrices. Now, we develop our approach for the more complicated cases of block TBT-matrices and 3-D Toeplitz matrices.

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“…Liu et al [20] extend the results of [7] to GI/M/1-type Markov chains. Furthermore, Bini et al [3] discuss a general solution of the Poisson equation for QBDs.…”

Section: Introductionmentioning

confidence: 99%

Specific bounds for a probabilistically interpretable solution of the Poisson equation for general state-space Markov chains with queueing applications

Masuyama

2019

Preprint

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This paper considers the Poisson equation for general state-space Markov chains in continuous time. The main purpose of this paper is to present specific bounds for the solutions of the Poisson equation for general state-space Markov chains. The solutions of the Poisson equation are unique in the sense that they are expressed in terms of a certain probabilistically interpretable solution (called the standard solution). Thus, we establish some specific bounds for the standard solution under the f -modulated drift condition (which is a kind of Foster-Lyapunov-type condition) and some moderate conditions. To demonstrate the applicability of our results, we consider the workload processes in two queues: MAP/GI/1 queue, and M/GI/1 queue with workload capacity limit.

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General Solution of the Poisson Equation for Quasi-Birth-and-Death Processes

Cited by 11 publications

References 17 publications

Generalization of the Brauer Theorem to Matrix Polynomials and Matrix Laurent Series

Generalization of the Brauer Theorem to Matrix Polynomials and Matrix Laurent Series

On the inversion of the block double-structured and of the triple-structured Toeplitz matrices and on the corresponding reflection coefficients

Specific bounds for a probabilistically interpretable solution of the Poisson equation for general state-space Markov chains with queueing applications

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