96 research outputs found
A Perron iteration for the solution of a quadratic vector equation arising in Markovian Binary Trees
We propose a novel numerical method for solving a quadratic vector equation
arising in Markovian Binary Trees. The numerical method consists in a fixed
point iteration, expressed by means of the Perron vectors of a sequence of
nonnegative matrices. A theoretical convergence analysis is performed. The
proposed method outperforms the existing methods for close-to-critical
problems
Shift techniques for Quasi-Birth and Death processes: canonical factorizations and matrix equations
We revisit the shift technique applied to Quasi-Birth and Death (QBD)
processes (He, Meini, Rhee, SIAM J. Matrix Anal. Appl., 2001) by bringing the
attention to the existence and properties of canonical factorizations. To this
regard, we prove new results concerning the solutions of the quadratic matrix
equations associated with the QBD. These results find applications to the
solution of the Poisson equation for QBDs
On Functions of quasi Toeplitz matrices
Let be a complex valued continuous
function, defined for , such that
. Consider the semi-infinite Toeplitz
matrix associated with the symbol
such that . A quasi-Toeplitz matrix associated with the
continuous symbol is a matrix of the form where
, , and is called a
CQT-matrix. Given a function and a CQT matrix , we provide conditions
under which is well defined and is a CQT matrix. Moreover, we introduce
a parametrization of CQT matrices and algorithms for the computation of .
We treat the case where is assigned in terms of power series and the
case where 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
The palindromic cyclic reduction and related algorithms
The cyclic reduction algorithm is specialized to palindromic matrix polynomials and a complete analysis of applicability and convergence is provided. The resulting iteration is then related to other algorithms as the evaluation/interpolation at the roots of unity of a certain Laurent matrix polynomial, the trapezoidal rule for a certain integral and an algorithm based on the finite sections of a tridiagonal block Toeplitz matrix
Palindromic matrix polynomials, matrix functions and integral representations
AbstractWe study the properties of palindromic quadratic matrix polynomials φ(z)=P+Qz+Pz2, i.e., quadratic polynomials where the coefficients P and Q are square matrices, and where the constant and the leading coefficients are equal. We show that, for suitable choices of the matrix coefficients P and Q, it is possible to characterize by means of φ(z) well known matrix functions, namely the matrix square root, the matrix polar factor, the matrix sign and the geometric mean of two matrices. Finally we provide some integral representations of these matrix functions
General solution of the Poisson equation for Quasi-Birth-and-Death processes
We consider the Poisson equation , where
is the transition matrix of a Quasi-Birth-and-Death (QBD) process with
infinitely many levels, is a given infinite dimensional vector and is the unknown. Our main result is to provide the general solution of this
equation. To this purpose we use the block tridiagonal and block Toeplitz
structure of the matrix to obtain a set of matrix difference equations,
which are solved by constructing suitable resolvent triples
From Algebraic Riccati equations to unilateral quadratic matrix equations: old and new algorithms
The problem of reducing an algebraic Riccati equation to a unilateral quadratic matrix equation (UQME) of the
kind is analyzed. New reductions are introduced
which enable one to prove some theoretical and computational properties.
In particular we show that the structure preserving doubling algorithm
of B.D.O. Anderson [Internat. J. Control, 1978] is nothing else but the
cyclic reduction algorithm applied to a suitable UQME. A new algorithm
obtained by complementing our reductions with the shrink-and-shift tech-
nique of Ramaswami is presented. Finally, faster algorithms which require
some non-singularity conditions, are designed. The non-singularity re-
striction is relaxed by introducing a suitable similarity transformation of
the Hamiltonian
On the tail decay of M/G/1-type Markov renewal processes
The tail decay of M/G/1-type Markov renewal processes is studied. The Markov
renewal process is transformed into a Markov chain so that the problem of
tail decay is reformulated in terms of the decay of the coefficients of a
suitable power series. The latter problem is reduced to analyze the
analyticity domain of the power series
Traffic lights, clumping and QBDs
In discrete time, -blocks of red lights are separated by -blocks
of green lights. Cars arrive at random. \ We seek the distribution of maximum
line length of idle cars, and justify conjectured probabilistic asymptotics
algebraically for and numerically for
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