Leonardo Robol

A Framework for Structured Linearizations of Matrix Polynomials in Various Bases

We present a framework for the construction of linearizations for scalar and matrix polynomials based on dual bases which, in the case of orthogonal polynomials, can be described by the associated recurrence relations. The framework provides an extension of the classical linearization theory for polynomials expressed in nonmonomial bases and allows us to represent polynomials expressed in product families, that is, as a linear combination of elements of the form

On quadratic matrix equations with infinite size coefficients encountered in QBD stochastic processes

\phi_i(\lambda) \psi_j(\lambda)

Quasiseparable Hessenberg reduction of real diagonal plus low rank matrices and applications

, where

Quasi-Toeplitz matrix arithmetic: a MATLAB toolbox

\{ \phi_i(\lambda) \}

On a class of matrix pencils and ℓ-ifications equivalent to a given matrix polynomial☆

and

Solving Rank-Structured Sylvester and Lyapunov Equations

\{ \psi_j(\lambda) \}

Low-rank updates and a divide-and-conquer method for linear matrix equations

can either be polynomial bases or polynomial families which satisfy some mild assumptions. We show that this general construction can be used for many different purposes. Among them, we show how to linearize sums of polynomials and rational functions expressed in different bases. As an example, this allows us to look for intersections of functions interpolated on different nodes without converting them to the same basis. We then provide some constructions ...

Solvability and uniqueness criteria for generalized Sylvester-type equations

Summary Matrix equations of the kind A1X2+A0X+A−1=X, where both the matrix coefficients and the unknown are semi-infinite matrices belonging to a Banach algebra, are considered. These equations, where coefficients are quasi-Toeplitz matrices, are encountered in certain quasi-birth–death processes as the tandem Jackson queue or in any other processes that can be modeled as a reflecting random walk in the quarter plane. We provide a numerical framework for approximating the minimal nonnegative solution of these equations that relies on semi-infinite quasi-Toeplitz matrix arithmetic. In particular, we show that the algorithm of cyclic reduction can be effectively applied and can approximate the infinite-dimensional solutions with quadratic convergence at a cost that is comparable to that of the finite case. This way, we may compute a finite approximation of the sought solution and of the invariant probability measure of the associated quasi-birth–death process, within a given accuracy. Numerical experiments, performed on a collection of benchmarks, confirm the theoretical analysis.

On a Class of Matrix Pencils Equivalent to a Given Matrix Polynomial

Abstract We present a novel algorithm to perform the Hessenberg reduction of an n × n matrix A of the form A = D + U V ⁎ where D is diagonal with real entries and U and V are n × k matrices with k ≤ n . The algorithm has a cost of O ( n 2 k ) arithmetic operations and is based on the quasiseparable matrix technology. Applications are shown to solving polynomial eigenvalue problems and some numerical experiments are reported in order to analyze the stability of the approach.

Efficient Ehrlich–Aberth iteration for finding intersections of interpolating polynomials and rational functions

A quasi-Toeplitz (QT) matrix is a semi-infinite matrix of the kind A=T(a)+E