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2018 Journal article Open Access

Corrigendum to "Solvability and uniqueness criteria for generalized Sylvester-type equations"
De Teran F, Iannazzo B, Poloni F, Robol L
We provide an amended version of Corollaries 7 and 9 in [De Teran, Iannazzo, Poloni, Robol, "Solvability and uniqueness criteria for generalized Sylvester-type equations"]. These results characterize the unique solvability of the matrix equation AXB + CX*D = E (where the coefficients need not be square) in terms of an equivalent condition on the spectrum of certain matrix pencils of the same size as one of its coefficients. (C) 2017 Elsevier Inc. All rights reserved.Source: LINEAR ALGEBRA AND ITS APPLICATIONS, vol. 542, pp. 522-526
DOI: 10.1016/j.laa.2017.10.018
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2018 Journal article Open Access

Solvability and uniqueness criteria for generalized Sylvester-type equations
De Teran F, Iannazzo B, Poloni F, Robol L
We provide necessary and sufficient conditions for the generalized (star operator)-Sylvester matrix equation, AXB+CX(star operator)D=E, to have exactly one solution for any right-hand side E. These conditions are given for arbitrary coefficient matrices A, B, C, D (either square or rectangular) and generalize existing results for the same equation with square coefficients. We also review the known results regarding the existence and uniqueness of solution for generalized Sylvester and (star operator)-Sylvester equations.Source: LINEAR ALGEBRA AND ITS APPLICATIONS, vol. 542, pp. 501-521
DOI: 10.1016/j.laa.2017.07.010
DOI: 10.48550/arxiv.1608.01183
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2018 Other Metadata Only Access

Progetto TITANIO - Sensori innovativi per il monitoraggio del patrimonio architettonico. Rapporto sull'attività svolta nel periodo 19 Agosto 2017 - 19 Agosto 2018
Azzara R. M., Barsocchi P., Cassarà P., Girardi M., Lucchesi D., Mavilia F., Padovani C., Pellegrini D., Robol L.
Il documento descrive le attività conclusive del progetto TITANIO (Sensori innovativi per il monitoraggio del patrimonio architettonico), finanziato dalla Fondazione Carilucca per il biennio 2016-2018.

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2018 Conference article Open Access

Fea for masonry structures and vibration-based model updating using NOSA-ITACA
Girardi M, Padovani C, Pellegrini D, Porcelli M, Robol L
NOSA-ITACA is a finite-element code developed by the Mechanics of Materials and Structures Laboratory of ISTI-CNR for the structural analysis of masonry constructions of historical interest via the constitutive equation of masonry-like materials. The latest improvements in the software allow applying model updating techniques to match experimentally measured frequencies in order to fine-tune calculation of the free parameters in the model. The numerical method is briefly presented and applied to two historical buildings in Lucca, the Church of San Francesco and the Clock Tower.

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2018 Other Open Access

Analyzing a security and reliability model using Krylov methods and matrix functions
Masetti G, Robol L
It has been recently shown how the computation of performability measures for Markov models can be recasted as the evaluation of a bilinear forminduced by appropriate matrix functions. In view of these results, we show how to analyze a security model, inspired by a real world scenario. The model describes a mobile cyber-physical system of communicating nodes which are subject to security attacks. We take advantage of the properties of matrix functions of block matrices, and provide efficient evaluation methods.Moreover, we show how this new formulation can be used to retrieve interesting theoretical results, which can also rephrased in probabilistic terms.

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2018 Journal article Open Access

Model parameter estimation using Bayesian and deterministic approaches: the case study of the Maddalena Bridge
De Falco A, Girardi M, Pellegrini D, Robol L, Sevieri G
Finite element modeling has become common practice for assessing the structural health of historic constructions. However, because of the uncertainties typically affecting our knowledge of the geometrical dimensions, material properties and boundary conditions, numerical models can fail to predict the static and dynamic behavior of such structures. In order to achieve more reliable predictions, important information can be obtained measuring the structural response under ambient vibrations. This wholly non-destructive technique allows obtaining very accurate information on the structure's dynamic properties (Brincker and Ventura (2015)).Source: PROCEDIA STRUCTURAL INTEGRITY, vol. 11, pp. 210-217
DOI: 10.1016/j.prostr.2018.11.028
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2018 Journal article Open Access

Efficient Ehrlich-Aberth iteration for finding intersections of interpolating polynomials and rational functions
Robol L, Vandebril R
We analyze the problem of carrying out an efficient iteration to approximate the eigenvalues of some rank structured pencils obtained as linearization of sums of polynomials and rational functions expressed in (possibly different) interpolation bases. The class of linearizations that we consider has been introduced by Robol, Vandebril and Van Dooren in [17]. We show that a traditional QZ iteration on the pencil is both asymptotically slow (since it is a cubic algorithm in the size of the matrices) and sometimes not accurate (since in some cases the deflation of artificially introduced infinite eigenvalues is numerically difficult). To solve these issues we propose to use a specifically designed Ehrlich-Aberth iteration that can approximate the eigenvalues in O(kn2) flops, where k is the average number of iterations per eigenvalue, and n the degree of the linearized polynomial. We suggest possible strategies for the choice of the initial starting points that make k asymptotically smaller than O(n), thus making this method less expensive than the QZ iteration. Moreover, we show in the numerical experiments that this approach does not suffer of numerical issues, and accurate results are obtained.Source: LINEAR ALGEBRA AND ITS APPLICATIONS, vol. 542, pp. 282-309
DOI: 10.1016/j.laa.2017.05.010
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2018 Journal article Open Access

On quadratic matrix equations with infinite size coefficients encountered in QBD stochastic processes
Bini Da, Massei S, Meini B, Robol L
Matrix equations of the kind $A_1 X^2 + A0 X + A_{-1} = X$, 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 approxi- mate 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.Source: NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS (ONLINE), vol. 25 (issue 6)
DOI: 10.1002/nla.2128
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2018 Journal article Open Access

Solving rank-structured Sylvester and Lyapunov equations
Massei S, Palitta D, Robol L
We consider the problem of efficiently solving Sylvester and Lyapunov equations of medium and large scale, in case of rank-structured data, i.e., when the coefficient matrices and the right-hand side have low-rank off-diagonal blocks. This comprises problems with banded data, recently studied in [A. Haber and M. Verhaegen, Automatica J. IFAC, 73 (2016), pp. 256-268; D. Palitta and V. Simoncini, Numerical Methods for Large-Scale Lyapunov Equations with Symmetric Banded Data, preprint, arxiv, 1711.04187, 2017], which often arise in the discretization of elliptic PDEs. We show that, under suitable assumptions, the quasiseparable structure is guaranteed to be numerically present in the solution, and explicit novel estimates of the numerical rank of the offdiagonal blocks are provided. Efficient solution schemes that rely on the technology of hierarchical matrices are described, and several numerical experiments confirm the applicability and efficiency of the approaches. We develop a MATLAB toolbox that allows easy replication of the experiments and a ready-to-use interface for the solvers. The performances of the different approaches are compared, and we show that the new methods described are efficient on several classes of relevant problems.Source: SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS, vol. 39 (issue 4), pp. 1564-1590
DOI: 10.1137/17m1157155
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2018 Journal article Open Access

Fast and backward stable computation of roots of polynomials. Part II: backward error analysis; companion matrix and companion pencil
Aurentz Jl, Mach T, Robol L, Vandebril R, Watkins Ds
This work is a continuation of work by J. L. Aurentz, T. Mach, R. Vandebril, and D. S. Watkins, J. Matrix Anal. Appl., 36 (2015), pp. 942--973. In that paper we introduced a companion QR algorithm that finds the roots of a polynomial by computing the eigenvalues of the companion matrix in O(n^2) time using O(n) memory. We proved that the method is backward stable. Here we introduce, as an alternative, a companion QZ algorithm that solves a generalized eigenvalue problem for a companion pencil. More importantly, we provide an improved backward error analysis that takes advantage of the special structure of the problem. The improvement is also due, in part, to an improvement in the accuracy (in both theory and practice) of the turnover operation, which is the key component of our algorithms. We prove that for the companion QR algorithm, the backward error on the polynomial coefficients varies linearly with the norm of the polynomial's vector of coefficients. Thus, the companion QR lgorithm has a smaller backward error than the unstructured QR algorithm (used by MATLAB's roots command, for example), for which the backward error on the polynomial coefficients grows quadratically with the norm of the coefficient vector. The companion QZ algorithm has the same favorable backward error as companion QR, provided that the polynomial coefficients are properly scaled.Source: SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS, vol. 39 (issue 3), pp. 1245-1269
DOI: 10.1137/17m1152802
DOI: 10.48550/arxiv.1611.02435
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2018 Book Restricted

Core-chasing algorithms for the eigenvalue problem
Aurentz Jl, Mach T, Robol L, Vandebril R, Watkins Ds
This monograph is about a class of methods for solving matrix eigenvalue problems. Of course the methods are also useful for computing related objects such as eigenvectors and invariant subspaces. We will introduce new algorithms along the way, but we are also advocating a new way of viewing and implementing existing algorithms, notably Francis's implicitly-shifted QR algorithm [36]. Our first message is that if we want to compute the eigenvalues of a matrix A, it is often advantageous to store A in QR-decomposed form. That is, we write A = QR, where Q is unitary and R is upper triangular, and we store Q and R instead of A. This may appear to be an inefficient approach but, as we shall see, it often is not. Most matrices that arise in applications have some special structures, and these often imply special structures for the factors Q and R. For example, if A is upper Hessenberg, then Q is also upper Hessenberg, and it follows from this that Q can be stored very compactly. As another example, suppose A is unitary. Then Q = A, and R is the identity matrix, so we don't have to store R at all. Every matrix can be transformed to upper Hessenberg form by a unitary similarity transformation. We will study this and related transformations in detail in Chapter 6, but for the early chapters of the book we will simply take the transformation for granted; we will assume that A is already in Hessenberg form. Thus we consider an arbitrary upper Hessenberg matrix in QR-decomposed form and show how to compute its eigenvalues. Our method proceeds by a sequence of similarity transformations that drive the matrix toward upper triangular form. Once the matrix is triangular, the eigenvalues can be read from the main diagonal. In fact our method is just a new implementation of Francis's implicitly-shifted QR algorithm. The storage space requirement is O(n^2) because we must store the upper-triangular matrix R, and the flop count is O(n^3). Once we know how to handle general matrices, we consider how the procedure can be simplified in special cases. The easiest is the unitary case, where R = I. This results in an immediate reduction in the storage requirement to O(n) and a corresponding reduction of the computational cost to O(n^2) flops. A similar case is that of a companion matrix, which is both upper Hessenberg and unitary-plus-rank-one. This results in an R that is unitary-plus-rank-one. Once we have figured out how to store R using only O(n) storage, we again get an algorithm that runs in O(n 2 ) flops. The unitary case is old [42], but our companion algorithm is new [7]. A structure that arises frequently in eigenvalue problems is symmetry: A = A^T . This very important structure does not translate into any obvious structure for the factors Q and R, so it does not fit into our framework in an obvious way. In Section 4.4 we show that the symmetric problem can be solved using our methodology: we turn it into a unitary problem by a Cayley transform [10]. We did not seriously expect this approach would be faster than all of the many other existing methods for the symmetric eigenvalue problem [73, § 7.2], but we were pleased to find that it is not conspicuously slow. Our solution to the symmetric problem serves as a stepping stone to the solution of the symmetric-plus-rank-one problem, which includes comrade and colleague matrices [14] as important special cases. If a polynomial p is presented as a linear combination of Chebyshev or Legendre polynomials, for example, the coefficients can be placed into a comrade matrix with eigenvalues equal to the zeros of p. A Cayley transform turns the symmetric-plus-rank-one problem into a unitary-plus-rank-one problem, which we can solve by our fast companion solver. Our O(n^2) algorithm gives us a fast way to compute the zeros of polynomials expressed in terms of these classic orthogonal polynomial bases. We also study the generalized eigenvalue problem, for which the Moler-Stewart QZ algorithm is the appropriate variant of Francis's algorithm. We show how to apply this to a companion pencil using the same approach as for the companion matrix, but utilizing two unitary-plus-rank-one upper triangular matrices instead of one [4]. We then extend this methodology to matrix polynomial eigenvalue problems. A block companion pencil is formed and then factored into a large number of unitary-plus-rank-one factors. The resulting algorithm is advantageous when the degree of the matrix polynomial is large [6]. The final chapter discusses the reduction to Hessenberg form and generalizations of Hessenberg form. We introduce generalizations of Francis's algorithm that can be applied to these generalized Hessenberg forms [64]. This monograph is a summary and report on a research project that the five of us (in various combinations) have been working on for several years now. Included within these covers is material from a large number of recent sources, including [4-10, 12, 54, 62-64, 74]. We have also included some material that has not yet been submitted for publication. This book exists because the senior author decided that it would be worthwhile to present a compact and unified treatment of our findings in a single volume. The actual writing was done by the senior author, who wanted to ensure uniformity of style and viewpoint (and, one could also say, prejudices). But the senior author is only the scribe, who (of course) benefitted from substantial feedback from the other authors. Moreover, the book would never have come into existence without the work of a team over the course of years. We are excited about the outcome of our research, and we are pleased to share it with you. The project is not finished by any means, so a second edition of this book might appear at some time in the future.

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