Dario A. Bini

Dario A. Bini
Università di Pisa | UNIPI · Department of Mathematics

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229
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January 1980 - December 2012

Publications

Publications (229)
Article
Full-text available
A quasi-Toeplitz (QT) matrix is a semi-infinite matrix of the form A=T(a)+E where T(a) is the Toeplitz matrix with entries (T(a))i,j=aj-i, for aj-i∈C, i,j≥1, while E is a matrix representing a compact operator in ℓ2. The matrix A is finitely representable if ak=0 for k<-m and for k>n, given m,n>0, and if E has a finite number of nonzero entries. Th...
Preprint
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A new measure $c(e)$ of the centrality of an edge $e$ in an undirected graph $G$ is introduced. It is based on the variation of the Kemeny constant of the graph after removing the edge $e$. The new measure is designed in such a way that the Braess paradox is avoided. A numerical method for computing $c(e)$ is introduced and a regularization techniq...
Preprint
Full-text available
A quasi-Toeplitz (QT) matrix is a semi-infinite matrix of the form $A=T(a)+E$ where $T(a)$ is the Toeplitz matrix with entries $(T(a))_{i,j}=a_{j-i}$, for $a_{j-i}\in\mathbb C$, $i,j\ge 1$, while $E$ is a compact correction. The matrix $A$ is finitely representable if $a_k=0$ for $k<-m$ or $k>n$, given $m,n>0$, and if $E$ has a finite number of non...
Preprint
Full-text available
Markov-modulated L\'evy processes lead to matrix integral equations of the kind $ A_0 + A_1X+A_2 X^2+A_3(X)=0$ where $A_0$, $A_1$, $A_2$ are given matrix coefficients, while $A_3(X)$ is a nonlinear function, expressed in terms of integrals involving the exponential of the matrix $X$ itself. In this paper we propose some numerical methods for the so...
Chapter
We provide algorithms for computing the Karcher mean of positive definite semi-infinite quasi-Toeplitz matrices. After showing that the power mean of quasi-Toeplitz matrices is a quasi-Toeplitz matrix, we obtain a first algorithm based on the fact that the Karcher mean is the limit of a family of power means. A second algorithm, that is shown to be...
Article
Full-text available
We consider the problem of computing the minimal non-negative solution $G$ of the nonlinear matrix equation $X=\sum _{i=-1}^\infty A_iX^{i+1}$ where $A_i$, for $i\geqslant -1$, are non-negative square matrices such that $\sum _{i=-1}^\infty A_i$ is stochastic. This equation is fundamental in the analysis of M/G/1-type Markov chains, since the matri...
Preprint
Full-text available
We study means of geometric type of quasi-Toeplitz matrices, that are semi-infinite matrices $A=(a_{i,j})_{i,j=1,2,\ldots}$ of the form $A=T(a)+E$, where $E$ represents a compact operator, and $T(a)$ is a semi-infinite Toeplitz matrix associated with the function $a$, with Fourier series $\sum_{\ell=-\infty}^{\infty} a_\ell e^{\mathfrak i \ell t}$,...
Preprint
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We consider the problem of computing the minimal nonnegative solution $G$ of the nonlinear matrix equation $X=\sum_{i=-1}^\infty A_iX^{i+1}$ where $A_i$, for $i\ge -1$, are nonnegative square matrices such that $\sum_{i=-1}^\infty A_i$ is stochastic. This equation is fundamental in the analysis of M/G/1-type Markov chains, since the matrix $G$ prov...
Preprint
Full-text available
The treatment of two-dimensional random walks in the quarter plane leads to Markov processes which involve semi-infinite matrices having Toeplitz or block Toeplitz structure plus a low-rank correction. Finding the steady state probability distribution of the process requires to perform operations involving these structured matrices. We propose an e...
Preprint
Full-text available
Quadratic matrix equations of the kind $A_1X^2+A_0X+A_{-1}=X$ are encountered in the analysis of Quasi--Birth-Death stochastic processes where the solution of interest is the minimal nonnegative solution $G$. In many queueing models, described by random walks in the quarter plane, the coefficients $A_1,A_0,A_{-1}$ are infinite tridiagonal matrices...
Article
Full-text available
A Quasi Toeplitz (QT) matrix is a semi-infinite matrix of the kind $A=T(a)+E$ where $T(a)=(a_{j-i})_{i,j\in\mathbb Z^+}$, $E=(e_{i,j})_{i,j\in\mathbb Z^+}$ is compact and the norms $\lVert a\rVert_{\mathcal W} = \sum_{i\in\mathbb Z}|a_i|$ and $\lVert E \rVert_2$ are finite. These properties allow to approximate any QT-matrix, within any given preci...
Article
Full-text available
Let $a(z)=\sum_{i\in\mathbb Z}a_iz^i$ be an analytic function on the annulus $\mathbb A(r,R)=\{z\in\CC,\quad r<|z|<R\}$, with $0<r<1<R$, and let $E=(e_{i,j})_{i,j\in\ZZ^+}$ be such that $\sum_{i,j\in\ZZ^+}|e_{i,j}|$ is finite. A semi-infinite quasi-Toeplitz matrix is a matrix of the kind $A=T(a)+E$, where $T(a)=(t_{i,j})_{i,j\in\ZZ^+}$ is the semi-...
Book
This book gathers selected contributions presented at the INdAM Meeting Structured Matrices in Numerical Linear Algebra: Analysis, Algorithms and Applications, held in Cortona, Italy on September 4-8, 2017. Highlights cutting-edge research on Structured Matrix Analysis, it covers theoretical issues, computational aspects, and applications alike. Th...
Article
In their 1960 book on finite Markov chains, Kemeny and Snell established that a certain sum is invariant. The value of this sum has become known as Kemeny’s constant . Various proofs have been given over time, some more technical than others. We give here a very simple physical justification, which extends without a hitch to continuous-time Markov...
Article
Full-text available
In their 1960 book on finite Markov chains, Kemeny and Snell established that a certain sum is invariant. This sum has become known as Kemeny's constant. Various proofs have been given over time, some more technical than others. We give here a very simple physical justification, which extends without a hitch to continuous-time Markov chains on a fi...
Article
Full-text available
Matrix equations of the kind $A_1X^2+A_0X+A_{-1}=X$, 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 (QBD) processes as the tandem Jackson queue or in any other p...
Chapter
Given a square matrix A, Brauer’s theorem [Duke Math. J. 19 (1952), 75–91] 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 gene...
Article
Full-text available
The class of semi-infinite Analytically Quasi-Toeplitz (AQT) matrices is introduced. This class is formed by matrices which can be written in the form $A=T(a)+E$, where $T(a)=(t_{i,j})_{i,j\in\Z^+}$ is the semi-infinite Toeplitz matrix associated with the symbol $a(z)=\sum_{i=-\infty}^{+\infty}a_iz^i$, that is $t_{i,j}=a_{j-i}$, for $i,j\in\mathbb...
Article
Full-text available
Let $a(z)=\sum_{i\in\mathbb Z}a_iz^i$ be an analytic function on an annulus containing the unit circle and let $\mathbb Z^+$ be the set of positive integers. An Analytically Quasi-Toeplitz (AQT) matrix $M=(m_{i,j})_{i,j\in\mathbb Z^+}$ is a semi-infinite matrix of the kind $M=T(a)+E$ where $T(a)=(t_{i,j})_{i,j\in\mathbb Z^+}$, $t_{i,j}=a_{j-i}$, an...
Article
Full-text available
It was recently observed in [9] that the singular values of the off-diagonal blocks of the matrix sequences generated by the Cyclic Reduction algorithm decay exponentially. This property was used to solve, with a higher efficiency, certain quadratic matrix equations encountered in the analysis of queueing models. In this paper, we provide a sharp t...
Article
We provide effective algorithms for solving block tridiagonal block Toeplitz systems with quasiseparable blocks, as well as quadratic matrix equations with quasiseparable coefficients, based on cyclic reduction and on the technology of rank-structured matrices. The algorithms rely on the exponential decay of the singular values of the off-diagonal...
Article
Full-text available
We consider the Poisson equation $(I-P)\boldsymbol{u}=\boldsymbol{g}$, where $P$ is the transition matrix of a Quasi-Birth-and-Death (QBD) process with infinitely many levels, $\bm g$ is a given infinite dimensional vector and $\bm u$ is the unknown. Our main result is to provide the general solution of this equation. To this purpose we use the blo...
Article
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 resu...
Article
Full-text available
We provide effective algorithms for solving block tridiagonal block Toeplitz systems with $m\times m$ quasiseparable blocks, as well as quadratic matrix equations with $m\times m$ quasiseparable coefficients, based on cyclic reduction and on the technology of rank-structured matrices. The algorithms rely on the exponential decay of the singular val...
Chapter
Matrix structures are ubiquitous in linear algebra problems stemming from scientific computing, engineering and from any mathematical models of the real world. They translate, in matrix language, the specific properties of the physical problem. Often, structured matrices reveal themselves in a clear form and apparently seem to show immediately all...
Book
Focusing on special matrices and matrices which are in some sense `near’ to structured matrices, this volume covers a broad range of topics of current interest in numerical linear algebra. Exploitation of these less obvious structural properties can be of great importance in the design of efficient numerical methods, for example algorithms for matr...
Article
Full-text available
Given a square matrix $A$, Brauer's theorem [Duke Math. J. 19 (1952), 75--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 wi...
Article
A new class of linearizations and ℓ-ifications for matrix polynomials of degree n is proposed. The ℓ-ifications in this class have the form where D is a block diagonal matrix polynomial with blocks of size m, W is an matrix polynomial and , for a suitable integer q. The blocks can be chosen a priori, subjected to some restrictions. Under additional...
Article
The Erlangian approximation of Markovian fluid queues leads to the problem of computing the matrix exponential of a subgenerator having a block-triangular, block-Toeplitz structure. To this end, we propose some algorithms which exploit the Toeplitz structure and the properties of generators. Such algorithms allow to compute the exponential of very...
Article
We present a novel algorithm to perform the Hessenberg reduction of an $n\times n$ matrix $A$ of the form $A = D + UV^*$ where $D$ is diagonal with real entries and $U$ and $V$ are $n\times k$ matrices with $k\le n$. The algorithm has a cost of $O(n^2k)$ arithmetic operations and is based on the quasiseparable matrix technology. Applications are sh...
Article
We present an algorithm for the solution of polynomial equations and secular equations of the form S(x)=0S(x)=0 for S(x)=∑i=1naix−bi−1=0, which provides guaranteed approximation of the roots with any desired number of digits. It relies on the combination of two different strategies for dealing with the precision of the floating point computation: t...
Article
Many applications of the real world are modelled by matrix polynomials [EQUATION] where Ai are m x m matrices, see for instance [2], [6]. A computational task encountered in this framework is computing the eigenvalues of P(x), that is, the solutions of the polynomial equation det P(x) = 0. This task is generally accomplished by reducing P(x) to a l...
Article
Full-text available
We say that an $m\times m$ matrix polynomial $P(x)=\sum_{i=0}^nP_i x^i$ is equivalent to an $mq\times mq$ matrix polynomial $A(x)$, and write $A(x)\approx P(x)$, if there exist $mq\times mq$ matrix polynomials $E(x)$, $F(x)$ such that $\det E(x)$ and $\det F(x)$ are nonzero constants and $E(x)A(x)F(x)=I_{m(q-1)}\oplus P(x)$. Given $P(x)$ of degree...
Article
Full-text available
The geometric mean of positive definite matrices is usually identified with the Karcher mean, which possesses all properties—generalized from the scalar case—a geometric mean is expected to satisfy. Unfortunately, the Karcher mean is typically not structure preserving, and destroys, e.g., Toeplitz and band structures, which emerge in many applicati...
Article
We present and analyze an iterative method for approximating the Karcher mean of a set of n×n positive definite matrices Ai, i=1,…,k, defined as the unique positive definite solution of the matrix equation .
Article
Given the $n\times n$ matrix polynomial $P(x)=\sum_{i=0}^kP_i x^i$, we consider the associated polynomial eigenvalue problem. This problem, viewed in terms of computing the roots of the scalar polynomial $\det P(x)$, is treated in polynomial form rather than in matrix form by means of the Ehrlich-Aberth iteration. The main computational issues are...
Article
Full-text available
Some known results for locating the roots of polynomials are extended to the case of matrix polynomials. In particular, a theorem by A.E. Pellet [Bulletin des Sciences Math\'ematiques, (2), vol 5 (1881), pp.393-395], some results of D.A. Bini [Numer. Algorithms 13:179-200, 1996] based on the Newton polygon technique, and recent results of M. Akian,...
Article
Full-text available
Matrix-analytic methods have advanced considerably since the pioneering work of Marcel Neuts [6, 5] on Quasi-Birth-Death (QBD), GI/M/1- and M/G/1- type Markov chains (MCs). Especially the algorithms involved to (iteratively) solve these structured Markov chains have matured a lot, which has resulted in more efficient, but also more complex algorith...
Article
Full-text available
In this chapter we consider quasi-birth and death processes with low rank downward and upward transitions. We show how such structure can be exploited to reduce the computational cost of the cyclic reduction iteration. The proposed algorithm saves computation by performing multiplications and inversions of matrices of small size (equal to the rank...
Article
Full-text available
A new definition is introduced for the matrix geometric mean of a set of k positive definite n×n matrices together with an iterative method for its computation. The iterative method is locally convergent with cubic convergence and requires O(n 3k 2) arithmetic operations per step whereas the methods based on the symmetrization technique of Ando et...
Article
We present some advances, both from a theoretical and from a computational point of view, on a quadratic vector equation (QVE) arising in Markovian Binary Trees. Concerning the theoretical advances, some irreducibility assumptions are relaxed, and the minimality of the solution of the QVE is expressed in terms of properties of the Jacobian of a sui...
Article
Full-text available
This initial online publication is an extract from the project forms submitted to Regione Toscana under the PAR FAS Regione Toscana Action Line 1.1.a.3. call. It has been partially reviewed and adapted for a less technical reading. We believe that it is important to publish the project extract in order to provide details on the project objectives,...
Chapter
This article reports the main properties of M/G/1-type Markov chains together with the classical and the most advanced algorithms for their analysis. Keywords: Markov chains; M/G/1-type; non-skip-free chain; QBD process
Article
Full-text available
After defining the area of investigation, the methods used for acquiring new data will be described. The archaeologists will address the problems encountered with the archives and the status oof documentation, the sedimentologists will identify the area for continuous coring, whilst the geomorphologists will base their analyses on micro-relief, pho...
Article
This concise and comprehensive treatment of the basic theory of algebraic Riccati equations describes the classical as well as the more advanced algorithms for their solution in a manner that is accessible to both practitioners and scholars. It is the first book in which nonsymmetric algebraic Riccati equations are treated in a clear and systematic...
Book
Linear algebra preliminaries.- Quadratic vector equations.- A Perron vector iteration for QVEs.- Unilateral quadratic matrix equations.- Nonsymmetric algebraic Riccati equations.- Transforming NAREs into UQMEs.- Storage optimal algorithms for Cauchy-like matrices.- Newton method for rank-structured algebraic Riccati equations.- Lur'e equations.- Ge...
Article
Full-text available
An integrated model for ranking scientific publications together with authors and journals recently presented in [Bini, Del Corso, Romani, ETNA 2008] is closely analyzed. The model, which relies on certain adjacency matrices H,K and F obtained from the relations of citation, authorship and publication, provides the ranking by means of the Perron ve...
Article
Full-text available
The problem of reducing an algebraic Riccati equation XCX − AX − XD+B=0 to a unilateral quadratic matrix equation (UQME) of the kind PX 2+QX+R=0 is analyzed. New transformations are introduced which enable one to prove some theoretical and computational properties. In particular we show that the structure preserving doubling algorithm (SDA) of Ande...
Article
SUMMARY An implicit version of the shifted QR eigenvalue algorithm given in [D.~A.~Bini, Y.~Eidelman, I.~Gohberg, L.~Gemignani, SIAM J. Matrix Anal. Appl. 29 (2007), no. 2, 566--585] is presented for computing the eigenvalues of an $n\times n$ companion matrix using $O(n^2)$ flops and $O(n)$ memory storage. Numerical experiments and comparisons con...
Article
Full-text available
We propose a new matrix geometric mean satisfying the ten prop- erties given by Ando, Li and Mathias (Linear Alg. Appl. 2004). This mean is the limit of a sequence which converges superlinearly with convergence of or- der 3 whereas the mean introduced by Ando, Li and Mathias is the limit of a sequence having order of convergence 1. This makes this...
Chapter
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An O(n 2) complexity algorithm for computing an ∈-greatest common divisor (gcd) of two polynomials of degree at most n is presented. The algorithm is based on the formulation of polynomial gcd given in terms of resultant (Bézout, Sylvester) matrices, on their displacement structure and on the reduction of displacement structured matrices to Cauchy-...
Article
The tail decay of the hitting probabilities on level zero in M/G/1-type Markov renewal processes (MRPs) 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 analyti...
Article
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Cyclic reduction is an algorithm invented by G.H.Golub and R.W.Hockney in the mid 1960s for solving linear systems related to the finite differences discretization of the Poisson equation over a rectangle. Among the algorithms of Gene Golub, it is one of the most versatile and powerful ever created. Recently, it has been applied to solve different...
Article
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We expand and update the software tool SMCSolver, pre-sented at the SMCTools workshop in 2006, for the numerical solution of structured Markov chains encountered in queuing models. In particular the new version of the package imple-ments different transformation techniques and different shift strategies which are combined in order to speed up and o...
Article
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We consider a special instance of the algebraic Riccati equation XCX − XE − AX + B = 0 encountered in transport theory, where the n × n matrix coefficients A, B, C, E are rank structured matrices. The equation is reduced to unilateral form A 1 X 2 + A 0 X + A −1 = 0 and solved by means of Cyclic Reduction (CR). It is shown that the matrices generat...
Article
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We provide a simple convergence proof for the cyclic reduction algorithm for M/G/1 type Markov chains together with a probabilistic interpretation which helps to understand better the relationships between logarithmic reduction and cyclic reduction.
Article
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Interpolation of smooth functions and the discretization of elliptic PDEs by means of radial functions lead to structured linear systems which, for equidistant grid points, have almost the (block) Toeplitz structure. We prove upper bounds for the condition numbers of the n×n Toeplitz matrices which discretize the model problem u″(x)=f(x), x∈(0,1),...
Article
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We consider the problem of completion of a matrix with a specified lower triangular part to a unitary matrix. In this paper we obtain the necessary and sufficient conditions of existence of a unitary completion without any additional constraints and give a general formula for this completion. The paper is mainly focused on matrices with the specifi...
Article
Full-text available
A special instance of the algebraic Riccati equation XCX XE AX + B = 0 encountered in transport theory is considered where the n × n matrix coefficientsA, B, C, E are rank structured matrices. Relying on the structural properties of Cauchy-like matrices, an algorithm is designed for performing the customary Newton iteration in O(n2) arithmetic oper...
Article
Full-text available
Some integrated models for ranking scientific publications together with authors and journals are presented and analyzed. The models rely on certain adjacency matrices obtained from the relationships between citations, authors and publications, which together give a suitable irreducible stochastic matrix whose Perron vector provides the ranking. So...