arXiv subjects

Dario A. Bini

Publications and source records attributed to Dario A. Bini.

At least 19 recordsLinked to original sources

arXiv2026

Component-wise accurate computation of the square root of an M-matrix

Component-wise accurate algorithms for computing the principal square root of an M-matrix are designed in terms of triplet representations. A triplet representation of an M-matrix $A$ is the triple $(P, {\bf u},{\bf v})$, where the matrix $P$ is such that $p_{ij}=-a_{ij}$ for $i\ne j$, $p_{ii}=0$, and ${\bf u}>0$, ${\bf v}\ge 0$ are two vectors such that $A{\bf u}={\bf v}$. It is shown that if $A$ is an M-matrix representable by a triplet, then its principal square root exists and is an M-matrix represented by a triplet as well. New versions of the Cyclic Reduction and the Incremental Newton iterations are provided in terms of triplets, to compute the principal matrix square root of $A$. It is shown that these algorithms are component-wise numerically stable independently of the singularity of $A$ and of its condition number. Numerical experiments are shown to confirm the component-wise stability.

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arXiv2025

Computing matrix functions associated with a Hermitian-definite pencil

We consider the numerical evaluation of the quantity $Af(A^{-1}B)$, where $A$ is Hermitian positive definite, $B$ is Hermitian, and $f$ is a function defined on the spectrum of $A^{-1}B$. This problem is related to the Hermitian-definite matrix pencil $B-\lambda A$. We study the conditioning of the problem, and we introduce several algorithms that combine the Schur decomposition with either the matrix square root or the Cholesky factorization. We study the numerical behavior of these algorithms in floating-point arithmetic, assess their computational costs, and compare their numerical performance. Our analysis suggests that the algorithms based on the Cholesky factorization will be more accurate and efficient than those based on the matrix square root. This is confirmed by our numerical experiments.

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arXiv2025

The Derivative of Kemeny's Constant as a Centrality Measure in Undirected Graphs

Kemeny's constant quantifies a graph's connectivity by measuring the average time for a random walker to reach any other vertex. We introduce two concepts of the directional derivative of Kemeny's constant with respect to an edge and use them to define centrality measures for edges and non-edges in the graph. Additionally, we present a sensitivity measure of Kemeny's constant. An explicit expression for these quantities involving the inverse of the modified graph Laplacian is provided, which is valid even for cut-edges. These measures are connected to the one introduced in [Altafini et al., SIMAX 2023], and algorithms for their computation are included. The benefits of these measures are discussed, along with applications to road networks and link prediction analysis. For one-path graphs, an explicit expression for these measures is given in terms of the edge weights.

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arXiv2023

Numerical computation of the roots of Mandelbrot polynomials: an experimental analysis

This paper deals with the problem of numerically computing the roots of polynomials $p_k(x)$, $k=1,2,\ldots$, of degree $n=2^k-1$ recursively defined by $p_1(x)=x+1$, $p_k(x)=xp_{k-1}(x)^2+1$. An algorithm based on the Ehrlich-Aberth simultaneous iterations complemented by the Fast Multi-pole Method and the fast search of near neighbors of a set of complex numbers is provided. The algorithm, which relies on a specific strategy of selecting initial approximations, costs $O(n\log n)$ arithmetic operations per step. A Fortran 95 implementation is given and numerical experiments are carried out. Experimentally, it turns out that the number of iterations needed to arrive at numerical convergence is $O(\log n)$. This allows us to compute the roots of $p_k(x)$ up to degree $n=2^{24}-1$ in about 16 minutes on a laptop with 16 GB RAM, and up to degree $n=2^{28}-1$ in about 69 minutes on a machine with 256 GB RAM. The case of degree $n=2^{30}-1$ would require higher memory and higher precision to separate the roots. With a suitable adaptation of FMM to the limit of 256 GB RAM and by performing the computation in extended precision (i.e. with 10-byte floating point representation) we were able to compute all the roots in about two weeks of CPU time for $n=2^{30}-1$. From the experimental analysis, explicit asymptotic expressions of the real roots of $p_k(x)$ and an explicit expression of $\min_{i\ne j}|\xi_i^{(k)}-\xi_j^{(k)}|$ for the roots $\xi_i^{(k)}$ of $p_k(x)$ are deduced. The approach is effectively applied to general classes of polynomials defined by a doubling recurrence.

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arXiv2021

Numerical solution of a matrix integral equation arising in Markov Modulated L\'evy processes

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 solution of this class of matrix equations, perform a theoretical convergence analysis and show the effectiveness of the new methods by means of a wide numerical experimentation.

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arXiv2021

Geometric means of quasi-Toeplitz matrices

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 $\sum_{\ell=-\infty}^{\infty} a_\ell e^{\mathfrak i \ell t}$, in the sense that $(T(a))_{i,j}=a_{j-i}$. If $a$ is \rv\ and essentially bounded, then these matrices represent bounded self-adjoint operators on $\ell^2$. We consider the case where $a$ is a continuous function, where quasi-Toeplitz matrices coincide with a classical Toeplitz algebra, and the case where $a$ is in the Wiener algebra, that is, has absolutely convergent Fourier series. We prove that if $a_1,\ldots,a_p$ are continuous and positive functions, or are in the Wiener algebra with some further conditions, then means of geometric type, such as the ALM, the NBMP and the Karcher mean of quasi-Toeplitz positive definite matrices associated with $a_1,\ldots,a_p$, are quasi-Toeplitz matrices associated with the geometric mean $(a_1\cdots a_p)^{1/p}$, which differ only by the compact correction. We show by numerical tests that these operator means can be practically approximated.

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arXiv2019

A computational framework for two-dimensional random walks with restarts

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 extension of the framework of [5] which allows to deal with more general situations such as processes involving restart events. This is motivated by the need for modeling processes that can incur in unexpected failures like computer system reboots. Algebraically, this gives rise to corrections with infinite support that cannot be treated using the tools currently available in the literature. We present a theoretical analysis of an enriched Banach algebra that, combined with appropriate algorithms, enables the numerical treatment of these problems. The results are applied to the solution of bidimensional Quasi-Birth-Death processes with infinitely many phases which model random walks in the quarter plane, relying on the matrix analytic approach. This methodology reduces the problem to solving a quadratic matrix equation with coefficients of infinite size. We provide conditions on the transition probabilities which ensure that the solution of interest of the matrix equation belongs to the enriched algebra. The reliability of our approach is confirmed by extensive numerical experimentation on some case studies.

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arXiv2019

Solving quadratic matrix equations arising in random walks in the quarter plane

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 with an almost Toeplitz structure. Here, we analyze some fixed point iterations, including Newton's iteration, for the computation of $G$ and introduce effective algorithms and acceleration strategies which fully exploit the Toeplitz structure of the matrix coefficients and of the current approximation. Moreover, we provide a structured perturbation analysis for the solution $G$. The results of some numerical experiments which demonstrate the effectiveness of our approach are reported.

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arXiv2018

Quasi-Toeplitz matrix arithmetic: a MATLAB toolbox

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 precision, by means of a finite number of parameters. QT-matrices, equipped with the norm $\lVert A \rVert_{\mathcal QT}=\alpha\lVert a\rVert_{\mathcal{W}} \lVert E \rVert_2$, for $\alpha = (1+\sqrt 5)/2$, are a Banach algebra with the standard arithmetic operations. We provide an algorithmic description of these operations on the finite parametrization of QT-matrices, and we develop a MATLAB toolbox implementing them in a transparent way. The toolbox is then extended to perform arithmetic operations on matrices of finite size that have a Toeplitz plus low-rank structure. This enables the development of algorithms for Toeplitz and quasi-Toeplitz matrices whose cost does not necessarily increase with the dimension of the problem. Some examples of applications to computing matrix functions and to solving matrix equations are presented, and confirm the effectiveness of the approach.

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arXiv2016

On the exponential of semi-infinite quasi-Toeplitz matrices

Let $a(z)=\sum_{i\in\mathbb Z}a_iz^i$ be a complex valued function defined for $|z|=1$, such that $\sum_{i\in\mathbb Z}|ia_i|<\infty$, and let $E=(e_{i,j})_{i,j\in\mathbb {Z}^+}$ be such that $\sum_{i,j\in\mathbb{Z}^+}|e_{i,j}|<\infty$. A semi-infinite quasi-Toeplitz matrix is a matrix of the kind $A=T(a)+E$, where $T(a)=(t_{i,j})_{i,j\in\mathbb{Z}^+}$ is the semi-infinite Toeplitz matrix associated with the symbol $a(z)$, that is, $t_{i,j}=a_{j-i}$ for $i,j\in\mathbb Z^+$. We analyze theoretical and computational properties of the exponential of $A$. More specifically, it is shown that $\exp(A)=T(\exp(a))+F$ where $F=(f_{i,j})_{i,j\in\mathbb{Z}^+}$ is such that $\sum_{i,j\in\mathbb{Z}^+}|f_{i,j}|$ is finite, i.e., $\exp(A)$ is a semi-infinite quasi-Toeplitz matrix as well, and an effective algorithm for its computation is given. These results can be extended from the function $\exp(z)$ to any function $f(z)$ satisfying mild conditions, and can be applied to finite quasi-Toeplitz matrices.

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arXiv2016

On Functions of quasi Toeplitz matrices

Let $a(z)=\sum_{i\in\mathbb Z}a_iz^i$ be a complex valued continuous function, defined for $|z|=1$, such that $\sum_{i=-\infty}^{+\infty}|ia_i|<\infty$. Consider the semi-infinite Toeplitz matrix $T(a)=(t_{i,j})_{i,j\in\mathbb Z^+}$ associated with the symbol $a(z)$ such that $t_{i,j}=a_{j-i}$. A quasi-Toeplitz matrix associated with the continuous symbol $a(z)$ is a matrix of the form $A=T(a)+E$ where $E=(e_{i,j})$, $\sum_{i,j\in\mathbb Z^+}|e_{i,j}|<\infty$, and is called a CQT-matrix. Given a function $f(x)$ and a CQT matrix $M$, we provide conditions under which $f(M)$ is well defined and is a CQT matrix. Moreover, we introduce a parametrization of CQT matrices and algorithms for the computation of $f(M)$. We treat the case where $f(x)$ is assigned in terms of power series and the case where $f(x)$ 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.

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arXiv2016

Semi-Infinite Quasi-Toeplitz Matrices with Applications to QBD Stochastic Processes

Denote by $\mathcal{W}_1$ the set of complex valued functions of the form $a(z)=\sum_{i=-\infty}^{+\infty}a_iz^i$ which are continuous on the unit circle, and such that $\sum_{i=-\infty}^{+\infty}|ia_i|<\infty$. We call CQT matrix a quasi-Toeplitz matrix $A$, associated with a continuous symbol $a(z)\in\mathcal W_1$, of the form $A=T(a)+E$, where $T(a)=(t_{i,j})_{i,j\in\mathbb{Z}^+}$ is the semi-infinite Toeplitz matrix such that $t_{i,j}=a_{j-i}$, for $i,j\in\mathbb Z^+$, and $E=(e_{i,j})_{i,j\in\mathbb{Z}^+}$ is a semi-infinite matrix such that $\sum_{i,j=1}^{+\infty}|e_{i,j}|$ is finite. We prove that the class of CQT matrices is a Banach algebra with a suitable sub-multiplicative matrix norm $\|\cdot\|$. We introduce a finite representation of CQT matrices together with algorithms which implement elementary matrix operations. An application to solving quadratic matrix equations of the kind $AX^2+BX+C=0$, encountered in the solution of Quasi-Birth and Death (QBD) stochastic processes with a denumerable set of phases, is presented where $A,B,C$ are CQT matrices.

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arXiv2016

General solution of the Poisson equation for Quasi-Birth-and-Death processes

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 block tridiagonal and block Toeplitz structure of the matrix $P$ to obtain a set of matrix difference equations, which are solved by constructing suitable resolvent triples.

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arXiv2016

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.

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arXiv2016

Efficient cyclic reduction for QBDs with rank structured blocks

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 values of the off-diagonal submatrices generated by cyclic reduction. We provide a formal proof of this decay in the Markovian framework. The results of the numerical experiments that we report confirm a significant speed up over the general algorithms, already starting with the moderately small size $m\approx 10^2$.

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arXiv2015

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

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 generalizations to shifting a set of eigenvalues. We provide conditions under which the modified function $\widetilde A(z)$ has a canonical factorization $\widetilde A(z)=\widetilde U(z)\widetilde L(z^{-1})$ and we provide explicit expressions of the factors $\widetilde U(z)$ and $\widetilde L(z)$. Similar conditions and expressions are given for the factorization of $\widetilde A(z^{-1})$. Some applications are discussed.

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arXiv2015

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

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 shown to solving polynomial eigenvalue problems and some numerical experiments are reported in order to analyze the stability of the approach

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arXiv2014

On a Class of Matrix Pencils and $\ell$-ifications Equivalent to a Given Matrix Polynomial

A new class of linearizations and $\ell$-ifications for $m\times m$ matrix polynomials $P(x)$ of degree $n$ is proposed. The $\ell$-ifications in this class have the form $A(x) = D(x) + (e\otimes I_m) W(x)$ where $D$ is a block diagonal matrix polynomial with blocks $B_i(x)$ of size $m$, $W$ is an $m\times qm$ matrix polynomial and $e=(1,\ldots,1)^t\in\mathbb C^q$, for a suitable integer $q$. The blocks $B_i(x)$ can be chosen a priori, subjected to some restrictions. Under additional assumptions on the blocks $B_i(x)$ the matrix polynomial $A(x)$ is a strong $\ell$-ification, i.e., the reversed polynomial of $A(x)$ defined by $A^\#(x) := x^{\mathrm{deg} A(x)} A(x^{-1})$ is an $\ell$-ification of $P^\#(x)$. The eigenvectors of the matrix polynomials $P(x)$ and $A(x)$ are related by means of explicit formulas. Some practical examples of $\ell$-ifications are provided. A strategy for choosing $B_i(x)$ in such a way that $A(x)$ is a well conditioned linearization of $P(x)$ is proposed. Some numerical experiments that validate the theoretical results are reported

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