arXiv subjects

Federico Poloni

Publications and source records attributed to Federico Poloni.

At least 19 recordsLinked to original sources

arXiv2026

The mmatrix toolbox: componentwise accurate algorithms for M-matrices with triplet representation

We introduce the mmatrix toolbox, a Matlab software package for componentwise accurate computations with M-matrices described through left or right triplet representations. The core of the toolbox is a Fortran implementation, in the Lapack style, of the unblocked, recursive, and blocked versions of the GTH algorithm and its applications for the accurate computation of the solution of linear systems with M-matrix coefficient and nonnegative right-hand side, the LU factorization of an M-matrix and its inverse. These algorithms avoid subtractive cancellation; this property ensures high componentwise accuracy even for ill-conditioned problems. The toolbox contains also accurate algorithms for related problems, such as computing the Schur complement, the singular values, the square root of an M-matrix, and the solution of nonsymmetric algebraic Riccati equations associated with M-matrices. The Matlab interface is based on an object-oriented implementation allowing one to use standard Matlab operations on M-matrices with triplet representation.

cs.MS↗

arXiv2026

Structured distance to singularity as a nonlinear system of equations

In this article we study the structured distance to singularity for a nonsingular matrix $A\in\mathbb{C}^{n\times n}$, with a prescribed linear structure $\mathcal{S}$ (for instance, a sparsity pattern, or a real Toeplitz structure), i.e., the norm of the smallest perturbation $\Delta \in \mathcal{S}$, such that $A + \Delta$ is singular. This is an example of structured matrix nearness problem: a family of problems that arise in control and systems theory and in numerical analysis, when characterizing the robustness of a certain property of a system with respect to perturbations that are constrained to a certain structure (for example the structure of the nominal system). We start by highlighting the parallelism between two main tools which have been proposed in the literature: a gradient system approach for a functional in the eigenvalues, which requires the solution of certain low-rank matrix differential equations (see [Guglielmi, Lubich, Sicilia, SINUM 2023]), and a two-level optimization approach in which the inner linear least-squares problem is solved explicitly (see [Usevich, Markovsky, JCAM 2014] and [Gnazzo, Noferini, Nyman, Poloni, FoCM 2025]). In particular, these articles underline the remarkable property that $\Delta$ is (at least generically) the orthogonal projection onto the structure $\mathcal{S}$ of a rank-1 matrix $uv^*$. This property and the parallelism suggest a new reformulation of the problem into a system of nonlinear equations in the two vector unknowns $u,v \in\mathbb{C}^n$. We study this new formulation, and propose an algorithm to solve these nonlinear equations directly with the multivariate Newton's method. We discuss how to avoid the singularity of such system of nonlinear equations, and how to ensure monotonic convergence. The resulting algorithm is faster than the existing ones for large matrices, and maintains comparable accuracy.

math.NA↗

arXiv2025

Error analysis of Abate--Whitt methods for Inverse Laplace Transforms and a new algorithm for queuing theory applications

We study the accuracy of a class of methods to compute the Inverse Laplace Transform, the so-called \emph{Abate--Whitt methods} [Abate, Whitt 2006], which are based on a linear combination of evaluations of $\widehat{f}$ in a few points. We provide error bounds which relate the accuracy of a method to the rational approximation of the exponential function. We specialize our analysis to applications in queuing theory, a field in which Abate--Whitt methods are often used; in particular, we study phase-type distributions and Markov-modulated fluid models (or \emph{fluid queues}). We use a recently developed algorithm for rational approximation, the AAA algorithm [Nakatsukasa, S\`ete, Trefethen 2018], to produce a new family of methods, which we call TAME. The parameters of these methods are constructed depending on a function-specific domain $\Omega$; we provide a quasi-optimal choice for certain families of functions. We discuss numerical issues related to floating-point computation, and we validate our results through numerical experiments which show that the new methods require significantly fewer function evaluations to achieve an accuracy that is comparable (or better) to that of the classical methods.

math.NA↗

arXiv2025

Nearest matrix with multiple eigenvalues by Riemannian optimization

Given a square complex matrix $A$, we tackle the problem of finding the nearest matrix with multiple eigenvalues or, equivalently when $A$ had distinct eigenvalues, the nearest defective matrix. To this goal, we extend the general framework described in [M. Gnazzo, V. Noferini, L. Nyman, F. Poloni, \emph{Riemann-Oracle: A general-purpose Riemannian optimizer to solve nearness problems in matrix theory}, Found. Comput. Math., To appear] and based on variable projection and Riemannian optimization, allowing the ambient manifold to simultaneously track left and right eigenvectors. Our method also allows us to impose arbitrary complex-linear constraints on either the perturbation or the perturbed matrix; this can be useful to study structured eigenvalue condition numbers. We present numerical experiments, comparing with preexisting algorithms.

math.NA↗

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.

math.NA↗

arXiv2025

Accuracy and componentwise accuracy in multilinear PageRank

We study the stability with respect to perturbations and the accuracy of numerical algorithms for computing solutions to the multilinear PageRank problem $\mathbf{x} = (1-\alpha)\mathbf{v} + \alpha \mathcal{P} \mathbf{x}^2$. Our results reveal that the solution can be more stable with respect to perturbations and numerical errors with respect to the classical bounds for nonlinear systems of equations (based on the norm of the Jacobian). In detail, one can obtain bounds for the minimal solution which ignore the singularity of the problem for $\alpha=1/2$, and one can show that the limiting accuracy of the Newton method depends not on the norm of the Jacobian but on a quantity that can be much smaller thanks to the nonnegativity structure of the equation. For the minimal solution, we also suggest subtraction-free modifications to the existing algorithms to achieve componentwise stability. Some of the theoretical results we obtain are interesting even outside the scope of this problem: bounds for more general quadratic vector equations, and a partial inverse for M-matrices which remains bounded when the matrix to invert approaches singularity.

math.NA↗

arXiv2024

Riemann-Oracle: A general-purpose Riemannian optimizer to solve nearness problems in matrix theory

We propose an extremely versatile approach to address a large family of matrix nearness problems, possibly with additional linear constraints. Our method is based on splitting a matrix nearness problem into two nested optimization problems, of which the inner one can be solved either exactly or cheaply, while the outer one can be recast as an unconstrained optimization task over a smooth real Riemannian manifold. We observe that this paradigm applies to many matrix nearness problems of practical interest appearing in the literature, thus revealing that they are equivalent in this sense to a Riemannian optimization problem. We also show that the objective function to be minimized on the Riemannian manifold can be discontinuous, thus requiring regularization techniques, and we give conditions for this to happen. Finally, we demonstrate the practical applicability of our method by implementing it for a number of matrix nearness problems that are relevant for applications and are currently considered very demanding in practice. Extensive numerical experiments demonstrate that our method often greatly outperforms its predecessors, including algorithms specifically designed for those particular problems.

math.NA↗

arXiv2023

Invariant subspaces of $T$-palindromic pencils and algebraic $T$-Riccati equations

By exploiting the connection between solving algebraic $\top$-Riccati equations and computing certain deflating subspaces of $\top$-palindromic matrix pencils, we obtain theoretical and computational results on both problems. Theoretically, we introduce conditions to avoid the presence of modulus-one eigenvalues in a $\top$-palindromic matrix pencil and conditions for the existence of solutions of a $\top$-Riccati equation. Computationally, we improve the palindromic QZ algorithm with a new ordering procedure and introduce new algorithms for computing a deflating subspace of the $\top$-palindromic pencil, based on quadraticizations of the pencil or on an integral representation of the orthogonal projector on the sought deflating subspace.

math.NA↗

arXiv2022

A general framework for the rigorous computation of invariant densities and the coarse-fine strategy

In this paper we present a general, axiomatical framework for the rigorous approximation of invariant densities and other important statistical features of dynamics. We approximate the system trough a finite element reduction, by composing the associated transfer operator with a suitable finite dimensional projection (a discretization scheme) as in the well-known Ulam method. We introduce a general framework based on a list of properties (of the system and of the projection) that need to be verified so that we can take advantage of a so-called ``coarse-fine'' strategy. This strategy is a novel method in which we exploit information coming from a coarser approximation of the system to get useful information on a finer approximation, speeding up the computation. This coarse-fine strategy allows a precise estimation of invariant densities and also allows to estimate rigorously the speed of mixing of the system by the speed of mixing of a coarse approximation of it, which can easily be estimated by the computer. The estimates obtained here are rigourous, i.e., they come with exact error bounds that are guaranteed to hold and take into account both the discretiazation and the approximations induced by finite-precision arithmetic. We apply this framework to several discretization schemes and examples of invariant density computation from previous works, obtaining a remarkable reduction in computation time. We have implemented the numerical methods described here in the Julia programming language, and released our implementation publicly as a Julia package.

math.DS↗

arXiv2022

On the use of Markov chains for epidemic modeling on networks

We discuss various models for epidemics on networks that rely on Markov chains. Random walks on graphs are often used to predict epidemic spread and to investigate possible control actions to mitigate them. In this study, we demonstrate that they do not fully reflect the dynamics of epidemics, as they overestimate infection times. Accordingly, we explain how Markov chains may still be used to accurately model the virus spread, and to correctly predict infection times. We also provide an algorithm that efficiently estimates infection times via a sampling strategy. Finally, we present a novel indicator based on infection times, and we compare its node ranking properties with other centrality measures based on random walks.

physics.soc-ph↗

arXiv2021

Comparison Theorems for Splittings of M-matrices in (block) Hessenberg Form

Some variants of the (block) Gauss--Seidel iteration for the solution of linear systems with $M$-matrices in (block) Hessenberg form are discussed. Comparison results for the asymptotic convergence rate of some regular splittings are derived: in particular, we prove that for a lower-Hessenberg M-matrix $ρ(P_{GS})\geq ρ(P_S)\geq ρ(P_{AGS})$, where $P_{GS}, P_S, P_{AGS}$ are the iteration matrices of the Gauss--Seidel, staircase, and anti-Gauss--Seidel method. This is a result that does not seem to follow from classical comparison results, as these splittings are not directly comparable. It is shown that the concept of stair partitioning provides a powerful tool for the design of new variants that are suited for parallel computation.

math.NA↗

arXiv2021

Efficient Analysis of Chemical Reaction Networks Dynamics based on Input-Output Monotonicity

Motivation: A Chemical Reaction Network (CRN) is a set of chemical reactions, which can be very complex and difficult to analyze. Indeed, dynamical properties of CRNs can be described by a set of non-linear differential equations that rarely can be solved in closed-form, but that can instead be used to reason on the system dynamics. In this context, one of the possible approaches is to perform numerical simulations, which may require a high computational effort. In particular, in order to investigate some dynamical properties, such as robustness or global sensitivity, many simulations have to be performed by varying the initial concentration of chemical species. Results: In order to reduce the computational effort required when many simulations are needed to assess a property, we exploit a new notion of monotonicity of the output of the system (the concentration of a target chemical species at the steady-state) with respect to the input (the initial concentration of another chemical species). To assess such monotonicity behavior, we propose a new graphical approach that allows us to state sufficient conditions for ensuring that the monotonicity property holds. Our sufficient conditions allow us to efficiently verify the monotonicity property by exploring a graph constructed on the basis of the reactions involved in the network. Once established, our monotonicity property allows us to drastically reduce the number of simulations required to assess some dynamical properties of the CRN.

cs.CE↗

arXiv2021

A continuation method for computing the multilinear Pagerank

The multilinear Pagerank model [Gleich, Lim and Yu, 2015] is a tensor-based generalization of the Pagerank model. Its computation requires solving a system of polynomial equations that contains a parameter $α\in [0,1)$. For $α\approx 1$, this computation remains a challenging problem, especially since the solution may be non-unique. Extrapolation strategies that start from smaller values of $α$ and `follow' the solution by slowly increasing this parameter have been suggested; however, there are known cases where these strategies fail, because a globally continuous solution curve cannot be defined as a function of $α$. In this paper, we improve on this idea, by employing a predictor-corrector continuation algorithm based on a more general representation of the solutions as a curve in $\mathbb{R}^{n+1}$. We prove several global properties of this curve that ensure the good behavior of the algorithm, and we show in our numerical experiments that this method is significantly more reliable than the existing alternatives.

math.NA↗

arXiv2021

Nearest $Ω$-stable matrix via Riemannian optimization

We study the problem of finding the nearest $Ω$-stable matrix to a certain matrix $A$, i.e., the nearest matrix with all its eigenvalues in a prescribed closed set $Ω$. Distances are measured in the Frobenius norm. An important special case is finding the nearest Hurwitz or Schur stable matrix, which has applications in systems theory. We describe a reformulation of the task as an optimization problem on the Riemannian manifold of orthogonal (or unitary) matrices. The problem can then be solved using standard methods from the theory of Riemannian optimization. The resulting algorithm is remarkably fast on small-scale and medium-scale matrices, and returns directly a Schur factorization of the minimizer, sidestepping the numerical difficulties associated with eigenvalues with high multiplicity.

math.NA↗

arXiv2020

Iterative and doubling algorithms for Riccati-type matrix equations: a comparative introduction

We review a family of algorithms for Lyapunov- and Riccati-type equations which are all related to each other by the idea of \emph{doubling}: they construct the iterate $Q_k = X_{2^k}$ of another naturally-arising fixed-point iteration $(X_h)$ via a sort of repeated squaring. The equations we consider are Stein equations $X - A^*XA=Q$, Lyapunov equations $A^*X+XA+Q=0$, discrete-time algebraic Riccati equations $X=Q+A^*X(I+GX)^{-1}A$, continuous-time algebraic Riccati equations $Q+A^*X+XA-XGX=0$, palindromic quadratic matrix equations $A+QY+A^*Y^2=0$, and nonlinear matrix equations $X+A^*X^{-1}A=Q$. We draw comparisons among these algorithms, highlight the connections between them and to other algorithms such as subspace iteration, and discuss open issues in their theory.

math.NA↗

arXiv2019

When is a matrix unitary or Hermitian plus low rank?

Hermitian and unitary matrices are two representatives of the class of normal matrices whose full eigenvalue decomposition can be stably computed in quadratic computing com plexity. Recently, fast and reliable eigensolvers dealing with low rank perturbations of unitary and Hermitian matrices were proposed. These structured eigenvalue problems appear naturally when computing roots, via confederate linearizations, of polynomials expressed in, e.g., the monomial or Chebyshev basis. Often, however, it is not known beforehand whether or not a matrix can be written as the sum of an Hermitian or unitary matrix plus a low rank perturbation. We propose necessary and sufficient conditions characterizing the class of Hermitian or unitary plus low rank matrices. The number of singular values deviating from 1 determines the rank of a perturbation to bring a matrix to unitary form. A similar condition holds for Hermitian matrices; the eigenvalues of the skew-Hermitian part differing from 0 dictate the rank of the perturbation. We prove that these relations are linked via the Cayley transform. Based on these conditions we are able to identify the closest Hermitian and unitary plus low rank matrix in Frobenius and spectral norm and a practical Lanczos iteration to detect the low rank perturbation is presented. Numerical tests prove that this straightforward algorithm is robust with respect to noise.

math.NA↗

arXiv2019

Nonsingular systems of generalized Sylvester equations: an algorithmic approach

We consider the uniqueness of solution (i.e., nonsingularity) of systems of $r$ generalized Sylvester and $\star$-Sylvester equations with $n\times n$ coefficients. After several reductions, we show that it is sufficient to analyze periodic systems having, at most, one generalized $\star$-Sylvester equation. We provide characterizations for the nonsingularity in terms of spectral properties of either matrix pencils or formal matrix products, both constructed from the coefficients of the system. The proposed approach uses the periodic Schur decomposition, and leads to a backward stable $O(n^3r)$ algorithm for computing the (unique) solution.

math.NA↗

arXiv2018

Doubling Algorithms for Stationary Distributions of Fluid Queues: A Probabilistic Interpretation

Fluid queues are mathematical models frequently used in stochastic modelling. Their stationary distributions involve a key matrix recording the conditional probabilities of returning to an initial level from above, often known in the literature as the matrix $\Psi$. Here, we present a probabilistic interpretation of the family of algorithms known as \emph{doubling}, which are currently the most effective algorithms for computing the return probability matrix $\Psi$. To this end, we first revisit the links described in \cite{ram99, soares02} between fluid queues and Quasi-Birth-Death processes; in particular, we give new probabilistic interpretations for these connections. We generalize this framework to give a probabilistic meaning for the initial step of doubling algorithms, and include also an interpretation for the iterative step of these algorithms. Our work is the first probabilistic interpretation available for doubling algorithms.

math.PR↗