arXiv subjects

Dario Andrea Bini

Publications and source records attributed to Dario Andrea Bini.

4 recordsLinked to original sources

arXiv2026

Component-wise accurate fixed point iterations for computing the square root of a singular M-matrix

We analyze two fixed-point iterations for computing the principal square root of an M-matrix $A$. Although these iterations, with customary initialization, converge sublinearly when $A$ is a singular M-matrix, we show that, under suitable mild conditions on the initial approximation, the convergence is linear. Moreover, we provide component-wise accurate versions of these iterations, which allow us to approximate the principal square root with a component-wise relative error uniformly bounded by a small multiple of the machine precision. Numerical experiments demonstrating the effectiveness of the proposed algorithms for certain classes of problems are presented.

math.NA↗

arXiv2024

On Kemeny's constant and stochastic complement

Given a stochastic matrix $P$ partitioned in four blocks $P_{ij}$, $i,j=1,2$, Kemeny's constant $κ(P)$ is expressed in terms of Kemeny's constants of the stochastic complements $P_1=P_{11}+P_{12}(I-P_{22})^{-1}P_{21}$, and $P_2=P_{22}+P_{21}(I-P_{11})^{-1}P_{12}$. Specific cases concerning periodic Markov chains and Kronecker products of stochastic matrices are investigated. Bounds to Kemeny's constant of perturbed matrices are given. Relying on these theoretical results, a divide-and-conquer algorithm for the efficient computation of Kemeny's constant of graphs is designed. Numerical experiments performed on real-world problems show the high efficiency and reliability of this algorithm.

math.NA↗

arXiv2021

A family of fast fixed point iterations for M/G/1-type Markov chains

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 $G$ provides probabilistic measures of interest. A new family of fixed point iterations for the numerical computation of $G$, that includes the classical iterations, is introduced. A detailed convergence analysis proves that the iterations in the new class converge faster than the classical iterations. Numerical experiments confirm the effectiveness of our extension.

math.NA↗

arXiv2016

On the decay of the off-diagonal singular values in cyclic reduction

It was recently observed 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 theoretical bound to the basis of this exponential decay together with a tool for its estimation based on a rational interpolation problem. Applications to solving $n\times n$ block tridiagonal block Toeplitz systems with $n\times n$ semiseparable blocks and certain generalized Sylvester equations in $O(n^2\log n)$ arithmetic operations are shown.

math.NA↗