arXiv subjects

Luca Gemignani

Publications and source records attributed to Luca Gemignani.

At least 19 recordsLinked to original sources

arXiv2026

Solving Shifted Systems for Quasiseparable Matrices

In this paper we develop fast numerical algorithms for solving shifted linear systems with semidefinite quasiseparable matrices. A combination of Givens and hyperbolic plane rotations is used to update the Cholesky-type factorization of the input quasiseparable matrix by determining a factorization of its shifted version of the form $LDL^T$, where $L$ is lower triangular and $D$ is a signature matrix. If the shifted matrix is also definite then the Cholesky factorization of the shifted matrix is computed in a stable way by using orthogonal transformations. Since quasiseparability is maintained under diagonal shifting, a fast variant of the updating procedure using computations with generators is also devised. Numerical experiments show the effectiveness and robustness of the proposed algorithm.

math.NA↗

arXiv2026

A Jacobi-Type Eigensolver for Diagonally Dominant Symmetric Matrices

This paper presents a Jacobi-type iteration for computing a given specified eigenpair of a symmetric matrix. For a certain class of diagonally dominant matrices, the procedure is shown to converge at a linear rate depending on how the matrix is significantly dominated. The cost per iteration is generally quadratic. Therefore, the proposed procedure can compute an approximation of the desired eigenpair in quadratic time.

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arXiv2025

Scaling-and-squaring method for computing the inverses of matrix $φ$-functions

This paper aims to develop efficient numerical methods for computing the inverse of matrix $φ$-functions, $ψ_\ell(A) := (φ_\ell(A))^{-1}$, for $\ell =1,2,\ldots,$ when $A$ is a large and sparse matrix with eigenvalues in the open left half-plane. While $φ$-functions play a crucial role in the analysis and implementation of exponential integrators, their inverses arise in solving certain direct and inverse differential problems with non-local boundary conditions. We propose an adaptation of the standard scaling-and-squaring technique for computing $ψ_\ell(A)$, based on the Newton-Schulz iteration for matrix inversion. The convergence of this method is analyzed both theoretically and numerically. In addition, we derive and analyze Padé approximants for approximating $ψ_1(A/2^s)$, where $s$ is a suitably chosen integer, necessary at the root of the squaring process. Numerical experiments demonstrate the effectiveness of the proposed approach.

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arXiv2024

Computing the Action of the Generating Function of Bernoulli Polynomials on a Matrix with An Application to Non-local Boundary Value Problems

This paper deals with efficient numerical methods for computing the action of the generating function of Bernoulli polynomials, say $q(τ,w)$, on a typically large sparse matrix. This problem occurs when solving some non-local boundary value problems. Methods based on the Fourier expansion of $q(τ,w)$ have already been addressed in the scientific literature. The contribution of this paper is twofold. First, we place these methods in the classical framework of Krylov-Lanczos (polynomial-rational) techniques for accelerating Fourier series. This allows us to apply the convergence results developed in this context to our function. Second, we design a new acceleration scheme. Some numerical results are presented to show the effectiveness of the proposed algorithms.

math.NA↗

arXiv2024

Numerical Solution of Nonclassical Boundary Value Problems

We provide a new approach to obtain solutions of certain evolution equations set in a Banach space and equipped with nonlocal boundary conditions. From this approach we derive a family of numerical schemes for the approximation of the solutions. We show by numerical tests that these schemes are numerically robust and computationally efficient.

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arXiv2022

Relaxed Fixed Point Iterations for Matrix Equations Arising in Markov Chains Modeling

We present some accelerated variants of fixed point iterations for computing the minimal non-negative solution of the unilateral matrix equation associated with an M/G/1-type Markov chain. These variants derive from certain staircase regular splittings of the block Hessenberg M-matrix associated with the Markov chain. By exploiting the staircase profile we introduce a two-step fixed point iteration. The iteration can be further accelerated by computing a weighted average between the approximations obtained at two consecutive steps. The convergence of the basic two-step fixed point iteration and of its relaxed modification is proved. Our theoretical analysis, along with several numerical experiments show that the proposed variants generally outperform the classical iterations.

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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.

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arXiv2021

Computing the Reciprocal of a $ϕ$-function by Rational Approximation

In this paper we introduce a family of rational approximations of the reciprocal of a $ϕ$-function involved in the explicit solutions of certain linear differential equations, as well as in integration schemes evolving on manifolds. The derivation and properties of this family of approximations applied to scalar and matrix arguments are presented. Moreover, we show that the matrix functions computed by these approximations exhibit decaying properties comparable to the best existing theoretical bounds. Numerical examples highlight the benefits of the proposed rational approximations w.r.t.~the classical Taylor polynomials and other rational functions.

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arXiv2021

Orthogonal iterations on Structured Pencils

We present a class of fast subspace tracking algorithms based on orthogonal iterations for structured matrices/pencils that can be represented as small rank perturbations of unitary matrices. The algorithms rely upon an updated data sparse factorization -- named LFR factorization -- using orthogonal Hessenberg matrices. These new subspace trackers reach a complexity of only $O(nk^2)$ operations per time update, where $n$ and $k$ are the size of the matrix and of the small rank perturbation, respectively.

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arXiv2020

Bezout-like polynomial equations associated with dual univariate interpolating subdivision schemes

The algebraic characterization of dual univariate interpolating subdivision schemes is investigated. Specifically, we provide a constructive approach for finding dual univariate interpolating subdivision schemes based on the solutions of certain associated polynomial equations. The proposed approach also makes possible to identify conditions for the existence of the sought schemes.

math.NA↗

arXiv2019

Fast QR iterations for unitary plus low rank matrices

Some fast algorithms for computing the eigenvalues of a block companion matrix $A = U + XY^H$, where $U\in \mathbb C^{n\times n}$ is unitary block circulant and $X, Y \in\mathbb{C}^{n \times k}$, have recently appeared in the literature. Most of these algorithms rely on the decomposition of $A$ as product of scalar companion matrices which turns into a factored representation of the Hessenberg reduction of $A$. In this paper we generalize the approach to encompass Hessenberg matrices of the form $A=U + XY^H$ where $U$ is a general unitary matrix. A remarkable case is $U$ unitary diagonal which makes possible to deal with interpolation techniques for rootfinding problems and nonlinear eigenvalue problems. Our extension exploits the properties of a larger matrix $\hat A$ obtained by a certain embedding of the Hessenberg reduction of $A$ suitable to maintain its structural properties. We show that $\hat A$ can be factored as product of lower and upper unitary Hessenberg matrices possibly perturbed in the first $k$ rows, and, moreover, such a data-sparse representation is well suited for the design of fast eigensolvers based on the QR/QZ iteration. The resulting algorithm is fast and backward stable.

math.NA↗

arXiv2019

Efficient Reduction of Compressed Unitary plus Low-rank Matrices to Hessenberg form

We present fast numerical methods for computing the Hessenberg reduction of a unitary plus low-rank matrix $A=G+U V^H$, where $G\in \mathbb C^{n\times n}$ is a unitary matrix represented in some compressed format using $O(nk)$ parameters and $U$ and $V$ are $n\times k$ matrices with $k< n$. At the core of these methods is a certain structured decomposition, referred to as a LFR decomposition, of $A$ as product of three possibly perturbed unitary $k$ Hessenberg matrices of size $n$. It is shown that in most interesting cases an initial LFR decomposition of $A$ can be computed very cheaply. Then we prove structural properties of LFR decompositions by giving conditions under which the LFR decomposition of $A$ implies its Hessenberg shape. Finally, we describe a bulge chasing scheme for converting the initial LFR decomposition of $A$ into the LFR decomposition of a Hessenberg matrix by means of unitary transformations. The reduction can be performed at the overall computational cost of $O(n^2 k)$ arithmetic operations using $O(nk)$ storage. The computed LFR decomposition of the Hessenberg reduction of $A$ can be processed by the fast QR algorithm presented in [8] in order to compute the eigenvalues of $A$ within the same costs.

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arXiv2016

Fast Hessenberg reduction of some rank structured matrices

We develop two fast algorithms for Hessenberg reduction of a structured matrix $A = D + UV^H$ where $D$ is a real or unitary $n \times n$ diagonal matrix and $U, V \in\mathbb{C}^{n \times k}$. The proposed algorithm for the real case exploits a two--stage approach by first reducing the matrix to a generalized Hessenberg form and then completing the reduction by annihilation of the unwanted sub-diagonals. It is shown that the novel method requires $O(n^2k)$ arithmetic operations and it is significantly faster than other reduction algorithms for rank structured matrices. The method is then extended to the unitary plus low rank case by using a block analogue of the CMV form of unitary matrices. It is shown that a block Lanczos-type procedure for the block tridiagonalization of $\Re(D)$ induces a structured reduction on $A$ in a block staircase CMV--type shape. Then, we present a numerically stable method for performing this reduction using unitary transformations and we show how to generalize the sub-diagonal elimination to this shape, while still being able to provide a condensed representation for the reduced matrix. In this way the complexity still remains linear in $k$ and, moreover, the resulting algorithm can be adapted to deal efficiently with block companion matrices.

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arXiv2016

Efficient Solution of Parameter Dependent Quasiseparable Systems and Computation of Meromorphic Matrix Functions

In this paper we focus on the solution of shifted quasiseparable systems and of more general parameter dependent matrix equations with quasiseparable representations. We propose an efficient algorithm exploiting the invariance of the quasiseparable structure under diagonal shifting and inversion. This algorithm is applied to compute various functions of matrices. Numerical experiments show that this approach is fast and numerically robust.

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arXiv2016

A Real QZ Algorithm for Structured Companion Pencils

We design a fast implicit real QZ algorithm for eigenvalue computation of structured companion pencils arising from linearizations of polynomial rootfinding problems. The modified QZ algorithm computes the generalized eigenvalues of an $N\times N$ structured matrix pencil using $O(N)$ flops per iteration and $O(N)$ memory storage. Numerical experiments and comparisons confirm the effectiveness and the stability of the proposed method.

math.NA↗

arXiv2014

Exponential Splines and Pseudo-Splines: Generation versus reproduction of exponential polynomials

Subdivision schemes are iterative methods for the design of smooth curves and surfaces. Any linear subdivision scheme can be identified by a sequence of Laurent polynomials, also called subdivision symbols, which describe the linear rules determining successive refinements of coarse initial meshes. One important property of subdivision schemes is their capability of exactly reproducing in the limit specific types of functions from which the data is sampled. Indeed, this property is linked to the approximation order of the scheme and to its regularity. When the capability of reproducing polynomials is required, it is possible to define a family of subdivision schemes that allows to meet various demands for balancing approximation order, regularity and support size. The members of this family are known in the literature with the name of pseudo-splines. In case reproduction of exponential polynomials instead of polynomials is requested, the resulting family turns out to be the non-stationary counterpart of the one of pseudo-splines, that we here call the family of exponential pseudo-splines. The goal of this work is to derive the explicit expressions of the subdivision symbols of exponential pseudo-splines and to study their symmetry properties as well as their convergence and regularity.

math.NA↗

arXiv2014

Implicit QR for Companion-like Pencils

A fast implicit QR algorithm for eigenvalue computation of low rank corrections of unitary matrices is adjusted to work with matrix pencils arising from polynomial zerofinding problems . The modified QZ algorithm computes the generalized eigenvalues of certain NxN rank structured matrix pencils using O(N^2) ops and O(N) memory storage. Numerical experiments and comparisons confirm the effectiveness and the stability of the proposed method.

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arXiv2013

Compression of unitary rank--structured matrices to CMV-like shape with an application to polynomial rootfinding

This paper is concerned with the reduction of a unitary matrix U to CMV-like shape. A Lanczos--type algorithm is presented which carries out the reduction by computing the block tridiagonal form of the Hermitian part of U, i.e., of the matrix U+U^H. By elaborating on the Lanczos approach we also propose an alternative algorithm using elementary matrices which is numerically stable. If U is rank--structured then the same property holds for its Hermitian part and, therefore, the block tridiagonalization process can be performed using the rank--structured matrix technology with reduced complexity. Our interest in the CMV-like reduction is motivated by the unitary and almost unitary eigenvalue problem. In this respect, finally, we discuss the application of the CMV-like reduction for the design of fast companion eigensolvers based on the customary QR iteration.

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