Yuli Eidelman

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Let Hn ⊂ Cn×n be the class of n × n Hessenberg matrices A which are rank-one modifications of a unitary matrix, that is, A = H + uwH , where H is unitary and u,w ∈ Cn. The class Hn includes three well-known subclasses: unitary Hessenberg matrices, companion (Frobenius) matrices, and fellow matrices. The paper presents some novel fast adaptations of the(More)
The QR iteration method for tridiagonal matrices is in the heart of one classical method to solve the general eigenvalue problem. In this paper we consider the more general class of quasiseparable matrices that includes not only tridiagonal but also companion, comrade, unitary Hessenberg and semiseparble matrices. A fast QR iteration method exploiting the(More)
An interplay between polynomials and dense structured matrices is a classical topic. The structure of these dense matrices is understood in the sense that their n2 entries can be “compressed” to a smaller number O(n) of parameters. Operating directly on these parameters allows one to design efficient fast algorithms for these matrices and for the related(More)
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 specified lower(More)
In this paper we address the problem of efficiently computing all the eigenvalues of a large N×N Hermitian matrix modified by a possibly non Hermitian perturbation of low rank. Previously proposed fast adaptations of the QR algorithm are considerably simplified by performing a preliminary transformation of the matrix by similarity into an upper Hessenberg(More)
In this paper we carry over the Björck-Pereyra algorithm for solving Vandermonde linear systems to what we suggest to call Szegö-Vandermonde systems VΦ(x), i.e., polynomialVandermonde systems where the corresponding polynomial system Φ is the Szegö polynomials. The properties of the corresponding unitary Hessenberg matrix allow us to derive a fast O(n2)(More)
In this paper we derive a fast O(n) algorithm for solving linear systems where the coefficient matrix is a polynomial-Vandermonde matrix VR(x) = [rj−1(xi)] with polynomials {rk(x)} related to a Hessenberg quasiseparable matrix. The result generalizes the well-known Björck-Pereyra algorithm for classical Vandermonde systems involving monomials. It also(More)
The expression structured matrices appeared for the first time in a conference title in 1995, specifically in the session “Algorithms for Structured Matrices”, organized within the SPIE conference, held in San Diego (USA) [1, Session 6] and in the “Minisymposium on Structured Matrices” within the ILAS conference, held in Atlanta (USA) [2]. These first(More)
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