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This paper presents explicit formulas and algorithms to compute the eigenvalues and eigenvectors of order-one-quasiseparable matrices. Various recursive relations for characteristic polynomials of their principal submatrices are derived. The cost of evaluating the characteristic polynomial of an N × N matrix and its derivative is only O(N). This leads(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 V Φ (x), i.e., polynomial-Vandermonde systems where the corresponding polynomial system Φ is the Szegö polyno-mials. The properties of the corresponding unitary Hessenberg matrix allow us to derive a fast O(n(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 n 2 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)
Although Gaussian elimination uses O(n 3) operations to invert an arbitrary matrix, matrices with a special Vandermonde structure can be inverted in only O(n 2) operations by the fast Traub algorithm. The original version of Traub algorithm was numerically unstable although only a minor modification of it yields a high accuracy in practice. The Traub(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 Her-mitian 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(More)
In this paper we derive a fast O(n 2) algorithm for solving linear systems where the coefficient matrix is a polynomial-Vandermonde VR(x) = [rj−1(xi)] matrix with polynomials R related to a qua-siseparable matrix. The result is a generalization of the well-known Björck-Pereyra algorithm for classical Vandermonde systems. As will be shown, many important(More)