A standard way to solve polynomial eigenvalue problems P (λ)x = 0 is to convert the matrix polynomial P (λ) into a matrix pencil that preserves its elementary divisors and, therefore, its eigenvalues. This process is known as linearization and is not unique, since there are infinitely many linearizations with widely varying properties associated with P (λ).… (More)
(Received 00 Month 200x; in final form 00 Month 200x) In this paper we obtain formulas for the left and right eigenvectors and minimal bases of some families of Fiedler-like linearizations of square matrix polynomials. In particular, for the families of Fiedler pencils , generalized Fiedler pencils, and Fiedler pencils with repetition. These formulas allow… (More)
The monic Jacobi matrix is a tridiagonal matrix which contains the parameters of the three-term recurrence relation satisfied by the sequence of monic polynomials orthogonal with respect to a measure. Darboux transformation without parameter changes a monic Jacobi matrix associated with a measure £ into the monic Jacobi matrix associated with ¤ ¦ ¥ § £.… (More)
In this paper we show that Doolittle's method to compute the LU factorization of Hessenberg matrices is mixed forward-backward stable and therefore, componentwise forward stable. We also conjecture that this algorithm for computing the LU factorization of dense matrices is forward stable.
The development of strong linearizations preserving whatever structure a matrix polynomial might possess has been a very active area of research in the last years, since such lin-earizations are the starting point of numerical algorithms for computing eigenvalues of structured matrix polynomials with the properties imposed by the considered structure. In… (More)
Strong linearizations of a matrix polynomial P(λ) that preserve some structure of P(λ) are relevant in many applications. In this paper we characterize all the pencils in the family of the Fiedler pencils with repetition, introduced by Antoniou and Vologiannidis, associated with a square matrix polynomial P(λ) that are symmetric when P(λ) is. When some… (More)
Given a family of polynomials orthogonal with respect to a measure dµ, it is often necessary to find another family of polynomials orthogonal with respect to the measure r(x)dµ where r(x) is a rational function. The basic Geronimus transformation computes the polynomials orthogonal with respect to 1 x−α dµ, where α is a real number, in terms of the… (More)