arXiv subjects

Dario A. Bini

Publications and source records attributed to Dario A. Bini.

22 records · Page 2Linked to original sources

arXiv2012

Locating the eigenvalues of matrix polynomials

Some known results for locating the roots of polynomials are extended to the case of matrix polynomials. In particular, a theorem by A.E. Pellet [Bulletin des Sciences Mathématiques, (2), vol 5 (1881), pp.393-395], some results of D.A. Bini [Numer. Algorithms 13:179-200, 1996] based on the Newton polygon technique, and recent results of M. Akian, S. Gaubert and M. Sharify (see in particular [LNCIS, 389, Springer p.p.291-303] and [M. Sharify, Ph.D. thesis, École Polytechnique, ParisTech, 2011]). These extensions are applied for determining effective initial approximations for the numerical computation of the eigenvalues of matrix polynomials by means of simultaneous iterations, like the Ehrlich-Aberth method. Numerical experiments that show the computational advantage of these results are presented.

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arXiv2012

Solving polynomial eigenvalue problems by means of the Ehrlich-Aberth method

Given the $n\times n$ matrix polynomial $P(x)=\sum_{i=0}^kP_i x^i$, we consider the associated polynomial eigenvalue problem. This problem, viewed in terms of computing the roots of the scalar polynomial $\det P(x)$, is treated in polynomial form rather than in matrix form by means of the Ehrlich-Aberth iteration. The main computational issues are discussed, namely, the choice of the starting approximations needed to start the Ehrlich-Aberth iteration, the computation of the Newton correction, the halting criterion, and the treatment of eigenvalues at infinity. We arrive at an effective implementation which provides more accurate approximations to the eigenvalues with respect to the methods based on the QZ algorithm. The case of polynomials having special structures, like palindromic, Hamiltonian, symplectic, etc., where the eigenvalues have special symmetries in the complex plane, is considered. A general way to adapt the Ehrlich-Aberth iteration to structured matrix polynomial is introduced. Numerical experiments which confirm the effectiveness of this approach are reported.

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arXiv2010

On the solution of a quadratic vector equation arising in Markovian Binary Trees

We present some advances, both from a theoretical and from a computational point of view, on a quadratic vector equation (QVE) arising in Markovian Binary Trees. Concerning the theoretical advances, some irreducibility assumptions are relaxed, and the minimality of the solution of the QVE is expressed in terms of properties of the Jacobian of a suitable function. From the computational point of view, we elaborate on the Perron vector-based iteration proposed in [http://arxiv.org/abs/1006.0577]. In particular we provide a condition which ensures that the Perron iteration converges to the sought solution of the QVE. Moreover we introduce a variant of the algorithm which consists in applying the Newton method instead of a fixed-point iteration. This method has the same convergence behaviour as the Perron iteration, since its convergence speed increases for close-to-critical problems. Moreover, unlike the Perron iteration, the method has a quadratic convergence. Finally, we show that it is possible to alter the bilinear form defining the QVE in several ways without changing the solution. This modification has an impact on convergence speed of the algorithms.

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arXiv2006

A note on the location of polynomial roots

We review some known inclusion results for the roots of a polynomial, and adapt them to a conjecture recently presented by S. A. Vavasis. In particular, we provide strict upper and lower bounds to the distance of the closest root of a polynomial p(z) from a given root of p'(z).

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