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An Algorithm for the Complete Solution of Quadratic Eigenvalue Problems
, 2012
"... We develop a new algorithm for the computation of all the eigenvalues and optionally the right and left eigenvectors of dense quadratic matrix polynomials. It incorporates scaling of the problem parameters prior to the computation of eigenvalues, a choice of linearization with favorable conditioning ..."
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Cited by 6 (2 self)
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We develop a new algorithm for the computation of all the eigenvalues and optionally the right and left eigenvectors of dense quadratic matrix polynomials. It incorporates scaling of the problem parameters prior to the computation of eigenvalues, a choice of linearization with favorable conditioning and backward stability properties, and a preprocessing step that reveals and deflates the zero and infinite eigenvalues contributed by singular leading and trailing matrix coefficients. The algorithm is backward stable for quadratics that are not too heavily damped. Numerical experiments show that our MATLAB implementation of the algorithm, quadeig, outperforms the MATLAB function polyeig in terms of both stability and efficiency.
Locating the eigenvalues of matrix polynomials
 SIAM J. Matrix Anal. Appl
"... Abstract. 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.393395], some results of D.A. Bini [Numer. Algorithms 13:179200, 1996] based on th ..."
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Cited by 5 (1 self)
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Abstract. 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.393395], some results of D.A. Bini [Numer. Algorithms 13:179200, 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.291303] 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 EhrlichAberth method. Numerical experiments that show the computational advantage of these results are presented. AMS classification: 15A22,15A80,15A18,47J10.
BOUNDS ON THE GENERALIZED AND THE JOINT SPECTRAL RADIUS OF HADAMARD PRODUCTS OF BOUNDED SETS OF POSITIVE OPERATORS ON SEQUENCE SPACES
"... Hadamard products of finite and infinite nonnegative matrices that define operators on sequence spaces and the spectral radius of their ordinary matrix product. We extend these results to the generalized and the joint spectral radius of bounded sets of such operators. Moreover, we prove new inequal ..."
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Hadamard products of finite and infinite nonnegative matrices that define operators on sequence spaces and the spectral radius of their ordinary matrix product. We extend these results to the generalized and the joint spectral radius of bounded sets of such operators. Moreover, we prove new inequalities even in the case of the usual spectral radius of nonnegative matrices. We also obtain related results in max algebra.
TROPICAL ROOTS AS APPROXIMATIONS TO EIGENVALUES OF MATRIX POLYNOMIALS∗
"... Abstract. The tropical roots of t×p(x) = max0≤j≤ ` ‖Aj‖xj are points at which the maximum is attained at least twice. These roots, which can be computed in only O(`) operations, can be good approximations to the moduli of the eigenvalues of the matrix polynomial P (λ) = j=0 λ jAj, in particular whe ..."
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Abstract. The tropical roots of t×p(x) = max0≤j≤ ` ‖Aj‖xj are points at which the maximum is attained at least twice. These roots, which can be computed in only O(`) operations, can be good approximations to the moduli of the eigenvalues of the matrix polynomial P (λ) = j=0 λ jAj, in particular when the norms of the matrices Aj vary widely. Our aim is to investigate this observation and its applications. We start by providing annuli defined in terms of the tropical roots of t×p(x) that contain the eigenvalues of P (λ). Our localization results yield conditions under which tropical roots offer order of magnitude approximations to the moduli of the eigenvalues of P (λ). Our tropical localization of eigenvalues are less tight than eigenvalue localization results derived from a generalized matrix version of Pellet’s theorem but they are easier to interpret. Tropical roots are already used to determine the starting points for matrix polynomial eigensolvers based on scalar polynomial root solvers such as the EhrlichAberth method and our results further justify this choice. Our results provide the basis for analyzing the effect of Gaubert and Sharify’s tropical scalings for P (λ) on (a) the conditioning of linearizations of tropically scaled P (λ) and (b) the backward stability of eigensolvers based on linearizations of tropically scaled P (λ). We anticipate that the tropical roots of t×p(x), on which the tropical scalings are based, will help designing polynomial eigensolvers with better numerical properties than standard algorithms for polynomial eigenvalue problems such as that implemented in the MATLAB function polyeig.