Journal article  Open Access

Fast and backward stable computation of the eigenvalues and eigenvectors of matrix polynomials

Aurentz J., Mach T., Robol L., Vandebril R., Watkins D. S.

Computational Mathematics  65L07  FOS: Mathematics  Product eigenvalue problem  eigenvector  Mathematics - Numerical Analysis  Algebra and Number Theory  Eigenvalues  65F15  Matrix polynomial  Eigenvectors  Core chasing algorithm  Applied Mathematics  Numerical Analysis (math.NA)  eigenvalue 

In the last decade matrix polynomials have been investigated with the primary focus on adequate linearizations and good scaling techniques for computing their eigenvalues and eigenvectors. In this article we propose a new method for computing a factored Schur form of the associated companion pencil. The algorithm has a quadratic cost in the degree of the polynomial and a cubic one in the size of the coefficient matrices. Also the eigenvectors can be computed at the same cost. The algorithm is a variant of Francis's implicitly shifted QR algorithm applied on the companion pencil. A preprocessing unitary equivalence is executed on the matrix polynomial to simultaneously bring the leading matrix coefficient and the constant matrix term to triangular form before forming the companion pencil. The resulting structure allows us to stably factor each matrix of the pencil as a product of k matrices of unitary-plus-rank-one form, admitting cheap and numerically reliable storage. The problem is then solved as a product core chasing eigenvalue problem. A backward error analysis is included, implying normwise backward stability after a proper scaling. Computing the eigenvectors via reordering the Schur form is discussed as well. Numerical experiments illustrate stability and efficiency of the proposed methods.

Source: Mathematics of computation (Online) 88 (2019): 313–347. doi:10.1090/mcom/3338

Publisher: American Mathematical Society, [Providence, R.I.] , Stati Uniti d'America

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BibTeX entry
	title = {Fast and backward stable computation of the eigenvalues and eigenvectors of matrix polynomials},
	author = {Aurentz J. and Mach T. and Robol L. and Vandebril R. and Watkins D.  S.},
	publisher = {American Mathematical Society, [Providence, R.I.] , Stati Uniti d'America},
	doi = {10.1090/mcom/3338 and 10.48550/arxiv.1611.10142},
	journal = {Mathematics of computation (Online)},
	volume = {88},
	pages = {313–347},
	year = {2019}

Numerical Computation with Functions Instead of Numbers