Journal of Computational and Applied Mathematics
Critical delays and polynomial eigenvalue problems
Journal of Computational and Applied Mathematics
A method for finding zeros of polynomial equations using a contour integral based eigensolver
Proceedings of the 2009 conference on Symbolic numeric computation
Journal of Computational and Applied Mathematics
Fiedler Companion Linearizations and the Recovery of Minimal Indices
SIAM Journal on Matrix Analysis and Applications
Palindromic companion forms for matrix polynomials of odd degree
Journal of Computational and Applied Mathematics
A Krylov Method for the Delay Eigenvalue Problem
SIAM Journal on Scientific Computing
Solving Rational Eigenvalue Problems via Linearization
SIAM Journal on Matrix Analysis and Applications
Numerical studies on structure-preserving algorithms for surface acoustic wave simulations
Journal of Computational and Applied Mathematics
An algorithm for the complete solution of quadratic eigenvalue problems
ACM Transactions on Mathematical Software (TOMS)
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The classical approach to investigating polynomial eigenvalue problems is linearization, where the polynomial is converted into a larger matrix pencil with the same eigenvalues. For any polynomial there are infinitely many linearizations with widely varying properties, but in practice the companion forms are typically used. However, these companion forms are not always entirely satisfactory, and linearizations with special properties may sometimes be required. Given a matrix polynomial $P$, we develop a systematic approach to generating large classes of linearizations for $P$. We show how to simply construct two vector spaces of pencils that generalize the companion forms of $P$, and prove that almost all of these pencils are linearizations for $P$. Eigenvectors of these pencils are shown to be closely related to those of $P$. A distinguished subspace is then isolated, and the special properties of these pencils are investigated. These spaces of pencils provide a convenient arena in which to look for structured linearizations of structured polynomials, as well as to try to optimize the conditioning of linearizations.