On Rank-Revealing Factorisations
SIAM Journal on Matrix Analysis and Applications
Matrix computations (3rd ed.)
Performance of the QZ Algorithm in the Presence of Infinite Eigenvalues
SIAM Journal on Matrix Analysis and Applications
The Quadratic Eigenvalue Problem
SIAM Review
Normwise Scaling of Second Order Polynomial Matrices
SIAM Journal on Matrix Analysis and Applications
Balancing Regular Matrix Pencils
SIAM Journal on Matrix Analysis and Applications
The Conditioning of Linearizations of Matrix Polynomials
SIAM Journal on Matrix Analysis and Applications
Structured Polynomial Eigenvalue Problems: Good Vibrations from Good Linearizations
SIAM Journal on Matrix Analysis and Applications
Vector Spaces of Linearizations for Matrix Polynomials
SIAM Journal on Matrix Analysis and Applications
Symmetric Linearizations for Matrix Polynomials
SIAM Journal on Matrix Analysis and Applications
Multishift Variants of the QZ Algorithm with Aggressive Early Deflation
SIAM Journal on Matrix Analysis and Applications
Backward Error of Polynomial Eigenproblems Solved by Linearization
SIAM Journal on Matrix Analysis and Applications
Optimal Scaling of Generalized and Polynomial Eigenvalue Problems
SIAM Journal on Matrix Analysis and Applications
NLEVP: A Collection of Nonlinear Eigenvalue Problems
ACM Transactions on Mathematical Software (TOMS)
NLEVP: A Collection of Nonlinear Eigenvalue Problems
ACM Transactions on Mathematical Software (TOMS)
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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.