An algorithm for the complete solution of quadratic eigenvalue problems. 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.

References in zbMATH (referenced in 34 articles , 1 standard article )

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  1. Dong, Bo: The homotopy method for the complete solution of quadratic two-parameter eigenvalue problems (2022)
  2. Sadkane, Miloud: Inexact inverse subspace iteration with preconditioning applied to quadratic matrix polynomials (2022)
  3. Tisseur, Françoise; Van Barel, Marc: Min-max elementwise backward error for roots of polynomials and a corresponding backward stable root finder (2021)
  4. Anguas, Luis Miguel; Bueno, Maria Isabel; Dopico, Froilán M.: Conditioning and backward errors of eigenvalues of homogeneous matrix polynomials under Möbius transformations (2020)
  5. Beltrán, Carlos; Kozhasov, Khazhgali: The real polynomial eigenvalue problem is well conditioned on the average (2020)
  6. De Terán, Fernando: Backward error and conditioning of Fiedler companion linearizations (2020)
  7. Hochstenbach, Michiel E.; Plestenjak, Bor: Computing several eigenvalues of nonlinear eigenvalue problems by selection (2020)
  8. Kapitula, Todd; Parker, Ross; Sandstede, Björn: A reformulated Krein matrix for star-even polynomial operators with applications (2020)
  9. Pandur, Marija Miloloža: Detecting a hyperbolic quadratic eigenvalue problem by using a subspace algorithm (2020)
  10. Wong, Clint Y. H.; Trinh, Philippe H.; Chapman, S. Jonathan: Shear-induced instabilities of flows through submerged vegetation (2020)
  11. Anguas, Luis M.; Dopico, FroiláN M.; Hollister, Richard; Mackey, D. Steven: Van Dooren’s index sum theorem and rational matrices with prescribed structural data (2019)
  12. Anguas, Luis Miguel; Bueno, María Isabel; Dopico, Froilán M.: A comparison of eigenvalue condition numbers for matrix polynomials (2019)
  13. Armentano, Diego; Beltrán, Carlos: The polynomial eigenvalue problem is well conditioned for random inputs (2019)
  14. Lê, Công-Trình; Du, Thi-Hoa-binh; Nguyen, Tran-Duc: On the location of eigenvalues of matrix polynomials (2019)
  15. Malyshev, Alexander; Sadkane, Miloud: On the distance to instability of quadratic matrix polynomials (2019)
  16. Bueno, M. I.; Dopico, F. M.; Furtado, S.; Medina, L.: A block-symmetric linearization of odd degree matrix polynomials with optimal eigenvalue condition number and backward error (2018)
  17. Dopico, Froilán M.; Lawrence, Piers W.; Pérez, Javier; Van Dooren, Paul: Block Kronecker linearizations of matrix polynomials and their backward errors (2018)
  18. Hook, James; Scott, Jennifer; Tisseur, Françoise; Hogg, Jonathan: A Max-plus approach to incomplete Cholesky factorization preconditioners (2018)
  19. Leclercq, T.; Peake, N.; de Langre, E.: Does flutter prevent drag reduction by reconfiguration? (2018)
  20. Meerbergen, Karl; Pérez, Javier: Mixed forward-backward stability of the two-level orthogonal Arnoldi method for quadratic problems (2018)

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