Tropical Roots as Approximations to Eigenvalues of Matrix Polynomials

DOI10.1137/14096637XzbMath1315.65038OpenAlexW1975795800MaRDI QIDQ5251754

Françoise Tisseur, Meisam Sharify, Vanni Noferini

Publication date: 21 May 2015

Published in: SIAM Journal on Matrix Analysis and Applications (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1137/14096637x

zbMATH Keywords

algorithmRouché's theoremconditioningscalingmatrix polynomialtropical algebrapolynomial eigenvalue problemtropical rootsEhrlich-Aberth methodlocalization of eigenvaluesNewton's polygonPellet's theoremtropical scaling

Mathematics Subject Classification ID

Numerical computation of eigenvalues and eigenvectors of matrices (65F15) Numerical computation of matrix norms, conditioning, scaling (65F35) Conditioning of matrices (15A12) Matrix pencils (15A22) Max-plus and related algebras (15A80)

Related Items (16)

Non-archimedean valuations of eigenvalues of matrix polynomials ⋮ A Max-Plus Approach to Incomplete Cholesky Factorization Preconditioners ⋮ Annulus containing all the eigenvalues of a matrix polynomial ⋮ Locating eigenvalues of quadratic matrix polynomials ⋮ Incomplete LU Preconditioner Based on Max-Plus Approximation of LU Factorization ⋮ Stability of matrix polynomials in one and several variables ⋮ The Deformed Graph Laplacian and Its Applications to Network Centrality Analysis ⋮ Polynomial eigenvalue solver based on tropically scaled Lagrange linearization ⋮ Backward error measures for roots of polynomials ⋮ Location of right eigenvalues of quaternionic matrix polynomials ⋮ Min-max elementwise backward error for roots of polynomials and a corresponding backward stable root finder ⋮ Log-majorization of the moduli of the eigenvalues of a matrix polynomial by tropical roots ⋮ On Descartes' rule of signs for matrix polynomials ⋮ On the conditioning for heavily damped quadratic eigenvalue problem solved by linearizations ⋮ Inertia laws and localization of real eigenvalues for generalized indefinite eigenvalue problems ⋮ Tropical Roots as Approximations to Eigenvalues of Matrix Polynomials

Uses Software

• Matlab
• NLEVP

Cites Work

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• Solving polynomial eigenvalue problems by means of the Ehrlich-Aberth method
• A numerical method for polynomial eigenvalue problems using contour integral
• Log-majorization of the moduli of the eigenvalues of a matrix polynomial by tropical roots
• Bounds for eigenvalues of matrix polynomials
• Backward error and condition of polynomial eigenvalue problems
• An efficient algorithm for determining the convex hull of a finite planar set
• Generalization and variations of Pellet's theorem for matrix polynomials
• Recherches sur la méthode de Graeffe et les zéros des polynômes et des séries de Laurent
• NLEVP
• Tropical Scaling of Polynomial Matrices
• Backward Error of Polynomial Eigenproblems Solved by Linearization
• Optimal Scaling of Generalized and Polynomial Eigenvalue Problems
• Tropical Roots as Approximations to Eigenvalues of Matrix Polynomials
• Locating the Eigenvalues of Matrix Polynomials
• The Conditioning of Linearizations of Matrix Polynomials
• AN OPERATOR GENERALIZATION OF THE LOGARITHMIC RESIDUE THEOREM AND THE THEOREM OF ROUCHÉ

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