A block-symmetric linearization of odd degree matrix polynomials with optimal eigenvalue condition number and backward error

DOI10.1007/s10092-018-0273-4zbMath1416.65091OpenAlexW2884510252WikidataQ114228551 ScholiaQ114228551MaRDI QIDQ1616095

R. Smith

Publication date: 31 October 2018

Published in: Calcolo (Search for Journal in Brave)

Full work available at URL: http://hdl.handle.net/10016/32176

zbMATH Keywords

eigenvaluelinearizationeigenvectormatrix polynomialstrong linearizationbackward error of an approximate eigenpairblock-symmetric linearizationconditioning of an eigenvalue

Mathematics Subject Classification ID

Numerical computation of eigenvalues and eigenvectors of matrices (65F15) Numerical computation of matrix norms, conditioning, scaling (65F35) Matrices over function rings in one or more variables (15A54)

Related Items (10)

Structured strong $\boldsymbol{\ell}$-ifications for structured matrix polynomials in the monomial basis ⋮ Unnamed Item ⋮ Strong linearizations of rational matrices with polynomial part expressed in an orthogonal basis ⋮ The infinite Lanczos method for symmetric nonlinear eigenvalue problems ⋮ A comparison of eigenvalue condition numbers for matrix polynomials ⋮ Factoring Block Fiedler Companion Matrices ⋮ Conditioning and backward errors of eigenvalues of homogeneous matrix polynomials under Möbius transformations ⋮ Backward error and conditioning of Fiedler companion linearizations ⋮ Explicit block-structures for block-symmetric Fiedler-like pencils ⋮ On why using \(\mathbb{DL}(P)\) for the symmetric polynomial eigenvalue problem might need to be reconsidered

Uses Software

• quadeig

Cites Work

• Unnamed Item
• Solving polynomial eigenvalue problems by means of the Ehrlich-Aberth method
• Skew-symmetric matrix polynomials and their Smith forms
• Spectral equivalence of matrix polynomials and the index sum theorem
• Hermitian matrix polynomials with real eigenvalues of definite type. I: Classification
• On the sign characteristic of Hermitian linearizations in \(\mathbb{DL}(P)\)
• Structured eigenvalue condition numbers and linearizations for matrix polynomials
• Jordan structures of alternating matrix polynomials
• A note on companion matrices
• Block Kronecker linearizations of matrix polynomials and their backward errors
• The eigenstructure of an arbitrary polynomial matrix: Computational aspects
• Backward error and condition of polynomial eigenvalue problems
• Polynomial eigenvalue solver based on tropically scaled Lagrange linearization
• Large vector spaces of block-symmetric strong linearizations of matrix polynomials
• On vector spaces of linearizations for matrix polynomials in orthogonal bases
• The Quadratic Eigenvalue Problem
• Backward Error Analysis of Polynomial Eigenvalue Problems Solved by Linearization
• Stability Analysis of the Two-level Orthogonal Arnoldi Procedure
• Parallel Krylov Solvers for the Polynomial Eigenvalue Problem in SLEPc
• A Backward Stable Algorithm for Quadratic Eigenvalue Problems
• Vector Spaces of Linearizations for Matrix Polynomials: A Bivariate Polynomial Approach
• Smith forms of palindromic matrix polynomials
• Recovery of Eigenvectors and Minimal Bases of Matrix Polynomials from Generalized Fiedler Linearizations
• NLEVP
• An algorithm for the complete solution of quadratic eigenvalue problems
• Backward Error of Polynomial Eigenproblems Solved by Linearization
• Structured Eigenvalue Condition Number and Backward Error of a Class of Polynomial Eigenvalue Problems
• The nonlinear eigenvalue problem
• Explicit block-structures for block-symmetric Fiedler-like pencils
• Tridiagonal-Diagonal Reduction of Symmetric Indefinite Pairs
• A new family of companion forms of polynomial matrices
• Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods
• Compact Rational Krylov Methods for Nonlinear Eigenvalue Problems
• The Ehrlich--Aberth Method for the Nonsymmetric Tridiagonal Eigenvalue Problem
• Matrix Polynomials
• Vector Spaces of Linearizations for Matrix Polynomials
• The Conditioning of Linearizations of Matrix Polynomials
• Structured Polynomial Eigenvalue Problems: Good Vibrations from Good Linearizations
• Symmetric Linearizations for Matrix Polynomials
• Memory-efficient Arnoldi algorithms for linearizations of matrix polynomials in Chebyshev basis
• An Algorithm for Generalized Matrix Eigenvalue Problems

This page was built for publication: A block-symmetric linearization of odd degree matrix polynomials with optimal eigenvalue condition number and backward error