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The Quadratic Eigenvalue Problem
TLDR
We survey the quadratic eigenvalue problem, treating its many applications, its mathematical properties, and a variety of numerical solution techniques. Expand
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NLEVP: A Collection of Nonlinear Eigenvalue Problems
TLDR
We present a collection of 52 nonlinear eigenvalue problems in the form of a MATLAB toolbox. Expand
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A Block Algorithm for Matrix 1-Norm Estimation, with an Application to 1-Norm Pseudospectra
TLDR
The matrix 1-norm estimation algorithm used in LAPACK and various other software libraries and packages has proved to be a valuable tool. Expand
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Perturbation theory for homogeneous polynomial eigenvalue problems
We consider polynomial eigenvalue problems P(A,alpha,beta)x=0 in which the matrix polynomial is homogeneous in the eigenvalue (alpha,beta)uC2. In this framework infinite eigenvalues are on the sameExpand
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Structured Pseudospectra for Polynomial Eigenvalue Problems, with Applications
TLDR
We extend the usual definitions in two respects, by treating the polynomial eigenvalue problem and by allowing structured perturbations of a type arising in control theory. Expand
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The Conditioning of Linearizations of Matrix Polynomials
TLDR
The standard way of solving the polynomial eigenvalue problem of degree $m$ in $n\times n$ matrices is to “linearize” to a pencil in vector space $\mathbb{DL}(P)$ of pencils recently identified and studied by Mackey and Mehrmann. Expand
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Symmetric Linearizations for Matrix Polynomials
TLDR
A standard way of treating the polynomial eigenvalue problem $P(\lambda)x = 0$ is to convert it into an equivalent matrix pencil—a process known as linearization. Expand
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Bounds for eigenvalues of matrix polynomials
Upper and lower bounds are derived for the absolute values of the eigenvalues of a matrix polynomial (or λ-matrix). The bounds are based on norms of the coefficient matrices and involve the inversesExpand
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Backward Error of Polynomial Eigenproblems Solved by Linearization
TLDR
The most widely used approach for solving the polynomial eigenvalue problem $P(\lambda)x = (\sum_{i=0}^m \l^i A_i) x = 0$ in $n\times n$ matrices $A_i$ is to linearize to produce a larger order pencil $L(\lambda X + Y$, whose eigensystem is then found by any method for generalized eigenproblems. Expand
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An algorithm for the complete solution of quadratic eigenvalue problems
TLDR
We develop a new algorithm for the computation of all the eigenvalues and optionally the right and left eigenvectors of dense quadratic matrix polynomials that is backward-stable for quadratics that are not too heavily damped. Expand
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