The Quadratic Eigenvalue Problem

By Françoise Tisseur, Karl Meerbergen. SIAM Review, 2001

1,469 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/5fe15561a9148bdee467a442e38ec5fa7a26ac16

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Citations in the corpus

A new family of companion forms of polynomial matrices (2004, neutral): “In view of the variety of forms arising from the proposed family of linearizations, one may choose particular ones that are better suited for specific applications (for instance when dealing with self-adjoint polynomial matrices [3], [4], [7], [5], [6] or the quadratic eigenvalue problem [10]).”
Active vibration suppression by pole-zero placement using measured receptances (2008, methodology): “From the point of view of the structural dynamics it is preferable to work with the second-order matrix pencil [23].”
On the quadratic two-parameter eigenvalue problem and its linearization☆ (2010, background): “, [11]), where we can linearize the problem to a generalized eigenvalue problem with matrices of double dimension, we can write (1) as a two-parameter eigenvalue problem”
Bounds for eigenvalues of matrix polynomials (2003, background): “With the growing interest in the numerical solution of polynomial eigenvalue problems [1,24] the derivation of eigenvalue bounds is a timely topic of investigation.”
Detecting a definite Hermitian pair and a hyperbolic or elliptic quadratic eigenvalue problem, and associated nearness problems (2002, background): “Now we turn our attention to the quadratic eigenvalue problem (QEP) [22] Q(λ)x = (λ2A + λB + C)x = 0, A, B,C ∈ Cn×n. (3.1) An important class of QEPs is defined as follows [15].”
Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods (2004, methodology): “A detailed survey has recently been given in [100].”
Self-Generating and Efficient Shift Parameters in ADI Methods for Large Lyapunov and Sylvester Equations (2014, neutral): “and x(t) = [q(t) , q̇(t) ] ; see [36].”
A numerical method for polynomial eigenvalue problems using contour integral (2010, background): “analysis with damping [9, 18] and stability problems in fluid dynamics [8], and the three-dimensional (3D) Schrödinger equation can result in a cubic eigenvalue problem [10].”
Symmetry preserving eigenvalue embedding in finite-element model updating of vibrating structures (2006, background): “The quadratic eigenvalue problem is much harder to solve numerically [17].”
Pseudospectra and stability radii for analytic matrix functions with application to time-delay systems (2006, background): “In general, one can formulate (22) as an abstract evolution equation over the Hilbert space X := C n (cid:2) L 2 ([ (cid:0) (cid:28) m ; 0] ; C n ) , equipped with the usual inner product ( y 0 ; y 1 ) ; ( z 0 ; z 1 ) X = h y 0 ; z 0 namely: where and the solutions of (27) and (22) are connected by ”
Simultaneous linear estimation of multiple view geometry and lens distortion (2001, methodology): “Gathering the three row vectors into three design matrices, we obtain the following quadratic eigenvalue problem (QEP) [17]:”