Structured Polynomial Eigenvalue Problems: Good Vibrations from Good Linearizations

By D. Steven Mackey, Niloufer Mackey, Christian Mehl, Volker Mehrmann, Mims EPrint. SIAM Journal on Matrix Analysis and Applications, 2006

330 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/c824316bfeeb65d8df6f31e06379fbc327e4a981

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Citations in the corpus

LINEARIZATIONS OF SINGULAR MATRIX POLYNOMIALS AND THE RECOVERY OF MINIMAL INDICES (2009, background): “This has motivated the recent development of new classes of linearizations in [25] that have been further analyzed in [19, 20, 21, 26].”
Jordan structures of alternating matrix polynomials (2010, background): “The (±)-pairing of nonzero eigenvalues and the pairing of their corresponding elementary divisors was already known for regular complex T -alternating polynomials, see [20].”
A framework for analyzing nonlinear eigenproblems and parametrized linear systems (2011, background): “The first test is nlevp(‘railtrack’), which is a badly scaled quadratic of dimension 1005 arising from a model of the vibration of rail tracks under the excitation of high speed trains [21,28].”
NLEVP: A Collection of Nonlinear Eigenvalue Problems (2013, background): “De.nitions of identi.ers for odd-even and palindromic-like square matrix polynomials, together with the special symmetry properties of their spectra [Mackey et al. 2006] are given in Table IV.”
Möbius transformations of matrix polynomials (2015, methodology): “When expressed in terms of the associated Cayley transformations, this yields the identity C+1(C−1(P )) = C−1(C+1(P )) = 2kP , which was derived in [44].”
Spectral equivalence of matrix polynomials and the index sum theorem (2014, background): “Therefore the development of structure-preserving algorithms for computing eigenvalues of structured matrix polynomials has become a hot area of research in numerical linear algebra in recent years and, as a consequence, considerable effort has been invested in devising linearizations of matrix poly”
Vibration of fast trains, palindromic eigenvalue problems and structure-preserving doubling algorithms (2008, background): “MSC: 43.40.+s; 02.60.x; 46.70.p; 46.40.f; 43.40.+s Keywords: Palindromic eigenvalue problem; Nonlinear matrix equation; Structure-preserving; Doubling algorithm”
Smith Forms of Palindromic Matrix Polynomials (2011, unknown): “The analysis in [21] revealed a disparity between T -palindromic and ∗-palindromic structure which arises even in the 1 × 1 case, as the following example illustrates.”
Linear Algebra and its Applications (2010, unknown): “We do allow u and v to have different dimensions which can occur with certain ﬁnite-element methods, see e.g. [40], and assume u ∈ R m , v ∈ R n with n (cid:2) m and M u ∈ R m × m , M v ∈ R n × n , K ∈ R m × n , S ∈ R n × n .”
A robust numerical method for the γ-iteration in H∞ control (2007, methodology): “Here, we embed the Hamiltonian eigenvalue problems into so called even generalized eigenvalue problems, see [15, 36], which generalize Hamiltonian matrices and skew-Hamiltonian/Hamiltonian pencils, see [4].”
Fiedler companion linearizations for rectangular matrix polynomials (2012, background): “At first, only linearizations for regular matrix polynomials were considered [1, 2, 23, 27, 28], while more recently square singular polynomials have also received attention [10, 11, 12, 34].”
The solution of the equation XA + AXT = 0 and its application to the theory of orbits (2011, background): “These eigenvalue problems arise in a number of applications and are receiving a considerable attention in the last years (see for instance [13, 19, 21, 22] and the references therein).”