Numerical methods for rectangular multiparameter eigenvalue problems, with applications to finding optimal ARMA and LTI models [PDF]
Numerical Linear Algebra with Applications, Volume 31, Issue 2, March 2024., 2022 Standard multiparameter eigenvalue problems (MEPs) are systems of k≥2$$ k\ge 2 $$ linear k$$ k $$ ‐parameter square matrix pencils. Recently, a new form of multiparameter eigenvalue problems has emerged: a rectangular MEP (RMEP) with only one ...
M. Hochstenbach +2 more
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Fiber product homotopy method for multiparameter eigenvalue problems [PDF]
Numerische Mathematik, 2018 We develop a new homotopy method for solving multiparameter eigenvalue problems (MEPs) called the fiber product homotopy method. For a k-parameter eigenvalue problem with matrices of sizes n1,…,nk=O(n)\documentclass[12pt]{minimal} \usepackage{amsmath ...
J. Rodriguez +3 more
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Comparison cones for multiparameter eigenvalue problems
Journal of Mathematical Analysis and Applications, 1980 AbstractWe consider the multiparameter eigenvalue problem (Tr + ∑s = 1k λs Vrs) xr = 0, xr ≠ 0, 1 ⩽ r ⩽ k, where Tr and Vrs are self-adjoint linear operators on Hilbert spaces Hr, the Vrs being bounded. The problem may be posed in either ⊕r = 1k Hr or ⊕r = 1k Hr and we develop variational approaches for both settings.
P. Binding, P. Browne
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A Newton method for solving locally definite multiparameter eigenvalue problems by multiindex
SIAM Journal on Matrix Analysis and ApplicationsWe present a new approach to compute eigenvalues and eigenvectors of locally definite multiparameter eigenvalue problems by its signed multiindex. The method has the interpretation of a semismooth Newton method applied to certain functions that have a ...
Eisenmann, Henrik
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Discretization of multiparameter eigenvalue problems
Numerische Mathematik, 1982 Although multiparameter eigenvalue problems, as for example Mathieu's differential equation, have been known for a long time, so far no work has been done on the numerical treatment of these problems. So in this paper we extend the spectral theory for one parameter (cf.
R. E. Müller
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Abstract oscillation theorems for multiparameter eigenvalue problems
Journal of Differential Equations, 1983 AbstractWe prove abstract analogous of Klein's oscillation theorem by demonstrating the existence (and in some cases uniqueness) of eigenpairs with a given index for the multiparameter problem Tmxm = ∑n = 1k λnVmnxm, 0 ≠ xm ϵ Hm, m = 1 … k. (∗) Here Tm and Vmn are self-adjoint operators on Hilbert spaces Hm. The index is based on the number of negative
P. Binding
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Two Double Recursive Block Macaulay Matrix Algorithms to Solve Multiparameter Eigenvalue Problems
IEEE Control Systems Letters, 2022 We present two double recursive block Macaulay matrix algorithms to solve multiparameter eigenvalue problems (MEPs). In earlier work, we have developed a non-recursive approach that finds the solutions of an MEP via a multidimensional realization problem
C. Vermeersch, B. De Moor
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Subspace method for multiparameter‐eigenvalue problems based on tensor‐train representations [PDF]
Numerical Linear Algebra with Applications, 2020 In this article, we solve m ‐parameter eigenvalue problems ( m EPs), with m any natural number by representing the problem using tensor‐trains (TTs) and designing a method based on this format.
Koen Ruymbeek +2 more
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Multiparameter eigenvalue problems and expansion theorems
, 1988 H. Volkmer
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Roots of bivariate polynomial systems via determinantal representations [PDF]
, 2015 We give two determinantal representations for a bivariate polynomial. They may be used to compute the zeros of a system of two of these polynomials via the eigenvalues of a two-parameter eigenvalue problem.
Hochstenbach, Michiel E. +1 more
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