Continuation of eigenvalues and invariant pairs for parameterized nonlinear eigenvalue problems

Beyn, Wolf–Jürgen; Effenberger, Cedric; Kreßner, Daniel

doi:10.1007/s00211-011-0392-1

Cited by 29 publications

(19 citation statements)

References 26 publications

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“…It has a wide range of applications, e.g., polynomial root finding via homotopy [3], nonlinear eigenvalue problems [4], biobjective optimization [18,33], and robot path planning [24]. The most simple and effective continuation method is the predictor corrector algorithm.…”

Section: Introduction Continuation Methodsmentioning

confidence: 99%

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Certified Parallelotope Continuation for One-Manifolds

Martin¹,

Goldsztejn²,

Granvilliers³

et al. 2013

SIAM J. Numer. Anal.

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Starting from an initial solution, continuation methods efficiently produce a sequence of points on a manifold typically defined as the solution set of an underconstrained system of equations. They have a wide range of applications ranging from curve plotting to polynomial root-finding by homotopy. However, classical methods cannot guarantee that the returned points all belong to the same connected component of the manifold, i.e., they may jump from one component to another. Trying to overcome this issue has given birth to several sophisticated heuristics on the one hand and to guaranteed methods based on rigorous computations on the other hand. In this paper we introduce a new rigorous predictor corrector continuation method based on interval computations. Its novelty lies in the fact that it uses parallelotopes as defined in A. Goldsztejn and L. Granvilliers, A new framework for sharp and efficient resolution of NCSP with manifolds of solutions, Constraints, 15 (2010), pp. 190-212, to enclose consecutive portions of the followed manifold. Though computationally more expensive than regular interval computations, the fact that their orientation can be automatically adapted to the local topology of the followed curve makes parallelotopes a very suitable tool for a certified continuation method, as shown by reported experimental results. Introduction.Continuation methods [1] allow exploring step by step a solution manifold, usually defined as the solution set of an underconstrained system of equations F (x) = 0 with F : R n → R m and m < n. This paper focuses on systems where n − m = 1, whose solution manifolds are curves, and proposes a robust singleparameter continuation method. It has a wide range of applications, e.g., polynomial root finding via homotopy [3], nonlinear eigenvalue problems [4], biobjective optimization [18,33], and robot path planning [24]. The most simple and effective continuation method is the predictor corrector algorithm. It includes the simple embedding method and several of its improvements such as the parameter switching [28,29] and the pseudo-arclength methods that have the definite advantage over the simple embedding method that they naturally track folds [6]. They are all, however, subject to jumping between disconnected components without any prompt. For a given system, there exist some theoretical upper bounds on the step size that enforce the connectivity of the continuation, e.g., Theorem 5.2.1 in [1], but they are impractical. While this may be acceptable in some contexts since solutions are eventually computed, such jumps contradict the essence of continuation. Furthermore, connectivity is mandatory in some applications, e.g., robot command synthesis, where such jumps would result in a nonfeasible command path, or in homotopy methods, where such jumps would prevent computing all the solutions of the original system.

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Section: Introduction Continuation Methodsmentioning

confidence: 99%

“…(iiib) If K > 1 and y 0 ⊆ x K , then γ is an injective loop. 4 Proof. Let μ k : v k → u k be the function given by Theorem 2.1 applied to the parallelotope x k , and defineγ k :…”

Section: Correctnessmentioning

confidence: 97%

Certified Parallelotope Continuation for One-Manifolds

Martin¹,

Goldsztejn²,

Granvilliers³

et al. 2013

SIAM J. Numer. Anal.

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“…The NLEP (1.1) has been studied extensively in the literature and there exist specialized methods for different families of A(λ); see, e.g., [28,35]. Popular approaches can be roughly classified as Newton-type methods [29,6,22], methods based on contour integration [8,7,10], and methods based on approximations of A(λ) [28,11,35,21,36]. The method proposed in this paper belongs to the last class.…”

Section: 1)mentioning

confidence: 97%

NLEIGS: A Class of Fully Rational Krylov Methods for Nonlinear Eigenvalue Problems

Güttel

Beeumen²,

Meerbergen³

et al. 2014

SIAM J. Sci. Comput.

215

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A new rational Krylov method for the efficient solution of nonlinear eigenvalue problems, A(λ)x = 0, is proposed. This iterative method, called fully rational Krylov method for nonlinear eigenvalue problems (abbreviated as NLEIGS), is based on linear rational interpolation and generalizes the Newton rational Krylov method proposed in [R. Van Beeumen, K. Meerbergen, and W. Michiels, SIAM J. Sci. Comput., 35 (2013), pp. A327-A350]. NLEIGS utilizes a dynamically constructed rational interpolant of the nonlinear function A(λ) and a new companion-type linearization for obtaining a generalized eigenvalue problem with special structure. This structure is particularly suited for the rational Krylov method. A new approach for the computation of rational divided differences using matrix functions is presented. It is shown that NLEIGS has a computational cost comparable to the Newton rational Krylov method but converges more reliably, in particular, if the nonlinear function A(λ) has singularities nearby the target set. Moreover, NLEIGS implements an automatic scaling procedure which makes it work robustly independently of the location and shape of the target set, and it also features low-rank approximation techniques for increased computational efficiency. Small-and large-scale numerical examples are included. From the numerical experiments we can recommend two variants of the algorithm for solving the nonlinear eigenvalue problem.

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“…This is a multi-valued function, that is, for a z ∈ C there are infinitely many solutions to (2). To identify these values a branch number is assigned, and we refer to W k (z) as the k-th branch of the Lambert W function of z.…”

Section: The Lambert W Functionmentioning

confidence: 99%

Some special cases in the stability analysis of multi-dimensional time-delay systems using the matrix Lambert W function

Cepeda-Gomez

Michiels

2015

Automatica

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This paper revisits a recently developed methodology based on the matrix Lambert W function for the stability analysis of linear time invariant, time delay systems. By studying a particular, yet common, second order system, we show that in general there is no one to one correspondence between the branches of the matrix Lambert W function and the characteristic roots of the system. Furthermore, it is shown that under mild conditions only two branches suffice to find the complete spectrum of the system, and that the principal branch can be used to find several roots, and not the dominant root only, as stated in previous works. The results are first presented analytically, and then verified by numerical experiments.

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Continuation of eigenvalues and invariant pairs for parameterized nonlinear eigenvalue problems

Cited by 29 publications

References 26 publications

Certified Parallelotope Continuation for One-Manifolds

Certified Parallelotope Continuation for One-Manifolds

NLEIGS: A Class of Fully Rational Krylov Methods for Nonlinear Eigenvalue Problems

Some special cases in the stability analysis of multi-dimensional time-delay systems using the matrix Lambert W function

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