Jacobi–Davidson methods for polynomial two-parameter eigenvalue problems

Hochstenbach, Michiel E.; Muhič, Andrej; Plestenjak, Bor

doi:10.1016/j.cam.2015.04.019

Cited by 11 publications

(9 citation statements)

References 27 publications

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“…Subspace methods for one-parameter eigenvalue problems are based around generating a series of linear spaces that eventually approximate one of the system's eigenspaces (the linear space of eigenvectors corresponding to a given eigenvalue). The Jacobi-Davidson and Rayleigh-Ritz methods are well-known one-parameter subspace methods, which can be generalized to apply to two-parameter systems [26,32,[49][50][51], though this is not without difficulties [49]. These methods do not invoke the operator determinants, and show potential for generalization both to 𝑁-parameter and to polynomial systems.…”

Section: Alternative Solution Methodsmentioning

confidence: 99%

Multiparameter Spectral Analysis for Aeroelastic Instability Problems

Pons

Gutschmidt

2018

Journal of Applied Mechanics

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This paper presents a novel application of multiparameter spectral theory to the study of structural stability, with particular emphasis on aeroelastic flutter. Methods of multiparameter analysis allow the development of new solution algorithms for aeroelastic flutter problems; most significantly, a direct solver for polynomial problems of arbitrary order and size, something which has not before been achieved. Two major variants of this direct solver are presented, and their computational characteristics are compared. Both are effective for smaller problems arising in reduced-order modelling and preliminary design optimization. Extensions and improvements to this new conceptual framework and solution method are then discussed. * Corresponding author, adp53@cam.ac.uk, tel. +44 755 366 3296

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Section: Alternative Solution Methodsmentioning

confidence: 99%

Multiparameter Spectral Analysis for Aeroelastic Instability Problems

Pons

Gutschmidt

2018

Journal of Applied Mechanics

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“…We present some numerical examples obtained with Matlab. Several successful experiments with several types of multiparameter eigenvalue problems have been carried out and described in [16,12,14,26,13]. Therefore, we concentrate ourselves mostly on the new use for polynomial eigenvalue problems.…”

Section: Comparison With Other Approaches a Good Comparison Of Variou...mentioning

confidence: 99%

“…6.1(b). The eigenvalues are detected after 10,12,14,16,78,96,105,118,133,147,165, and 178 iterations. Elegantly, when we sort the eigenvalues with respect to distance to the target, these are eigenvalues number 1 through 12, in this order!…”

Section: Comparison With Other Approaches a Good Comparison Of Variou...mentioning

confidence: 99%

“…This work contains contributions in three directions. Although we have already successfully used some of these selection techniques in our work on linear multi-parameter eigenvalue problems ( [16,12], followed by nonlinear two-parameter eigenproblems [14,26] and linear three-parameter eigenvalue problems [13] very recently), we will present an improvement on these criteria for these problems. Secondly, to the best of our knowledge, the use of selection techniques to compute several eigenvalues in the form as described in this paper is new for one-parameter nonlinear eigenproblems: the QEP (1.4), PEP (1.3), and general NEP (1.2).…”

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Computing several eigenvalues of nonlinear eigenvalue problems by selection

Hochstenbach

Plestenjak

2020

Calcolo

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Computing more than one eigenvalue for (large sparse) one-parameter polynomial and general nonlinear eigenproblems, as well as for multiparameter linear and nonlinear eigenproblems, is a much harder task than for standard eigenvalue problems. We present simple but efficient selection methods based on divided differences to do this. In contrast to locking techniques, it is not necessary to keep converged eigenvectors in the search space, so that the entire search space may be devoted to new information. The techniques are applicable to many types of matrix eigenvalue problems; standard deflation is possible only for linear one-parameter problems. The methods are easy to understand and implement. Although divided differences are well-known in the context of nonlinear eigenproblems, the proposed selection techniques are new for one-parameter problems. For multiparameter problems, we improve on and generalize our previous work. We also show how to use divided differences in the framework of homogeneous coordinates, which may be appropriate for generalized eigenvalue problems with infinite eigenvalues.While the approaches are valuable alternatives for one-parameter nonlinear eigenproblems, they seem the only option for multiparameter problems.

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“…We compare our method with two existing approaches, Mathematica's NSolve [48] and PHCpack [46] in Section 7, and show that our approach is competitive for polynomials up to degree < ∼ 10. Let us mention that another advantage of writing the system of bivariate polynomials as a twoparameter eigenvalue problem is that then we can apply iterative subspace numerical methods such as the Jacobi-Davidson method and compute just a small part of zeros close to a given target (x 0 , y 0 ) [18]; we will not pursue this approach in this paper.…”

Section: Introductionmentioning

confidence: 99%

Roots of Bivariate Polynomial Systems via Determinantal Representations

Plestenjak

Hochstenbach

2016

SIAM J. Sci. Comput.

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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. The first determinantal representation is suitable for polynomials with scalar or matrix coefficients, and consists of matrices with asymptotic order n 2 /4, where n is the degree of the polynomial. The second representation is useful for scalar polynomials and has asymptotic order n 2 /6. The resulting method to compute the roots of a system of two bivariate polynomials is competitive with some existing methods for polynomials up to degree 10, as well as for polynomials with a small number of terms.

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Jacobi–Davidson methods for polynomial two-parameter eigenvalue problems

Cited by 11 publications

References 27 publications

Multiparameter Spectral Analysis for Aeroelastic Instability Problems

Multiparameter Spectral Analysis for Aeroelastic Instability Problems

Computing several eigenvalues of nonlinear eigenvalue problems by selection

Roots of Bivariate Polynomial Systems via Determinantal Representations

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