Jacobi–Davidson methods for cubic eigenvalue problems

Hwang, Tsung Min; Lin, Wen Wei; Liu, Jinn Liang; Wang, Weichung

doi:10.1002/nla.423

Cited by 40 publications

(27 citation statements)

References 18 publications

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“…The preference of the JDM is based on the successful experience on solving the eigenvalue systems arising in various QD models detailed in [8,9,18].…”

Section: The Eigenvalue Problem Solvermentioning

confidence: 99%

“…While the framework of the JDM used in this paper is similar to the ones in [8,9,18], we can further improve the algorithm. The idea is to solve the correct…”

Section: The Eigenvalue Problem Solvermentioning

confidence: 99%

“…In short, standard methods like [1,2,6] can not be recommended due to the inefficiency and instability. See [8] for detail.…”

Section: Introductionmentioning

confidence: 99%

“…• Modifying the Jacobi-Davidson large-scale eigenvalue solver [8] by suggesting an adaptive scheme for approximating the span vectors. The adaptive scheme carefully incorporates several linear system solvers, preconditioning strategies, and dynamic stopping criteria.…”

Section: Introductionmentioning

confidence: 99%

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Numerical schemes for three-dimensional irregular shape quantum dots over curvilinear coordinate systems

Hwang

Wang

2007

Journal of Computational Physics

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In this article, we present efficient and stable numerical schemes to simulate threedimensional quantum dot with irregular shape, so that we can compute all the bound state energies and associated wave functions. A curvilinear coordinate system that fits the target quantum dot shape is first determined. Three finite difference discretizations of the Schrödinger equation are then developed on the original and the skewed curvilinear coordinate system. The resulting large-scale generalized eigenvalue systems are solved by a modified Jacobi-Davidson method. Intensive numerical experiments show that the scheme using both grid points on the original and skewed curvilinear coordinate system can converge to the eigenpairs quickly and stably with second order accuracy.

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“…The preference of the JDM is based on the successful experience on solving the eigenvalue systems arising in various QD models detailed in [8,9,18].…”

Section: The Eigenvalue Problem Solvermentioning

confidence: 99%

“…While the framework of the JDM used in this paper is similar to the ones in [8,9,18], we can further improve the algorithm. The idea is to solve the correct…”

Section: The Eigenvalue Problem Solvermentioning

confidence: 99%

“…In short, standard methods like [1,2,6] can not be recommended due to the inefficiency and instability. See [8] for detail.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Numerical schemes for three-dimensional irregular shape quantum dots over curvilinear coordinate systems

Hwang

Wang

2007

Journal of Computational Physics

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“…For example, quadratic eigenvalue problems arise in oscillation 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]. Similarly, the study of higher-order systems of differential equations leads to a matrix polynomial of degree greater than one [5].…”

Section: Introductionmentioning

confidence: 99%

A numerical method for polynomial eigenvalue problems using contour integral

Asakura¹,

Sakurai

Tadano

et al. 2010

Japan J. Indust. Appl. Math.

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We propose a numerical method using contour integral to solve polynomial eigenvalue problems (PEPs). The method finds eigenvalues contained in a certain domain which is defined by a surrounding integral path. By evaluating the contour integral numerically along the path, the method reduces the original PEP into a small generalized eigenvalue problem, which has the identical eigenvalues in the domain. Error analysis indicates that the error of the eigenvalues is not uniform: inner eigenvalues are less erroneous. Four numerical examples are presented, which confirm the theoretical predictions.

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On the solution of generalized non‐linear complex‐symmetric eigenvalue problems

Dumont

2007

Numerical Meth Engineering

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SUMMARYThis paper brings an attempt toward the systematic solution of the generalized non-linear, complexsymmetric eigenproblem, which are associated with the dynamic governing equations of a structure submitted to viscous damping, as laid out in the frame of an advanced mode superposition technique. The problem can be restated as (are complex-symmetric matrices given as power series of the complex eigenfrequencies , such that, if ( , /) is a solution eigenpair,The traditional Rayleigh quotient iteration and the more recent JacobiDavidson method are outlined for complex-symmetric linear problems and shown to be mathematically equivalent, both with asymptotically cubic convergence. The Jacobi-Davidson method is more robust and adequate for the solution of a set of eigenpairs. The non-linear eigenproblem subject of this paper can be dealt with in the exact frame of the linear analysis, thus also presenting cubic convergence. Two examples help us to visualize some of the basic concepts developed. Three more examples illustrate the applicability of the proposed algorithm to solve non-linear problems, in the general case of underdamping, but also for overdamping combined with multiple and close eigenvalues.

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Jacobi–Davidson methods for cubic eigenvalue problems

Cited by 40 publications

References 18 publications

Numerical schemes for three-dimensional irregular shape quantum dots over curvilinear coordinate systems

Numerical schemes for three-dimensional irregular shape quantum dots over curvilinear coordinate systems

A numerical method for polynomial eigenvalue problems using contour integral

On the solution of generalized non‐linear complex‐symmetric eigenvalue problems

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