A homotopy algorithm for a symmetric generalized eigenproblem

Li, Kuiyuan; Li, Tien Yien

doi:10.1007/bf02142745

Cited by 9 publications

(5 citation statements)

References 9 publications

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“…However, such projection algorithm is not a unique method for the solution of eigenvalue problem. There exists other procedures, for instance, in [39], the authors considered the solution of a symmetric generalized eigenvalue problem by using the homotopy algorithm. For a description of other methods, interested readers may refer to [39,45,46, and references therein].…”

Section: Resultsmentioning

confidence: 99%

Krylov subspace method for fuzzy eigenvalue problem

Kanaksabee

Dookhitram

Bhuruth

2014

Journal of Intelligent &Amp; Fuzzy Systems

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The eigenvalue problem arises in many application areas and in the fuzzy setting, focus has always been geared towards the finding of solution for the whole set of eigenvalues and corresponding eigenvectors. This paper introduces the computation of a few eigenpairs of a matrix with triangular fuzzy numbers as elements, where the modal matrix is assumed to be sparse and real symmetric. A two-step procedure is developed for the solution of this type of fuzzy eigenvalue problem. The first step solves the 1-cut of the problem, where the well-known Krylov subspace method, implicitly restarted Lanczos algorithm, is employed to approximate part of the spectrum with respect to either smallest or largest eigenvalues. The second step assigns unknown symmetric linear spreads to the approximate eigenpairs. Numerical experiments are provided to illustrate the efficiency of the proposed scheme for determining fuzzy symmetric eigenpairs of a fuzzy matrix.

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Section: Resultsmentioning

confidence: 99%

Krylov subspace method for fuzzy eigenvalue problem

Kanaksabee

Dookhitram

Bhuruth

2014

Journal of Intelligent &Amp; Fuzzy Systems

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“…Since samples are taken only at some discrete time stages, the existing drawback may be overcome by use of smooth mappings that link the time stages. Homotopy continuation [6,7] is one such mapping method. Hence, for 0 ≤ x ≤ h, we construct a homotopy continuation as follows:…”

Section: Mathematical Methodsmentioning

confidence: 99%

“…For the second method, the boundary conditions suffice to obtain parameter values. Furthermore, we utilize the well-known homotopy continuation method [6,7] to connect the time stages so as to provide concentration estimates at any time and any depth. Homotopy continuation produces smooth curves that bridge the gap between two points with known data.…”

Section: Introductionmentioning

confidence: 99%

Mathematical Methods for Dissolved Phosphorus Concentrations

Li¹,

Uvah²

2019

JAAM

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Phosphorus is a nutrient contained in fertilizer that can run off from lawns and crops. Predicting the phosphorus concentrations is an important component in determining how phosphorus is cycled in the surrounding ecosystem. In this paper, we present the development of two mathematical methods. The first is a steady-state diffusion ordinary differential equation with given initial data. For this method, we estimate the unknown parameter values using least squares approximations for the data set without the boundary values. The second method is identical to the first except that the boundary values are imposed. We then distinguish our methods from existing approaches by deploying homotopy continuation to connect all time stages. With this approach, phosphorus concentrations can be estimated at all times and any depth. Using real-life data, we give an example to show that the methods are not only easy to use, but also provide estimates between any time stages and at any depth.

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“…Recently, the homotopy (continuation) method has been proposed to develop series solutions for various problems in physics and mechanics.' [18][19][20][21] ! A nice feature of the method is that it is applicable to systems with no small parameters.…”

Section: Introductionmentioning

confidence: 99%

“…Due to the simplicity of the method, it has been widely used to give all zeros of nonlinear algebraic equations, for example eigenvalues.! [18][19][20][21] ' It is thus tempting to extend the method to solve the electrostatic boundary-value problems of nonlinear composite media.…”

Section: Introductionmentioning

confidence: 99%

Homotopy Continuation Approach to Electrostatic Boundary-Value Problems of Nonlinear Media

Gu¹,

Yu²

1998

Commun. Theor. Phys.

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The homotopy continuation method is employed to solve electrostatic boundaryvalue problems of nonlinear media. The difficulty associated with matching-the inherently nonlinear boundary conditions on the interface is overcome by the mode expansion method, by which the nonlinear partial differential equations of the original problem are transformed into an infinite set of nonlinear ordinary differential equations. In this regard, the homotopy method has to be modiSed to handle the nonlinear boundary conditions. As an illustration, we study two cases: (a) nonlinear inclusion in linear host and (b) linear inclusion in nonlinear host, both in two dimensions. The homotopy method is validated by comparing the results with the exact solution of case (a) and the results derived by perturbation method in case (b).

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A homotopy algorithm for a symmetric generalized eigenproblem

Cited by 9 publications

References 9 publications

Krylov subspace method for fuzzy eigenvalue problem

Krylov subspace method for fuzzy eigenvalue problem

Mathematical Methods for Dissolved Phosphorus Concentrations

Homotopy Continuation Approach to Electrostatic Boundary-Value Problems of Nonlinear Media

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