A Nonlinear QR Algorithm for Banded Nonlinear Eigenvalue Problems

Garrett, Charles Kristopher; Bai, Zhaojun; Li, Ren-Cang

doi:10.1145/2870628

Cited by 5 publications

(11 citation statements)

References 17 publications

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“…This strategy has been shown to work well in practice (Garrett et al 2016). When the derivative of F is not available, it can be replaced by a finite difference approximation, leading to a quasi-Newton method.…”

Section: Newton's Methods For Scalar Functionsmentioning

confidence: 99%

“…The choice of an initial guess is typically the only crucial parameter of a Newton-type method, which is a great advantage over other NEP solution approaches. As a result, many algorithmic variants have been developed and applied over the years, including the Newton-QR iteration of Kublanovskaya (1970) and its variant in Garrett, Bai and Li (2016), the Newton-trace iteration of Lancaster (1966), nonlinear inverse iteration (Unger 1950), residual inverse iteration (Neumaier 1985), Rayleigh functional iterations (Lancaster 1961, Schreiber 2008), the block Newton method of Kressner (2009), and for large sparse NEPs, Jacobi–Davidson-type methods (Betcke and Voss 2004, Sleijpen, Booten, Fokkema and van der Vorst 1996).…”

Section: Solvers Based On Newton’s Methodsmentioning

confidence: 99%

“…In a similar way, they show that for inexact versions of the residual inverse iteration and the twosided Rayleigh functional iteration, the same order of convergence as for the exact iterations can be achieved if an appropriate sequence of tolerances is used for the inner solves. Garrett et al (2016) propose a variation of Kublanovskaya's Newton-QR method for either large banded NEPs with narrow bandwidths relative to the size n of the NEP, or large sparse NEPs that can be reduced to such banded NEPs. They make use of a special data structure for storing the matrices to keep memory and computational costs low.…”

Section: Large Sparse Nepsmentioning

confidence: 99%

“…This NEP has real eigenvalues, and we are interested in the five eigenvalues in the interval [4,296], all of which are given in (3.5). The MATLAB code in Figure 4.17 calls the function NQR4UB from Garrett and Li (2013) which, compared with (3.5), are excellent approximations to the eigenvalues λ 1 , λ 3 , λ 4 , λ 5 , and to another eigenvalue λ 9 outside the interval of interest. Note that the eigenvalue λ 2 is missed.…”

Section: Large Sparse Nepsmentioning

confidence: 99%

See 3 more Smart Citations

The nonlinear eigenvalue problem

Güttel¹,

Tisseur²

2017

Acta Numerica

183

169

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Nonlinear eigenvalue problems arise in a variety of science and engineering applications and in the past ten years there have been numerous breakthroughs in the development of numerical methods. This article surveys nonlinear eigenvalue problems associated with matrix-valued functions which depend nonlinearly on a single scalar parameter, with a particular emphasis on their mathematical properties and available numerical solution techniques. Solvers based on Newton's method, contour integration, and sampling via rational interpolation are reviewed. Problems of selecting the appropriate parameters for each of the solver classes are discussed and illustrated with numerical examples. This survey also contains numerous MATLAB code snippets that can be used for interactive exploration of the discussed methods.

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Section: Newton's Methods For Scalar Functionsmentioning

confidence: 99%

Section: Solvers Based On Newton’s Methodsmentioning

confidence: 99%

Section: Large Sparse Nepsmentioning

confidence: 99%

Section: Large Sparse Nepsmentioning

confidence: 99%

See 2 more Smart Citations

The nonlinear eigenvalue problem

Güttel¹,

Tisseur²

2017

Acta Numerica

183

169

View full text Add to dashboard Cite

show abstract

“…There is a summary of many methods [18] of which many are Newton methods or can be interpreted as flavors of Newton's method. The QR-approach for banded matrices in [14] is based on Kublanovskaya's approach [28] which is also a Newton method applied to the (n, n)-element of the R-matrix in the QR-factorization of M (λ). Two-sided Newton approaches and Jacobi-Davidson approaches have been studied in [37].…”

mentioning

confidence: 99%

Broyden's Method for Nonlinear Eigenproblems

Jarlebring¹

2019

SIAM J. Sci. Comput.

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Broyden's method is a general method commonly used for nonlinear systems of equations, when very little information is available about the problem. We develop an approach based on Broyden's method for nonlinear eigenvalue problems. Our approach is designed for problems where the evaluation of a matrix vector product is computationally expensive, essentially as expensive as solving the corresponding linear system of equations. We show how the structure of the Jacobian matrix can be incorporated into the algorithm to improve convergence. The algorithm exhibits local superlinear convergence for simple eigenvalues, and we characterize the convergence. We show how deflation can be integrated and combined such that the method can be used to compute several eigenvalues. A specific problem in machine tool milling, coupled with a PDE is used to illustrate the approach. The simulations are done in the julia programming language, and are provided as publicly available module for reproducability.(2.1)where the next approximation is computed with a damped update equationThe choice of the damping parameter γ k will be tuned to our setting, essentially to avoid taking too large steps (as we shall further describe in Remark 3.3). The next matrix J k+1 will satisfy (what is commonly called) the secant conditionwhere J k+1 is a rank-one modification of J k . We will focus on updates of the form,(2.4) J k+1 = J k + 1 ∆x k 2 z k+1 ∆x H .

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A rational‐Chebyshev projection method for nonlinear eigenvalue problems

Tang,

Saad

2024

Numerical Linear Algebra App

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This article describes a projection method based on a combination of rational and polynomial approximations for efficiently solving large nonlinear eigenvalue problems. In a first stage the nonlinear matrix function under consideration is approximated by a matrix polynomial in . The error resulting from this polynomial approximation is in turn approximated by rational functions with the help of the Cauchy integral formula. The two approximations are combined and a linearization is performed. A key ingredient of the proposed approach is a projection method that uses subspaces spanned by vectors of the same dimension as that of the original problem instead of that of the linearized problem. A procedure is also presented to automatically select shifts and to partition the region of interest into a few subregions. This allows to subdivide the problem into smaller subproblems that are solved independently. The accuracy of the proposed method is theoretically analyzed and its performance is illustrated with a few test problems that have been discussed in the literature.

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A Nonlinear QR Algorithm for Banded Nonlinear Eigenvalue Problems

Cited by 5 publications

References 17 publications

The nonlinear eigenvalue problem

The nonlinear eigenvalue problem

Broyden's Method for Nonlinear Eigenproblems

A rational‐Chebyshev projection method for nonlinear eigenvalue problems

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