Finding the Zeros of a Univariate Equation: Proxy Rootfinders, Chebyshev Interpolation, and the Companion Matrix

Boyd, John P.

doi:10.1137/110838297

Cited by 40 publications

(47 citation statements)

References 43 publications

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“…This large scaling of the polynomial over the entire interval relative to its size on the half of the interval containing the spectrum causes its roots to be badly conditioned, resulting in the poor approximations to the eigenvalues we see in the table. In the polynomial rootfinding literature, this is sometimes referred to as a "dynamic range" problem [3].…”

Section: Instability Of Rational Interpolation For Finding Eigenvaluesmentioning

confidence: 99%

Computing Eigenvalues of Real Symmetric Matrices with Rational Filters in Real Arithmetic

Austin¹,

Trefethen²

2015

SIAM J. Sci. Comput.

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Powerful algorithms have recently been proposed for computing eigenvalues of large matrices by methods related to contour integrals; best known are the works of Sakurai and coauthors and Polizzi and coauthors. Even if the matrices are real symmetric, most such methods rely on complex arithmetic, leading to expensive linear systems to solve. An appealing technique for overcoming this starts from the observation that certain discretized contour integrals are equivalent to rational interpolation problems, for which there is no need to leave the real axis. Investigation shows that using rational interpolation per se suffers from instability; however, related techniques involving real rational filters can be very effective. This article presents a technique of this kind that is related to previous work published in Japanese by Murakami. Introduction. LetA ∈ C N ×N be a large square matrix, and consider the problem of computing the eigenvalues of A that lie within a given region of the complex plane. Some of the most successful techniques for attacking this problem are based on projecting the matrix A onto an approximately invariant subspace associated with the eigenvalues of interest and computing the eigenvalues of the projection. Of these, perhaps the best known are the Krylov subspace techniques such as the implicitly restarted Arnoldi method [39], which is implemented in the widely used software package ARPACK [20].Recently, a new class of algorithms has been proposed which derive their projections from complex contour integrals. Though early traces of these ideas can be found in the works of Goedecker on linear-scaling methods for electronic structure calculation [11,12], the best-known algorithms of this type are the Sakurai-Sugiura (SS) method [37] and the FEAST algorithm, due to Polizzi [32]. A major computational advantage offered by these algorithms is that they are very easily parallelizable.Some of the most important eigenvalue problems involve matrices A that are real and symmetric. For large such problems, it is natural to want to take advantage of the parallelism offered by contour integral methods; however, by their very nature, these methods require complex arithmetic even though the sought-after eigenvalues are real. Aside from the fact that the need to use complex arithmetic to solve a real problem is conceptually jarring, this means that methods based on contour integrals suffer roughly a factor of two penalty in both time and storage costs, compared with approaches that rely only on real arithmetic.

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Section: Instability Of Rational Interpolation For Finding Eigenvaluesmentioning

confidence: 99%

Computing Eigenvalues of Real Symmetric Matrices with Rational Filters in Real Arithmetic

Austin¹,

Trefethen²

2015

SIAM J. Sci. Comput.

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“…, σ N ∈ Σ, and then to compute the roots of P N (λ) on Σ. See [9] for a recent review of polynomial root finding. Let λ * be such a root, i.e., P N (λ * ) = 0.…”

Section: Motivating Examplementioning

confidence: 99%

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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“…We chose this one because in [14] the method is extended to achieve slightly complex roots too, whereby fits perfectly with our requirements.…”

Section: Chebyshev's Root Findermentioning

confidence: 99%

“…Then, applying the method proposed in [11] and [14], the zeros of the function f (x) in the interval [a, b] are calculated.…”

Section: Chebyshev's Root Findermentioning

confidence: 99%

Fast and Accurate Determination of the Complex Resonant Frequency of a Multilayer Circular Cavity Using Chebyshev's Root-Finder

Marques-Villarroya¹,

Peñaranda‐Foix²,

García-Baños³

et al. 2017

PIER M

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Abstract-In this paper, a general multilayer circular cavity with N slabs is analyzed analytically, obtaining characteristic equations for TE and TM modes to compute the complex resonant frequency efficiently using an algorithm based on Chebyshev's root finder. The accuracy of the solutions is compared with full-wave circuit method, and the computational speed to achieve the roots of the characteristic equations is also compared with Cauchy Integral Method, which is commonly used to obtain complex roots. Furthermore, the relationship between the amplitudes of the different regions is obtained, whereby the whole structure can be analyzed as a single one from now on.

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Finding the Zeros of a Univariate Equation: Proxy Rootfinders, Chebyshev Interpolation, and the Companion Matrix

Cited by 40 publications

References 43 publications

Computing Eigenvalues of Real Symmetric Matrices with Rational Filters in Real Arithmetic

Computing Eigenvalues of Real Symmetric Matrices with Rational Filters in Real Arithmetic

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

Fast and Accurate Determination of the Complex Resonant Frequency of a Multilayer Circular Cavity Using Chebyshev's Root-Finder

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