Rayleigh quotient iteration for nonsymmetric matrices

Batterson, Steve; Smillie, John

doi:10.1090/s0025-5718-1990-1023041-4

Cited by 21 publications

(8 citation statements)

References 10 publications

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“…At this point, the vocabulary and techniques from dynamical systems are natural (another application of dynamical systems to numerical spectral theory is the work of Batterson and Smillie [1] on the Rayleigh quotient iteration). In Sect.…”

Section: Theorem 11mentioning

confidence: 99%

The Asymptotics of Wilkinson’s Shift: Loss of Cubic Convergence

Leite¹,

Saldanha

Tomei

2009

Found Comput Math

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One of the most widely used methods for eigenvalue computation is the QR iteration with Wilkinson's shift: Here, the shift s is the eigenvalue of the bottom 2 × 2 principal minor closest to the corner entry. It has been a long-standing question whether the rate of convergence of the algorithm is always cubic. In contrast, we show that there exist matrices for which the rate of convergence is strictly quadratic. More precisely, let T X be the 3 × 3 matrix having only two nonzero entries (T X ) 12 = (T X ) 21 = 1 and let T Λ be the set of real, symmetric tridiagonal matrices with the same spectrum as T X . There exists a neighborhood U ⊂ T Λ of T X which is invariant under Wilkinson's shift strategy with the following properties. For T 0 ∈ U , the sequence of iterates (T k ) exhibits either strictly quadratic or strictly cubic convergence to zero of the entry (T k ) 23 . In fact, quadratic convergence occurs exactly when lim T k = T X . Let X be the union of such quadratically convergent sequences (T k ): The set X has Hausdorff dimension 1 and is a union of disjoint arcs X σ meeting at T X , where σ ranges over a Cantor set.

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Section: Theorem 11mentioning

confidence: 99%

The Asymptotics of Wilkinson’s Shift: Loss of Cubic Convergence

Leite¹,

Saldanha

Tomei

2009

Found Comput Math

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“…For example:• The unshifted QR algorithm terminates with probability 1 but is probably infinite average cost if approximations to the eigenvectors are to be output (see [29]). • The QR algorithm with Rayleigh Quotient shift fails for open sets of real input matrices (see [6,7]).…”

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confidence: 99%

A stable, polynomial-time algorithm for the eigenpair problem

et al. 2018

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We describe algorithms for computing eigenpairs (eigenvalueeigenvector pairs) of a complex n × n matrix A. These algorithms are numerically stable, strongly accurate, and theoretically efficient (i.e., polynomialtime). We do not believe they outperform in practice the algorithms currently used for this computational problem. The merit of our paper is to give a positive answer to a long-standing open problem in numerical linear algebra. So the problem of devising an algorithm [for the eigenvalue problem] that is numerically stable and globally (and quickly!) convergent remains open.Our initial quotation followed these words in Demmel's text. It asked for an algorithm which will be numerically stable and for which, convergence, and if possible small complexity bounds, can be established. Today, 17 years after Demmel's text, this demand retains all of its urgency: it is not known if any of the standard numerical linear algebra algorithms satisfies the properties above. For example:• The unshifted QR algorithm terminates with probability 1 but is probably infinite average cost if approximations to the eigenvectors are to be output (see [29]).• The QR algorithm with Rayleigh Quotient shift fails for open sets of real input matrices (see [6,7]).

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“…In order to reduce this effort, a special Rayleigh quotient iteration for non-symmetric matrices can be applied [4]. This method converges towards the nearest eigenvalue.…”

Section: Hints For Microcontroller Implementationmentioning

confidence: 99%

Constrained Ellipse Fitting with Center on a Line

2015

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Fitting an ellipse to given data points is a common optimization task in computer vision problems. However, the possibility of incorporating the prior constraint "the ellipse's center is located on a given line" into the optimization algorithm has not been examined so far. This problem arises, for example, by fitting an ellipse to data points representing the path of the image positions of an adhesion inside a rotating vessel whose position of the rotational axis in the image is known. Our new method makes use of a constrained algebraic cost function with the incorporated "ellipse center on given line"-prior condition in a global convergent onedimensional optimization approach. Further advantages of the algorithm are computational efficiency and numerical stability.

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Rayleigh quotient iteration for nonsymmetric matrices

Cited by 21 publications

References 10 publications

The Asymptotics of Wilkinson’s Shift: Loss of Cubic Convergence

The Asymptotics of Wilkinson’s Shift: Loss of Cubic Convergence

A stable, polynomial-time algorithm for the eigenpair problem

Constrained Ellipse Fitting with Center on a Line

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