On Isospectral Gradient Flows — Solving Matrix Eigenprdblems using Differential Equations

Driessel, Kenneth R.

doi:10.1007/978-3-0348-7014-6_5

Cited by 7 publications

(8 citation statements)

References 9 publications

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“…Further applications to the travelling salesman problem and to digital quantization of continuous-time signals have been described, see Brockett (1989a; and Brockett and Wong (1991). Note also the parallel efforts by Chu (1984b;1984a), Driessel (1986), Chu and Driessel (1990) with applications to structured inverse eigenvalue problems and matrix least squares estimation. The motivation for studying the double bracket equation (1.1) comes from the fact that it provides a solution to the following matrix least squares approximation problem.…”

Section: Double Bracket Flows For Diagonalizationmentioning

confidence: 99%

Optimization and Dynamical Systems

Helmke¹,

Moore²

1994

Communications and Control Engineering

546

419

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PrefaceThis work is aimed at mathematics and engineering graduate students and researchers in the areas of optimization, dynamical systems, control systems, signal processing, and linear algebra. The motivation for the results developed here arises from advanced engineering applications and the emergence of highly parallel computing machines for tackling such applications. The problems solved are those of linear algebra and linear systems theory, and include such topics as diagonalizing a symmetric matrix, singular value decomposition, balanced realizations, linear programming, sensitivity minimization, and eigenvalue assignment by feedback control.The tools are those, not only of linear algebra and systems theory, but also of differential geometry. The problems are solved via dynamical systems implementation, either in continuous time or discrete time , which is ideally suited to distributed parallel processing. The problems tackled are indirectly or directly concerned with dynamical systems themselves, so there is feedback in that dynamical systems are used to understand and optimize dynamical systems. One key to the new research results has been the recent discovery of rather deep existence and uniqueness results for the solution of certain matrix least squares optimization problems in geometric invariant theory. These problems, as well as many other optimization problems arising in linear algebra and systems theory, do not always admit solutions which can be found by algebraic methods. Even for such problems that do admit solutions via algebraic methods, as for example the classical task of singular value decomposition, there is merit in viewing the task as a certain matrix optimization problem, so as to shift the focus from algebraic methods to geometric methods. It is in this context that gradient flows on manifolds appear as a natural approach to achieve construction methods that complement the existence and uniqueness results of geometric invari- There has been an attempt to bridge the disciplines of engineering and mathematics in such a way that the work is lucid for engineers and yet suitably rigorous for mathematicians. There is also an attempt to teach, to provide insight, and to make connections, rather than to present the results as a fait accompli.The research for this work has been carried out by the authors while visiting each other's institutions. Some of the work has been written in conjunction with the writing of research papers in collaboration with PhD students Jane Perkins and Robert Mahony, and post doctoral fellow Weiyong Yan. Indeed, the papers benefited from the book and vice versa, and consequently many of the paragraphs are common to both. Uwe Helmke has a background in global analysis with a strong interest in systems theory, and John Moore has a background in control systems and signal processing. AcknowledgementsThis work was partially supported by grants from Boeing Commercial Airplane Company, the Cooperative Research Centre for Robust and Adaptive Systems, and the German-Israel...

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Section: Double Bracket Flows For Diagonalizationmentioning

confidence: 99%

Optimization and Dynamical Systems

Helmke¹,

Moore²

1994

Communications and Control Engineering

546

419

View full text Add to dashboard Cite

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“…Insert T L −T T L T in the centre of each side of (7). Premultiply the result by T R −1 and post-multiply it by T R .…”

Section: Structure-preserving Transformationsmentioning

confidence: 99%

“…Isospectral flows have been investigated extensively to gain insight into the structure of numerical algorithms for solving the standard eigenvalue problem [4,5,21] and they have been used directly in some cases as solution methods for eigenvalue problems (for example [2,3,7,18]). …”

Section: Definition 8 (Isospectral Flows) An Isospectral Flow Is a Tmentioning

confidence: 99%

General isospectral flows for linear dynamic systems

Garvey

Prells

Friswell

et al. 2004

Linear Algebra and its Applications

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The λ-matrix A(λ) = (A 0 + λA 1 + λ 2 A 2 + · · · λ k A k · · · + λ l A l ) with matrix coefficients {A 0 , A 1 , A 2 . . . A } ∈ C m×n defines a linear dynamic system of dimension (m × n). When m = n, and when det(A(λ)) / = 0 for some values of λ, the eigenvalues of this system are welldefined. A one-parameter trajectory of such a system {A 0 (σ ), A 1 (σ ), A 2 (σ ) . . . A (σ )} is an isospectral flow if the eigenvalues and the dimensions of the associated eigenspaces are the same for all parameter values σ ∈ IR. This paper presents the most general form for isospectral flows of linear dynamic systems of orders ( = 2, 3, 4), and the forms for isospectral flows for even higher order systems are evident from the patterns emerging. Based on the definition of a class of coordinate transformations called structure-preserving transformations, the concept of isospectrality and the associated flows is seen to extend to cases where (m / = n).

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“…In independent work by Driessel [7], Chu and Driessel [5], Smith [12] and Helmke and Ivfoore [8], a similar gradient flow approach is developed for the task of computing the singular values of a general nonsymmetric, nonsquare matrix. The differential equation obtained in these approaches is almost identical to the double-bracket equation.…”

Section: Introductionmentioning

confidence: 99%

“…By estimating the error term we obtain a mathematically simple function A@u (Hk, T), which is an upper bound to A@(H~,~) for all T. Then, choosing a suitable time-step~k based on minimising A@u, we guarantee that the aCtUal change in pOtentia~,~@(Hk, ok) 5~$u (Hk, ok) < 0, satisfies (7). Due to the simple nature of the function A+u, there is an explicit form for the time-step~k depending only on Hk and N. We begin by deriving an expression for the error term.…”

mentioning

confidence: 99%

Numerical Gradient Algorithms for Eigenvalue and Singular Value Calculations

Moore¹,

Mahony²,

Helmke³

1994

SIAM J. Matrix Anal. & Appl.

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Abstract.Recent work has shown that the algebraic question of determining the eigenvalues, or singular values, of a matrix can be answered by solving certain continuous-time gradient flows on matrix manifolds, To obtain computational methods based on this theory, it is reasonable to develop algorithms that iteratively approximate the continuous-time flows. In this paper the authors propose two algorithms, based on a double Lie-bracket equation recently studied by Brockett, that appear to be suitable for implementation in parallel processing environments. The algorithms presented achieve, respectively, the eigenvalue decomposition of a symmetric matrix and the singular value decomposition of an arbitrary matrix. The algorithms have the same equilibria as the continuoustime flows on wh~ch they are based and inherit the exponential convergence of the continuous-time solutions.

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On Isospectral Gradient Flows — Solving Matrix Eigenprdblems using Differential Equations

Cited by 7 publications

References 9 publications

Optimization and Dynamical Systems

Optimization and Dynamical Systems

General isospectral flows for linear dynamic systems

Numerical Gradient Algorithms for Eigenvalue and Singular Value Calculations

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