Structured tools for structured matrices

Mackey, D. Steven; Mackey, Niloufer; Tisseur, Françoise

doi:10.13001/1081-3810.1101

Cited by 55 publications

(39 citation statements)

References 42 publications

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“…We remark that the automorphism group G M associated with a non-degenerate bilinear form x, y M = x ⊤ M y contains as special cases the orthogonal group and the symplectic group (Mackey, Mackey and Tisseur 2003). Table 7.3 typifies some group actions that have been commonly used in numerical linear algorithm algorithms.…”

Section: Group Actions and Canonical Formsmentioning

confidence: 99%

“…Standard simplicity preserving allows a doubling algorithm to effectively separate stable and unstable eigenvalues when solving the discrete algebraic Riccati equation (Lin and Xu 2006). See also an interesting discussion by Mackey et al (2003) for structured matrices arising in the context of a bilinear or sesqui-linear form. A quick search of the key word "structure preserving" over the internet brings up a wide range of applications across multiple disciplines.…”

Section: Structure Preserving Dynamical Systemsmentioning

confidence: 99%

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Linear algebra algorithms as dynamical systems

Chu

2008

Acta Numerica

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Any logical procedure that is used to reason or infer either deductively or inductively so as to draw conclusions or make decisions can be called, in a broad sense, a realization process. A realization process usually assumes the recursive form that one state gets developed into another state by following a certain specific rule. Such an action is qualified as what is generally known as a dynamical system. In mathematics, especially for existence questions, a realization process often appears in the form of an iterative procedure or a differential equation. For years researchers have taken great effort to describe, analyze, and modify realization processes for various applications.The thrust in this exposition is to exploit the notion of dynamical systems as a special realization process for problems arising from the field of linear algebra. Several differential equations whose solutions evolve in some submanifolds of matrices are cast in fairly general frameworks of which special cases have been found to afford unified and fundamental insights into the structure and behavior of existing discrete methods and, now and then, suggest new and improved numerical methods. In some cases, there are remarkable connections between smooth flows and discrete numerical algorithms. In other cases, the flow approach seems advantageous in tackling very difficult open problems. Various aspects of the recent development and application in this direction are discussed in this paper.

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Section: Group Actions and Canonical Formsmentioning

confidence: 99%

Section: Structure Preserving Dynamical Systemsmentioning

confidence: 99%

Linear algebra algorithms as dynamical systems

Chu

2008

Acta Numerica

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“…In [8], the problem of finding good numerical methods to compute Hamiltonian square roots for general skew-Hamiltonian matrices has been left as an open problem which, to our knowledge, has not been addressed until now. It is a basic tenet in numerical analysis that structure should be exploited allowing, in general, the development of faster and/or more accurate algorithms [4,24].…”

Section: Introduction Given a ∈ Cmentioning

confidence: 99%

Structure-preserving Schur methods for computing square roots of real skew-Hamiltonian matrices

Liu¹,

Zhang²,

Ferreira³

et al. 2012

ELA

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Abstract. The contribution in this paper is two-folded. First, a complete characterization is given of the square roots of a real nonsingular skew-Hamiltonian matrix W . Using the known fact that every real skew-Hamiltonian matrix has infinitely many real Hamiltonian square roots, such square roots are described. Second, a structure-exploiting method is proposed for computing square roots of W , skew-Hamiltonian and Hamiltonian square roots. Compared to the standard real Schur method, which ignores the structure, this method requires significantly less arithmetic.

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“…Introduction. Orthosymmetric scalar products, introduced in [9] by Mackey, Mackey and Tisseur, still do not belong to the standard vocabulary of numerical linear algebra, even though they provide a unified setting for many modern structure preserving matrix tools (see for example [2,8,9,10]). …”

mentioning

confidence: 99%

Orthosymmetric block rotations

Singer¹

2012

ELA

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Abstract. Rotations are essential transformations in many parts of numerical linear algebra. In this paper, it is shown that there exists a family of matrices unitary with respect to an orthosymmetric scalar product J, that can be decomposed into the product of two J-unitary matrices-a block diagonal matrix and an orthosymmetric block rotation. This decomposition can be used for computing various one-sided and two-sided matrix transformations by divide-and-conquer or treelike algorithms. As an illustration, a blocked version of the QR-like factorization of a given matrix is considered.

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Structured tools for structured matrices

Cited by 55 publications

References 42 publications

Linear algebra algorithms as dynamical systems

Linear algebra algorithms as dynamical systems

Structure-preserving Schur methods for computing square roots of real skew-Hamiltonian matrices

Orthosymmetric block rotations

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