Matrix factorizations for symplectic QR-like methods

Bunse-Gerstner, Angelika

doi:10.1016/0024-3795(86)90265-x

Cited by 94 publications

(87 citation statements)

References 4 publications

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“…Eigenvalues and eigenvectors of symplectic butterfly matrices can be computed efficiently by the SR algorithm (see [7]), which is a QR-like algorithm in which the QR decomposition is replaced by the SR decomposition. Almost every matrix A ∈ R 2n×2n can be decomposed into a product A = SR where S is symplectic and R is J-triangular.…”

Section: Introductionmentioning

confidence: 99%

The parameterized 𝑆𝑅 algorithm for symplectic (butterfly) matrices

Faßbender

2000

Math. Comp.

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Abstract. The SR algorithm is a structure-preserving algorithm for computing the spectrum of symplectic matrices. Any symplectic matrix can be reduced to symplectic butterfly form. A symplectic matrix B in butterfly form is uniquely determined by 4n − 1 parameters. Using these 4n − 1 parameters, we show how one step of the symplectic SR algorithm for B can be carried out in O(n) arithmetic operations compared to O(n 3 ) arithmetic operations when working on the actual symplectic matrix. Moreover, the symplectic structure, which will be destroyed in the numerical process due to roundoff errors when working with a symplectic (butterfly) matrix, will be forced by working just with the parameters.

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Section: Introductionmentioning

confidence: 99%

The parameterized 𝑆𝑅 algorithm for symplectic (butterfly) matrices

Faßbender

2000

Math. Comp.

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“…Suppose that a matrix G ∈ F n×k is given, with n ≥ k. Let G (1) be a submatrix of G that consists of the first m columns of G, where m ≤ k and m < n. If G (1) has the full column rank, there exists a nonsingular square submatrix G 1 (of order m) of G (1) . By row permutations, this square submatrix can be brought to the top of G (1) . The submatrix G 2 then contains the remaining rows of G (1) .…”

mentioning

confidence: 99%

“…By row permutations, this square submatrix can be brought to the top of G (1) . The submatrix G 2 then contains the remaining rows of G (1) .…”

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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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“…A decomposition of this form is called symplectic QR decomposition [Bunse-Gerstner, 1986] and can be computed by the following algorithm.…”

Section: Outputmentioning

confidence: 99%

“…Other algorithms based on orthogonal transformations are the Hamiltonian Jacobi algorithm [Byers, 1990;Bunse-Gerstner and Faßben-der, 1997], its variants for Hamiltonian matrices that have additional structure ] and the multishift algorithm [Ammar et al, 1993]. Algorithms based on symplectic but non-orthogonal transformations include the SR algorithm [Bunse-Gerstner and Mehrmann, 1986;Bunse-Gerstner, 1986;Mehrmann, 1991] as well as related methods [Bunse-Gerstner et al, 1989;Raines and Watkins, 1994]. A completely different class of algorithms is based on the matrix sign function, see, e.g., [Benner, 1999;Mehrmann, 1991;Sima, 1996] and the references therein.…”

Section: Outputmentioning

confidence: 99%

Skew-Hamiltonian and Hamiltonian Eigenvalue Problems: Theory, Algorithms and Applications

Benner

Kreßner

Mehrmann

Proceedings of the Conference on Applied Mathematics and Scientific Computing

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Skew-Hamiltonian and Hamiltonian eigenvalue problems arise from a number of applications, particularly in systems and control theory. The preservation of the underlying matrix structures often plays an important role in these applications and may lead to more accurate and more efficient computational methods. We will discuss the relation of structured and unstructured condition numbers for these problems as well as algorithms exploiting the given matrix structures. Applications of Hamiltonian and skew-Hamiltonian eigenproblems are briefly described.

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Matrix factorizations for symplectic QR-like methods

Cited by 94 publications

References 4 publications

The parameterized 𝑆𝑅 algorithm for symplectic (butterfly) matrices

The parameterized 𝑆𝑅 algorithm for symplectic (butterfly) matrices

Orthosymmetric block rotations

Skew-Hamiltonian and Hamiltonian Eigenvalue Problems: Theory, Algorithms and Applications

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