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SIAM REVIEW 2005年第1期47卷 3-40页

作者： Watkins, DS Washington State Univ Dept Math Pullman WA 99164 USA

Many eigenvalue problems are most naturally viewed as product eigenvalue problems. The eigenvalues of a matrix A are wanted, but A is not given explicitly. Instead it is presented as a product of several factors: A = A(k)A(k-1)... A(1). Usually more accurate results are obtained by working with the factors rather than forming A explicitly. For example, if we want eigenvalues/vectors of B-T B, it is better to work directly with B and not compute the product. The intent of this paper is to demonstrate that the product eigenvalue problem is a powerful unifying concept. Diverse examples of eigenvalue problems are discussed and formulated as product eigenvalue problems. For all but a couple of these examples it is shown that the standard algorithms for solving them are instances of a generic gr algorithm applied to a related cyclic matrix.

关键词： eigenvalue QR algorithm gr algorithm generalized eigenvalue problem SVD symmetric skew symmetric positive definite totally positive pseudosymmetric Hamiltonian symplectic unitary

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QR-like algorithms for eigenvalue problems

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2000年第1-2期123卷 67-83页

作者： Watkins, DS Washington State Univ Dept Math Pullman WA 99164 USA

In the year 2000 the dominant method for solving matrix eigenvalue problems is still the QR algorithm. This paper discusses the family of gr algorithms, with emphasis on the QR algorithm. Included are historical remarks, an outline of what gr algorithms are and why they work, and descriptions of the latest, highly parallelizable, versions of the QR algorithm. Now that we know how to parallelize it, the QR algorithm seems likely to retain its dominance for many years to come. (C) 2000 Elsevier Science B.V. All rights reserved. MSC: 65F15.

关键词： eigenvalue QR algorithm gr algorithm

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BIDIRECTIONAL CHASING algorithmS FOR THE EIGENVALUE PROBLEM

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 1993年第1期14卷 166-179页

作者： WATKINS, DS

The QR eigenvalue algorithm and its variants are usually implemented implicitly as chasing algorithms. The matrix whose eigenvalues are sought is first reduced to Hessenberg form by a similarity transformation, then the chasing iterations are begun. Each iteration is a sequence of similarity transformations that create a bulge in the Hessenberg form at one corner of the matrix, then chase the bulge along the diagonal to the opposite corner and finally off the edge of the matrix. Byers's Hamiltonian QR algorithm is an example of a chasing algorithm that has a special feature. In order to preserve the Hamiltonian structure of the matrices, the algorithm forms and chases two bulges simultaneously. Viewed in the appropriate coordinate system, this process can be seen as one in which two bulges are created at opposite comers of the matrix and chased toward each other. They collide in the middle, they interact, and then they continue to the corners of the matrix, where they vanish. In this paper it is shown that the single-shift Hamiltonian QR algorithm is just one of a family of bidirectional chasing algorithms that can be applied to arbitrary matrices. Thus the basic mechanism underlying the Hamiltonian QR algorithm does not rely on the Hamiltonian structure in any way. The structure is exploited to make the algorithm more efficient and to improve its numerical stability.

关键词： EIGENVALUE QR algorithm gr algorithm CHASING THE BULGE HAMILTONIAN MATRIX

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QR-LIKE algorithmS FOR THE NONSYMMETRIC EIGENVALUE PROBLEM

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ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE 1993年第3期19卷 407-418页

作者： HAAG, JB WATKINS, DS WASHINGTON STATE UNIV DEPT PURE & APPL MATH PULLMAN WA 99164 USA

Hybrid codes that combine elements of the QR and LR algorithms are described. The codes can calculate the eigenvalues and, optionally, eigenvectors of real, nonsymmetric matrices. Extensive tests are presented as evidence that, for certain choices of parameters, the hybrid codes possess the same high reliability as the QR algorithm and are significantly faster. The greatest success has been achieved with the codes that calculate eigenvalues only. These can do the task in 15% to 50% less time than the QR algorithm.

关键词： algorithmS PERFORMANCE CHASING THE BULGE EIGENVALUE EIGENVECTOR gr algorithm LR algorithm QR algorithm

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CHASING algorithmS FOR THE EIGENVALUE PROBLEM

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 1991年第2期12卷 374-384页

作者： WATKINS, DS ELSNER, L UNIV BIELEFELD FAK MATHW-4800 BIELEFELD 1GERMANY

A generic chasing algorithm for the matrix eigenvalue problem is introduced and studied. This algorithm includes, as special cases, the implicit, multiple-step QR and LR algorithms and similar bulge-chasing algorithms for the standard eigenvalue problem. The scope of the generic chasing algorithm is quite broad;it encompasses a number of chasing algorithms that cannot be analyzed by the traditional (e.g., implicit Q theorem) approach. These include the LR algorithm with partial pivoting and other chasing algorithms that employ pivoting for stability, as well as hybrid algorithms that combine elements of the LR and QR algorithms. The main result is that each step of the generic chasing algorithm amounts to one step of the generic gr algorithm. Therefore the convergence theorems for gr algorithms that were proven in a previous work [D. S. Watkins and L. Elsner, Linear Algebra Appl., 143 (1991), pp. 19-47] also apply to the generic chasing algorithm.

关键词： EIGENVALUE QR algorithm gr algorithm CHASING THE BULGE SUBSPACE ITERATION

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