Eigenvalues of Ax = λBx for real symmetric matrices A and B computed by reduction to a pseudosymmetric form and the HR process

Brebner, M. A.; Grad, Janez

doi:10.1016/0024-3795(82)90246-4

Cited by 42 publications

(30 citation statements)

References 4 publications

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“…In the class of GR algorithms, the one that preserves pseudosymmetric structure is called HR [22,24]. The H stands for hyperbolic.…”

Section: Pseudosymmetric Matrices a Real Tridiagonal Matrixmentioning

confidence: 99%

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Product Eigenvalue Problems

Watkins¹

2005

SIAM Rev.

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Abstract. 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: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.

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“…In the class of GR algorithms, the one that preserves pseudosymmetric structure is called HR [22,24]. The H stands for hyperbolic.…”

Section: Pseudosymmetric Matrices a Real Tridiagonal Matrixmentioning

confidence: 99%

“…This is a laudable goal. A more aggressive approach is to choose the entries to make the condensed form (22) as simple as possible. If one decides to do this, then the logical choice for E is the zero matrix.…”

Section: Pseudosymmetric Matrices a Real Tridiagonal Matrixmentioning

confidence: 99%

Product Eigenvalue Problems

Watkins¹

2005

SIAM Rev.

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“…Another motivation for this work is that the first step in most natural frequency or eigenvalue computations is the reduction, in a finite number of operations, to a simple form such as the tridiagonal reduction (1). Then an iterative procedure can be applied to compute the eigensystem efficiently.…”

Section: Computation Of the Transient Response Q As A Function Of Timmentioning

confidence: 99%

“…If M is indefinite, that is, M has both positive and negative eigenvalues, (K, M ) can be reduced to tridiagonal-diagonal form using one of the procedures described by Brebner and Grad [1] or by Zurmühl and Falk [10]. However, these reductions require M to be nonsingular.…”

Section: Review Of Existing Reductionsmentioning

confidence: 99%

Simultaneous tridiagonalization of two symmetric matrices

Garvey¹,

Tisseur²,

Friswell³

et al. 2003

Numerical Meth Engineering

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SUMMARYWe show how to simultaneously reduce a pair of symmetric matrices to tridiagonal form by congruence transformations. No assumptions are made on the nonsingularity or definiteness of the two matrices. The reduction follows a strategy similar to the one used for the tridiagonalization of a single symmetric matrix via Householder reflectors. Two algorithms are proposed, one using non-orthogonal rank-one modifications of the identity matrix and the other, more costly but more stable, using a combination of Householder reflectors and non-orthogonal rank-one modifications of the identity matrix with minimal condition numbers. Each of these tridiagonalization processes requires O(n 3 ) arithmetic operations and respects the symmetry of the problem. We illustrate and compare the two algorithms with some numerical experiments.

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“…For this case, we have implemented a variant of Algorithm 3 where W = V , the orthogonalization is performed with respect to the B inner product (that is an indefinite inner product or a pseudo-inner product), and the resulting vectors are normalized so that |x T Bx| = 1. The corresponding projected problem is a generalized symmetric-indefinite eigenproblem and can be solved with a structure-preserving method like the one proposed in [Brebner and Grad 1982]. However, since this solver is not available as a LAPACK subroutine, we currently use the Schur decomposition instead.…”

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confidence: 99%

A parallel implementation of Davidson methods for large-scale eigenvalue problems in SLEPc

Romero

Román

2014

ACM Trans. Math. Softw.

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In the context of large-scale eigenvalue problems, methods of Davidson type such as Jacobi-Davidson can be competitive with respect to other types of algorithms, especially in some particularly difficult situations such as computing interior eigenvalues or when matrix factorization is prohibitive or highly inefficient. However, these types of methods are not generally available in the form of high-quality parallel implementations, especially for the case of non-Hermitian eigenproblems. We present our implementation of various Davidsontype methods in SLEPc, the Scalable Library for Eigenvalue Problem Computations. The solvers incorporate many algorithmic variants for subspace expansion and extraction, and cover a wide range of eigenproblems including standard and generalized, Hermitian and non-Hermitian, with either real or complex arithmetic. We provide performance results on a large battery of test problems.

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Eigenvalues of Ax = λBx for real symmetric matrices A and B computed by reduction to a pseudosymmetric form and the HR process

Cited by 42 publications

References 4 publications

Product Eigenvalue Problems

Product Eigenvalue Problems

Simultaneous tridiagonalization of two symmetric matrices

A parallel implementation of Davidson methods for large-scale eigenvalue problems in SLEPc

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