Computing Eigenvalues of Non-Hermitian Matrices by Methods of Jacobi Type

Causey, Robert L.

doi:10.1137/0106010

Cited by 16 publications

(11 citation statements)

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“…In this paper, we will focus on a particular cyclic Jacobi-like algorithm for the computation of the Schur form of a complex matrix. The basic idea of the algorithm is a direct adaption of Jacobi's method to the nonsymmetric case which has been proposed 1955 by Greenstadt [13] and has later been taken up and modified by various authors [4,10,14,19,24,32]. Given a matrix M = (m ij ) ∈ C n×n , the algorithm selects in each step a pivot element m kl , k < l in the strict lower triangular part.…”

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confidence: 99%

On Asymptotic Convergence of Nonsymmetric Jacobi Algorithms

Mehl¹

2008

SIAM J. Matrix Anal. & Appl.

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The asymptotic convergence behavior of cyclic versions of the nonsymmetric Jacobi algorithm for the computation of the Schur form of a general complex matrix is investigated. Similar to the symmetric case, the nonsymmetric Jacobi algorithm proceeds by applying a sequence of rotations that annihilate a pivot element in the strict lower triangular part of the matrix until convergence to the Schur form of the matrix is achieved. In this paper, it is shown that the cyclic nonsymmetric Jacobi method converges locally and asymptotically quadratically under mild hypotheses if special ordering schemes are chosen, namely ordering schemes that lead to so-called northeast directed sweeps. The theory is illustrated by the help of numerical experiments. In particular, it is shown that there are ordering schemes that lead to asymptotic quadratic convergence for the cyclic symmetric Jacobi method, but only to asymptotic linear convergence for the cyclic nonsymmetric Jacobi method. Finally, a generalization of the nonsymmetric Jacobi method to the computation of the Hamiltonian Schur form for Hamiltonian matrices is introduced and investigated.

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On Asymptotic Convergence of Nonsymmetric Jacobi Algorithms

Mehl¹

2008

SIAM J. Matrix Anal. & Appl.

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“…It is more convenient to select the pairs (i,j) in some cyclic order. We here consider two cyclic orders: (i) cyclic by rows, indicated by the scheme (to, jo) -(1,2), (ik,jk + 1), if ik < n -l,jk < n, (IHr) (ik+i,jk+i) = (ik + 1, ik + 2), ii ik < n -1, jk = n, (1,2), if ik = n -l,jk = n;…”

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confidence: 99%

“…Let A = (apf) be a real symmetric matrix of order «, and let Xi, Xa, • • • , X" be its eigenvalues. It is well known that if U is an orthogonal matrix such that (1) A = UA UT is diagonal (T denotes the transpose), then the main diagonal of A is made up of the numbers X< in some order. If it is desired to compute the Xj numerically, this result is of no immediate use, since for w>2 there exists no manageable expression for the general orthogonal matrix of order n. However, Jacobi [6] suggested the computation of the set of X,-as the limiting set of diagonal elements of a sequence of matrices which are generated from A recursively by plane rotations.…”

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confidence: 99%

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1960

Trans. Amer. Math. Soc.

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“…Again let us consider matrix A as defined by (13), now subject to the following conditions: has been mentioned as one which defies direct treatment by Greenstadt's method [2], as well as by the method of reference [1]. However, it is stated in [3] that by applying a transformation to B which effects a rotation through 8 = 7r/4, the transformed matrix becomes tractable by Greenstadt's method, and that in twelve cyclically executed annihilations of the respective pivot the matrix B becomes triangularized to a sufficient degree of accuracy.…”

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“…These are symmetric matrices whose elements are [Vol. XVII,No. 3 a,-,-= (i + j -l)1, i, j = 1, 2, 3, • ■ • .…”

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confidence: 99%