On quadratic convergence bounds for theJ-symmetric Jacobi method

Drmač, Zlatko; Hari, Vjeran

doi:10.1007/bf01388685

Cited by 20 publications

(31 citation statements)

References 13 publications

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“…The quasi-cyclic strategy means, choosing pivot positions in row-cyclic fashion inside the submatrices, respectively, H [33] , H [44] , H [34] , H [33] , H [44] , H [11] , H [22] , H [12] , H [11] , H [22] , H [12] . Let H be the matrix computed after such quasi-cycle.…”

Section: Review Of Asymptotic Convergencementioning

confidence: 99%

New Fast and Accurate Jacobi SVD Algorithm. II

Drmač¹,

Veselić²

2008

SIAM J. Matrix Anal. & Appl.

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Abstract. This paper presents new implementation of one-sided Jacobi SVD for triangular matrices and its use as the core routine in a new preconditioned Jacobi SVD algorithm, recently proposed by the authors. New pivot strategy exploits the triangular form and uses the fact that the input triangular matrix is the result of rank revealing QR factorization. If used in the preconditioned Jacobi SVD algorithm, it delivers superior performance leading to the currently fastest method for computing SVD decomposition with high relative accuracy. Furthermore, the efficiency of the new algorithm is comparable to the less accurate bidiagonalization based methods. The paper also discusses underflow issues in floating point implementation, and shows how to use perturbation theory to fix the imperfectness of machine arithmetic on some systems.

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Section: Review Of Asymptotic Convergencementioning

confidence: 99%

New Fast and Accurate Jacobi SVD Algorithm. II

Drmač¹,

Veselić²

2008

SIAM J. Matrix Anal. & Appl.

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“…The technique described here recently found application in proving sharp quadratic convergence bounds of the J-symmetric Jacobi method (see [4,22]). …”

Section: Introductionmentioning

confidence: 99%

On sharp quadratic convergence bounds for the serial Jacobi methods

Hari

1991

Numer. Math.

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“…Indeed, if ]a,, t < a Ixf~,l [a,~ I for all r = s then by [2] the eigenvalues lie in the union of the intervals Although there is little doubt that our algorithm converges under this strategy as well (our experiments confirm that) we have a proof only for the important special case where A itself is positive definite (case d). The quadratic convergence of the method has recently been proved in [9] in full analogy to the standard Jacobi algorithm. If the method is convergent then the value tro from Lemma 2.2 is given by…”

Section: ~ T Ma~sign T [T] > Tmaxmentioning

confidence: 98%

A Jacobi eigenreduction algorithm for definite matrix pairs

Veselié

1993

Numer. Math.

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Summary.We propose a Jacobi eigenreduction algorithm for symmetric definite matrix pairs A, J of small to medium-size real symmetric matrices with j2 = I, J diagonal (neither J nor A itself need be definite). Our Jacobi reduction works only on one matrix and uses J-orthogonal elementary congruences which include both trigonometric and hyperbolic rotations and preserve the symmetry throughout the process. For the rotation parameters only the pivotal elements of the current matrix are needed which facilitates parallelization. We prove the global convergence of the method; the quadratic convergence was observed in all experiments. We apply our method in two situations: (i) eigenreducing a single real symmetric matrix and (ii) eigenreducing an overdamped quadratic matrix pencil. In both cases our method is preceded by a symmetric indefinite decomposition and performed in its "one-sided" variant on the thus obtained factors. Our method outdoes the standard methods like standard Jacobi or qr/qI in accuracy in spite of the use of hyperbolic transformations which are not orthogonal (a theoretical justification of this behaviour is made elsewhere). The accuracy advantage of our method can be particularly drastic if the eigenvalues are of different order. In addition, in working with quadratic pencils our method is shown to either converge or to detect non-overdampedness.

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On quadratic convergence bounds for theJ-symmetric Jacobi method

Cited by 20 publications

References 13 publications

New Fast and Accurate Jacobi SVD Algorithm. II

New Fast and Accurate Jacobi SVD Algorithm. II

On sharp quadratic convergence bounds for the serial Jacobi methods

A Jacobi eigenreduction algorithm for definite matrix pairs

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