A proof of convergence for two parallel Jacobi SVD algorithms

Luk, Franklin T.; Park, H.

doi:10.1109/12.24289

Cited by 66 publications

(21 citation statements)

References 10 publications

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“…In such a case we shall also call the pairs (i,,j,) and (i,+ 1,J,+l) disjoint. This fact has been widely used in defining special orderings that make Jacobi methods amenable for parallel processing, and also for proving the global convergence of such processes (see [17,16] etc.). Here we shall use similar ideas for proving sharp quadratic convergence bounds of the serial processes.…”

Section: ) Takes Form a (K+ I) A(k)~ =mentioning

confidence: 98%

On sharp quadratic convergence bounds for the serial Jacobi methods

Hari

1991

Numer. Math.

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Section: ) Takes Form a (K+ I) A(k)~ =mentioning

confidence: 98%

On sharp quadratic convergence bounds for the serial Jacobi methods

Hari

1991

Numer. Math.

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“…The convergence of the cyclic-by-row method is analyzed in [14]. More recently, [15,16] demonstrate that the Jacobi method converges with other ordering schemes when the cyclic-by-row method is convergent. These orderings are intended to simplify parallelization of the method.…”

Section: Jacobi's Methodsmentioning

confidence: 98%

A block Jacobi method on a mesh of processors

Giménez

Hernández

Geijn

et al. 1997

Concurrency: Pract. Exper.

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“…For implementation on multiprocessor arrays cyclic strategies different from the row (or column) cyclic strategy are used ( [3,4,21,25,29,31]). The convergence proofs for these strategies should carry over without difficulties (the proofs are essentially a matter of combinatorics and are independent of the nature of the elementary matrices C used in the transformation).…”

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

confidence: 99%

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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A proof of convergence for two parallel Jacobi SVD algorithms

Cited by 66 publications

References 10 publications

On sharp quadratic convergence bounds for the serial Jacobi methods

On sharp quadratic convergence bounds for the serial Jacobi methods

A block Jacobi method on a mesh of processors

A Jacobi eigenreduction algorithm for definite matrix pairs

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