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Globally convergent Jacobi methods for positive definite matrix pairs

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Abstract

The paper derives and investigates the Jacobi methods for the generalized eigenvalue problem A x = λ B x, where A is a symmetric and B is a symmetric positive definite matrix. The methods first “normalize” B to have the unit diagonal and then maintain that property during the iterative process. The global convergence is proved for all such methods. That result is obtained for the large class of generalized serial strategies from Hari and Begović Kovač (Trans. Numer. Anal. (ETNA) 47, 107–147, 2017). Preliminary numerical tests confirm a high relative accuracy of some of those methods, provided that both matrices are positive definite and the spectral condition numbers of Δ A AΔ A and Δ B BΔ B are small, for some nonsingular diagonal matrices Δ A and Δ B .

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Acknowledgements

The author is indebted to an anonymous reviewer for providing insightful comments and remarks. He is also thankful to V. Novaković for reading and improving the text of the manuscript.

Funding

This work has been fully supported by Croatian Science Foundation under the project: IP_09_2014_3670.

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Correspondence to Vjeran Hari.

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Hari, V. Globally convergent Jacobi methods for positive definite matrix pairs. Numer Algor 79, 221–249 (2018). https://doi.org/10.1007/s11075-017-0435-5

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  • DOI: https://doi.org/10.1007/s11075-017-0435-5

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