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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References
de Rijk, P. P. M.: A one-sided Jacobi algorithm for computing the singular value decomposition on a vector computer. SIAM J. Sci. Stat. Comput. 10, 359–371 (1989)
Drmač, Z.: A tangent algorithm for computing the generalized singular value decomposition. SIAM J. Numer. Anal. 35(5), 1804–1832 (1998)
Falk, S., Langemeyer, P.: Das Jacobische Rotations-Verfahren für reel symmetrische Matrizen-Paare I, II. Elektronische Datenverarbeitung 30–43 (1960)
Gose, G.: Das Jacobi Verfahren für A x = λ B x. Zamm 59, 93–101 (1979)
Hari, V.: On cyclic Jacobi methods for the positive definite generalized eigenvalue problem. Ph.D. thesis, University of Hagen (1984)
Hari, V.: On pairs of almost diagonal matrices. Linear Algebra Appl. 148, 193–223 (1991)
Hari, V., Drmač, Z: On scaled almost diagonal hermitian matrix pairs. SIAM J. Matrix Anal. Appl. 18(4), 1000–1012 (1997)
Hari, V., Singer, S., Singer, S.: Block-oriented J-Jacobi methods for Hermitian matrices. Linear Algebra Appl. 433(8–10), 1491–1512 (2010)
Hari, V., Singer, S., Singer, S.: Full block J-Jacobi method for Hermitian matrices. Linear Algebra Appl. 444, 1–27 (2014)
Hari, V.: Convergence to diagonal form of block Jacobi-type methods. Numer. Math. 129(3), 449–481 (2015)
Hari, V., Begović Kovač, E.: Convergence of the cyclic and quasi-cyclic block Jacobi methods. Electron. Trans. Numer. Anal. (ETNA) 47, 107–147 (2017)
Matejaš, J.: Accuracy of the Jacobi method on scaled diagonally dominant symmetric matrices. SIAM J. Matrix Anal. Appl. 31(1), 133–153 (2009)
Matejaš, J.: Accuracy of one step of the Falk-Langemeyer method. Numer. Algorithms 68(4), 645–670 (2015)
Novaković, V., Singer, S., Singer, S.: Blocking and parallelization of the Hari–Zimmermann variant of the Falk–Langemeyer algorithm for the generalized SVD. Parallel Comput. 49, 136–152 (2015)
Rutishauser, H.: The Jacobi method for real symmetric matrices. Handbook for Automatic Computation Series, Volum 2, Linear Algebra, 202–211 (1969). Numer. Math. 9(1), 1–10 (1966). https://doi.org/10.1007/BF02165223. MR 1553948
Shroff, G., Schreiber, R.: On the convergence of the cyclyc Jacobi method for parallel block orderings. SIAM J. Matrix Anal. Appl. 10(3), 326–346 (1989)
Slapničar, I., Hari, V.: On the quadratic convergence of the Falk-Langemeyer method for definite matrix pairs. SIAM J. Matrix Anal. Appl. 12(1), 84–114 (1991)
van der Sluis, A.: Condition numbers and equilibration of matrices. Numer. Math. 14(1), 14–23 (1969)
Veseliċ, K.: A Jacobi eigenreduction algorithm for definite matrix pairs. Numer. Math. 64(1), 241–269 (1993)
Zimmermann, K.: On the convergence of the Jacobi process for ordinary and generalized eigenvalue problems. Ph.D. Thesis, Dissertation No. 4305. ETH, Zürich (1965)
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.
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This work has been fully supported by Croatian Science Foundation under the project: IP_09_2014_3670.
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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