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On the global convergence of the block Jacobi method for the positive definite generalized eigenvalue problem

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Abstract

The paper proves the global convergence of a general block Jacobi method for the generalized eigenvalue problem \(\mathbf {A}x=\lambda \mathbf {B}x\) with symmetric matrices \(\mathbf {A}\), \(\mathbf {B}\) such that \(\mathbf {B}\) is positive definite. The proof is made for a large class of generalized serial strategies that includes important serial and parallel strategies. The sequence of matrix pairs generated by the block method converges to \((\varvec{\varLambda } , \mathbf {I})\) where \(\varvec{\varLambda }\) is a diagonal matrix of the eigenvalues of the initial matrix pair \((\mathbf {A},\mathbf {B})\) and \(\mathbf {I}\) is the identity matrix. First, the convergence to diagonal form is proved. After that several conditions are imposed to ensure the global convergence of the block method.

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References

  1. Hari, V., Begović Kovač, E.: Convergence of the cyclic and quasi-cyclic block Jacobi methods. Electron. Trans. Numer. Anal. 46, 107–147 (2017)

    MathSciNet  MATH  Google Scholar 

  2. Demmel, W.J., Veselić, K.: Jacobi’s method is more accurate than QR. SIAM J. Matrix Anal. Appl. 13, 1204–1245 (1992)

    Article  MathSciNet  Google Scholar 

  3. Slapničar, I.: Accurate symmetric eigenreduction by a Jacobi method. Ph.D. thesis, University of Hagen (1992)

  4. Drmač, Z.: Implementation of Jacobi rotations for accurate singular value computation in floating-point arithmetic. SIAM J. Sci. Stat. Comp. 18(4), 1200–1222 (1997)

    Article  MathSciNet  Google Scholar 

  5. Demmel, W.J., Gu, M., Eisenstat, S., Slapničar, I., Veselić, K., Drmač, Z.: Computing the singular value decomposition with high relative accuracy. Linear Algebra Appl. 299, 21–80 (1999)

    Article  MathSciNet  Google Scholar 

  6. Slapničar, I.: Highly accurate symmetric eigenvalue decomposition and hyperbolic SVD. Linear Algebra Appl. 358, 387–424 (2002)

    Article  MathSciNet  Google Scholar 

  7. Dopico, F.M., Molera, J.M., Moro, J.: An orthogonal high relative accuracy algorithm for the symmetric eigenproblem. SIAM J. Matrix Anal. Appl. 25(2), 301–351 (2003)

    Article  MathSciNet  Google Scholar 

  8. Matejaš, J.: Accuracy of the Jacobi method on scaled diagonally dominant symmetric matrices. SIAM J. Matrix Anal. Appl. 31(1), 133–153 (2009)

    Article  MathSciNet  Google Scholar 

  9. Matejaš, J., Hari, V.: On high relative accuracy of the Kogbetliantz method. Linear Algebra Appl. 464, 100–129 (2015)

    Article  MathSciNet  Google Scholar 

  10. Hari, V.: Globally convergent Jacobi methods for positive definite matrix pairs. Numer. Algor. 79(1), 221–249 (2018). https://doi.org/10.1007/s11075-017-0435-5

    Article  MathSciNet  MATH  Google Scholar 

  11. de Rijk, P.P.M.: A one-sided Jacobi algorithm for computing the singular-value decomposition on a vector computer. SIAM J. Sci. Stat. Comp. 10(2), 359–371 (1989)

    Article  MathSciNet  Google Scholar 

  12. Sameh, A.H.: On Jacobi and Jacobi-like algorithms for parallel computer. Math. Comp. 25, 579–590 (1971)

    Article  MathSciNet  Google Scholar 

  13. Hari, V., Veselić, K.: On Jacobi methods for singular value decompositions. SIAM J. Sci. Stat. Comput. 8(5), 741–754 (1987)

    Article  MathSciNet  Google Scholar 

  14. Eberlein, P.J., Park, H.: Efficient implementation of Jacobi algorithms and Jacobi sets on distributed memory architectures. J. Parallel Distrib. Comput. 8, 358–366 (1990)

    Article  Google Scholar 

  15. Hari, V., Zadelj-Martić, V.: Parallelizing the Kogbetliantz method: a first attempt. JNAIAM 2(1–2), 49–66 (2007)

    MathSciNet  MATH  Google Scholar 

  16. Drmač, Z.: A posteriori computation of the singular vectors in a preconditioned Jacobi SVD algorithm. IMA J. Numer. Anal. 19, 191–213 (1999)

    Article  MathSciNet  Google Scholar 

  17. Hari, V.: Accelerating the SVD block-Jacobi method. Computing 75, 27–53 (2005)

    Article  MathSciNet  Google Scholar 

  18. Drmač, Z., Veselić, K.: New fast and accurate Jacobi SVD algorithm I. SIAM J. Matrix Anal. Appl. 29(4), 1322–1342 (2007)

    Article  MathSciNet  Google Scholar 

  19. Drmač, Z., Veselić, K.: New fast and accurate Jacobi SVD algorithm II. SIAM J. Matrix Anal. Appl. 29(4), 1343–1362 (2007)

    Article  MathSciNet  Google Scholar 

  20. Bečka, M., Okša, G., Vajtersic, M.: Parallel Block-Jacobi SVD Methods. In: Berry, M., et al. (eds.) High-Performance Scientific Computing, pp. 185–197. Springer, London (2012)

    Chapter  Google Scholar 

  21. Bečka, M., Okša, G., Vajtersic, M.: New dynamic orderings for the parallel one-sided block-Jacobi SVD algorithm. Parallel Process. Lett. 25(2), 1–19 (2015)

    Article  MathSciNet  Google Scholar 

  22. Novaković, J., 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)

    Article  MathSciNet  Google Scholar 

  23. Novaković, J., Singer, S.: Implicit Hari-Zimmermann algorithm for the generalized SVD on the GPUs. Int. J. High. Perform. Comput. Appl. 35(2), 170–205 (2021). https://doi.org/10.1177/1094342020972772

    Article  Google Scholar 

  24. Singer, S., Di Napoli, E., Novaković, V., Čaklović, G.: The LAPW method with eigendecomposition based on the Hari-Zimmermann generalized hyperbolic SVD. SIAM J. Sci. Comput. 42(5), C265–C293 (2020). https://doi.org/10.1137/19M1277813

    Article  MathSciNet  MATH  Google Scholar 

  25. Hari, V.: Convergence of a block-oriented quasi-cyclic Jacobi method. SIAM J. Matrix Anal. Appl. 29(2), 349–369 (2007)

    Article  MathSciNet  Google Scholar 

  26. Drmač, Z.: A global convergence proof for cyclic Jacobi methods with block rotations. SIAM J. Matrix Anal. Appl. 31(3), 1329–1350 (2009)

    Article  MathSciNet  Google Scholar 

  27. Hari, V., Singer, S., Singer, S.: Block-oriented \(J\)-Jacobi methods for hermitian matrices. Lin. Alg. and Its Appl. 433(8–10), 1491–1512 (2010)

    Article  MathSciNet  Google Scholar 

  28. Bujanović, Z.: Drmač, Z: A contribution to the theory and practice of the block Kogbetliantz method for computing the SVD. BIT 52(4), 827–849 (2012)

    Article  MathSciNet  Google Scholar 

  29. Hari, V., Singer, S., Singer, S.: Full block \(J\)-Jacobi method for hermitian matrices. Linear Algebra Appl. 444, 1–27 (2014)

    Article  MathSciNet  Google Scholar 

  30. Hari, V.: Convergence to diagonal form of block Jacobi-type methods. Numer. Math. 129(3), 449–481 (2015)

    Article  MathSciNet  Google Scholar 

  31. Okša, G., Yamamoto, Y., Vajteršic, M.: Asymptotic quadratic convergence of the serial block-Jacobi EVD algorithm for Hermitian matrices. Numer. Math. 136(4), 1071–1095 (2017)

    Article  MathSciNet  Google Scholar 

  32. Hansen, E.R.: On cyclic Jacobi methods. SIAM J. Appl. Math. 11, 449–459 (1963)

    Article  MathSciNet  Google Scholar 

  33. Luk, F., Park, H.: On the equivalence and convergence of parallel Jacobi SVD algorithms. IEEE Tran. Comp. 38(6), 806–811 (1989)

    Article  Google Scholar 

  34. Luk, F., Park, H.: On parallel Jacobi orderings. SIAM J. Sci. Stat. Comput. 10, 18–26 (1989)

    Article  MathSciNet  Google Scholar 

  35. Shroff, G., Schreiber, R.: On the convergence of the cyclic Jacobi method for parallel block orderings. SIAM J. Matrix Anal. Appl. 10, 326–346 (1989)

    Article  MathSciNet  Google Scholar 

  36. van der Sluis, A.: Condition numbers and equilibration of matrices. Numer. Math. 14(1), 14–23 (1969)

    Article  MathSciNet  Google Scholar 

  37. Drmač, Z.: A Tangent Algorithm for Computing the Generalized Singular Value Decomposition. SIAM J. Numer. Anal. 35(5), 1804–1832 (1998)

    Article  MathSciNet  Google Scholar 

  38. Falk, S., Langemeyer, P.: Das Jacobische Rotations-Verfahren für reel symmetrische Matrizen-Paare I. II. Elektronische Datenverarbeitung 7, 30–34 (1960)

    MATH  Google Scholar 

  39. 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)

    Article  MathSciNet  Google Scholar 

  40. Forsythe, G.E., Henrici, P.: The cyclic Jacobi method for computing the principal values of a complex matrix. Trans. Am. Math. Soc. 94, 1–23 (1960)

    Article  MathSciNet  Google Scholar 

  41. Eberlein, P.J.: A Jacobi method for automatic computation of eigenvalues and eigenvectors of an arbitrary matrix. Siam J. 10, 74–88 (1962)

    MathSciNet  MATH  Google Scholar 

  42. Eberlein, P.J., Boothroyd, J.: Solution to the eigenproblem by a norm-reducing Jacobi-type method. Numer. Math. 11, 1–12 (1968)

    Article  Google Scholar 

  43. Hari, V., Begović Kovač, E.: On the convergence of complex Jacobi methods. Linear Multilinear Algebra 69(3), 489–514 (2019). https://doi.org/10.1080/03081087.2019.1604622

    Article  MathSciNet  MATH  Google Scholar 

  44. Hari, V.: On pairs of almost diagonal matrices. Linear Algebra Appl. 148, 193–223 (1991)

    Article  MathSciNet  Google Scholar 

  45. Hari, V.: On cyclic Jacobi methods for the positive definite generalized eigenvalue problem. Ph.D.  thesis, University of Hagen (1984)

  46. Mascarenhas, W.: On the convergence of the Jacobi method for arbitrary orderings. SIAM J. Matrix Anal. Appl. 16(4), 1197–1206 (1995)

    Article  MathSciNet  Google Scholar 

  47. Hari, V.: On the global convergence of the complex HZ method. SIAM J. Matrix Anal. Appl. 40(4), 1291–1310 (2019). https://doi.org/10.1137/19M1265594

  48. Hari, V.: Complex Cholesky–Jacobi algorithm for PGEP. AIP Conf. Proc. 2116, 450011 (2019). https://doi.org/10.1063/1.5114478

    Article  Google Scholar 

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Acknowledgements

The author is grateful to anonymous referees and editor Raf Vandebril for very helpful comments, including a suggestion on how to organize and improve the paper. He is thankful to Sanja Singer for reading the paper.

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  1. Department of Mathematics, Faculty of Science, University of Zagreb, Zagreb, Croatia

    Vjeran Hari

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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. On the global convergence of the block Jacobi method for the positive definite generalized eigenvalue problem. Calcolo 58, 24 (2021). https://doi.org/10.1007/s10092-021-00415-8

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