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A new reprojection of the conjugate directions

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  • In the paper we apply a Parlett–Kahan's "twice is enough" type algorithm to conjugate directions. We give a lower bound of the digits of precision of the conjugate directions, too.

    Mathematics Subject Classification: Primary: 15A06, 15A18; Secondary: 15A23.

    Citation:

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  • Figure 1.  (A) S2Hssz without CDRA and with reprojections in the dyads. (B) S2rsz without CDRA and without reprojections in the dyads

    Figure 2.  (A) S2HSsz with CDRA and reprojections in the dyads. (B) S2rsz with CDRA and without reprojections in the dyads

    Figure 3.  (A) S2asz with CDRA and reprojections in the dyads. (B) S2a with CDRA and without reprojections in the dyads

    Figure 4.  (A) S2LU with CDRA and reprojections in the dyads. (B) S2esz with CDRA and without reprojections in the dyads

    Figure 5.  (A) S2Hpsz with CDRA and reprojections in the dyads. (B) S2psz with CDRA and without reprojections in the dyads

    Figure 6.  (A) S2HSsz with CDRA and reprojections in the dyads. (B) S2rsz with CDRA and without reprojections in the dyads

    Figure 7.  (A) S2asz with CDRA and reprojections in the dyads. (B) S2a with CDRA and without reprojections in the dyads

    Figure 8.  (A) S2LU with CDRA and reprojections in the dyads. (B) S2esz with CDRA and without reprojections in the dyads

    Figure 9.  (A) S2Hpsz with CDRA and reprojections in the dyads. (B) S2psz with CDRA and without reprojections in the dyads

    Figure 10.  (A) S2HSsz with CDRA and reprojections in the dyads. (B) S2rsz with CDRA and without reprojections in the dyads

    Figure 11.  (A) S2asz with CDRA and reprojections in the dyads. (B) S2a with CDRA and without reprojections in the dyads

    Figure 12.  (A) S2LU with CDRA and reprojections in the dyads. (B) S2esz with CDRA and without reprojections in the dyads

    Figure 13.  (A) S2Hpsz with CDRA and reprojections in the dyads. (B) S2psz with CDRA and without reprojections in the dyads

  • [1] J. Abaffy and E. Spedicato, ABS Projections Algorithms: Mathematical Techniques for Linear and Nonlinear Algebraic Equations, Ellis Horwood Ltd, Chichester, England, 1989.
    [2] J. Abaffy, Reprojection of the conjugate directions in ABS classes Part I, Acta Polytechnica Hungarica, 13 (2016), 7-24. 
    [3] J. Abaffy, Reprojection of the Conjugate Directions in the ABS Class: Part Ⅱ, https://aisberg.unibg.it/handle/10446/86246.
    [4] H. Golub and  C. F. Van LoanMatrix Computation, North Oxford Academic Press, Oxford, 1983. 
    [5] C. S. Hegedűs, Reorthogonalization Methods Revisited, Acta Polytechnica Hungarica, 12 (2015), 7-26. 
    [6] M. R. Hestenes and E. L. Stiefel, Methods of conjugate gradients for solving linear systems, J. Res. Nat. Bur. Standards Sect, 5 (1952), 409-436. 
    [7] B. N. Parlett, The symmetric Eigenvalue Problem, Englewood Cliffs, N. J. Prentice-Hall, 1980.
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