A new reprojection of the conjugate directions

József Abaffy

doi:10.3934/naco.2019012

Article Contents

2019, Volume 9, Issue 2: 157-171. Doi: 10.3934/naco.2019012

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A new reprojection of the conjugate directions

József Abaffy^*,

Óbuda University, Institute of Applied Mathematics, H-1034 Budapest, Bécsi út 96/b, Hungary

Received: June 2017

Revised: June 2018

Published: January 2019

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Abstract

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.

Keywords:

Parlett–Kahan's "twice is enough" method,
conjugate directions,
reprojections,
ABS methods.

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

Citation:

Full Text(HTML)

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

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

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

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

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

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

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

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

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

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

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

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

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

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References

[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 Loan, Matrix 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.