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June 2019, 9(2): 157-171. doi: 10.3934/naco.2019012

A new reprojection of the conjugate directions

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

Received  June 2017 Revised  June 2018 Published  January 2019

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.

Citation: József Abaffy. A new reprojection of the conjugate directions. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 157-171. doi: 10.3934/naco.2019012
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.

show all references

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.

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
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