Convergence of The Barzilai-Borwein Method for Solving Slightly Unsymmetric Linear Systems
Keywords:
Unsymmetric linear algebraic equations, Barzilai and Borwein gradient method, symmetric and skew-symmetric matrices, eigenvalues, condition numberAbstract
Due to its simplicity and numerical efficiency, the Barzilai and Borwein (BB) gradient method has received numerous attentions in different scientific fields. In this paper, the sufficient condition for convergence of the BB method when the coefficient matrix of linear algebraic equations is slightly unsymmetric with positive definite symmetric part is presented
References
Barzilai, J. and Borwein, J. M.: Two-point step size gradient methods. IMA Journal of Numerical Analysis, 8, pp. 141–148, 1988.
Dai, Y. H. and Fletcher, R.: On the asymptotic behaviour of some new gradient methods. Mathematical Programming, 103(3), pp541–559, 2005.
Dai, Y. H. and Liao, L. Z.: R-linear convergence of the Barzilai and Borwein method for unconstrained optimization. IMA Journal of Numerical Analysis, 1, pp. 1–10, 2002.
Dennis, J. E. and Schnabel, R. B: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs, NJ, 1983.
Fletcher, R.: Practical Methods of Optimization. Wiley, 2nd edition, 2000.
Raydan, M. M.: On the Barzilai and Borwein choice of steplength for the gradient method. IMA Journal of Numerical Analysis, 13, pp. 321–326, 1993.
Raydan, M. M.: Convergence propreties of the Barzilai and Borwein gradient method. PhD thesis, Rice University, 1991.
Dai, Y. H. and Fletcher, R.: On the asymptotic behaviour of some new gradient methods. Mathematical Programming, 103(3), pp541–559, 2005.
Dai, Y. H. and Liao, L. Z.: R-linear convergence of the Barzilai and Borwein method for unconstrained optimization. IMA Journal of Numerical Analysis, 1, pp. 1–10, 2002.
Dennis, J. E. and Schnabel, R. B: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs, NJ, 1983.
Fletcher, R.: Practical Methods of Optimization. Wiley, 2nd edition, 2000.
Raydan, M. M.: On the Barzilai and Borwein choice of steplength for the gradient method. IMA Journal of Numerical Analysis, 13, pp. 321–326, 1993.
Raydan, M. M.: Convergence propreties of the Barzilai and Borwein gradient method. PhD thesis, Rice University, 1991.
Downloads
Published
2015-06-30
How to Cite
Fathi, B. G. (2015). Convergence of The Barzilai-Borwein Method for Solving Slightly Unsymmetric Linear Systems. Science Journal of University of Zakho, 3(1), 140–144. Retrieved from https://sjuoz.uoz.edu.krd/index.php/sjuoz/article/view/118
Issue
Section
Science Journal of University of Zakho
License
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License [CC BY-NC-SA 4.0] that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work, with an acknowledgment of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online.
