A generalized projection iterative method for solving non-singular linear systems

Ashif Mustafa; Manideepa Saha

doi:10.3934/mfc.2022009

Article Contents

2022, Volume 5, Issue 4: 343-350. Doi: 10.3934/mfc.2022009

This issue Previous Article On approximation of Bernstein-Durrmeyer operators in movable interval Next Article CNN models for readability of Chinese texts

A generalized projection iterative method for solving non-singular linear systems

Ashif Mustafa and
Manideepa Saha^,

Department of Mathematics, National Institute of Technology Meghalaya, Shillong 793003, India

^*Corresponding author: Manideepa Saha
^*Corresponding author: Manideepa Saha

Received: June 2021

Revised: January 2022

Early access: April 2022

Published: November 2022

The work was supported by Department of Science and Technology-Science and Engineering Research Board (grant no. ECR/2017/002116)

Abstract / Introduction Full Text(HTML) Figure(0) / Table(3) Related Papers Cited by

Abstract

In this paper, we propose and analyze iterative method based on projection techniques to solve a non-singular linear system $ Ax = b $. In particular, for a given positive integer $ m $, $ m $-dimensional successive projection method ($ m $D-SPM) for symmetric positive definite matrix $ A $, is generalized for non-singular matrix $ A $. Moreover, it is proved that $ m $D-SPM gives better result for large values of $ m $. Numerical experiments are carried out to demonstrate the superiority of the proposed method in comparison with other schemes in the scientific literature.

Keywords:

Mathematics Subject Classification: Primary: 15A06, 65F10.

Citation:

Full Text(HTML)

Table 1. Results for 4.1

Iteration Process	No of Iterations	Relative Residual
GMRES	9	$ 1.2 \times 10^{-7} $
LSQR	7	$ 5.5 \times 10^{-8} $
$ 3 $D-OPM	20	$ 6.7857 \times 10^{-7} $
$ 10 $D-OPM	6	$ 5.9894\times 10^{-7} $
$ 20 $D-OPM	3	$ 5.0539\times 10^{-7} $
$ 50 $D-OPM	2	$ 2.0235\times 10^{-11} $

| Show Table

DownLoad: CSV

Table 2. Results for 4.2

Iteration Process	No of Iterations	Relative Residual
GMRES	9	$ 1.9 \times 10^{-7} $
BiCG	9	$ 2.0 \times 10^{-7} $
LSQR	2	$ 3.2 \times 10^{-16} $
$ 3 $D-OPM	3	$ 4.5922 \times 10^{-7} $
$ 10 $D-OPM	1	$ 9.5332\times 10^{-8} $
$ 20 $D-OPM	1	$ 9.1\times 10^{-15} $
$ 50 $D-OPM	1	$ 6.3231\times 10^{-17} $

| Show Table

DownLoad: CSV

Table 3. Results for 4.3

Iteration Process	No of Iterations	Relative Residual
GMRES	10	$ 2.7 \times 10^{-7} $
BiCG	10	$ 2.8 \times 10^{-7} $
LSQR	13	$ 7.4 \times 10^{-7} $
$ 3 $D-OPM	4	$ 4.6042 \times 10^{-7} $
$ 10 $D-OPM	2	$ 2.62\times 10^{-11} $
$ 20 $D-OPM	1	$ 1.9751\times 10^{-11} $

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511840371.
[2]	G. Hou and L. Wang, A generalized iterative method and comparision results using projection techniques for solving linear systems, Appl. Math. Comput., 215 (2009), 806-817. doi: 10.1016/j.amc.2009.06.004.
[3]	Y.-F. Jing and T.-Z. Huang, On a new iterative method for solving linear systems and comparision results, J. Comput. Appl. Math., 220 (2008), 74-84. doi: 10.1016/j.cam.2007.07.035.
[4]	N. M. Nachtigal, S. C. Reddy and L. N. Trefethen, How fast are nonsymmetric matrix iterations?, SIAM J. Matrix Anal. Appl., 13 (1992), 778-795. doi: 10.1137/0613049.
[5]	C. C. Paige and M. A. Saunders, LSQR: An algorithm for sparse linear equations and sparse least squares, ACM Trans. Math. Software, 8 (1982), 43-71. doi: 10.1145/355984.355989.
[6]	Y. Saad, Iterative Methods for Sparse Linear Systems, 2$^nd$ edition, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2003. doi: 10.1137/1.9780898718003.
[7]	D. K. Salkuyeh, A generalization of the 2D-DSPM for solving linear system of equations, prperint, 2009, arXiv: 0906.1798.
[8]	X. Sheng, Y. Su and G. Chen, A modification of minimal residual iterative method to solve linear systems, Math. Probl. Eng., 2009 (2009), 9pp. doi: 10.1155/2009/794589.
[9]	N. Ujević, A new iterative method for solving linear systems, Appl. Math. Comput., 179 (2006), 725-730. doi: 10.1016/j.amc.2005.11.128.