doi: 10.3934/mfc.2022009
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A generalized projection iterative method for solving non-singular linear systems

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

*Corresponding author: Manideepa Saha

Received  June 2021 Revised  January 2022 Early access April 2022

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

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.

Citation: Ashif Mustafa, Manideepa Saha. A generalized projection iterative method for solving non-singular linear systems. Mathematical Foundations of Computing, doi: 10.3934/mfc.2022009
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. NachtigalS. 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.

show all references

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. NachtigalS. 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.

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} $
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} $
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} $
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} $
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} $
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} $
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