# American Institute of Mathematical Sciences

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. 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.

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##### 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.
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 $2.0235\times 10^{-11}$
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 $6.3231\times 10^{-17}$
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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