Convergence of Triple Accelerated Over-Relaxation (TAOR) Method for M-Matrix Linear Systems

Subject Areas : Numerical Analysis

KHADEEJAH  AUDU ¹ (Department of Mathematics, School of Physical Sciences, Federal University of Technology, Minna, Nigeria)
Yusuph  Yahaya ² (Department of Mathematics, School of Physical Sciences, Federal University of Technology, Minna, Nigeria)
Rufus  Adeboye ³ (Department of Mathematics, School of Physical Sciences, Federal University of Technology, Minna, Nigeria)
Usman  Abubakar ⁴ (Department of Mathematics, School of Physical Sciences, Federal University of Technology, Minna, Nigeria)

Received: 2021-02-04 Accepted : 2021-09-19 Published : 2021-06-01

Keywords: TAOR Method, $ M- $matrix, Linear systems, Spectral Radius, Convergence Conditions,

Abstract :

In this paper, we propose some necessary conditions for convergence of Triple Accelerated Over-Relaxation (TAOR) method with respect to $M-$ coefficient matrices. The theoretical approach for the proofs is analyzed through some standard procedures in the literature. Some numerical experiments are performed to show the efficiency of our approach, and the results obtained compared favourably with those obtained through the existing methods in terms of spectral radii of their iteration matrices.

References:

Audu, K. J., Yahaya, Y. A., Adeboye, K. R., Abubakar, U. Y., & Ndanusa, A., (2020). Triple Accelerated Over-relaxation Method for System of Linear Equations. Journal of Science, Technology, Mathematics and Education, 16 (2), 137-146.

Axelsson, O., (1996). Iteration Solution Methods. Cambridge University Press, USA.

Berman, A., & Plemmon, R. J., (1994).  Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, USA.

Hadjidimos, A., (1978). Accelerated Over-relaxation Method. Mathematics of Computation, 32, 149-57.

Hadjidimos, A., (1980). Some basic results on -matrices in connection with the Accelerated Over-relaxation method. Computing, 24, 259-268.

Liu, Q., & Chen, G., (2010). Convergence Analysis of Preconditioned AOR iterative method for Linear System. Mathematical Problems in Engineering, 341982, 1-14.

Mayaki, Z. & Ndanusa, A., (2019). Modified Successive Over-Relaxation (SOR) Type method for M- matrices. Science World Journal, 14 (4), 1-5.

Saad, Y. (1995). Iterative Method for Sparse Linear Systems. PWS Press, New York.

Salkuyeh, D. K., (2011). Generalised AOR Method for solving System of Linear Equations. Australian Journal of Basic and Applied Sciences. 5(3), 351-358.

Shi-Guang, Z., & Ting, Z. (2017). Comparison Theorems between the Preconditioned Gauss-Seidel method and the AOR method for -matrices. Advance in Engineering Research, 136, 446-449.

 Varga, R. S., (2000). Matrix Iterative Analysis. Springer Berlin, Germany.

  Vatti, V. B. K., Rao, G. C., & Srinesh, P. S., (2019). Parametric Accelerated Over-Relaxation (PAOR) Method. Numerical Optimization in Engineering and Sciences, 979, 283-288.

Wang, L., & Song, Y., (2009). Preconditioned AOR Iterative methods for M-matrices. Journal of Computational and Applied Numerical Mathematics, 226(1) 114 -124.

Wu, M., Wang, L., & Song, Y., (2007). Preconditioned AOR Iterative method for Linear System. Applied Numerical Mathematics, 57(5-7), 672-685.

Wu, S., & Liu, Y., (2014). A New Version of Accelerated Over-relaxation iterative method. Journal of Applied Mathematics, 725360, 1-6.

Xue, Q., Gao, X., & Liu, X. (2013, December). Preconditioned AOR Iterative method for -matrix. In 2013 Ninth International Conference on Computational Intelligence and Security (pp. 372-376). IEEE.

 Young, D. M., (2014). Iterative Solutions of Large Linear Systems, Elsevier.

Youssef, I. K., & Farid, M. M., (2015). On the Accelerated Over-relaxation method. Pure and Applied Mathematics Journal, 4, 26-31.