Pure and Applied Mathematics Journal
Volume 4, Issue 1, February 2015, Pages: 26-31
Received: Dec. 29, 2014;
Accepted: Jan. 25, 2015;
Published: Feb. 2, 2015
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I. K. Youssef, Math. Dept. Faculty of Science Ain Shams Uni., Cairo, Egypt
M. M. Farid, Basic Science Dept. British University in Egypt, Elshorouk City, Cairo, Egypt
A new variant of the accelerated over relaxation (AOR) method for solving systems of linear algebraic equations, the KAOR method is established. The treatment depends on the use of the extrapolation techniques introduced by Hadjidimos (1978) and the KSOR introduced by Youssef (2012) and its modified version published in (2013). The KAOR is an extrapolation version of the KSOR. This approach has enables us in establishing eigenvalue functional relations concerning the eigenvalues of the iteration matrices and their spectral radii. Moreover, our eigenvalue functional relation depends directly on the eigenvalues of Jacobi iteration matrix. The proposed method considers the advantages of the AOR in addition to those of the KSOR. A graphical representation of the behavior of the spectral radius near the minimum value illustrates the smoothness in the choice of optimum parameters.
I. K. Youssef,
M. M. Farid,
On the Accelerated Overrelaxation Method, Pure and Applied Mathematics Journal.
Vol. 4, No. 1,
2015, pp. 26-31.
A. Hadjidimos, Accelerated Overelaxation Method, Math. Comput. , 32, pp149-157, 1978.
D.M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, 1971.
R.S. Varga, Matrix Iterative Analysis, prentice-Hall, Englewood cliffs, N. J., 1962.
A. Hadjidimos, A. Yeyios, The Principale of Extrapolation in Connection With the Accelerated Overrelaxation Method, Linear Algebra and its Applications 30, 115-128 , 1980.
N. M. Missirlis, D. J. Evans, The Extrapolated Successive Overrelaxation (ESOR) Method for Consistently Ordered Matrices, Internat. J. Math. & Math. sci. Vol. 7, No. 2, pp. 361-370, 1984.
I.K. Youssef, On the Successive Overrelaxation method, J. Math. Stat., 8 pp.176-184, 2012.
I.K. Youssef, A.A. Taha, On the Modified Successive Overrelaxation Method, Appl. Math.Comput., 219, pp. 4601-4613, 2013.
G. Avdelas and A. Hadjidimos, Optimum Accelerated Overrelaxation Method in a Special Case, Mathematics of computation Vol. 36, No. 153, 1981.
M. Madalena Martins, On an Accelerated Over relaxation Iterative Method for Linear Systems With Strictly Diagonally Dominant Matrix, Mathematics of Computation, Vol. 35, No. 152, pp. 1269-1273, 1980.
A.J. Hughes Hallet, The Convergence of Accelerated Overrelaxation Iterations, Mathematics of Computation, Vol. 47, No. 175, pp. 219-223, 1986.
Shi-liang Wu, Yu-Jun Liu, A new Version of the Accelerated Overrelaxation Iterative Method, Journal of Applied Mathematics, 2014.