Semiconvergence of Alternating Iterative Methods for Singular Linear Systems

Authors: Jing Wu

Abstract:

In this paper, we discuss semiconvergence of the alternating iterative methods for solving singular systems. The semiconvergence theories for the alternating methods are established when the coefficient matrix is a singular matrix. Furthermore, the corresponding comparison theorems are obtained.

Keywords: Alternating iterative method, Semiconvergence, Singular matrix.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1088186

References:

[1] M. Benzi, D.B. Szyld, Existence and uniqueness of splittings for stationary iterative methods with application to alternating methods, Numer. Math, 76 (1997): 39-321.
[2] A.C. Aitken, On the iterative solution of a system of linear equations, Proceedings of the Royal Society of Edinburgh, Sec. A 63(1950): 52-60.
[3] D.M. Yong, D.R. Kincaid,A new class of parallel alternatingtype iterative methods, J.Comput. Appl. Math., 74 (1996): 331-344.
[4] G. I. Marchuk, Splittting and alternating direction methods, in: P.G. Ciarlet, J.L. Lions (Eds), Handbook of Numerical Analysis, vol. I, North Holland, New York, NY, 1990, PP. 197-462.
[5] J.-J. Climent, C. Perea, Convergence and comparison theorems for a generalized alternating iterative method, Appl. Math. Comput., 143 (2003): 1-14.
[6] J.-J. Climent, C. Perea, L. Tortosa, A. Zamora, Convergence theorems for parallel alternating iterative methods, Appl. Math. Comput., 148 (2004):497-517.
[7] C.L. Wang, T.Z. Huang, New convergence results for alternating methods, J.Comput. Appl. Math., 135 (2001): 325-333.
[8] Y. Song, Semiconvergence of extrapolated iterative methods for singular linear systems,J.Comput. Appl. Math., 106 (1999): 117-129.
[9] Y. Song, Semiconvergence of nonnegative splittings for singular matrices, Numer. Math., 85 (2000): 109-127.
[10] Y. Huang, Y. Song, Semiconvergence of block AOR method for singular. Appl. Math. Comput., 189 (2007): 1637-1647.
[11] A. Berman, R.J. Plemmons, Nonnegative matrices in the mathematical sciences, Academic Press, New York.
[12] R.S. Varga, Matrix iterative analysis, Prentice-Hall, Englewood Cliffs, N.J.
[13] Y. Song, Comparisons of nonnegative splittings of matrices, Lin. Alg. Appl., 154-156 (1991): 433-455.
[14] M. Neumann, R.F. Plemmons, Convergent nonnegative matrices and iterative methods for consisternt linear systems, Numer. Math.,31 (1978): 265-279.
[15] Z.H. Cao, Variational iterative methods, www.sciencep.com., (2006).
[16] P.J. Lanzkron, D.J. Rose, D.B. Szyld, Convergence of nested classical iterative mehods for linear systems, Numer. Math., 58 (1991): 685-702.
[17] J.M. Ortega, Numerical Analysis, A Second Course, Academic Press.,(1972).
[18] L. Elsner, Comparisons of weak regular splittings and multisplitting methods. Numer. Math., 56 (1989): 283-289.
[19] L. Wang, Semiconvergence of two-stage iterative methods for singular linear systems, Lin. Alg. Appl., 422 (2007): 824-838.