{"title":"On Algebraic Structure of Improved Gauss-Seidel Iteration","authors":"O. M. Bamigbola, A. A. Ibrahim","country":null,"institution":"","volume":94,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":1296,"pagesEnd":1302,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/9999476","abstract":"

Analysis of real life problems often results in linear
\r\nsystems of equations for which solutions are sought. The method to
\r\nemploy depends, to some extent, on the properties of the coefficient
\r\nmatrix. It is not always feasible to solve linear systems of equations
\r\nby direct methods, as such the need to use an iterative method
\r\nbecomes imperative. Before an iterative method can be employed
\r\nto solve a linear system of equations there must be a guaranty that
\r\nthe process of solution will converge. This guaranty, which must
\r\nbe determined apriori, involve the use of some criterion expressible
\r\nin terms of the entries of the coefficient matrix. It is, therefore,
\r\nlogical that the convergence criterion should depend implicitly on the
\r\nalgebraic structure of such a method. However, in deference to this
\r\nview is the practice of conducting convergence analysis for Gauss-
\r\nSeidel iteration on a criterion formulated based on the algebraic
\r\nstructure of Jacobi iteration. To remedy this anomaly, the Gauss-
\r\nSeidel iteration was studied for its algebraic structure and contrary
\r\nto the usual assumption, it was discovered that some property of the
\r\niteration matrix of Gauss-Seidel method is only diagonally dominant
\r\nin its first row while the other rows do not satisfy diagonal dominance.
\r\nWith the aid of this structure we herein fashion out an improved
\r\nversion of Gauss-Seidel iteration with the prospect of enhancing
\r\nconvergence and robustness of the method. A numerical section is
\r\nincluded to demonstrate the validity of the theoretical results obtained
\r\nfor the improved Gauss-Seidel method.<\/p>\r\n","references":null,"publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 94, 2014"}