Page | 51 Tridiagonal Iterative Method for Linear Systems Jinrui Guan 1 , Zubair Ahmed 2 , Aftab Ahmed Chandio 2 1 Department of Mathematics, Taiyuan Normal University, China 2 Institute of Mathematics and Computer Science, University of Sindh, Jamshoro, Pakistan guanjinrui2012@163.com, zubairabassi@gmail.com, chandio.aftab@usindh.edu.pk Abstract: In this study, we propose a tridiagonal iterative method to solve linear systems based on dominant tridiagonal entries. For solving a tridiagonal system, we incorporated the proposed method with Thomas algorithm in each step of the method. Moreover, this paper presents a comprehensive theoretical analysis, wherein we choose two well-known methods for comparison i.e., the Gauss-Seidel and Jacobi. The numerical experiment shows that our proposed iterative method is a feasible and effective method than the studied methods. Keywords: Iterative method; tridiagonal system; Thomas algorithm, Jacobi and Gauss-Seidel I. INTRODUCTION AND PRELIMINARIES Consider the linear system Ax b  , (1) where nn A R   is a non-singular matrix with dominant tridiagonal parts, i.e., the entries in the tridiagonal parts are very large compared with other entries. In some applications, such as numerical solution of differential equations [4, 6], we encounter such type of the problem in linear systems. The well-known iterative method, i.e., Gauss-Seidel and Jacobi iterative methods are not very effective for such type of systems due to special structure of the nonsingular matrix. In this study, we present an updated version of the iterative method for tridiagonal linear systems. Each step of this method is required for solving a tridiagonal system by Thomas algorithm. We provide some theoretical analysis for this new iterative method. The numerical experiment shows that our proposed iterative method is a feasible and effective method. The following are some notations and preliminaries. Definition 1.1. Let nn A R   . If | | | | ii ij j i a a    for all 1, 2, , i n  L , then A is a strictly diagonally dominant matrix (SDD). If there is a positive diagonal matrix D so AD is a SDD matrix, then A is a generalized strictly diagonally dominant matrix, denoted by GDDM. Lemma 1.1. (see [5, 14]) If A is a GDDM, then A is nonsingular and 0 ii a  for 1, 2, , i n  L . A group of numerical methods for solving linear system Ax b  is the splitting methods as follows [6, 8, 14]. Let A M N   , where M is a non-singular matrix, then we have the iterative form, 1 k k Mx Nx b    , 0,1, k  L or 1 1 1 k k x M Nx M b      , 0,1, k  L (2) where 0 x is a given initial vector. Different splitting of A induce different iterative methods. The classical iterative methods include: a) Jacobi method: M D  , N D A   , where D is the diagonal part of A . b) Gauss-Seidel method: M D L   , N U  , here D is diagonal part of A, U is strictly upper part and L is strictly lower part of triangular matrix A respectively. c) SOR method: 1 M D L    , 1 N D U      , where  is a parameter and DLU be as above. Other iterative methods include AOR, two-stage iterative methods, multisplitting iterative methods, HSS method, QR method, and etc. For more details we refer to [1, 7, 9, 11, 15, 17]. We have the following results for the convergence of the iterative method (2). University of Sindh Journal of Information and Communication Technology (USJICT) Volume 1, Issue 1, October 2017 ISSN: 2521-5582 Website: http://sujo.usindh.edu.pk/index.php/USJICT/ © Published by University of Sindh, Jamshoro