(c
University of Sindh Journal of Information
and Communication Technology (USJICT) Volume 1, Issue 1, October 2017 ISSN: 25215582 Website: http://sujo.usindh.edu.pk/index.php/USJICT/
© Published
by University of Sindh, Jamshoro. 

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
[email protected], [email protected], [email protected]
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 wellknown methods
for comparison i.e., the GaussSeidel 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 GaussSeidel
I.
Introduction and
Preliminaries
Consider the linear system
_{}, (1)
where _{} is a nonsingular 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 wellknown iterative method, i.e., GaussSeidel 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 _{}.
If _{} for all _{},
then _{} is a strictly diagonally dominant matrix (SDD).
If there is a positive diagonal matrix _{} so _{} is a SDD matrix, then _{} is a generalized strictly diagonally dominant
matrix, denoted by GDDM.
Lemma 1.1. (see [5, 14]) If _{} is a GDDM, then _{} is nonsingular and_{} for _{}.
A group
of numerical methods for solving linear system _{} is the splitting methods as follows [6, 8,
14]. Let _{},
where _{} is a nonsingular matrix, then we have the iterative
form,
_{}, _{}
or
_{}, _{} (2)
where _{} is a given initial
vector.
Different
splitting of _{} induce different iterative methods. The
classical iterative methods include:
a)
Jacobi method: _{}, _{},
where _{} is the diagonal part of _{}.
b)
GaussSeidel method: _{}, _{},
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: _{}, _{}, where _{} is a parameter and _{} be as above.
Other iterative methods include AOR,
twostage 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).
Lemma 1.2. (see [5, 6, 8, 14]) The iterative
method (2) is converge for any initial vector _{} if _{}.
Consider the tridiagonal linear
system_{},
where
_{}
and_{}.
The Thomas algorithm for solving such a system is as follows [12, Chapter 3.7]
. Let the LU decomposition of _{} be as:
_{}, _{}.
By
using following relations, the coefficients _{} and _{} can be computed easily.
_{}, _{}, _{}, _{}.
Then
the given tridiagonal system _{} can be reduced into two bidiagonal systems _{} and _{}.
For _{} we have
_{}, _{}, _{},
and for _{} we have
_{}, _{}, _{}.
The algorithm
involves only _{} flops: _{} flops for generate
the LU decomposition and _{}flops for solving the two bidiagonal systems. It is showed in
[12, 13] that when _{} is a DDM or SPD the
algorithm is very stable.
We organize the rest of the paper as follows.
Section 2 gives updated version of iterative method and then some convergence
analysis. In section 3 we use some numerical experiments to show the efficiency
of the new iterative method. The conclusion is drawn in section 4.
Refbacks
 There are currently no refbacks.
Copyright (c) 2017 Dr. Jinrui Guan, Dr. Zubair Ahmed Kalhoro, Dr. Aftab Ahmed Chandio
ISSNE: 25231235, ISSNP: 25215582
Â Copyright Â© University of Sindh, Jamshoro. 2017 All Rights Reserved.Â Â
Printing and Publication by: Sindh University Press.Â
Journal Office, Institute of Information and Communication Technology,Â
University of Sindh, Jamshoro, Sindh, Pakistan. 76080Â
Â Â