Research Article
Refinement of Multiparameters Overrelaxation (RMPOR) Method
Gashaye Dessalew , Tesfaye Kebede , Gurju Awgichew , and Assaye Walelign
Department of Mathematics, Bahir Dar University, Bahir Dar, Ethiopia
Correspondence should be addressed to Tesfaye Kebede; tk_ke@yahoo.com
Received 31 May 2021; Accepted 12 August 2021; Published 23 August 2021
Academic Editor: Jia-Bao Liu
Copyright © 2021 Gashaye Dessalew et al. is is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
In this paper, we present refinement of multiparameters overrelaxation (RMPOR) method which is used to solve the linear system
of equations. We investigate its convergence properties for different matrices such as strictly diagonally dominant matrix,
symmetric positive definite matrix, and M-matrix. e proposed method minimizes the number of iterations as compared with
the multiparameter overrelaxation method. Its spectral radius is also minimum. To show the efficiency of the proposed method,
we prove some theorems and take some numerical examples.
1. Introduction
Large and sparse linear systems of the form
Ax � b, (1)
can be solved using iterative methods. One of the techniques
of obtaining iterative methods is splitting the coefficient
matrix. A � M − N and A � D −
k
i�1
E
i
− F are some of the
splitting of the matrix A. Using either of splittings, we can
derive iterative methods which can solve the system of linear
equation (1).
Recently, many researchers used the splitting
A � D −
k
i�1
E
i
− F, where D is diagonal matrix of A,
k
i�1
E
i
is strictly lower triangular matrix of D − A, and F is
strictly upper triangular matrix of D − A, to get the new
iterative methods. Song and Dai [1] introduced multipa-
rameters overrelaxation (MPOR) method, whose special
cases involve many classical iterative methods. e con-
vergence conditions with A being a Hermitian matrix, an
L-matrix, an M-matrix, an H-matrix, and a strictly diago-
nally dominant matrix are derived. Kuang and Ji [2] pre-
sented a two-parameter iterative method called TOR
method, which is effective to give the numerical solution of
partial differential equations. Wang extended the TOR
method to the GTOR method and improved some results of
Ju, Wang, and Zeng. In [3], O’Leary and White proposed
multisplitting methods which are based on several splittings
of the matrix A. More precisely, a multisplitting of A
is defined as a collection of triples (M
k
,N
k
,E
k
),
k � 1, 2, ... ,K, such that for all k, M
k
, N
k
, and E
k
are n × n
matrices, each M
k
is nonsingular, A � M
k
− N
k
, and E
k
is a
diagonal matrix with nonnegative entries satisfying
k
i�1
E
k
� I. e corresponding multisplitting method to
solve (1) is given by the iteration x
m+1
�
k
i�1
E
k
y
m,k
,
m � 0, 1, 2, ..., where M
k
y
m,k
� N
k
x
m
+ b, k � 1, 2, ... ,K.
Similarly, the convergence of the parallel multisplitting TOR
method is studied for M-matrix in [4]. Chang studied the
convergence of the parallel multisplitting TOR method for
H-matrices, but Zhang et al. [5] found some gaps in the
proof of theorem.
ere are other many techniques to get new iterative
methods such as extrapolation, overrelaxation, and accel-
eration. Up to now, a lot of first-order stationary iterative
methods are proposed. ey include the well-known
methods as Jacobi, JOR, Gauss–Seidel, SOR, AOR, GAOR,
TOR, and so on [1].
Partial differential equations (PDEs) can be solved using
finite difference and finite element methods. ese two
methods have deficiencies in accuracy and code imple-
mentation but there is novel efficient matrix approach for
solving the second-order linear matrix partial differential
equations (MPDEs) under given initial conditions which are
Hindawi
Journal of Mathematics
Volume 2021, Article ID 2804698, 10 pages
https://doi.org/10.1155/2021/2804698