Convergence on Gauss-Seidel iterative methods for linear systems with general H-matrices
Volume
1081-3810
Convergence on Gauss-Seidel iterative methods for linear systems with general H-matrices
Cheng-yi Zhang 0 1 2
Dan Ye 0 1 2
Cong-Lei Zhong 0 1 2
0 Xi'an Polytechnic University
1 Henan University of Science and Technology
2 School of Science, Xi'an Polytechnic University
SHUANGHUA SHUANGHUA Xi'an Polytechnic University Follow this and additional works at: https://repository.uwyo.edu/ela Part of the Numerical Analysis and Computation Commons Recommended Citation
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http://math.technion.ac.il/iic/ela
CONVERGENCE ON GAUSS-SEIDEL ITERATIVE METHODS FOR
LINEAR SYSTEMS WITH GENERAL H −MATRICES∗
CHENG-YI ZHANG† , DAN YE‡ , CONG-LEI ZHONG§ , AND SHUANGHUA LUO¶
AMS subject classifications. 15A15, 15F10.
1. Introduction. In this paper, we consider the solution methods for the system
of n linear equations
(1.1)
Ax = b,
where A = (aij ) ∈ Cn×n and is nonsingular, b, x ∈ Cn and x unknown. Let us recall
the standard decomposition of the coefficient matrix A ∈ Cn×n,
A = DA − LA − UA,
where DA = diag(a11, a22, . . . , ann) is a diagonal matrix, LA and UA are strictly
lower and strictly upper triangular matrices, respectively. If aii 6= 0 for all i ∈ hni =
∗Received by the editors on October 13, 2014. Accepted for publication on September 12, 2015.
Handling Editor: James G. Nagy.
†Department of Mathematics and Mechanics of School of Science, Xi’an Polytechnic
University, Xi’an, Shaanxi, 710048, P.R. China (). Supported by the Scientific
Research Foundation (BS1014), the Education Reform Foundation of Xi’an Polytechnic University
(2012JG40), and National Natural Science Foundations of China (11201362 and 11271297).
‡School of Science, Xi’an Polytechnic University, Xi’an, Shaanxi, 710048, P.R. China.
§School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang,
Henan, 417003, P.R. China.
¶Department of Mathematics and Mechanics of School of Science, Xi’an Polytechnic University,
Xi’an, Shaanxi, 710048, P.R. China. Supported by the Science Foundation of the Education
Department of Shaanxi Province of China (14JK1305).
{1, 2, . . . , n}, the Jacobi iteration matrix associated with the coefficient matrix A is
the forward, backward and symmetric Gauss-Seidel (FGS-, BGS- and SGS-) iteration
matrices associated with the coefficient matrix A are
C.-Y. Zhang, D. Ye, C.-L. Zhong, and S. Luo
HJ = DA−1(LA + UA);
HF GS = (DA − LA)−1UA,
HBGS = (DA − UA)−1LA,
HSGS
=
=
HBGS HF GS
(DA − UA)−1LA(DA − LA)−1UA,
844
(1.2)
(1.3)
and
(1.4)
(1.5)
respectively. Then, the Jacobi, FGS, BGS and SGS iterative method can be denoted
the following iterative scheme:
x(i+1) = Hx(i) + f,
i = 0, 1, 2, . . .
where H denotes iteration matrices HJ , HF GS , HBGS and HSGS, respectively,
correspondingly, f is equal to DA−1b, (DA − LA)−1b, (DA − UA)−1b and (DA −
UA)−1DA(DA − LA)−1b, respectively. It is well-known that (1.5) converges for any
given x(0) if and only if ρ(H) < 1 (see [
11
]), where ρ(H) denotes the spectral radius of
the iteration matrix H. Thus, to establish the convergence results of iterative scheme
(1.5), we mainly study the spectral radius of the iteration matrix in the iterative
scheme (1.5).
As is well known in some classical textbooks and monographs, see [
11
], Jacobi
and Gauss-Seidel iterative methods for linear systems with Hermitian positive
definite matrices, strictly or irreducibly diagonally dominant matrices and invertible
H−matrices (generalized strictly diagonally dominant matrices) are convergent.
Recently, the class of strictly or irreducibly diagonally dominant matrices and invertible
H−matrices has been extended to encompass a wider set, known as the set of
general H−matrices. In a recent paper, Ref. [
2, 3, 4
], a partition of the n × n general
H−matrix set, Hn, into three mutually exclusive classes was obtained: The Invertible
class, HnI , where the comparison matrices of all general H−matrices are nonsingular,
the Singular class, HnS , formed only by singular H−matrices, and the Mixed class,
HnM , in which singular and nonsingular H−matrices coexist. Lately, Zhang in [
16
]
proposed some necessary and sufficient conditions for convergence on Jacobi iterative
methods for linear systems with general H−matrices.
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A problem has to be proposed, i.e., whether Gauss-Seidel iterative methods for
linear systems with nonstrictly diagonally dominant matrices and general H−matrices
are convergent or not. Let us investigate the following examples.
Example 1.1. Assume that either A or B is the coefficient matrix of linear
2 1 1 2 −1 −1
system (1.1), where A = −1 2 1 and B = 1 2 −1 . It is verified
−1 −1 2 1 1 2
that both A and B are nonstrictly diagonally dominant and nonsingular. Direct
computations yield that ρ(HFAGS ) = ρ(HBBGS ) = 1, while ρ(HBAGS ) = ρ(HFBGS ) =
0.3536 < 1 and ρ(HSAGS ) = ρ(HSBGS ) = 0.5797 < 1. This shows that BGS and
SGS iterative methods for the matrix A are convergent, while the same is not FGS
iterative method for A. However, (...truncated)