1. Introduction
Solving a large linear system is one of the challenges of most modeling problems today. A linear system can be expressed in the format:
where
is a matrix of coefficient,
is a column of constants and
is an unknown vector. The relation
denotes the exact solution of problem (1) in a situation where the matrix of coefficients is not singular. It is common knowledge that direct methods for solving such systems require around
operations, which makes them unsuitable for large sparse systems. It appears that iterative approaches are the best option, especially when convergence with the requisite precision is attained within
steps [
1,
2,
3,
4]. The partitioning of
gives
in which
is the diagonal component,
and
are the strictly upper and bottom triangular constituents of
respectively. A matrix reformulation of conventional stationary iterative methods [
5] is employed. Three of these standard iterative approaches are of particular relevance to our current endeavor:
Gauss–Seidel Technique [
6,
7,
8]
Gauss–Seidel Refinement Technique
Gauss–Seidel Second Refinement Technique
Iteration matrices in stationary iterative methods are always the same. From the second step onwards, the computational expenses per iteration are at most
because the iteration matrix is only calculated once and then reused (much smaller for sparse matrices). Any iterative method’s convergence speed can be increased by employing the idea of refinement of an iterative process [
9,
10,
11].
Iterative approaches are unquestionably the most effective approach to employ when solving huge sparse linear systems. However, such an approach may require several rounds to converge, which may reduce computer storage and computing performance [
12]. In such cases, it is necessary to modify or accelerate existing methods in order to achieve approximate answers with rapid convergence. This motivated the current study to offer an improved technique capable of providing better estimated solutions quickly. A new Gauss–Seidel refinement method is presented in this paper. The pace of convergence and the influence of the proposed refinement technique on certain matrices are investigated. As we will see, the spectral radius of the iteration matrix decreased while the convergence rate increased.
2. Methodology
Considering large linear systems of (1), combination of (1) and (2) process gives the classical first-degree Gauss–Seidel iteration method
The general Refinement approach is expressed as
Substitution of (6) into
in (7), gives the Refinement of Gauss–Seidel [
4] as
Modification of (8) results into (9) [
5]
We remodel (9) to obtain (10)
Next, (10) is simplified to obtain
Equation (11) is called Third Refinement of Gauss–Seidel (TRGS) technique. The iteration matrix of TRGS is denoted as . The method converges if its spectral radius is less than one, represented as . In addition, the closer the spectral radius is to one, the faster the convergence.
2.1. Convergence of Third-Refinement of Gauss–Seidel (TRGS)
Theorem 1. If is strictly diagonally dominant matrix {SDD}, then the third refinement of Gauss–Seidel (TRGS) method converges for any choice of the initial approximation .
Proof. Applying the idea of [
4,
13], let
be the exact solution of the linear system of the form (1). We know that if
is SDD matrix and
. The TRGS method can be written as
Hence, taking the norms of both sides results into
Therefore, , implying that TRGS method is convergent. □
Theorem 2. If is an , then the third-refinement of Gauss–Seidel (TRGS) technique converges for any preliminary guess .
Proof. We employed a similar procedure to that in works of [
5,
13]. Therefore, we can show that TRGS converges by using the spectral radius of the iterative matrix. If
is an M-matrix, then the spectral radius of Gauss–Seidel is less than 1. Thus,
. Since the spectral radius of TRGS is less than 1, as such TRGS is convergent. □
2.2. Algorithm for Third Refinement of Gauss–Seidel (TRGS) Technique
- (i)
Input the coefficients of , indicate a preliminary estimation , maximum iteration quantity tolerance .
- (ii)
Obtain the partition matrices and from .
- (iii)
Create inverse of and obtain .
- (iv)
Create .
- (v)
Establish .
- (vi)
Iterate and stop if .
3. Results and Discussion
In this section, an ideal numerical experiment is observed to test the performance of the proposed technique with respect to its initial refinements.
Applied problem [14]: Consider the linear system of equations;
The true solution of the applied problem is
From
Table 1, it can be clearly observed that the proposed method has a greatly minimized spectral radius, with respect to the iteration matrix, and also shows that the convergence rate is very high.
Table 1 also shows that the TRGS reduced the number of iterations to one-fourth of GS, half of RGS and a few steps of SRGS. Based on how near their spectral radii are to zero, it is inferred that the TRGS has a faster rate of convergence than the techniques compared
.
4. Conclusions
In this study, an accelerated iterative technique named “Third-Refinement of Gauss-Seidel (TRGS) technique” is proposed. The TRGS algorithm is very appropriate in solving a large system of linear equations, as it shows a significant improvement in the reduction in iteration step and increase in convergence rate. The analysis from Theorems 1 and 2 verify that the proposed technique is convergent and the efficiency of TRGS is illustrated through the applied problem as shown in
Table 2. It can be deduced from our analysis that the proposed technique achieved a qualitative and quantitative shift in solving linear systems of equations and is more efficient than existing refinements of Gauss–Seidel techniques.
Author Contributions
Conceptualization, methodology, software, validation and formal analysis, K.J.A.; investigation, data curation, writing—original draft preparation, writing—review and editing, J.N.E.; visualization, supervision and project administration, K.J.A. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
All data are calculated and generated from the proposed method.
Acknowledgments
The authors would like to appreciate Abdgafar Tunde Tiamiyu, Department of Mathematics, The Chinese University of Hong Kong (CUHK), Hong Kong, for his academic support and valuable advice.
Conflicts of Interest
The authors declare no conflict of interest.
References
- Audu, K.J. Extended accelerated overrelaxation iteration techniques in solving heat distribution problems. Songklanakarin J. Sci. Technol. 2022, 44, 1232–1237. [Google Scholar]
- Audu, K.J.; Yahaya, Y.A.; Adeboye, K.R.; Abubakar, U.Y. Convergence of Triple Accelerated Over-Relaxation (TAOR) Method for M-Matrix Linear Systems. Iran. J. Optim. 2021, 13, 93–101. [Google Scholar]
- Vatti, V.B.K. Numerical Analysis Iterative Methods; I.K International Publishing House: New Delhi, India, 2016. [Google Scholar]
- Vatti, V.B.K.; Tesfaye, E.K. A refinement of Gauss-Seidel method for solving of linear system of equations. Int. J. Contemp. Math. Sci. 2011, 63, 117–127. [Google Scholar]
- Tesfaye, E.K.; Awgichew, G.; Haile, E.; Gashaye, D.A. Second refinement of Gauss-Seidel iteration method for solving linear system of equations. Ethiop. J. Sci. Technol. 2020, 13, 1–15. [Google Scholar]
- Tesfaye, E.K.; Awgichew, G.; Haile, E.; Gashaye, D.A. Second refinement of Jacobi iterative method for solving linear system of equations. Int. J. Comput. Sci. Appl. Math. 2019, 5, 41–47. [Google Scholar]
- Kumar, V.B.; Vatti; Shouri, D. Parametric preconditioned Gauss-Seidel iterative method. Int. J. Curr. Res. 2016, 8, 37905–37910. [Google Scholar]
- Saha, M.; Chakrabarty, J. Convergence of Jacobi, Gauss-Seidel and SOR methods for linear systems. Int. J. Appl. Contemp. Math. Sci. 2020, 6, 77. [Google Scholar] [CrossRef]
- Genanew, G.G. Refined iterative method for solving system of linear equations. Am. J. Comput. Appl. Math. 2016, 6, 144–147. [Google Scholar]
- Quateroni, A.; Sacco, R.; Saleri, F. Numerical Mathematics; Springer: New York, NY, USA, 2000. [Google Scholar]
- Constantlnescu, R.; Poenaru, R.C.; Popescu, P.G. A new version of KSOR method with lower number of iterations and lower spectral radius. Soft Comput. 2019, 23, 11729–11736. [Google Scholar] [CrossRef]
- Muhammad, S.R.B.; Zubair, A.K.; Mir Sarfraz, K.; Adul, W.S. A new improved Classical Iterative Algorithm for Solving System of Linear Equations. Proc. Pak. Acad. Sci. 2021, 58, 69–81. [Google Scholar]
- Lasker, A.H.; Behera, S. Refinement of iterative methods for the solution of system of linear equations Ax=b. ISOR-J. Math. 2014, 10, 70–73. [Google Scholar] [CrossRef]
- Meligy, S.A.; Youssef, I.K. Relaxation parameters and composite refinement techniques. Results Appl. Math. J. 2022, 15, 100282. [Google Scholar] [CrossRef]
Table 1.
Comparison of Spectral radius and Convergence rate for the Applied Problem.
Table 1.
Comparison of Spectral radius and Convergence rate for the Applied Problem.
Technique | Iteration Step | Spectral Radius | Execution Time (s) | Convergence Rate |
---|
GS | 88 | 0.89530 | 6.70 | 0.04803 |
RGS | 44 | 0.80157 | 5.53 | 0.09606 |
SRGS | 30 | 0.71765 | 5.00 | 0.14408 |
TRGS | 22 | 0.64251 | 4.10 | 0.19212 |
Table 2.
The Iterate Solution of Applied Problem.
Table 2.
The Iterate Solution of Applied Problem.
Technique | | | | | | | | | |
---|
GS | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 1.47620 | 1.63720 | −1.44920 | 0.04476 | 1.14180 | 0.91970 | −1.95840 | 0.79746 |
2 | 0.86540 | 1.96410 | −1.02590 | −0.02391 | 0.94982 | 0.90209 | −2.07580 | 0.94392 |
| | | | | | | | |
87 | 0.99999 | 2.00000 | −1.00000 | 0.00000 | 1.00000 | 1.00000 | −2.00000 | 1.00000 |
88 | 1.00000 | 2.00000 | −1.00000 | 0.00000 | 1.00000 | 1.00000 | −2.00000 | 1.00000 |
RGS | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0.86540 | 1.96410 | −1.02590 | −0.23917 | 0.949820 | 0.90209 | −2.07580 | 0.94392 |
2 | 0.94909 | 1.94710 | −1.05170 | −0.04878 | 0.95370 | 0.95407 | −2.04670 | 0.95306 |
| | | | | | | | |
43 | 0.99999 | 2.00000 | −1.00000 | 0.00000 | 0.99999 | 0.99999 | −2.00000 | 0.99999 |
44 | 1.00000 | 2.00000 | −1.00000 | 0.00000 | 1.00000 | 1.00000 | −2.00000 | 1.00000 |
SRGS | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0.95672 | 1.95870 | −1.05540 | −0.06439 | 0.94007 | 0.94674 | −2.04710 | 0.95308 |
2 | 0.95952 | 1.96010 | −1.03950 | −0.03950 | 0.96145 | 0.96206 | −2.03740 | 0.96315 |
| | | | | | | | |
29 | 0.99999 | 2.00000 | −1.00000 | 0.00000 | 1.00000 | 1.00000 | −2.00000 | 1.00000 |
30 | 1.00000 | 2.00000 | −1.00000 | 0.00000 | 1.00000 | 1.00000 | −2.00000 | 1.00000 |
TRGS | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0.94909 | 1.94710 | −1.05170 | −0.04878 | 0.95370 | 0.95407 | −2.04670 | 0.95306 |
2 | 0.96741 | 1.96790 | −1.03170 | −0.03170 | 0.96917 | 0.96959 | −2.03000 | 0.97042 |
| | | | | | | | |
21 | 0.99999 | 2.00000 | −1.00000 | 0.00000 | 0.99999 | 0.99999 | −2.00000 | 0.99999 |
22 | 1.00000 | 2.00000 | −1.00000 | 0.00000 | 1.00000 | 1.00000 | −2.00000 | 1.00000 |
| Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).