## Taiwanese Journal of Mathematics

### The Weaker Convergence of Non-stationary Matrix Multisplitting Methods for Almost Linear Systems

#### Abstract

In 1999, Arnal et al. [Numerical linear algebra and its applications, 6(1999): 79-92] introduced the non-stationary matrix multisplitting algorithms for almost linear systems and studied the convergence of them. In this paper, we generalize Arnal’s algorithms and study the non-stationary matrix multisplitting multi-parameters methods for almost linear systems. The parameters can be adjusted suitably so that the convergence property of methods can be substantially improved. Furthermore, the convergence results of our new method in this paper are weaker than those of Arnal’s. Finally, numerical examples show that our new convergence results are better and more efficient than Arnal’s.

#### Article information

Source
Taiwanese J. Math., Volume 15, Number 4 (2011), 1423-1436.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500406354

Digital Object Identifier
doi:10.11650/twjm/1500406354

Mathematical Reviews number (MathSciNet)
MR2848964

Zentralblatt MATH identifier
1233.65030

#### Citation

Zhang, Li-Tao; Huang, Ting-Zhu; Cheng, Shao-Hua; Gu, Tong-Xiang. The Weaker Convergence of Non-stationary Matrix Multisplitting Methods for Almost Linear Systems. Taiwanese J. Math. 15 (2011), no. 4, 1423--1436. doi:10.11650/twjm/1500406354. https://projecteuclid.org/euclid.twjm/1500406354

#### References

• J. Arnal, V. Migall$\rm\acute{o}$n and Jos$\rm\acute{e}$ Penad$\rm\acute{e}$, Non-stationary parallel multisplitting algorithms for almost linear systems, Numerical linear algebra and its applications, 6 (1999), 79-92.
• Z. Z. Bai, Parallel nonlinear AOR method and its convergence, Computers and Mathematics with Applications, 31(2) (1996), 21-31.
• A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Science, Academic Press, New York, third edition, 1979, Reprinted by SIAM, Philadelphia, 1994.
• G. Birkhoff, Numerical Solution of Elliptic Equation, Vol. 1 of CBMS Regional Conference Series in Applied Mathematics, Society for industrial and Applied Mathematics, Philadelphia, 1970.
• J. J. Climent and C. Perea, Convergence and comparison theorems for multisplittings, Numerical linear algebra and its applications, 6 (1999), 93-107.
• A. Frommer and G. Mayer, Convergence of relaxed parallel multisplitting methods, Linear Algebra and Its Applications, 119 (1989), 141-152.
• A. Frommer and G. Mayer, On the theory and practice of multisplitting methods in parallel computation, Computing, 49 (1992), 62-74.
• R. Fuster, V. Migall$\rm\acute{o}$n and J. Penad$\rm\acute{e}$s, Non-stationary parallel multisplitting AOR methods, Electronic Transaction on Numerical Analysis, 4 (1996), 1-13.
• Y. C. Kuo, W. W. Lin, S. F. Shieh and W. C. Wang, A minimal energy tracking method for non-radially symmetric solutions of coupled nonlinear Schrödinger equations, Journal of Computational Physics, 228 (2009), 7941-7956.
• J. Mas et al., Nonstationary parallel relaxed multisplitting methods, Linear Algebra and Its Applications, 241/243 (1992), 733-747.
• J. Mas, V. Migall$\rm\acute{o}$n, J. Penad$\rm\acute{e}$s and D. B. Szyld, Non-stationary parallel relaxed multisplitting methods, Linear Algebra and Its Applications, 241/243 (1996), 733-748,.
• J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, San Diego, 1970.
• F. Robert, M. Charnay and F. Musy, Iterations chaotiques serie-parallel pour des equations non-lineaires de point fixe, Aplikace Matematiky, 20 (1975), 1-38.
• Y. Z. Song, Convergence of parallel multisplitting methods, Linear Algebra and Its Applications, 50 (1994), 213-232.
• D. R. Wang, On the convergence of parallel multisplitting AOR algorithm, Linear Algebra and Its Applications, 154/156 (1991), 473-486.
• R. E. White, Parallel algorithms for nonlinear problems, SIAM Journal on Algebraic and Discrete Methods, 7 (1986), 137-149.
• J. H. Yun, Convergence of SSOR multisplitting method for an $H$-matrix, Journal of Computational and Applied Mathematics, 217 (2008), 252-258.
• J. H. Yun, Performance of ILU factorization perconditioners based on multisplitting, Numerische Mathematik, 95 (2003), 761-779.
• D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, 1972.
• L. T. Zhang, T. Z. Huang and T. X. Gu, Global relaxed non-stationary multisplitting multi-parameters methods, International Journal of Computer Mathematics, 85(2) (2008), 211-224.