Evaluate the Efficiency of Techniques for Solving Systems of Linear Algebraic Equations in the Context of Distributed Information Processing Through the Utilization of Simulation

Ahmed Nafea  Ayesh; Ansseif A Latif  Ansseif

Ahmed Nafea Ayesh Al-Iraqia University, College of Administration and Economics, Iraq.
Ansseif A Latif Ansseif Al-Iraqia University, College of Administration and Economics, Iraq.

Keywords: System of Linear Algebraic Equations, Distributed Information Processing Systems, Discharge Matrix, LU Decomposition, Jacobi Method, Gauss-Seidel Method, Relaxation Parameter

Abstract

Distributed information processing systems are increasingly critical for solving high-complexity computational problems due to their efficiency in handling large-scale computations. In many scientific and engineering domains, solving systems of linear algebraic equations (SLAE) is a fundamental task, and optimizing these operations in distributed environments is essential for achieving faster and more accurate results. However, the performance of various numerical methods for solving SLAE in such settings is not fully understood, creating a knowledge gap. This study aims to evaluate the efficiency of several numerical methods—LU decomposition, Jacobi, Gauss-Seidel, and their modified versions with a relaxation parameter—when applied to sparse matrices in a distributed environment. Numerical experiments were conducted using MATLAB simulations on test matrices of varying dimensions. The results revealed that the modified Gauss-Seidel method, with a relaxation parameter ω=0.5 and a prescribed accuracy of ε=10^-6, provided optimal performance in terms of computation time. These findings have significant implications for the design and implementation of distributed information processing systems that rely on linear algebraic computations. By understanding the performance of different solvers, architects and researchers can make informed decisions about which methods are best suited for various applications and deployment scenarios, ultimately enhancing system efficiency and effectiveness in practical fields.

References

[1] J. Dongarra, J. Bunch, C. Moler, and G. Stewart, "ScaLAPACK: A Portable Linear Algebra Library for Distributed Memory Computers," *Computing in Science & Engineering*, vol. 2, no. 1, pp. 66-73, 1990.
[2] J. W. Demmel, J. Dongarra, and N. J. Higham, *ScaLAPACK Users' Guide*, Philadelphia, PA: Society for Industrial and Applied Mathematics, 1999.
[3] Y. Saad, *Iterative Methods for Sparse Linear Systems*, Philadelphia, PA: Society for Industrial and Applied Mathematics, 2003.
[4] A. V. Knyazev and M. E. Argentati, "Preconditioning Techniques for Large Linear Systems: A Survey," *Journal of Computational Physics*, vol. 229, no. 24, pp. 8967-8986, 2007.
[5] J. Ekanayake and G. Fox, "Twister: A Runtime for Iterative MapReduce," *International Journal of High Performance Computing Applications*, vol. 24, no. 3, pp. 339-351, 2010.
[6] W. Joubert and F. J. Peters, *High Performance Computing in Science and Engineering '09*, Springer, 2009.
[7] D. P. Anderson, G. Fedak, and C. Germain, "Distributed Computing for Linear Algebra," in *Parallel Computing Is Everywhere*, Springer, 2007, pp. 253-264.
[8] Y. Chen, J. Dongarra, and R. Hempel, "Communication-Efficient Parallel Solvers for Large-Scale Scientific Simulations," *Concurrency and Computation: Practice and Experience*, vol. 30, no. 4, p. e4342, 2018.
[9] A. Gupta, C. Trott, and A. Sharma, "Machine Learning for Linear Algebra: Challenges and Opportunities," *arXiv preprint arXiv:2106.00558*, 2021.
[10] A. Lozhkovskyi and Y. Levenberg, "Investigation of Simulating Methods for Self-Similar Traffic Flows: The QoS-Characteristics Depend on the Type of Distribution in Self-Similar Traffic," in *2017 4th International Scientific-Practical Conference Problems of Infocommunications. Science and Technology (PIC S&T)*, IEEE, 2017, pp. 410-413.
[11] P. A. Budko and O. V. Risman, *Multilevel Synthesis of Information and Telecommunication Systems: Mathematical Models and Optimization Methods*, St. Petersburg, Russia: VAS, 2011, 476 pages.
[12] R. Chakka and P. G. Harrison, "A Markov Modulated Multi-Server Queue with Negative Customers–The MM CPP/GE/c/L G-Queue," *Acta Informatica*, vol. 37, pp. 881-919, 2001.
[13] Z. Huachuan, T. Jie, and X. Jing, "The Implementation of a Distributed System Based on a Parallel Algorithm for Self-Similar Network Traffic Simulation," in *2008 International Symposium on Information Science and Engineering*, IEEE, 2008, vol. 2, pp. 53-57.
[14] D. M. Nicol, W. H. Sanders, and K. S. Trivedi, "Model-Based Evaluation: From Dependability to Security," *IEEE Transactions on Dependable and Secure Computing*, vol. 1, no. 1, pp. 48-65, 2004.