EFFICIENT ALGORITHMS FOR PARALLELIZING TRIDIAGONAL SYSTEMS OF EQUATIONS

Abstract

The article is devoted to the development of the maximal parallel forms of mathematical models with a tridiagonal structure. The example of solving the Dirichlet and Neumann problems by the method of straight lines and the sweep method for the heat equation illustrates the direct fundamental features of constructing parallel algorithms. It is noted that the study of the heat and mass transfer processes is run through their numerical modeling based on modern computer technology.
It is shown that with the multiprocessor computing systems’ development, there disappear the problems of increasing their peak performance. On the other hand, building such systems, as a rule, requires standard network technologies, mass-produced processors, and free software. The noted circumstances aim at solving the so-called big problems.
It should be borne in mind that the classical approach to solving the tridiagonal structure models based on multiprocessor computing systems is far more time-consuming compared to single-processor computing facilities. That is explained by the recurrence relations that make the basis of classical methods. Therefore, the proposed studies are relevant and aim at the distributed algorithms development for solving applied problems.
The proposed research aims to construct the maximal parallel forms of mathematical models with a tridiagonal structure.
The paper proposes the schemes to implement parallelization algorithms for applied problems and their mapping to parallel computing systems.
Parallelization of tridiagonal mathematical models by the method of straight lines and the sweeping method allows designing absolutely stable algorithms with the maximum parallel form and, therefore, the minimum possible time for their implementation on parallel computing devices. It is noteworthy that in the proposed algorithms, the computational errors of the input data are separated from the round-off errors inherent in a PC.
The proposed approach can be used in various branches of metallurgical, thermal physics, economics, and ecology problems in the metallurgical industry.