Software Q-system for the Research of the Resource of Numerical Algorithms Parallelism

Abstract

The paper describes software Q-system for research of the resource of numerical algorithms parallelism. The theoretical basis of the Q-system is the concept of Q-determinant where Q is the set of operations used by the algorithm. The Q-determinant consists of Q-terms. Their number is equal to the number of output data items. Each Q-term describes all possible ways to calculate one of the output data items based on the input data. Any numerical algorithm has a Q-determinant and can be represented in the form of a Q-determinant. Such a representation is a universal description of numerical algorithms. It makes the algorithm transparent in terms of structure and implementation. The software Q-system enables to calculate the parallelism resource of any numerical algorithm, and also to compare the parallelism resources of two algorithms that solve the same algorithmic problem. In the paper we show the application of the Q-system on the example of numerical algorithms with different structures of Q-determinants. Among them, we have the matrix multiplication algorithm, methods of Gauss–Jordan, Jacobi, Gauss–Seidel for solving systems of linear equations, and other algorithms. The paper continues the research begun in the previous papers of the authors. The results of the research can be used to increase the efficiency of implementing numerical algorithms on parallel computing systems.