High-Performance Computing Using Application of Q-determinant of Numerical Algorithms

The conception of Q-determinant is one of the approaches to parallelizing numerical algorithms. The basic notion of the conception is Q-determinant of the algorithm. Here 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. Each Q-term describes all possible ways to calculate one of the output data based on the input data. Any numerical algorithm has a Q-determinant and can be represented in the form of a Q-determinant. This representation is a universal description of numerical algorithms. The representation algorithm in the form of Q-determinant makes the numerical algorithm in terms of the structure and implementation clearer. Although Q-determinant contains only machine-independent properties of the algorithm it can be used to implement algorithms on parallel computing systems. The paper describes the application of the Q-determinant to determine the parallelism resource of numerical algorithms and to develop Q-effective programs. A Q-effective program uses the parallelism resource of algorithm completely. At the present time the available theoretical results have been tested for numerical algorithms with various structures of Q-determinants. For example, they include multiplication algorithms for dense and sparse matrices, Gauss-Jordan, Jacobi, Gauss-Seidel methods for solving the systems of linear equations, the sweep method and the Fourier method for solving grid equations, and others. The results of the research can be used to increase the efficiency of implementing numerical algorithms on parallel computing systems.