Application of the $Q$-determinant concept for efficient implementation of numerical algorithms by the example of the conjugate gradient method for solving systems of linearequations

The problem of improving the efficiency of parallel computing is very topical. The article demonstrates the application of the concept of $Q$-determinant for the effective implementation of numerical algorithms by the example of the conjugate gradient method for solving systems of linear equations. The concept of the $Q$-determinant is based on a unified representation of numerical algorithms in the form of the $Q$-determinant. Any numerical algorithm has a $Q$-determinant. 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 compute one of the output data items based on the input data. The $Q$-determinant allows you to express and evaluate the internal parallelism of the algorithm, as well as to show the method of its parallel execution. The article gives the main notions of the $Q$-determinant concept necessary for better understanding of our research. Also, we describe a method of designing effective programs for numerical algorithms on the base of the concept of the $Q$-determinant. As a result, we obtain the program which uses the parallelism resource of the algorithm completely, and this program is called $Q$-effective. As application of the method for design of $Q$-effective programs, we describe the designing programs for conjugate gradient method for implementation on parallel computing systems with shared and distributed memory. Finally, for developed programs we present the results of experiments on a supercomputer "Tornado SUSU".