  The paper is devoted to an approach to solving a problem of the efficiency of
parallel computing. The theoretical basis of this approach is the concept of a
$Q$-determinant. Any numerical algorithm has a $Q$-determinant. The
$Q$-determinant of the algorithm has clear structure and is convenient for
implementation. 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.
  We also describe a software $Q$-system for studying the parallelism resource
of numerical algorithms. This system enables to compute and compare the
parallelism resources of numerical algorithms. The application of the
$Q$-system is shown on the example of numerical algorithms with different
structures of $Q$-determinants. Furthermore, we suggest a method for designing
of parallel programs for numerical algorithms. This method is based on a
representation of a numerical algorithm in the form of a $Q$-determinant. As a
result, we can obtain the program using the parallelism resource of the
algorithm completely. Such programs are called $Q$-effective.
  The results of this research can be applied to increase the implementation
efficiency of numerical algorithms, methods, as well as algorithmic problems on
parallel computing systems.

Computational Complexity;Numerical Analysis;Numerical Analysis