Advancement of research for the parallelism resource of numerical algorithms with the help of software $Q$-system

The software $Q$-system is designed to study the parallelism resource of numerical algorithms. By use of $Q$-system, we can calculate the parallelism resource of any numerical algorithm. Also, for the set of algorithms solving a given algorithmic problem, we can find an algorithm with the best parallelism resource. The theoretical basis of the $Q$-system is the concept of $Q$-determinant where $Q$ is the set of operations used by the algorithm. 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. 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. $Q$-determinant makes the algorithm transparent in terms of structure and implementation. The $Q$-system can be used for the improvement of parallel computing efficiency. In this paper we describe new features of the $Q$-system. With the new functionality, we can use the classification of algorithms when studying them. Also with the new functionality, we can calculate the functions for approximating the height and width of the algorithms more precisely and create a graphical representation of them. As a result, we have a more convenient user interface of the $Q$-system and more opportunities to study the parallelism resource of algorithms.