Parallelism Resource of Numerical Algorithms. Version 1

Abstract

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 Qdeterminant 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. CCS Concepts: • Theory of computation → Models of computation; Concurrency; Parallel computing models; Design and analysis of algorithms; Parallel algorithms; • Software and its engineering → Software creation and management; Software development techniques; Flowcharts; • Computing methodologies → Parallel computing methodologies; Parallel algorithms; Symbolic and algebraic manipulation; Symbolic and algebraic algorithms; Linear algebra algorithms;Additional Key Words and Phrases: Q-term of algorithm, Q-determinant of algorithm, representation of algorithm in form of Q-determinant, Q-effective implementation of algorithm, parallelism resource of algorithm, software Q-system, Q-effective program, Q-effective programming