Parallelism Resource of Numerical Algorithms. Version 1

TL;DR

This paper introduces a $Q$-determinant framework for analyzing and designing parallel algorithms, enabling efficient implementation by fully utilizing the parallelism resource in numerical computations.

Contribution

The paper presents the $Q$-determinant concept, a software $Q$-system for studying parallelism, and a method for designing $Q$-effective parallel programs for numerical algorithms.

Findings

The $Q$-system effectively compares parallelism resources of algorithms.

Application to various algorithms demonstrates the framework's utility.

Proposes a method for creating fully parallelized, $Q$-effective programs.

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 $Q$-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.