[Paper Review] The basic principles and the structure and algorithmically software of computing by hypercomplex number

Yakiv O. Kalinovsky, Yuliya E. Boyarinova+2 more|arXiv (Cornell University)|Aug 14, 2017

Advanced Data Processing Techniques1 references3 citations

TL;DR

This paper presents a novel software architecture and algorithmic framework for computing with hypercomplex numbers, including quaternions and higher-dimensional algebras. It details the design of functional subsystems, provides program listings, and demonstrates practical applications through concrete examples, establishing a foundation for robust, extensible hypercomplex number computation in mathematical software.

ABSTRACT

In article the basic principles put in a basis of algorithmicallysoftware of hypercomplex number calculations, structure of a software, structure of functional subsystems are considered. The most important procedures included in subsystems are considered, program listings and examples of their application are given.

Motivation & Objective

• To establish foundational principles for algorithmic software handling hypercomplex numbers.
• To design a modular software structure supporting hypercomplex arithmetic operations.
• To implement and document key procedures within functional subsystems for numerical computation.
• To provide program listings and application examples to demonstrate software usability.
• To enable efficient and extensible computation in hypercomplex number systems for mathematical and engineering applications.

Proposed method

• The authors define a software architecture based on functional subsystems tailored for hypercomplex number operations.
• Core components include data structures for representing hypercomplex numbers, such as quaternions and their higher-dimensional extensions.
• The system implements standard algebraic operations (addition, multiplication, conjugation) with optimized algorithms.
• The software is structured to support extensibility for additional hypercomplex algebras beyond quaternions.
• Program listings and code examples are integrated into the design to illustrate correct implementation and usage.
• The approach emphasizes modularity, reusability, and correctness through documented procedures.

Experimental results

Research questions

• RQ1 How can a consistent and extensible software architecture be designed for hypercomplex number computation?
• RQ2 What are the essential functional subsystems required to support hypercomplex arithmetic?
• RQ3 How can core operations like multiplication and conjugation be algorithmically implemented for hypercomplex numbers?
• RQ4 What role do code examples and program listings play in validating and demonstrating the software's functionality?
• RQ5 How can the software support future extensions to higher-dimensional hypercomplex algebras?

Key findings

• The proposed software architecture successfully supports fundamental hypercomplex number operations through well-defined functional subsystems.
• The implementation includes detailed program listings that illustrate correct usage of core procedures.
• The system demonstrates feasibility for handling quaternions and related hypercomplex algebras in a structured, reusable manner.
• Functional subsystems are designed to be extensible, enabling future support for additional hypercomplex number systems.
• The paper provides concrete examples of application, validating the practicality and correctness of the algorithmic approach.
• The work establishes a foundation for integrating hypercomplex number computation into mathematical and scientific software.