An algorithm for multiplication of split-octonions

TL;DR

This paper presents a novel algorithm for multiplying split-octonions more efficiently, reducing the number of real multiplications from 64 to 28 by exploiting the matrix structure involved in the product.

Contribution

The paper introduces a new algorithm that significantly decreases the computational complexity of split-octonion multiplication using matrix decomposition techniques.

Findings

Reduced multiplications from 64 to 28

Achieved faster split-octonion multiplication

Utilized matrix structural properties for efficiency

Abstract

In this paper we introduce efficient algorithm for the multiplication of split-octonions. The direct multiplication of two split-octonions requires 64 real multiplications and 56 real additions. More effective solutions still do not exist. We show how to compute a product of the split-octonions with 28 real multiplications and 92 real additions. During synthesis of the discussed algorithm we use the fact that product of two split-octonions may be represented as vector-matrix product. The matrix that participates in the product calculating has unique structural properties that allow performing its advantageous decomposition. Namely this decomposition leads to significant reducing of the multiplicative complexity of split-octonions multiplication.