Low complexity algorithm for multiplying octonions

Abstract

We propose an original algorithmic solution for multiplication of octonions. In previously published algorithms for computing the product of octonions the number of multiplications has been reduced by significantly increasing number of additions and shifts. A dignity of the proposed solutions is to reduce by 25% the number of multiplications needed to calculate the product of octonions compared with naive method. At the same time the number of additions is the same as in the naive way of calculations. During synthesis of the discussed algorithm we use a fact that octonion product may be represented as a matrix-vector product. Such representation provides a possibility to discover repeating elements in the matrix structure and to use specific properties of their mutual placement for reducing the number of real multiplications needed to calculate the octonion product. Streszczenie. W artykule przedstawiono szybki algorytm wyznaczania iloczynu oktonionow. Algorytm ten cechuje sie zredukowaną o 25% liczbą operacji mnozenia w porownaniu do algorytmu naiwnego przy zachowaniu takiej samej liczby dodawan liczb rzeczywistych. (Zracjonalizowany algorytm mnozenia oktonionow). Slowa kluczowe: mnozenie oktonionow, oktoniony, szybki algorytm, notacja macierzowa.