An unified approach for developing rationalized algorithms for hypercomplex number multiplication

Abstract

In this article we present a common approach for the development of algorithms for calculating products of hypercomplex numbers. The main idea of the proposed approach is based on the representation of hypernumbers multiplying via the matrix-vector products and further creative decomposition of the matrix, leading to the reduction of arithmetical complexity of calculations. The proposed approach allows the construction of sufficiently well algorithms for hypernumbers multiplication with reduced computational complexity. If the schoolbook method requires N 2 real multiplications and N(N-1) real additions, the proposed approach allows to develop algorithms, which take only [N(N-1)/2]+2 real multiplications and 3Nlog 2 N+[N(N-3)+4]/2 real additions. Streszczenie. W artykule zostało przedstawione uogólnione podejście do syntezy algorytmów wyznaczania iloczynów liczb hiperzespolonych. Główna idea proponowanego podejścia polega na reprezentacji operacji mnożenia liczb hiperzespolonych w formie iloczynu wektorowomacierzowego i dalszej możliwości kreatywnej dekompozycji czynnika macierzowego prowadzącej do redukcji złożoności obliczeniowej. Proponowane podejście pozwala zbudować algorytmy wyróżniające się w porównaniu do metody naiwnej zredukowaną złożonością obliczeniową. Jeśli metoda naiwna wymaga wykonania N 2 mnożeń oraz N(N-1) dodawań liczb rzeczywistych to proponowane podejście pozwala syntetyzować algorytmy wymagające tylko [N(N-1)/2]+2 mnożeń oraz 3Nlog 2 N+[N(N-3)+4]/2 dodawań. (Uogólnione podejście do konstruowania zracjonalizowanych algorytmów mnożenia liczb hiperzespolonych-tytuł polski artykułu).

Cost-effective quaternion minimum mean square error estimation: From widely linear to four-channel processing

“…However, for a large step-size µ, the term 2µE x n 2 |e a n | 2 is not negligible. Assuming that x n 2 is independent of |e a n | 2 , we arrive at Table 1: The number of real-valued operations per iteration for the QLMS algorithms, for an adaptive filter of length L. The number of real-valued operations for quaternion-quaternion multiplications is based on [28].…”

“…Compared with existing widely linear and strictly quaternion estimation approaches, the proposed four-channel estimation technique requires less computation cost. This is because the proposed estimation algorithm avoids the quaternionquaternion addition in (9) and (10) while the proposed weight update rule in (14) replaces quaternion-quaternion multiplications, which cost 8 real multiplications and 28 real additions per multiplication [28], with real-quaternion multiplications costing only 4 real multiplications per multiplication. Table 1 and Figure 5…”

“…The number of real-valued operations per iteration for the QLMS algorithms, for an adaptive filter of length L. The number of real-valued operations for quaternion-quaternion multiplications is based on[28]. NMSE curves of the QLMS algorithms for the estimation of MA systems.…”

Hyperbolic k-Fibonacci and k-Lucas octonions

“…In [3,4], A. Cariow, G. Cariow and J. Knapiski defined hyperbolic octonions. A hyperbolic octonion O is an expression of the form…”

“…The bases of hyperbolic octonion O appeared in [3,4] have multiplication rules as in Table 1. Table 1.…”