Abstract. In this paper we introduce an efficient algorithm for the multiplication of Pauli numbers. The direct multiplication of two Pauli numbers requires 64 real multiplications and 56 real additions. More effective solutions still do not exist. We show how to compute a product of the Pauli numbers with 24 conventional multiplications, 8 multiplications by 1/2 and 56 real additions.
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).
In this work a rationalized algorithm for Dirac numbers multiplication is presented. This algorithm has a low computational complexity feature and is well suited to FPGA implementation. The computation of two Dirac numbers product using the naïve method takes 256 real multiplications and 240 real additions, while the proposed algorithm can compute the same result in only 88 real multiplications and 256 real additions. During synthesis of the discussed algorithm we use the fact that Dirac numbers product may be represented as vector-matrix product. The matrix participating in the product has unique structural properties that allow performing its advantageous decomposition. Namely this decomposition leads to significant reducing of the computational complexity.
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