An Algorithm for Fast Multiplication of Pauli Numbers

Cariow, Aleksandr; Cariowa, Galina

doi:10.1007/s00006-014-0466-0

Cited by 6 publications

(10 citation statements)

References 11 publications

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“…Several efficient algorithms for the multiplication of hypercomplex numbers have been reported in the literature [12][13][14][15][16][17]. Our previous work [12] proposed an algorithm for computing product of two Dirac numbers which has lower computational complexity compared with the direct (schoolbook) method of computations. In this paper we propose another algorithm for this purpose.…”

Section: Introductionmentioning

confidence: 99%

Algorithm for multiplying two octonions

Cariow

Cariowa

2012

Radioelectron.Commun.Syst.

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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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Section: Introductionmentioning

confidence: 99%

Algorithm for multiplying two octonions

Cariow

Cariowa

2012

Radioelectron.Commun.Syst.

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“…In our previous work [15][16][17], we have applied the unified approach proposed here for the synthesis of fast algorithms for the multiplication of quaternions, octonions and sedenions. However, if the specific properties of the matrix are used, even more interesting solutions may be found [18][19]. We will try to develop the ideas raised here in our future publications, as far as possible.…”

Section: Discussionmentioning

confidence: 99%

An unified approach for developing rationalized algorithms for hypercomplex number multiplication

Cariow

Cariowa

2015

ELECTROTECHNICAL REVIEW

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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).

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“…At first, we rearrange the rows of the matrix in the following order {1, 2, 3, 7, 5, 9, 4, 8, 6, 10, 11,17,13,19,15,21,12,18,14,20,16,22,23,27,25,29,24,28,26,30, 31, 32}. Next, we rearrange the columns of obtained matrix in the same manner.…”

Section: Synthesis Of a Rationalized Algorithm For Computing Kaluza N...mentioning

confidence: 99%

“…Efficient algorithms for the multiplication of various hypercomplex numbers already exist [12,13,14,15,16,17,18,19,20,21,22,23,24]. No such algorithms for the multiplication of Kaluza numbers have been proposed.…”

Section: Introductionmentioning

confidence: 99%

An algorithm for multipication of Kaluza numbers

Cariow,

Cariowa,

Łentek

2015

Preprint

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This paper presents the derivation of a new algorithm for multiplying of two Kaluza numbers. Performing this operation directly requires 1024 real multiplications and 992 real additions. The proposed algorithm can compute the same result with only 512 real multiplications and 576 real additions. The derivation of our algorithm is based on utilizing the fact that multiplication of two Kaluza numbers can be expressed as a matrixvector product. The matrix multiplicand that participates in the product calculating has unique structural properties. Namely exploitation of these specific properties leads to significant reducing of the complexity of Kaluza numbers multiplication.

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An Algorithm for Fast Multiplication of Pauli Numbers

Cited by 6 publications

References 11 publications

Algorithm for multiplying two octonions

Algorithm for multiplying two octonions

An unified approach for developing rationalized algorithms for hypercomplex number multiplication

An algorithm for multipication of Kaluza numbers

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