Algorithm for multiplying two octonions

Abstract

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.

On Generalized Tribonacci Sedenions

“…The computation of a sedenion multiplication using the naive method requires 240 additions and 256 multiplications, while an algorithm which is given in [8] can compute the same result in only 298 additions and 122 multiplications, see [8] for details. Moreover, efficient algorithms for the multiplication of quaternions, octonions and trigintaduonions with reduced number of real multiplications is already exist in literature, see [33], [7] and [9], respectively.…”

An algorithm for multipication of Kaluza numbers

“…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.…”

“…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.…”

Solution of an inverse kinematics problem using dual quaternions

“…Sariyildiz et al (2011) analyzed forward and inverse kinematics problems for a 6DOF manipulator using three methods: quaternions algebra, exponential mapping method and dual quaternions. Cariow et al (2015) elaborated and implemented a method which improves the computational efficiency of dual quaternions operations. The obtained results indicated that dual quaternions provide a compact and computationally efficient solution.…”