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Pré-Publication, Document De Travail Année : 2023

Partial derivatives over hypercomplex variables for non-analytic functions in eight-dimensional algebra

Résumé

This paper provides the formulation of partial derivatives over hypercomplex variables for functions of octonions or split-biquaternions. The proposed partial derivatives are an extension to the case of octonions and split-biquaternions algebras of the well-known Wirtinger/CR-calculus for complex-numbers algebra, and HR-calculus or generalized HR-calculus for quaternion algebra. The octonionic functions considered in this paper are not constrained by analyticity, holomorphy, harmonicity, conformality, regularity or monogenic property constraints. Instead, the only constraint for the octonionic functions considered in this paper is that each element of an octonionic function be real-differentiable over the (real-valued) variable of each element in an octonion or split-biquaternion variable. We first show that the approach based on rotations over basis components used in the development of HR-calculus or generalized-HR calculus is not suitable for the two considered 8-dimensional hypercomplex algebras. Instead, we justify the use of transformations based on Hadamard matrices, which allows a common framework for the development of partial derivatives for complex-numbers, quaternion, octonion and split-biquaternion algebras. The results of this paper can be an important step towards the development of optimization algorithms and adaptive filtering algorithms of hypercomplex variables from 8-dimensional hypercomplex algebras.
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Dates et versions

hal-03558753 , version 1 (04-02-2022)
hal-03558753 , version 2 (02-12-2022)
hal-03558753 , version 3 (20-02-2023)

Identifiants

  • HAL Id : hal-03558753 , version 3

Citer

Martin Bouchard, Juhi Khalid. Partial derivatives over hypercomplex variables for non-analytic functions in eight-dimensional algebra. 2023. ⟨hal-03558753v3⟩
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