Linear algebra and numerical algorithms using dual numbers

Abstract

Dual number algebra is a powerful mathematical tool for the kinematic and dynamic analysis of spatial mechanisms. With the purpose of exploiting new applications, in this paper are presented the dual version of some classical linear algebra algorithms. These algorithms have been tested for the position analysis of the RCCC mechanism and computational improvements over existing methods obtained

Dual number algebra method for Green’s function derivatives in 3D magneto-electro-elasticity

“…A dual number z is an ordered pair of real numbers (x, y) associated with the real unit 1 and the dual unit ε, hence [6,7]:…”

“…The representation (15) is called Gaussian representation [7]. It is worth to notice, that the dual unit ε is an nilpotent number, which means that ε 2 = 0 and ε ≠ 0.…”

“…The finite difference (FD) scheme could be adopted to calculate the derivatives of Green's functions; after several numerical tests it can be established the appropriate value of the step size, however the value of the optimal step-size is problem-dependent. Therefore, in the present work the dual number algebra method (DNAM) [6,7] is applied to calculate numerically the derivatives of 3D Green's functions for the MEE materials. The DNAM is independent on the step size as shown in [7] and it can be treated as a special case of the automatic differentiation method.…”

“…Therefore, in the present work the dual number algebra method (DNAM) [6,7] is applied to calculate numerically the derivatives of 3D Green's functions for the MEE materials. The DNAM is independent on the step size as shown in [7] and it can be treated as a special case of the automatic differentiation method. These properties make the DNAM an easy-to-implement and highly accurate computational method for the numerical calculation of first-order derivatives of the real functions given by the implicit computational mapping.…”

A Polar Representation of a Regularity of a Dual Quaternionic Function in Clifford Analysis

“…A dual number is denoted in the form z = x + εy. Thus, the dual numbers are elements of the two dimensional real algebra D = R[ε] = {z = x + εy | x, y ∈ R, ε 2 = 0, ε = 0} generated by 1 and ε (see [17]). …”