Abstract
Dual numbers and their higher-order version are important tools for numerical computations, and in particular for finite difference calculus. Based on the relevant algebraic rules and matrix realizations of dual numbers, we present a novel point of view, embedding dual numbers within a formalism reminiscent of operational umbral calculus.
Highlights
Dual numbers (DNs), introduced during the second half of the 19th century [1,2,3,4,5], can be viewed as abstract entities, similar to ordinary complex numbers, and are defined as ( x, y) ∈ R z = x + ey, (1)where the corresponding “imaginary” unit or dual number unit (DNU) e is a nilpotent number, e2 = 0 and e 6= 0. (2)Dual numbers were originally introduced within the context of geometrical studies, and later exploited to deal with problems in pure and applied mechanics [6,7]
Based on the relevant algebraic rules and matrix realizations of dual numbers, we present a novel point of view, embedding dual numbers within a formalism reminiscent of operational umbral calculus
It has been demonstrated in [8,9,10] how to formulate the equations of rigid body motion in terms of just three “dual”. Equations instead of their six “real” counterparts. This approach has been extended in [11,12,13] to a treatment of rigid body motion in terms of a certain variant of “hyper-dual” numbers, implemented in contrast to our approach via sets of “ordinary” dual numbers together with certain algebraic relations that are motivated from the specific requirements within the relevant field of robotics and of mechanics
Summary
Dual numbers were originally introduced within the context of geometrical studies, and later exploited to deal with problems in pure and applied mechanics [6,7] It has been demonstrated in [8,9,10] how to formulate the equations of rigid body motion in terms of just three “dual”. Equations instead of their six “real” counterparts (thereby realizing an equivalence between spherical and spatial kinematics) This approach has been extended in [11,12,13] to a treatment of rigid body motion in terms of a certain variant of “hyper-dual” numbers, implemented in contrast to our approach via sets of “ordinary” dual numbers together with certain algebraic relations that are motivated from the specific requirements within the relevant field of robotics and of mechanics.
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