In this study, we present higher order Jacobsthal numbers. Then define quaternions by using We give the concept of norm and conjugate for these quaternions. express prove some propositions related to Also, find recurrence relation, Binet formula generating function Finally, calculate Cassini, Catalan, Vajda d’Ocagne identities

The known Complex Step Derivative (CSD) method allows easy and accurate differentiation up to machine precision of real analytic functions by evaluating them with a small imaginary step next the number line. current paper proposes that derivatives holomorphic can be calculated in similar fashion taking quaternionic direction instead. It is demonstrated so doing CSD properties high accuracy conv...

Structured real canonical forms for matrices in Rn×n that are symmetric or skew-symmetric about the anti-diagonal as well as the main diagonal are presented, and Jacobi algorithms for solving the complete eigenproblem for three of these four classes of matrices are developed. Based on the direct solution of 4× 4 subproblems constructed via quaternions, the algorithms calculate structured orthog...

This paper introduced the application of a more efficient mathematical representation of the kinematics of avatars, or digital human beings, in telecollaborative virtual reality environments (VRE). The human head, torso and arms were modeled as a redundant eight-degree-of-freedom kinematics structure using an excellent alternative tool to transformation matrices, called dual quaternions. This a...

The classical matrix groups are of fundamental importance in many parts of geometry and algebra. Some of them, like Sp.n/, are most conceptually defined as groups of quaternionic matrices. But, the quaternions not being commutative, we must reconsider some aspects of linear algebra. In particular, it is not clear how to define the determinant of a quaternionic matrix. Over the years, many peopl...

The four-component rotational operators called quaternions, which represent eye rotations in terms of their axes and angles, have several advantages over other representations of eye position (such as Fick coordinates): they provide easy computations, symmetry, a simple form for Listing's law, and useful three-dimensional plots of eye movements. In this paper we present algorithms for computing...