Quaternionic Equation Research Articles

Some new linear representations of matrix quaternions with some applications

In this paper, we construct several new attractive and interested linear representations of matrix quaternions by using Kronecker structures in order to obtain the general partitioned linear representation form of matrix quaternions. In addition, we present the general solutions of three important partitioned quaternions systems by using our new representations and Kronecker structure. These systems are: the partitioned linear quaternion equations, general linear matrix quaternion system and coupled Sylvester matrix quaternion system.

Quaternionic Wiener Algebras, Factorization and Applications

We define an almost periodic extension of the Wiener algebras in the quaternionic setting and prove a Wiener-Levy type theorem for it, as well as extending the theorem to the matrix-valued case. We prove a Wiener-Hopf factorization theorem for the quaternionic matrix-valued Wiener algebras (discrete and continuous) and explore the connection to the Riemann-Hilbert problem in that setting. As applications, we characterize solvability of two classes of quaternionic functional equations and give an explicit formula for the canonical factorization of quaternionic rational matrix functions via realization.

Algorithms for the orientation of a moving object with separation of the integration of fast and slow motions

Abstract Equations and algorithms for determining the orientation of a moving object in inertial and normal geographic coordinate systems are considered with separation of the integration of the fast and slow motions into ultrafast, fast and slow cycles. Ultrafast cycle algorithms are constructed using a Riccati-type kinematic quaternion equation and the Picard method of successive approximations, and the increments in the integrals of the projections of the absolute angular velocity vector of the object onto the coordinate axes (quasicoordinates) associated with them are used as input information. The fast cycle algorithm realizes the calculation of the classical rotation quaternion of an object on a step of the fast cycle in an inertial system of coordinates. The slow cycle algorithm is used in calculating the orientation quaternion of an object in the normal geographic coordinate system and aircraft angles. Results of modelling different versions of the fast and ultrafast cycle algorithms for calculating the inertial orientation of an object are presented and discussed. The experience of the authors in developing algorithms for determining the orientation of moving objects using a strapdown inertial navigation system is described and results obtained by them earlier in this field are developed and extended.

Geometry of Riccati equations over normed division algebras

This work introduces and studies Riccati equations over finite-dimensional normed division algebras. We prove that a Riccati equation over a finite-dimensional normed division algebra A is a particular case of conformal Riccati equation on a Euclidean space and it can be considered as a curve in a Lie algebra of vector fields V≃so(dim⁡A+1,1). Previous results on known types of Riccati equations are recovered from a new viewpoint. A new type of Riccati equations, the octonionic Riccati equations, are extended to the octonionic projective line OP1. As a new physical application, quaternionic Riccati equations are applied to study quaternionic Schrödinger equations on 1+1 dimensions.

Algorithms of increasing the calculation accuracy for an aircraft’s orientation angle

Algorithms for calculating an aircraft's orientation angles based on the results of the numerical integration of Poisson's and quaternion equations are proposed. The algorithms in the presence of random errors of the matrix elements of direction cosines and quaternions are characterized by a significantly higher accuracy in comparison to the formulas for solving the formulated problem. The results of testing the mathematical modeling data under random errors, which confirm the increased accuracy of the calculation of the orientation angles, are given.

Kinematic equations of a rigid body in four-dimensional skew-symmetric operators and their application in inertial navigation

Abstract New dual matrix and biquaternion kinematic equations of motion of a free rigid body in dual four-dimensional matrix and biquaternion skew-symmetric operators are proposed. The equations are constructed using dual matrix and biquaternion analogues of Cayley's formulae, which are used to match a dual four-dimensional matrix operator and a biquaternion skew-symmetric operator to a classical biquaternion four-dimensional matrix and the biquaternion of a finite screw displacement of a free rigid body. New quaternion and biquaternion formulae for the summation of finite rotations and finite screw displacements of a free rigid body in four-dimensional skew-symmetric operators are also proposed. The proposed equations and formulae are constructed using the Kotelnikov–Study transference principle. The use of the proposed real and dual matrix kinematic equations of motion, as well as quaternion and biquaternion kinematic equations of motion, of a rigid body to construct new high-precision algorithms for operating strapdown inertial navigation systems is discussed.

Cauchy representation formulas for Maxwell equations in 3-dimensional domains with fractal boundaries

The aim of this paper is to offer a new definition of Cauchy integral associated with Maxwell equations on 3-dimensional domains with fractal boundaries. This new Cauchy integral leads to several types of integral representation formulas, including the Cauchy representation formula for time-harmonic electromagnetic fields. Our approach is based on the framework of exploiting the close and well-defined connection between the quaternionic analysis and Maxwell equations.

The Cauchy problem for quaternion fuzzy fractional differential equations

We discuss the existence of a solution to the Cauchy problem for fuzzy fractional differential equations that accommodates the notion of fuzzy sets defined by quaternion-valued membership functions. We first propose definitions of quaternion fuzzy sets and discuss entailed results which parallel th ose of regular fuzzy numbers. Two existence results, relevant to the Cauchy problem for fuzzy fractional differential equations in the case of bounded integral operators, are given. The results require either Hölder continuous or Lipschitz continuous response functions.

Consimilarity of quaternions and coneigenvalues of quaternion matrices

First of all, by characterizing solutions of the quaternion equation ax=x˜b, this paper studies consimilarity of quaternions and some related consequences. For the important role of coneigenvalues in consimilarity tranformations of quaternion matrices, this paper further derives the relations between principle right coneigenvalues of a quaternion matrix and eigenvalues of the corresponding real representation matrix. Then, based on the real representation matrix, an effective algorithm is presented to calculate all coneigenvalues and the associated coneigenvectors of a quaternion matrix. Finally, two numerical examples are given to verify the effectiveness of the proposed algorithm.

Tensor-product surface patches with Pythagorean-hodograph isoparametric curves

Tensor-product surface patches with Pythagorean-hodograph isoparametric curves Rida T. Farouki Department of Mechanical and Aerospace Engineering, University of California, Davis, CA 95616, USA. Francesca Pelosi Dipartimento di Matematica, Universit`a di Roma “Tor Vergata,” Via della Ricerca Scientifica, 00133 Roma, Italy. Maria Lucia Sampoli Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Universit`a di Siena, San Niccol`o, Via Roma 56, 53100 Siena, Italy. Alessandra Sestini Dipartimento di Matematica e Informatica, Universit`a di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy. Abstract The construction of tensor–product surface patches with a family of Pythagorean–hodograph (PH) isoparametric curves is investigated. The simplest non–trivial instances, interpolating four prescribed patch boundary curves, involve degree (5, 4) tensor–product surface patches x(u, v) whose v = constant isoparametric curves are all spatial PH quintics. It is shown that the construction can be reduced to solving a novel type of quadratic quaternion equation, in which the quaternion unknown and its conjugate exhibit left and right coefficients, while the quadratic term has a coefficient interposed between them. A closed– form solution for this type of equation is derived, and conditions for

Solution of a quadratic quaternion equation with mixed coefficients

A comprehensive analysis of the morphology of the solution space for a special type of quadratic quaternion equation is presented. This equation, which arises in a surface construction problem, incorporates linear terms in a quaternion variable and its conjugate with right and left quaternion coefficients, while the quadratic term has a quaternion coefficient placed between the variable and its conjugate. It is proved that, for generic coefficients, the equation has two, one, or no solutions, but in certain special instances the solution set may comprise a circle or a 3-sphere in the quaternion space H. The analysis yields solutions for each case, and intuitive interpretations of them in terms of the four-dimensional geometry of the quaternion space H.

An algebraic method for quaternion and complex Least Squares coneigen-problem in quantum mechanics

In the study of theory and numerical computations of quantum theory, in order to well understand the perturbation theory, experimental proposals and theoretical discussions underlying the quaternion and complex formulations, one meets problems of approximate solutions of quaternion and complex problems, such as approximate solution of quaternion and complex linear equations Aα∼≈αλ or Aα‾≈αλ that is appropriate when there are errors in the vector α and λ, i.e. quaternion and complex Least Squares coneigen-problem in quantum mechanics. This paper, by means of representation of quaternion matrices and complex matrices, studies the problems of quaternion and complex Least Squares coneigen-problem, and give practical algebraic methods of computing approximate coneigenvalues and coneigenvectors for quaternion and complex matrices in quantum mechanics.

On two functional equations originating from number theory

Reducing the functional equations introduced in Proc. Indian Acad. Sci. (Math. Sci.) 113(2) (2003) 91–98 and in Appl. Math. Lett. 21 (2008) 974–977 to equations in complex variables and quaternions, we find general solutions of the equations. We also obtain the stability of the equations.

Construction of optimum controls and trajectories of motion of the center of masses of a spacecraft equipped with the solar sail and low-thrust engine, using quaternions and Kustaanheimo-Stiefel variables

The problem of optimum rendezvous of a controllable spacecraft (SC) with an uncontrollable spacecraft, moving over a Keplerian elliptic orbit in the gravitational field of the Sun, is considered. Control of the SC is performed using a solar sail and low-thrust engine. For solving the problem, the regular quaternion equations of the two-body problem with the Kustaanheimo-Stiefel variables and the Pontryagin maximum principle are used. The combined integral quality functional, which characterizes energy consumption for controllable SC transition from an initial to final state and the time spent for this transition, is used as a minimized functional. The differential boundary-value optimization problems are formulated, and their first integrals are found. Examples of numerical solution of problems are presented. The paper develops the application [1–6] of quaternion regular equations with the Kustaanheimo-Stiefel variables in the space flight mechanics.

Quaternion regularization and trajectory motion control in celestial mechanics and astrodynamics: II

Problems of regularization in celestial mechanics and astrodynamics are considered, and basic regular quaternion models for celestial mechanics and astrodynamics are presented. It is shown that the effectiveness of analytical studies and numerical solutions to boundary value problems of controlling the trajectory motion of spacecraft can be improved by using quaternion models of astrodynamics. In this second part of the paper, specific singularity-type features (division by zero) are considered. They result from using classical equations in angular variables (particularly in Euler variables) in celestial mechanics and astrodynamics and can be eliminated by using Euler (Rodrigues-Hamilton) parameters and Hamilton quaternions. Basic regular (in the above sense) quaternion models of celestial mechanics and astrodynamics are considered; these include equations of trajectory motion written in nonholonomic, orbital, and ideal moving trihedrals whose rotational motions are described by Euler parameters and quaternions of turn; and quaternion equations of instantaneous orbit orientation of a celestial body (spacecraft). New quaternion regular equations are derived for the perturbed three-dimensional two-body problem (spacecraft trajectory motion). These equations are constructed using ideal rectangular Hansen coordinates and quaternion variables, and they have additional advantages over those known for regular Kustaanheimo-Stiefel equations.

Optimal Reorientation Of Spacecraft Orbit

Abstract The problem of optimal reorientation of the spacecraft orbit is considered. For solving the problem we used quaternion equations of motion written in rotating coordinate system. The use of quaternion variables makes this consideration more efficient. The problem of optimal control is solved on the basis of the maximum principle. An example of numerical solution of the problem is given.

An iterative method for quaternionic linear equations

By the real representation of the quaternionic matrix, an iterative method for quaternionic linear equations Ax = b is proposed. Then the convergence conditions are obtained. At last, a numerical example is given to illustrate the efficiency of this method. Keywords—Quaternionic linear equations, Real representation, Iterative algorithm.

A NOTE ON THE HYPERCOMPLEX RIEMANN-CAUCHY LIKE RELATIONS FOR QUATERNIONS AND LAPLACE EQUATIONS

In this Note it is worked out a new set of Laplace-Like equations for quaternions through Riemann-Cauchy hypercomplex relations otained earlier [1]. As in the theory of functions of a complex variable, it is expected that this new set of Laplace-Like equations might be applied to a large number of Physical problems, providing new insights in the Classical Fields Theory. AMS Subject Classification: 30G99, 30E99

On the solutions of some linear complex quaternionic equations.

Some complex quaternionic equations in the type AX − XB = C are investigated. For convenience, these equations were called generalized Sylvester-quaternion equations, which include the Sylvester equation as special cases. By the real matrix representations of complex quaternions, the necessary and sufficient conditions for the solvability and the general expressions of the solutions are obtained.

Two-sided linear split quaternionic equations with n unknowns

In this paper, we investigate linear split quaternionic equations with the terms of the form axb. We give a new method of solving general linear split quaternionic equations with one, two and n unknowns. Moreover, we present some examples to show how this procedure works.