ALGORITHM FOR FINDING THE INVERSE ELEMENTS AND SOLUTION OF THE SILVESTER EQUATION IN THE CLIFFORD ALGEBRAS R4,0, R1,3, R5,0

Abstract

The purpose of the work is to find an algorithm for finding inverse elements in the Clifford algebras R4,0, R1,3, R5,0 and to solve the nonlinear Sylvester equation .
 
 Materials and methods. Using the basic conjugation operations in Clifford algebras, finding an algorithm for finding inverse elements. Application of this algorithm to solve the Sylvester equation.
 
 Results of the work. In Clifford algebras R4,0, R1,3, R5,0, which have a great application in physics, a method for finding inverse elements and equations for finding zero divisors were found. The found algorithm is used to solve the Sylvester equation. For Clifford algebras of even dimension R4,0, R1,3 an algorithm for finding inverse elements is given. Finding inverse elements is closely related to the concept of zero divisors in these algebras. The inverse element method is used to solve the Sylvester equation, using even conjugation, reverse conjugation and Clifford conjugation. For the odd Clifford algebra R5,0, a conjugation is found that can be used to apply the algorithm for finding the inverse element. The method of finding the inverse element is used to solve the Sylvester equation, which, in particular, is used to ensure the robustness of the piezodrive using the controlled relative interval method.
 
 Findings. An algorithm for finding inverse elements is constructed and the Sylvester equation is solved in the Clifford algebras R4,0, R1,3, R5,0.