A constructive method for solving the equation Xa=b in Rn: A generalization of division in Rn

摘要

In this paper, we present a new approach of defining division of vectors in Rn for arbitrary dimension n. Our approach is based on constructing a class of solutions of the equation Xa=b for any two known vectors a,b∈Rn for arbitrary dimension n by using some basic properties of octonions. The defined procedure differs depending on the dimension of the vectors, being analyzed the cases 1 < n < 7, n=7 and n > 7. We present an algorithm for computing divisions of vectors in multidimension space, and some numerical examples are given to confirm the presented theoretical results. Lastly, our algorithm is applied to extend several known methods of scalar case to multidimensional cases, and some numerical tests are made to compare the performance of the nonlinear system solvers.