Matrix methods for finding

Abstract

An iterative algorithm for finding , (m > 0, u < n), is developed which involves generating a sequence of approximations to using the concept of eigenvectors. The convergence of this method is then established by studying the eigenvalues and eigenvectors of a matrix A_n, directly related to the algorithm itself. The matrix A_n is constructed using the eigenvalues and eigenvectors, applying the concepts of diagonalization. An algorithm for finding higher powers of A_n is explained. Using these higher powers of A_n, a direct method is also derived. Two numerical examples explaining the methods are given.

Keywords:

• diagonalization
• nth roots of unity
• linearly independent eigenvectors
• spanning space
• auto-correlation
• convolution

AMS Classifications:

• 15A03
• 15A09
• 15A18
• 15A23