Extracting P-th Root of a Matrix with Positive Eigenvalues via Newton and Halley’s Methods

Abstract

This paper presents a class of methods for finding zeros of a polynomial of single variable as a prelude to deriving  Newton and Halley’s iteration methods for computing the p-th root of a matrix where . We use the LU Factorization and Givens QR–Cholesky Decomposition to invert the matrix which appears in the Newton and Halley’s methods. The givens QR algorithm when runs to completion could provide the eigenvalues of the symmetric matrices free of charge without additional task. The nature of distribution for the eigenvalues of matix A during iteration phases is discussed.  Numerical illustration is demonstrated with the described procedures which simultaneously provide the p-th root of a matrix and its inverse whose eigenvalues are not in the left-hand side in the extended real line. We compare note with results obtained from Guo and Higham who used polar decomposition in their approach for the cases p=12 and 52 respectively. It was observed that for for , the p-th root tends to a diagonal matrix. As a special case for p = 2,  we computed  Square root of Newton iteration with Euler-Chevbyshev method due to Lakic and Petkovic  for the lower bound of this class of methods.