Halley 's Method for the Matrix Sector Function

The matrix n -sector function is a generalization of the matrix sign function; it can be used to determine the number of eigenvalues of a matrix in a specific sector of the complex plane and to extract the eigenpairs belonging to this sector without explicitly computing the eigenvalues. It is known that Newton’s method, which can be used for computing the matrix sign function, is not globally convergent for the matrix sector function. The only existing algorithm for computing the matrix sector function is based on the continued fraction expansion approximation to the principal nth root of an arbitrary complex matrix. In this paper, we introduce a new algorithm based on Halley’s generalized iteration formula for solving nonlinear equations. It is shown that the iteration has good error propagation properties and high accuracy. Finally, we give two application examples and summarize the results of our numerical experiments comparing Newton's the continued fraction, and Halley's method.