HALLEYS METHOD FOR THE MATRIX SECTOR FUNCTION

The matrix n-sector function is a generalization of the matrix sign fu nction; it can be used to determine the number of eigenvalues of a mat rix in a specific sector of the complex plane and to extract the eigen pairs belonging to this sector without explicitly computing the eigenv alues. It is known that Newton's method, which can be used for computi ng the matrix sign function, is not globally convergent for the matrix sector function. The only existing algorithm fdr computing the matrix sector function is based on the continued fraction expansion approxim ation to the principal nth root of an arbitrary complex matrix, In thi s paper, we introduce a new algorithm based on Halley's generalized it eration formula for solving nonlinear equations. It is shown that the iteration has good error propagation properties and high accuracy. Fin ally, we give two application examples and summarize the results of ou r numerical experiments comparing Newton's, the continued fraction, an d Halley's method.