New integral representations and algorithms for computing nth roots and the matrix sector function of nonsingular complex matrices

It is known that sector switching is a problem of many locally convergent methods for computing the matrix sector function such as Newton's and Halley's methods. In this paper, fast convergent and stable algorithms for approximating the matrix sector function and the principal nth root of complex matrices which avoid these problems are presented. These methods are based on new integral representations of the matrix sector function and the principal nth root of a complex matrix. The new representations are based on Cauchy integral formula and the residue theorem in analytic function theory. The generalized Householder method for computing the matrix sector function and the principal nth root of a complex matrix are introduced. Finally, a new matrix decomposition called "sector factorization" is defined.