A Family of Rational Iterations and Its Application to the Computation of the Matrix pth Root

Matrix fixed-point iterations zn+1 = ψ(zn) defined by a rational function ψ are considered. For these iterations a new proof is given that matrix convergence is essentially reduced to scalar convergence. It is shown that the principal Padé family of iterations for the matrix sign function and the matrix square root is a special case of a family of rational iterations due to Ernst Schröder. This characterization provides a family of iterations for the matrix pth root which preserve the structure of a group of automorphisms associated with a bilinear or a sesquilinear form. The first iteration in that family is the Halley method for which a convergence result is proved. Finally, new algorithms for the matrix pth root based on the Newton and Halley iterations are designed using the idea of the Schur–Newton method of Guo and Higham.