Affine Invariant Convergence Theorems for Newton’s Method and Extensions to Related Methods

For a given starting point, the sequence of Newton iterates is well known to be invariant under affine transformation of the operator equation to be solved. This property, however, is not sufficiently reflected in most convergence theorems that are presently in common use. For this reason, certain affine invariant convergence theorems are given in this paper. The new theorems can be understood as refined versions of the Newton–Mysovskii theorem, the Newton–Kantorovitch theorem (including optimal error bounds) and a convergence theorem for approximate Newton processes. In addition, the concept of affine invariance associated with Newton’s method is extended to S-invariance associated with related iterative methods where S is some set of linear transformations not changing the iterates. As an application of these considerations, a new convergence theorem for a class of generalized Gauss–Newton methods is given. This theorem is S-invariant with an S containing all unitary transformations. Unlike previous competing theorems, the new one reduces to the Newton–Mysovskii theorem when the Gauss–Newton method reduces to Newton’s method.