Newton-like methods for the computation of fixed points

AbstractThe celebrated Banach fixed point theorem provides conditions which assure that the method of successive substitution is convergent; the convergence, however, may take place very slowly so that it may be desirable to use a Newton-like method for the computation of the fixed point. If Newton's method itself is applied one ignores the additional information that the problem arises from a frixed point problem with a contraction mapping. In the present note some variants of Newton's method are discussed which make use of this contraction information; it turns out that the convergence of Newton's method can be accelerated without any relevant additional computational labour

An improved Newton iteration for the generalized inverse of a matrix, with applications

The purpose here is to clarify and illustrate the potential for the use of variants of Newton's method of solving problems of practical interest on highly personal computers. The authors show how to accelerate the method substantially and how to modify it successfully to cope with ill-conditioned matrices. The authors conclude that Newton's method can be of value for some interesting computations, especially in parallel and other computing environments in which matrix products are especially easy to work with

A family of root-finding methods with accelerated convergence

AbstractA parametric family of iterative methods for the simultaneous determination of simple complex zeros of a polynomial is considered. The convergence of the basic method of the fourth order is accelerated using Newton's and Halley's corrections thus generating total-step methods of orders five and six. Further improvements are obtained by applying the Gauss-Seidel approach. Accelerated convergence of all proposed methods is attained at the cost of a negligible number of additional operations. Detailed convergence analysis and two numerical examples are given

Examination of accelerated first order methods for aircraft flight path optimization

Accelerated first order methods for aircraft flight path optimizatio

nth Root extraction: Double iteration process and Newton's method

AbstractA double iteration process already used to find the nth root of a positive real number is analysed and showed to be equivalent to the Newton's method. These methods are of order two and three. Higher-order methods for finding the nth root are also mentioned