A new class of third-order methods in Banach spaces.

Resumen

Convergence properties of a family of iterative methods for solving nonlinear operator equations are investigated. The methods discussed have the formxαn+1:=xαn−[I+0.5L(xαn)J(xαn)Tα(xαn)]f′(xαn)−1f(xαn),whereL(x):=f′(x)−1f′′(x)f′(x)−1f(x), J(x):=(I−0.5L(x))−1,Tα(x):=I+0.5αL(x)H(x), H(x):=(I−L(x))−1.They are a combination of the Halley (α=0) and the super-Halley (α=1) methods which are popular subjects of recent research. In the first part of the paper the authors concentrate on the scalar case. Theorems proved here establish sufficient conditions for convergence when α∈[0,1], α<0, α>1, and make it possible to determine the value of α promising the fastest rate. These results are illustrated by numerical examples. In the second part, which is devoted to the general case, only convex combinations (0≤α≤1) are considered. The main result is a Kantorovich-type theorem. For operators with Lipschitz continuous second derivative f′′, it provides a convergence condition in terms of the starting point x0, existence and uniqueness radii, and an upper bound for the error ∥x∞−xαn∥. This bound is shown to be the least for α=1, which fact leads the authors to conclude that the super-Halley method is faster than the Halley method. This conclusion is confirmed by several numerical examples.