Pseudodifference operators and uniform convergence of

divided differences. Sbornik: Mathematics, 193(2):205-230, 2002.

Secant method [1, 2], which uses

divided differences instead of the first derivative of the nonlinear operator, is one of the most famous iterative methods for solving the nonlinear equation.

A two-step secant iteration with order of convergence same as (5) with its semilocal and local convergence under combination of Lipschitz and center-Lipschitz continuous

divided differences of order one using majorizing sequences for solving (1) is described in Banach space setting in [17].

Our approach is based on Newton's

divided differences interpolation formula.

where an empty product of

divided differences stands for the function evaluated in [x.sub.1].

The confluent

divided differences involved here are defined as

This condition guarantees the existence of a related subdivision scheme for the

divided differences of the original control points and the existence of an associated Laurent polynomial

We define the complex

divided differences for the knot sequence [N.sub.0] := N [union] {0} via

A radius estimate of the convergence ball of such a method is obtained for the nonlinear systems with Lipschitz continuous

divided differences of the first order.

Among those are the endpoint interpolation property, the shape-preserving properties in the case 0 < q < 1, and the representation via

divided differences. Just as the classical Bernstein polynomials, the q-Bernstein polynomials reproduce linear functions, and they are degree-reducing on the set of polynomials.

This question was first considered by Ullrich [21], who introduced the exponential

divided differences (EDD) for the case when [[LAMBDA].sub.j] consist of equal number of points close to j.

In their algorithms, instead of derivatives,

divided differences are always used.