# divided differences

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## divided differences

[də′vīd·əd ′dif·rən·səs]
(mathematics)
Quantities which are used in the interpolation or numerical calculation or integration of a function when the function is known at a series of points which are not equally spaced, and which are formed by various operations on the difference between the values of the function at successive points.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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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].
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.

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