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.
References in periodicals archive ?
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.