Divided Difference

The divided difference f[x_0,x_1,x_2,...,x_n], sometimes also denoted [x_0,x_1,x_2,...,x_n] (Abramowitz and Stegun 1972), on n+1 points x_0, x_1, ..., x_n of a function f(x) is defined by f[x_0]=f(x_0) and


for n>=1. The first few differences are




and taking the derivative


gives the identity


Consider the following question: does the property


for n>=2 and h(x) a given function guarantee that f(x) is a polynomial of degree <=n? Aczél (1985) showed that the answer is "yes" for n=2, and Bailey (1992) showed it to be true for n=3 with differentiable f(x). Schwaiger (1994) and Andersen (1996) subsequently showed the answer to be "yes" for all n>=3 with restrictions on f(x) or h(x).

See also

Horner's Method, Interpolation, Newton's Divided Difference Interpolation Formula, Reciprocal Difference

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Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 877-878, 1972.Aczél, J. "A Mean Value Property of the Derivative of Quadratic Polynomials--Without Mean Values and Derivatives." Math. Mag. 58, 42-45, 1985.Andersen, K. M. "A Characterization of Polynomials." Math. Mag. 69, 137-142, 1996.Bailey, D. F. "A Mean-Value Property of Cubic Polynomials--Without Mean Values." Math. Mag. 65, 123-124, 1992.Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 439-440, 1987.Jeffreys, H. and Jeffreys, B. S. "Divided Differences." §9.012 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 260-264, 1988.Schwaiger, J. "On a Characterization of Polynomials by Divided Differences." Aequationes Math. 48, 317-323, 1994.Sauer, T. and Xu, Y. "On Multivariate Lagrange Interpolation." Math. Comput. 64, 1147-1170, 1995.Whittaker, E. T. and Robinson, G. "Divided Differences" and "Theorems on Divided Differences." §11-12 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 20-24, 1967.

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Divided Difference

Cite this as:

Weisstein, Eric W. "Divided Difference." From MathWorld--A Wolfram Web Resource.

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