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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

f[x_0,x_1,...,x_n]=(f[x_0,...,x_(n-1)]-f[x_1,...,x_n])/(x_0-x_n)
(1)

for n>=1. The first few differences are

f[x_0,x_1]=(f_0-f_1)/(x_0-x_1)
(2)
f[x_0,x_1,x_2]=(f[x_0,x_1]-f[x_1,x_2])/(x_0-x_2)
(3)
f[x_0,x_1,...,x_n]=(f[x_0,...,x_(n-1)]-f[x_1,...,x_n])/(x_0-x_n).
(4)

Defining

pi_n(x)=(x-x_0)(x-x_1)...(x-x_n)
(5)

and taking the derivative

pi_n^'(x_k)=(x_k-x_0)...(x_k-x_(k-1))(x_k-x_(k+1))...(x_k-x_n)
(6)

gives the identity

f[x_0,x_1,...,x_n]=sum_(k=0)^n(f_k)/(pi_n^'(x_k)).
(7)

Consider the following question: does the property

f[x_1,x_2,...,x_n]=h(x_1+x_2+...+x_n)
(8)

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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References

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. https://mathworld.wolfram.com/DividedDifference.html

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