Let
 |
(1)
|
then
![f(x)=f_0+sum_(k=1)^npi_(k-1)(x)[x_0,x_1,...,x_k]+R_n,](/images/equations/NewtonsDividedDifferenceInterpolationFormula/NumberedEquation2.svg) |
(2)
|
where ![[x_1,...]](/images/equations/NewtonsDividedDifferenceInterpolationFormula/Inline1.svg)
is a divided difference,
and the remainder is
![R_n(x)=pi_n(x)[x_0,...,x_n,x]=pi_n(x)(f^((n+1))(xi))/((n+1)!)](/images/equations/NewtonsDividedDifferenceInterpolationFormula/NumberedEquation3.svg) |
(3)
|
for 
.
Newton's Divided Difference Interpolation Formula
See also
Divided Difference, Finite Difference, Hermite's Interpolating Polynomial, Interpolation, Lagrange Interpolating PolynomialExplore with Wolfram|Alpha
References
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 880, 1972.Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 43-44 and 62-63, 1956.Whittaker, E. T. and Robinson, G. "Newton's Formula for Unequal Intervals." §13 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 24-26, 1967.Referenced on Wolfram|Alpha
Newton's Divided Difference Interpolation FormulaCite this as:
Weisstein, Eric W. "Newton's Divided Difference Interpolation Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NewtonsDividedDifferenceInterpolationFormula.html