A form of writing the Lagrange interpolation formula by using divided differences:
 | (1) |
where
are the divided differences of order
; it was treated by I. Newton in 1687. Formula (1) is called Newton's interpolation formula for unequal differences. When the
are equidistant, that is, if
then by introducing the notation
and expressing the divided differences
in terms of the finite differences
according to the formula
one obtains a way of writing the polynomial
in the form
 | (2) |
which is called Newton's interpolation formula for forward interpolation. If the same change of variables is made in the interpolation polynomial
with nodes
, where
,
then one obtains Newton's interpolation formula for backward interpolation:
 | (3) |
Formulas (2) and (3) are convenient for computing tables of a given function
if the point
is at the beginning or the end of the table, since in this case the addition of one or several nodes caused by the wish to increase the accuracy of the approximation does not lead to a repetition of the whole work done as in computations with Lagrange's formula.
References
[1] | I.S. Berezin, N.P. Zhidkov, "Computing methods" , 1 , Pergamon (1973) (Translated from Russian) |
[2] | N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian) |
The divided differences
are defined by:
or
where the prime in
means that the factor
is to be omitted. Formula (1) is also known as the finite Newton series for a function
.
References
[a1] | K.E. Atkinson, "An introduction to numerical analysis" , Wiley (1978) |
[a2] | P.J. Davis, "Interpolation and approximation" , Dover, reprint (1975) |
[a3] | F.B. Hildebrand, "Introduction to numerical analysis" , McGraw-Hill (1974) |
How to Cite This Entry:
Newton interpolation formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Newton_interpolation_formula&oldid=15016
This article was adapted from an original article by M.K. Samarin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article