Numerical Methods Course Notes Version 0.1 (UCSD Math 174, Fall ...

5.1. POLYNOMIAL INTERPOLATION 65
Supposing for an instant that the constants c k were known, this provides a better way of
calculating p n (t) at arbitrary t. By “better” we mean requiring few multiplications and additions.
This nested calculation is performed iteratively:
v 0 = c n
v 1 = c n−1 + (t − x n−1 )v 0
v 2 = c n−2 + (t − x n−2 )v 1
.
v n = c 0 + (t − x 0 )v n−1
This requires only n multiplications and 2n additions. Compare this with the number required
for using the Lagrange form: at least n 2 additions and multiplications.
5.1.4 Divided Differences
It turns out that the coefficients c k for Newton’s nested form can be calculated relatively easily by
using divided differences. We assume, for the remainder of this section, that we are considering
interpolating a function, that is, we have values of f(x i ) at the nodes x i .
Definition 5.5 (Divided Differences). For a given collection of nodes {x i } n i=0
and values
{f(x i )} n x=0 , a kth order divided difference is a function of k + 1 (not necessarily distinct) nodes,
written as
f [x i0 , x i1 , . . . , x ik ]
The divided differences are defined recursively as follows:
• The 0 th order divided differences are simply defined:
f [x i ] = f(x i ).
• Higher order divided differences are the ratio of differences:
f [x i0 , x i1 , . . . , x ik ] = f [x i 1
, x i2 , . . . , x ik ] − f [ x i0 , x i1 , . . . , x ik−1
]
x ik − x i0
We care about divided differences because coefficients for the Newton nested form are divided
differences:
c k = f [x 0 , x 1 , . . . , x k ] . (5.3)
Because we are only interested in the Newton method coefficients, we will only consider divided
differences with successive nodes, i.e., those of the form f [x j , x j+1 , . . . , x j+k ]. In this case the
higher order differences can more simply be written as
f [x j , x j+1 , . . . , x j+k ] = f [x j+1, x j+2 , . . . , x j+k ] − f [x j , x j+1 , . . . , x j+k−1 ]
x j+k − x j