Numerical Analysis II
Dr Abigail Wacher
Nov 18, 2010
1
Two drawbacks of Lagrange formula for obtaining the interpolating polynomial.
1) To evaluate the polynomial there is a lot of arithmetic.
2) If a new node is added to the date say
x
n
C
1
and we wish to include it in the interpolation, then
we would need to construct a new set of Langrange coefficients.
We now consider a different Formula to obtain the unique polynomial.
The Newton Formula
In the Lagrange formula the polynomials
L
0
,
1
,...,
are chosen as the basis for the linear space of
polynomials of degree at most
, and
p
is taken to be a linear combination of these basis elements.
The Newton polynomials
w
0
,
i
,...,
defined as
0
d
1
d
K
0
K
1
...
K
K
1
=
?
j
= 0
K
1
K
= 1, 2,.
..,
form a basis for the polynomials of degree at most
.
The Newton Formula is:
=
fx
0
0
C
0
,
1
1
C
...
C
0
,
1
,...,
where the Newton polynomials are meant to interpolate the given data at nodes
0
,
1
,...,
We need to find
0
,...,
0
=
0
=
0
1
=
1
=
0
C
0
,
1
1
K
0
2
=
2
=
0
C
0
,
1
2
K