We start with the interpolating polynomial of degree n given in Newton form:
where the coefficients 
are to be determined. If all the 
, then we
have the Maclaurin expansion of the polynomial; if all 
for some
constant 
, then we have the Taylor series expansion of 
about
c. In either event, the coefficients will be scaled derivatives at a single
point.
We're now interested in the case where the 
are different (possibly simply
equally spaced points); the coefficients will still be related to derivative
information, but that information will be distributed across the , rather
than focused on a point. The key to computing them will be divided differences.
Divided differences are basically approximations to derivatives, as one can see from Theorem 3.6.
This is a somewhat ``classical'' subject (old-fashioned?): one used to consult
tables to evaluate functions, whereas today we've got computers which have been
programmed to do the job for us. Textbooks used to include lots of tables
(e.g. of trigonometric functions), and if you wanted 
, you'd look
in the table, find the values of 
and 
, and
interpolate! Nowadays, this is rarer, but you may have still had to do such
things in a statistics class, for example, where tables of normal probabilities
or t-distribution values are still used....
-
zeroth divided difference:
-
first divided difference:
-
second divided difference:
-
kth divided difference:
-
Newton's interpolatory divided difference formula: using these divided
differences, we can demonstrate that
is the
degree interpolating polynomial.
Theorem 3.6: Suppose that 
and 
are distinct numbers in [a,b]. Then 
Derivation of the coefficients of the Newton polynomial: Rather than define the divided differences as above, we could generate a recursive definition for them.
Define the term 
as the leading coefficient of 
:
Since
where 
is the 
interpolating polynomial to
, and 
is the interpolating
polynomial to 
,
we can compute the leading coefficient of as a function of the leading
coefficients of 
and 
:
Hence,
When you have a recursive definition, you need a basement: it is the leading coefficient of the constant interpolating function:
Conclusion: the Newton interpolating polynomial is given by
To get the polynomial's coefficients, you simply look along the diagonal of the divided difference table. Computation of the coefficients costs
operations, whereas evaluation involves 4n-1 operations. While this seems expensive compared to Horner's method, we generate a succession of estimates (using the 0th through nth degree polynomials). So there is bang for the buck....
This form of the interpolating polynomial is nice because we can easily
increase the degree by adding an additional knot, without much additional
work. We can reuse the interpolating polynomial of previous
degree. Furthermore, the knots don't have to be added at the end or beginning:
this process was independent of the order of the .
By following the table of divided differences up from the function values, we get a look at the estimated value of the function using higher and higher powered polynomials. We hope that if the values are settling down, that we're doing a pretty good job of approximating it (and that we needn't proceed to even higher degree). If we're not settling down, however, we know that with the Newton formulation, we can do even better.
If you're at the ``left hand side'' (near 
) of the table, then it makes
sense to make use of the forward-differences; if you're at the right of the
table, then use the backward-differences. The only advantage of using one side
versus the other is in watching the value of P(x) stabilize as additional
degreed polynomials are used; using either form gives the same value for the
highest degreed approximation. So the authors' injunction that ``The Newton
formulas are not appropriate for approximating f(x) when x lies near the
center of the table....'' are a little draconian.... Use them!