Frequently we need an approximation of a function 
given at discrete
points 
. From basic calculus we know that a reasonably smooth function
can be written in a Taylor series expansion with some truncation
term, thus, is equal to a polynomial plus a correction term. Divided
differences can be used to compute such a polynomial most elegantly. The
first and second divided differences are defined by
(3.4-1) and (3.4-2), respectively, and in general the
divided difference is defined by (3.4-3).
The polynomial approximation 
of degree 
of is then given
by (3.4-4) with an error (3.4-5), where 
. Note, that the divided difference
is a symmetric function of its 
arguments.
As an example we compute the second order polynomial approximation for the
function whose values at 
, 
and 
are known to be
, 
and 
, respectively. Using (3.4-4) we get
the approximation (3.4-6).
Figure 3.4-2 shows the linear and quadratic approximations of a function given discrete values.
