The basic idea behind numerical
differentiation is very simple. If given the set of values
i=0,1,...,n, we determine the interpolating polynomial
through these points. We then differentiate this polynomial
to obtain
whose values for any
is taken as an
approximation to
. Let us very briefly describe this
interpolating polynomial.
Let
be
distinct points on an interval
I and let
be a real valued function which takes on the
values
, at these n+1 points. To construct
a polynomial of degree not exceeding n which passes through the
n+1 points
we use the method of
Lagrange. We begin by expressing the desired polynomial as