8.1 Polynomial interpolation
Although interpolation is not strictly a finite differencing topic, we briefly present it here because it is
used below and in Section 9, when discussing spectral methods.
Given a set of
distinct points
(sometimes referred to as nodal points or nodes) and
arbitrary associated function values
, the interpolation problem amounts to finding (in this case) a
polynomial
of degree less than or equal to
such that
for
.
It can be shown that there is one and only one such polynomial. Existence can be shown by explicit
construction: suppose one had
where, for each
,
is a polynomial of degree less than or equal to
such that
Then
as given by Eq. (8.1) would interpolate
at the
nodal points
. The
Lagrange polynomials, defined as
indeed do satisfy Eq. (8.2). Uniqueness of the interpolant can be shown by using the property that
polynomials of order
can have at most
roots, applied to the difference between any two
interpolants.
Defining the interpolation error by
and assuming that
is differentiable enough, it can be seen that
satisfies
where
is called the nodal polynomial of degree
, and
is in the
smallest interval
containing
and
. In other words, if we assume the ordering
, then
can actually be outside
. For example, if
, then
. Sometimes, approximating
by
when
is called extrapolation,
and interpolation only if
, even though an interpolating polynomial is used as
approximation.