In part II, we look at the error we're making in estimating our function (which looks familiar, actually), and examine a method for calculating the value of the interpolating polynomial at a single point (Neville's method).
-
Lagrange interpolating polynomial: the unique polynomial of
degree n that interpolates (that is, passes through) the n+1 data points
.
Theorem 3.3: Let 
be n+1 distinct numbers in [a,b],
and 
. Then 
The error term looks a lot like the error term of the Taylor
polynomial, except that it replaces 
with 
: i.e., it distributes the pain across the interpolation points
(sometimes called the nodes, or knots) 
.
Neville's method: is useful for calculating a value of the interpolating polynomial of degree n to n+1 data points at a single value of x: it is based on successive linear approximation to higher powered interpolating functions to more and more points.