The main idea of Neville's algorithm is to approximate the value of a polynomial at a particular point without having to first find all of the coefficients of the polynomial. The following examples and exercise illustrate how to use this method.
Approximate the function
at
using
,
, and
.
We begin by finding the value of the function at the given points,
, and
. We obtain

and
.
Since, we know from the Wikipedia page on Neville's Algorithm that
, the approximations for
,
and
are

and
.
Using Neville's Algorithm we can now calculate
and
. We find
and
to be

and

From these two values we now find
to be

Thus, our approximation for the function
at
using
, and
is
. We know the actual value of the function evaluated at
is
or
. Therefore, our approximation within
of the actual value.
For this example, we will use the points given in the example of Newton form to approximate the function
at
. The given points are

and
.
Using
, the approximations for
,
and
are

and
.
Using Neville's Algorithm we now calculate
and
to be equal to


From these two values we find
to be

Try this one on your own before revealing the answer. You can reveal one step at a time.
Approximate the function
at
using
, and
.
We begin by evaluating the function at four given points and obtain


and
.
Thus,
and
.
We can calculate
and
to be



From these values we now find
, and
and get

.
Finally, we can find
to be

http://people.math.sfu.ca/~kevmitch/teaching/316-10.09/neville.pdf