Numerical Analysis/Neville's algorithm examples

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.

Example 1 edit

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.

Example 2 edit

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

 

Exercise edit

Try this one on your own before revealing the answer. You can reveal one step at a time.

Approximate the function   at   using  , and  .

References edit

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