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

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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

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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

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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

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