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 | edit source]
Approximate the function at using , , and .
We begin by finding the value of the function at the given points, , and . We obtain
Since, we know from the Wikipedia page on Neville's Algorithm that , the approximations for , and are
Using Neville's Algorithm we can now calculate and . We find and to be
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 | edit source]
For this example, we will use the points given in the example of Newton form to approximate the function at . The given points are
Using , the approximations for , and are
Using Neville's Algorithm we now calculate and to be equal to
From these two values we find to be
Exercise[edit | edit source]
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
Thus, and .
We can calculate and to be
From these values we now find , and and get
Finally, we can find to be