Numerical Analysis/Neville's algorithm examples

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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