Example 1.   Form the Newton polynomials of degree  n = 1,2, 3, 4, and 5  for the function    over the interval    using equally spaced nodes selected from the following list  
  
Solution 1 (g).

Comparison with the Lagrange polynomial.

    When we constructed the Lagrange polynomial of degree n = 5 through the six equally spaced nodes we obtained  

        

which involves 30 multiplications and  possibly 35 additions or subtractions (for this example 30 because the first node was 0).

     The Newton polynomial is
  
        

and it involves 15 multiplications and possibly 20 additions or subtractions (for this example 15 because the first node was 0).

     The Newton polynomial is
  
        

and it involves 15 multiplications and possibly 20 additions or subtractions (for this example 15 because the first node was 0).

    The nested Newton polynomial obtained with Mathematica's InterpolatingPolynomial procedure is

        

and it involves 5 multiplications and possibly 10 additions or subtractions ( for this example 9 because the first node was 0).

In general the nth degree Lagrange polynomial requires    multiplications and   additions and subtractions.

The nth degree Newton polynomial requires    multiplications and   additions and subtractions.

And the nth degree Mathematica InterpolatingPolynomial requires multiplications and   additions and subtractions.

Observation.  Consider the Newton polynomial    and Lagrange polynomial of degree 5 that use six equally spaced nodes in the interval  [0,1].  Are they the same ?  The answer is "yes"  even though there were two "routes" for constructing the polynomial.  It is a fact that the polynomial of degree n that passes through n+1 points is "unique."  

Return to the Newton Polynomial Module

(c) John H. Mathews 2004