Module for the Newton Interpolation Polynomial

Check out the new Numerical Analysis Projects page.

 

Background. Newton Interpolation Polynomial.   

 

Definition. Divided Differences.

 

Animations (Newton InterpolationPolynomial  Newton InterpolationPolynomial).  Internet hyperlinks to animations.


Algorithm (Newton Interpolation Polynomial).   To construct and evaluate the Newton polynomial of  degree  [Graphics:Images/NewtonPolyMod_gr_16.gif]  that passes through the n+1 points  [Graphics:Images/NewtonPolyMod_gr_17.gif],  for  [Graphics:Images/NewtonPolyMod_gr_18.gif]     

    [Graphics:Images/NewtonPolyMod_gr_19.gif]  

where  
    [Graphics:Images/NewtonPolyMod_gr_20.gif]  
and  
    [Graphics:Images/NewtonPolyMod_gr_21.gif]  


Remark 1.  Newton polynomials are created "recursively."

    [Graphics:Images/NewtonPolyMod_gr_22.gif].

Remark 2.  Mathematica's arrays are lists and the subscripts must start with  1  instead of  0.

 

Mathematica Subroutine (Newton Polynomial).

[Graphics:Images/NewtonPolyMod_gr_23.gif]

Example 1.   Form several Newton polynomials of degree  n = 1,2, 3, 4, and 5  for the function  [Graphics:Images/NewtonPolyMod_gr_24.gif]  over the interval  [Graphics:Images/NewtonPolyMod_gr_25.gif]  using equally spaced nodes selected from the following list  
[Graphics:Images/NewtonPolyMod_gr_26.gif]  

Solution 1.

 

Example 2.  Error Analysis.  Investigate the error for the Newton polynomial approximations in Example 1.

Solution 2.

 

Example 3.  What is the maximum over the interval  [Graphics:Images/NewtonPolyMod_gr_169.gif]  for each of the quantities  
3 (a).  Find  [Graphics:Images/NewtonPolyMod_gr_170.gif].  
3 (b).  Find  [Graphics:Images/NewtonPolyMod_gr_171.gif].  
3 (c).  Find  [Graphics:Images/NewtonPolyMod_gr_172.gif].  
3 (d).  Find  [Graphics:Images/NewtonPolyMod_gr_173.gif].  
3 (e).  Find  [Graphics:Images/NewtonPolyMod_gr_174.gif].  

Solution 3.

 

Example 4.  Application to number theory.
4 (a).  Find the formula for the sum of the first  n  integers.
4 (b).  Find the formula for the sum of the squares of the first  n integers.  

Solution 4.

 

Old Lab Project (Newton Interpolation Polynomial  Newton Interpolation Polynomial).  Internet hyperlinks to an old lab project.  

 

Research Experience for Undergraduates

Newton Interpolation Polynomial  Newton Interpolation Polynomial  Internet hyperlinks to web sites and a bibliography of articles.  

  

Downloads (Newton Polynomial Newton Polynomial).  Download this Mathematica notebook.  

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2003