Iterated Interpolation

    We now discuss some heuristic methods of constructing interpolation polynomials recursively.  The methods of Aitken and Neville are examples of how iteration is used to construct a sequence of polynomial approximating of increasing order.

Definition (Selected Interpolation).  Given the function    that is to be approximated, and the set of nodes:  

        .

For any subset of  k  nodes

          

the polynomial that agrees with  f[x]  at the points is denoted  

        .

The polynomial    of degree  k-1  agrees with  f[x]  at these knots   

            for    .  

Exploration.

Exploration 1.  Given the six nodes  .  
1 (a).  Investigate  .   
1 (b).  Investigate  .    
1 (c).  Investigate  .    

Investigation 1.

Exploration 2.  Given the ten nodes  .  
Investigate  ,  ,  ,  ,  ,  and  .   

Investigation 2.

(c) John H. Mathews 2005