Example 1.  Given the three distinct nodes    and the function  .  
Use recursion to construct the polynomial  .

Solution 1.

Remark.  There are three ways to construct    and we will explore all three of them.  To prevent confusion, the latter two will be named   and  .

Define    

Define    using   and .  

Define    using   and .  

Define    using   and .  

1 (a).  Define    using    and  .   

1 (b).  Define    using    and  .   

1 (c).  Define    using    and  .   

Conclusion.

The three different methods of constructing   produce algebraically equivalent answers.  Only the first two methods are used in practice.  They are the methods of Aitken and Neville, respectively.

(c) John H. Mathews 2005