Example 3.  Given    and the nodes compute the entries in Aitken's tableau and Neville's tableau for evaluating    at    
Use the recursive subroutines.

Solution 3.

Enter the function and nodes and point of evaluation.  

3 (a).  Compute the values for Neville's tableau.

Neville's tableau is

    The accuracy of the computations in Neville's tableau  improves as you go down the table and to the right. Without knowing where lies in relation to the nodes   it is difficult to determine whether the diagonal approximations are converging faster or the bottom row.  For illustration, consider the sequence on the diagonal.

The Neville polynomials    are converging to  .

3 (b).  Compute the values for Aitken's tableau.

    The accuracy of the computations in Aitken's tableau  improves as you go down the table and to the right. Without knowing where lies in relation to the nodes   it is difficult to determine whether the diagonal approximations are converging faster or the bottom row.  For illustration, consider the sequence on the diagonal.

The Aitken polynomials    are converging to  .

Observation.  

It is easy verify that the entries on the diagonal are the same  
, , ,, and .

Observation.  

It is easy verify that the entries on the diagonal are the same  , , ,, , and .

(c) John H. Mathews 2005