Example 5.  Given    and the nodes compute the entries in Aitken's tableau and Neville's tableau for evaluating    at    
Use the rearrangement of the nodes and the recursive subroutines.

Solution 5.

Enter the function and nodes and point of evaluation.  

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

Neville's tableau is

Remark.  Notice the ordering of the nodes in the second column.

    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.

Note.  The convergence is slightly faster than in Example 3 (a) where the nodes were chosen in ascending order    and the  entries on the diagonal were:  

The Neville polynomials    are converging to  .

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

Remark.  Notice the ordering of the nodes in the second column.

    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.

Note.  The convergence is slightly faster than in Example 3 (b) where the nodes were chosen in ascending order    and the  entries on the diagonal were:  

The Aitken polynomials    are converging to  .

(c) John H. Mathews 2005