Example 8.  Consider the linear system AC=B in Example 7, i.e.  .

Solution 8.

(a). Enter the new matrix A and vector B.

(b). Solve the linear system for the coefficients using our  LUfactor[n]  and  SolveLU[n]  subroutines.

Remark.  The solution is quite different from that of Example 7.  This is not a problem in our subroutines, Mathematica will also get this answer.

    These solution values are so far from the previous ones in Example 7 that they appear to be a worthless answer, and this type of ill-conditioning is not easy to detect.  One would normally expect that a small change in a coefficient in the matrix would result in a small change in the computed solution.  We changed the element   a small fraction,  ,  and the changes in the coefficients went

from       to    .   

    The element    was replaced with in the matrix ?  This change is merely .  The percentage change in the solution is 36%.  

Continue with the explorations.

(c). Construct the polynomial  p[x].  The coefficients are stored in the array  c  and the elements are .

We are done.

We can graph the polynomial, this is just for fun !

Behold, do our eyes see anything significantly different from the previous graph in Example 7 ?

If you were looking for the polynomial   and found the polynomial    would you be happy ?

Warning.  Remember to check the condition number of your matrix, beware if it is large and seek alternate ways to find compute a solution.  In approximation theory there are other methods and algorithms which can be used.

(c) John H. Mathews 2004