Example 7.  Find the "least squares cubic" that for the four data points  .  

Solution 7.

(a). Write down the linear system AC = B to be solved.

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

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

Of course we could do all this work in two lines by using Mathematica's built in    procedure.

Notice that there appears to be a little "round-off" error creeping into the computation.  Lets "chop" it off.

We are done.

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

Why is the "least-squares" polynomial a "perfect" power   ?  Because the four data points are "perfect powers."
Of course we could do all this work in two lines by using Mathematica's built in  InterpolatingPolynomial[XY,x]  procedure.

The  InterpolatingPolynomial[XY,x]  procedure produces more accurate results if the answer should be an interpolating polynomial instead of a "least-squares" fit polynomial.

Caveat.  

    In numerical analysis we are concerned with round off error.  In the next example we will explore the "what if" scenario.  What will happen if the element    is replaced with in the matrix ?  This change is merely .  We will see that this will result is tremendous changes in the coefficients .  The numerical algorithms are correct, it is a problem with the matrix, it has a large condition number and is ill conditioned.  We need to be aware of these pitfalls.

(c) John H. Mathews 2004