Example 3.  Find the "least squares cubic" that for the four data points.   

Solution 3.

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

(b).  Construct the Cholesky factorization of matrix A.

Invoke the subroutine Cholesky.  

Verify the factorization.

(c). Solve the linear system for the coefficients using our  ForeSub[n]  and  [BackSub[n]  subroutines.

First, solve the lower-triangular system    LY = B  for  Y.

Verify that  LY = B.

Second, solve the upper-triangular system    UX = Y  for  X.

Verify that  UX = Y.  

Therefore X is the solution to  LUX = B. and hence AX = B
And we can verify that it is the solution.

Now use the solution to X make the coefficients  .

(d). 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.

We are done.

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

(c) John H. Mathews 2004