Example 4.  Find the continuous least squares polynomial of degree n=4 that approximates the function    over the interval  .  

Solution 4.

The set of functions is .  

The inner product

Since  i  and  j  are positive integers, this can be simplified with the command

Therefore, the Gram matrix G is the 5×5 Hilbert matrix, which is a real, symmetric and positive definite matrix.  

(a).  Form the matrix A and vector B.

Enter the function  ,  and the set of functions  , and compute   for   , and 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  .

Solve the linear system for the coefficients using   and the computation  .

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 !

(c) John H. Mathews 2004