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.

The inverse   is the matrix  M.

Enter the function  ,  and the set of functions  , and compute   for   , and write down the linear system GC = Y to be solved.

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