Approximation: Smoothing natural spline

We consider the following:

where pk_1(x0) is a (k-1)^th polynomial of x0. That is,

Here, c can be calculated from A*c=b by .

When it is natural spline,

where

and

When it is smoothing natural spline, A2 is only different. That is,

where its diagonal elements are from i=1 to n respectively.

We set n x 1 vector of w and scalar g to smooth out.