What is the derivation of cubic spline equations?

[fishingspree2]

I am trying to understand these equations:
http://en.wikipedia.org/wiki/Spline_interpolation#Interpolation_using_natural_cubic_spline

Unfortunately there is no explanation for where S_{i}\left ( x \right ) and the system of equations come from.

Can anyone help me understand? Thank you very much

 

[fishingspree2]

Ok, I have found a document that explains but there is something I don't understand. Heres a screenshot of the problem:

http://img412.imageshack.us/img412/2223/integraton22.th.jpg

When I integrate S''_i(x), I get products of x and x_subscripts. For example, when I integrate once I get
[PLAIN]http://img717.imageshack.us/img717/8044/firstintegral.gif

Can anyone please help me?

(the document I found is here http://www.cs.mcgill.ca/~dtitle/cs350/notes/21_cubsplin.pdf)

 

[hotvette]

A complete derivation of cubic spline equations is here:

https://www.physicsforums.com/showthread.php?t=178945