From: spellucci@mathematik.tu-darmstadt.de (Peter Spellucci)
Newsgroups: sci.math.num-analysis
Subject: Re: splines!
Date: 26 Nov 1998 17:58:10 GMT

In article <365D764E.B814E0B0@vasteras.mpe.se>,
 Michael Berglund <michael@vasteras.mpe.se> writes:
|> How do I calculate a spline-curve!??!?!!?
|> 
depends on which one you want. 
a very readable first approach is in forsythe&malcolm&moler:
computer methods for mathematical computations (with f77 source)
see also the corresonding chapter in "numerical recipes in c" (with c-source).
these two cover the interpolating cubic polynomial spline.
for more see
de boor: a practical guide to splines.
for software see 
http://www.netlib.org -> browse directory ->
fitpack, diercks.
definition:
the cubic interpolating spline s is a  function which is a cubic polynomial
on each [x_i,x_{i+1}], which is 
in the class C2 and interpolates the given data (x_i,y_i), i=0,..,n+1 (say).
basic idea:
the polynomial spline s is a piecewise cubic C2-function, hence s'' is
piecewise linear continuous. a piecewise linear and continuous function 
is uniquely represented by its values at the abscissae x_i, say M_i.
(the bending moments of the physical spline model).
now integrate s'' two times. this introduces two integration constants 
say c_i and d_i for every interval [x_i,x_{i+1}]. If you require 
that every piece interpolates the data y_i and y_{i+1}, this 
determines c_i and d_i uniquely (from x_j, y_j and the not yet determined M_j)
this also makes the piecewise defined function itself continuous. 
now write down the condition that the piecewise defined function is one times 
continuously differentiable. this gives conditions of the form
   s_i'(x_{i+1}) = s_{i+1}'(x_{i+1}) i=0,..,n-1 ,
where x_1 ,...,x_n are the inner nodes of the grid and s_i is the name of the
piece of s on [x_i,x_{i+1}]. a rather simple calculation results in a
tridiagonal linear system of n equations for the n+2 moments M_0 , .., M_{n+1}.
therefore there are many possibilities to define a spline.
the reuirement M_0=M_{n+1}=0 is the simplest one to make the
construction unique. this is the so called "natural" spline, already used 
by the Viking (but they didn't use this theory , I believe).
hope this helps
peter