(Comp.sys.handhelds)
Item: 2376 by rouben at math14.math.umbc.edu
Author: [Rouben Rostamian]
  Subj: SPLINE v2.0
  Date: Sun Mar 10 1991

This is version 2.0 of SPLINE.  SPLINE generates a piecewise cubic and twice
continuously differentiable interpolation y(x) of a set of points (x_i,y_i),
i=1,2,...,n.  It is assumed throughout that x_1 < x_2 < ... < x_n.

SPLINE's default action is to generate a _natural_ cubic spline, i.e., the
second derivative y'' vanishes at the end points x_1 and x_n.  The default
action of SPLINE may be modified by specifying optional switches which are
described later.

------- AUTHOR --------------------------------------------------------------
Rouben Rostamian                          Telephone: (301) 455-2458
Department of Mathematics and Statistics  e-mail:
University of Maryland Baltimore County   bitnet: rostamian@umbc.bitnet
Baltimore, MD 21228,  U.S.A.              internet: rouben@math9.math.umbc.edu

------- INPUT ---------------------------------------------------------------
SPLINE reads its input from the stack.  The n coordinate points may be
specified in two different formats:

THE COORDINATE-PAIRS INPUT FORMAT:
n+1: (x_1,y_1)
n:   (x_2,y_2)
     ...
2:   (x_n,y_n)
1:           n

THE ARRAY INPUT FORMAT:
2:   [ y_1 y_2 ... y_n ]
1:           [ x_1 x_n ]

In the ARRAY input format the interval [x_1 x_n] is automatically divided
into n equally spaced nodes.  The COORDINATE-PAIRS format is useful if
the nodes are not equally spaced.

----------- OUTPUT ----------------------------------------------------
The output of SPLINE is a *program* which can be used as a user-defined
function.  The program, which is placed on level 1 of the stack, has the 
following general format:

1: << -> X 
      << 
          Description of the spline curve here
      >>
   >>

This program may be stored in a variable, say TRY, and may be
evaluated as "TRY(X)" (algebraic mode) or as "X TRY" (RPN mode.)
TRY(X) may be plotted with the usual plotting commands.

-------- OPTIONAL SWITCHES ---------------------------------------------
SPLINE by default imposes the natural boundary conditions y''(x_1) = y''(x_n)
= 0.  It is possible to specify instead the first derivatives a := y'(x_1) and
b := y'(x_n) as boundary conditions. For this, enter the data in one of the
two formats described before, then push the list { a b } into the level 1 of
stack.

SPLINE can also return the first derivative y'(x) and the second derivative
y''(x) of the cubic spline interpolant.  To get these, enter data as before,
optionally enter the list { a b } from the previous paragraph, then push the
characters 'D1' into level 1 to compute y'(x), or 'D2' to compute y''(x).

---------- EXAMPLES --------------------------------------------------
Example 1:
    5:  (0,0)
    4:  (1,1)
    3:  (2,4)
    2: (4,16)
    1:     4

SPLINE returns the program:

1: << -> X
  <<
    CASE 'X' 2 >=
      THEN '2.60869565217*(4-X)^3/2/6+6.86956521739*(X-2)+2.26086956522'
      END 'X' 1 >=
      THEN '(2.34782608696*(2-X)^3+2.60869565217*(X-1)^3)/6
                         +2.95652173913*(X-1)+.608695652173'
      END  '2.34782608696X^3/6+.608695652173*X'
    END EVAL
  >>
>>

Example 2a:
    2: [ 0 1 0 1 0 1 0 ]
    1:           [ 0 9 ]

The second derivative is zero at the end points.

Example 2b:
    3: [ 0 1 0 1 0 1 0 ]
    2:           [ 0 9 ]
    1:           { 0 0 }

The first derivative is zero at the ends.   Replacing { 0 0 } by { 1 -1 }
makes the first derivative equal 1 and -1 at the ends.

Example 2c:
    4: [ 0 1 0 1 0 1 0 ]
    3:           [ 0 9 ]
    2:           { 0 0 }
    1:              'D1'

Computes the first derivative y'(x) of the spline y(x) of example 2b.
Replacing 'D1' with 'D2' will computes the second derivative.   It is
instructive to save y(x), y'(x), and y''(x) into variables Y0, Y1, and Y2,
and PLOT { 'Y2(X)' 'Y1(X)' Y0(X)' } with XRANGE set to 0,9 and AUTO.

-------- REFERENCE -----------------------------------------------------
Stoer and Bulirsch, Numerical Analysis

-------- REMARKS -------------------------------------------------------
SPLINE does not use, change, create or modify any global variables.
It does not modify parts of the stack it does not own and does not alter any
system flags, although the calculator has to be in the symbolic mode for
SPLINE to operate.   It clears _user_ flags 6,7,8,9.

-------- NOTES ---------------------------------------------------------
SPLINE V2.0 is completely different from an earlier version (no version
number) that I posted to comp.sys.handhelds a few weeks ago.  This new version
generates code which executes about 4 times faster than the previous version. 
It also has many additional features.  I will not describe the differences
here because because I consider the previous version obsolete.