Polyfit v1.0: produce polynomial given N data points
From: dfk@hp-lsd.cos.hp.com (David F. Kurth)
Newsgroups: comp.sources.hp48
Date: 12 Aug 91 01:49:19 GMT

I've always thought there should be a good use for a program like this,
but I still haven't found it.  It was a good challenge to program the
algorithm, and maybe someone else can find a good use for it.

POLYFIT produces the coefficients of an N-1 degree polynomial given
N data points that lie on the curve of that polynomial.
For example, consider the following table of data points:

   X     Y
-----------
  -2     1
  -1    -1
   0     1
   1    -1
   2     1

There are 5 data points, and the Y value a very symmetric.  POLYFIT
produces the following polynomial:

  f(x) = 2/3*x^4 + 0*x^3 - 8/3*x^2 + 0*x^1 + 1*x^0

If you plot this with an X range between -2 and 2, you will see the
curve "oscillate" between +1 and -1 as it hits all five data points.

POLYFIT is not a least square's type curve fit program.  There is no
minimization of errors.  The polynomial coefficients found by POLYFIT
will exactly hit the given data points (to within the accuracy of the 
calculator).

[Note: If you want a least-squares polyfit, see LEASTSQ on this disk.
-jkh-]

Using POLYFIT
-------------
Enter the following code into your calculator.  After loading, enter
the new directory if not already there.  Push the CST key load the
custom menu which should reveal the following keys:

  POLY - Executes the main program which takes the input array
         of X-Y points from the stack and produces the K array of 
         polymonial coefficients.
  VEIW - Displays the coefficients of the resulting polynomial.
  FX   - Calculates f(x) using level 1 of the stack for x.
  X-Y  - Recalls the input data points to the stack for editing
         or viewing.

To begin, you must first enter the input data points array.  This
is easiest to do by using the Matrix Writer (Right-shift ENTER).
Enter your first X-Y pair of points, then push the down arrow
which informs the calculator that the array is only 2 elements wide.
Continue entering the rest of your points until done.  Then just
push ENTER to return the array to the stack.  Now push POLY to
calculate the coefficients.

The display should produce "DOING L:" which can be ignored (the
program calculates a triangular array of difference values used
for the rest of the computations).  After this, the Ki coefficient
values are calculated and briefly displayed.

When done, the K values can be displayed again by using the VIEW
key.  Output values can be calculated by entering an X value on
the stack and pressing the FX key.  Plotting the equation can
be done by pressing Left-shift PLOT, PLOTR, and then AUTO.

For a small number of data points, the calculations are fairly
quick.  For each additional data point entered, calculation time
roughly doubles.  I've calculated 15 data points, but it took the
calculator about 7.5 hours to finish.  However, with sufficient
memory and batteries, have at it.

Dave Kurth
dfk@hp-lsd.cos.hp.com