GenPoly is a simple program to generate polynomials with certain
properties.
Note: I am talking about polynomials and not splines.
GenPoly can create polynomials of any order (dependent on the number
of points you specify). However, polynomials are not as easy to handle
as splines. If you ever tried to create a nice polynomial with certain
properties you know what I mean: The spline joins some points in a
natural and smooth way; the polynomial goes through the points, too, but
it may need to have large extrema in between them.
Sedgewick made some dark allusions about that but I did not believe it
before actually seeing it myself. So, try out but don't be surprised by
strange results.
Some maths:
Weierstrass (18151897) proved that it is possible to approximate any steady
function (in a compact interval) using polynomials.
This means, he proved that for a given tolerance, a polynomial
exists which does not vary from the original function by an
amount larger than the tolerance.
Finding such a polynomial is not trivial however. You can take samples
from the original function and make the polynomial go through these points
but the result will look completely different than the original function.
(This is what I meant above.)
That is why people are using splines and why polynomials are of limited
use.
Note, however, that you can get a fairly well polynomial approximation
of a given function using Tschebyschoff interpolation. (That method chooses
itself where to take samples for the interpolation.)
For more information, see the GenPoly manual further down on this page.
I provide only the source code but compiling it should be trivial.
Requirements:
Just a C++ compiler (GCC recommended).
Platform:
GenPoly works fine on my Linuxix86 box but it should be fairly
platformindependent. It may even compile on Window$ without tweaking.
Compiling:
Just call gcc for the one source file:
gcc genpoly.cpp o genpoly lm fnortti fnoexceptions O2
Install:
Simply copy the genpoly to the location you desire.
Bugs:
The matrix inversion is probably numerically instable. I have the
impression that it worked fine, though.
Other bugs may be reported to me (email on bottom of this
page and in source file).
GenPoly is a command line utility. You specify some properties of the
polynomial on the command line and hit enter to have the polynomial
calculated and dumped to the terminal.
GenPoly understands the following command line options:
help 
Print some usage information.

c 
Print polynomial coefficients only.

p 
Use pow(x,y) in output
instead of x^y.
Useful if you want to insert the polynomial into C/C++ code or
POVRay scenes.

a 
Use x**y instead of
x^y.
Useful when you want to plot the resulting polynomial using GnuPlot.

q
The following command line options (without leading dash) can be used
to specify the polynomial properties:
Remember to use quotes to prevent the shell from interpreting
the brackets.
p(x,y) 
Make the polynimial go through the point at position (x,y).

a(x,m) 
Make the polynimial have ascent m at Xposition x.

e(x) 
Make the polynimial have a firstorder extremum at Xposition x.

e(x,y) 
Make the polynimial have a firstorder extremum at position
(x,y).

w(x) 
Make the polynimial have a firstorder wending point at Xposition
x.

w(x,y) 
Make the polynimial have a firstorder wending point at position
(x,y).

i(x,y,a) 
This is the general property specification, all those above are just
special shorthands of this one:
Specify that the polynomial's ath derivation at Xposition
x shall have the value y.
