Hi Reuel;

Sir Arthur Conan Doyle wrote:

Nothing clears up a case so much as stating it to another person.

To digress for a bit!

Your post is somewhat enigmatic to me.

I am quite an amateur in all that I do and even then mathematics is not my strong point so please forgive any ignorant remarks on my part.

If one uses Thiele's Interpolation Formula to fit a curve that passes through three symmetrical points such as

Can I ask what you are trying to accomplish by using Thiele's Interpolation to begin with? If you can answer that you certainly are not ignorant or amateurish at all. Also you use Maple which places you ahead of lots of mathematicians.

Andrew Wiley wrote:

I never use a computer.

Well goody for you Andy! Read Paul Nahins or Doron Zeilberger books to get idea about what I mean, what I am saying...

I am not saying I can solve your problem. But I am willing to play with it with you. Who knows what will happen. At present though I do not have more than a tiny inkling as to what you are trying to do. Also I have no data.

Sir Arthur Conan Doyle wrote:

It is a capital mistake to theorize before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts.

Alright, here are some images from Maple I made, color-coded in hopes to communicate effectively. On my monitor they look as though they exported with low detail to quality, but they are still readable.

Thanks!

Uploaded Images

Correct - h(x), the unknown blue curve, is a piecewise function of f(x) from 0 to 75 and g(x) from 75 to 150. The ultimate goal is to fit a single curve defined by a single function.

The reason for using Thiele's Interpolation Formula was that it was a feature in Maple, yes. However, Maple also features polynomial, splines, linear, and so on but Thiele's formula does a perfect job on the regular curves shown in the majority of those pictures I uploaded.

The problem arises when the three primary points of the h(x) - the blue curve - are attempted to be fit for the points - (0,0), (75,25), and (150,50) - fit a perfectly linear line. But if more points from the original two functions f and g are chosen, say, the points of mean value, Thiele's formula does not fit very accurately if it fits at all for it is easy to hit vertical asymptotes as the formula divides by zero.

Polynomial interpolation does a terrible job of fitting the ideal h(x) perfectly.

I have read online about various interpolation methods but most of them are beyond me. One that at least sounded promising is found in this link:

http://en.wikipedia.org/wiki/Whittaker%E2%80%93Shannon_interpolation_formula

However it is quite beyond me. Do you happen to know if that is an idea worth trying? I am willing to learn anything new, I am just out of ideas on my end.

I will be taking differential equations starting next month and I know the course has something to do with finding unknown functions. Since you have probably already taken such a course, do you happen to know if there is anything similar to this in said class?

Thanks a lot for getting back to me as quickly as you did.

I will need some time to think.

Thank you in advance.

Exactly. A piecewise made into a single function h(x) or whatever we were calling it.

Part of my obsession with having one function instead of piecewise functions was experimentation I am doing in 3D surfaces which don't look so great when having to work with splines.

But, yes, a single function is the goal. One that matches the piecewise perfectly or at least extremely closely.

I am a perfectionist. If there was something I could do over and over and get a better fit every time, I wouldn't stop doing it until I had it perfect. By near perfect I meant thousandths of a decimal off.

Okay, but remember, that is why they invented piecewise functions and splines. Because it was often impossible to fit one curve through some set of ordered pairs.

I will do what I can but it may take a while. Please be patient.