Math Is Fun Forum

Reuel

Member

Registered: 2010-11-28

Posts: 178

Interpolation - Fitting specific curves

Thank you in advance to any who offer their knowledge to my aid.

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

the result will be the function

where the derivative of f(75) is exactly 1 and all derivatives on the interval x=[0,75] have a reciprocal set on x=[75,100].

My difficult problem is this: I need to fit a curve that is the above function f on the interval of x=[0,75] and then is the inverse of f on the interval [75,150] such that the domain of the function (in so far as we are concerned with) is [0,150], the range is [0,50], the derivative at x=75 is still 1 and all derivatives from opposite directions and toward x=75 are equal. That is, the derivative of this new function at x=0 is 1/9 and the derivative of this new function at x=150 is 1/9 and all derivatives from the two end points toward the middle at x=75 are also equal.

That may be confusing and so, to restate it, the function f, above, is the portion of a new function (say, h(x)) on its own interval x=[0,75] and at that point, where the derivative of the function is 1, the function needs to transition into its own inverse at the same point. The inverse of the function f, dislocated so as to connect to the function at the point of x=75, is

and the desired new function is, as stated, f on the interval from 0 to 75 and then its "dislocated" inverse on its interval from x=75 to x=150.

The complex problems arise right from the start when the first thing you try is to fit the points

which is a linear line.

I would provide you all with graphic representation to make clearer the objective but I do not have anywhere to host such a picture at this time. Plotting the function and its inverse and on said boundaries in Maple and using the multiplot feature display(p1,p2,etc) will show the desired curve.

I have no idea if this is something that has an easy solution or not. I am more than willing to learn new things in order to carry it out, however, and I welcome any and all advice. So far all I have managed to do is fit curves that are imperfect... and I am frustrated.

Thank you for your time.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Interpolation - Fitting specific curves

Hi Reuel;

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.

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.

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.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: Interpolation - Fitting specific curves

bobbym,

1) Data? The two curves - f(x) and its inverse and on the stated boundaries - provide an infinite number of data points. Take your pick.

2)

Can I ask what you are trying to accomplish by using Thiele's Interpolation to begin with?

Seeing as I have already performed an application of it, it should be fairly obvious "what I am trying to accomplish".

3.

...you use Maple which places you ahead of lots of mathematicians.

My calculus class in junior college just happens to use Maple. It doesn't affect my amateurish status for good or not. Does it matter?

...

I apologize if my response is annoying, I simply didn't realize the question would be so difficult when it is the problem that has troubled me so. Perhaps it would help make the scenario, which is abstract in itself, easier for you if you imagine, say, a new road is being built that is half of an old function and half of the old function's inverse.

Perhaps I have simply made a mess of everything in attempting to explain it. An easier approach might be to simply start learning about new interpolation methods. I have tried reading about them online but it isn't the same as having someone explain it with the chance for questions and answers.

Also, I will try to get some images to support the question in the hopes that pictures will make the idea easier to understand. I will post again soon with said pictures.

Thanks...

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: Interpolation - Fitting specific curves

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!

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Interpolation - Fitting specific curves

Hi Reuel;

Please bear with me. It looks like to me that you have a piecewise function f(x) from 0 to 75? g(x) from 75 to 150?

What I meant was you are using thiele's interpolation presumably because maple has a task for that. Is there some math reason why you chose it? Perhaps and osculating polynomial would be better...

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: Interpolation - Fitting specific curves

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.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Interpolation - Fitting specific curves

Hi Reuel;

I do not think shannon whittaker applies here. The more points you sample it is not unusual for any type of fit to become just an approximation. You may have to settle for that.

There is nothing so far that I know of yet in a DE class for this. I will need some time to think.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: Interpolation - Fitting specific curves

I will need some time to think.

Thank you in advance.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Interpolation - Fitting specific curves

Hi;

I am playing around with some graphs here. If I understand correctly now, you have two piecewise functions that you want a single exact fit for.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: Interpolation - Fitting specific curves

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.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Interpolation - Fitting specific curves

Okay, extremely closely? How big is the error you can stand?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: Interpolation - Fitting specific curves

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.

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: Interpolation - Fitting specific curves

With Maple you can plot a couple of points more along the curve than just the three linear points, as I said, I was trying out the mean values and quarter and three-quarter lengths of the arcs, but the resultant curves were really bad. I figured I could either fit a ton of points or try to figure something else out more logical.

Fitting a thousand different points seemed like cheating... and it created gigantic functions, anyway. I was hoping the solution would be something simplistic and beautiful in that simplicity.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Interpolation - Fitting specific curves

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.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: Interpolation - Fitting specific curves

Yeah. I suppose I may have to settle for a half-accurate curve when dealing with graphic representations of 3D surfaces. I appreciate the value of splines, I just preferred a single function. It seemed cleaner.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Interpolation - Fitting specific curves

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

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.