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•  » Thiele's Interpolation Formula - Inverse Phenomenon

## #1 2011-01-29 07:23:55

Reuel
Full Member

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### Thiele's Interpolation Formula - Inverse Phenomenon

Hello!

I am curious and trying to learn more about Thiele's Interpolation Formula, the mechanics of how it works, and understanding, in general, how it is derived, among other things.

That which fascinates me about the formula is an odd inversion of the function within the function itself that seems to appear with any set of three points friendly enough with the function to not divide by zero. For example, if you solve the formula for the three symmetric points

you get the formula

which, on the domain from 0 to 100 creates a pretty graph whose derivatives are 9 when x = 0, 1 when x = 25, and 1/9 when x = 100, simply using the values of x from the points given above.

What's bizarre is that the function has an inverse of itself across the vertical asymptotic line and in the second quadrant which is shifted over by a value equal but negative to the second x value (-25)and up by the second y value (75). The inverse in the second quadrant therefore has derivatives equal to and equidistant to those of the function's curve in the first quadrant, proving they are the same.

How does this work? How can a function contain its own inverse?

My observation of this goings on stems from my attempts at curve fitting which can be found elsewhere in the "Help Me!" portion of the forum, though by now I believe it must be buried several pages back.

Thank you for your input!

## #2 2011-01-29 13:36:08

bobbym
Administrator

Offline

### Re: Thiele's Interpolation Formula - Inverse Phenomenon

In mathematics, you don't understand things. You just get used to them.
Some cause happiness wherever they go; others, whenever they go.
If you can not overcome with talent...overcome with effort.

## #3 2011-01-29 16:32:46

Reuel
Full Member

Offline

### Re: Thiele's Interpolation Formula - Inverse Phenomenon

Yep. You never got back to me. The question I have now is about deriving the formula and, hopefully, learning about the formula itself. I thought it more appropriate, therefore, in the formula forum.

If anyone has a proof/derivation of this formula, I thank ye. I do not have a lot of books on statistics or physics.

## #4 2011-01-29 16:49:52

bobbym
Administrator

Offline

### Re: Thiele's Interpolation Formula - Inverse Phenomenon

Hi Reule;

I am pretty sure that the reason I did not get back to you is that I was unable to find a single smooth curve to join those 2 piecewise functions. Although they look fine graphed they are not one function but 2 very close together. You see interpolation like numerical integration is sensitive to the singularites or discontinuities in the derivative of the function. The two functions have different derivatives at the same point. That is why a fit was not possible in my opinion. I do not know of any rational or polynomial function that have 2 different derivative values at the same point.

In mathematics, you don't understand things. You just get used to them.
Some cause happiness wherever they go; others, whenever they go.
If you can not overcome with talent...overcome with effort.

## #5 2011-01-29 16:54:01

Reuel
Full Member

Offline

### Re: Thiele's Interpolation Formula - Inverse Phenomenon

No, the derivatives would be the same.

Anyway, this question was about deriving the formula itself.

## #6 2011-01-29 17:01:48

bobbym
Administrator

Offline

### Re: Thiele's Interpolation Formula - Inverse Phenomenon

No, the derivatives would be the same.

Yes, in the fit, but a fit was impossible, I think because of the problem with the derivatives being different at the same x value for the two piecewise functions. That is why they could not be joined into one function. You were also asking for a interpolating fit ( exact ). For many discontinous functions that is not possible. Just explaining why there was no reply, I failed in the attempt even though I used thousands of points and 3 computers! Mathematically, I suspect the above reason will not allow the problem to be solved.

Now getting to the Thiele fit that would require reciprocal differences and a continued fraction.

That is how it is done. The p() are the reciprocal differences, very hard to compute by hand. This is clearly a job for a computer.

In mathematics, you don't understand things. You just get used to them.
Some cause happiness wherever they go; others, whenever they go.
If you can not overcome with talent...overcome with effort.

## #7 2011-01-30 10:17:50

Reuel
Full Member

Offline

### Re: Thiele's Interpolation Formula - Inverse Phenomenon

All respect, working in as many "last words" as possible and copying and pasting, so to speak, from Wikipedia isn't helping. I taught myself how to do the formula by hand a year ago when my math teacher refused to teach it to me. What I am looking for is a derivation of the formula.

You know... a proof.

I have contacted a mathematics professor about it from a statistics class. When I obtain the derivation process I shall post it here to benefit others who may also be interested in learning about where the formula comes from.

Thanks...

## #8 2011-01-30 10:24:57

bobbym
Administrator

Offline

### Re: Thiele's Interpolation Formula - Inverse Phenomenon

I have contacted a mathematics professor about it from a statistics class.

That is a good idea because as I understand it Thiele and Fisher ( the statistician ) were coworkers. I have never seen the derivation of it. So I thank you beforehand.

year ago when my math teacher refused to teach it to me.

Maybe he was right? From a Numerical Analyst point of view I am forced to agree. As a practical method it is a hopeless jumble and there are better ways.

In mathematics, you don't understand things. You just get used to them.
Some cause happiness wherever they go; others, whenever they go.
If you can not overcome with talent...overcome with effort.

## #9 2012-03-21 19:17:07

kathrina
Novice

Offline

### Re: Thiele's Interpolation Formula - Inverse Phenomenon

hi can i ask about how does the formula of linear interpolation derived from?

## #10 2012-03-21 19:24:29

kathrina
Novice

Offline

### Re: Thiele's Interpolation Formula - Inverse Phenomenon

help me please i cant understand how does it derived from the only fact i know is that it is connected to the two point slope form

## #11 2012-03-21 19:25:10

bobbym
Administrator

Offline

### Re: Thiele's Interpolation Formula - Inverse Phenomenon

Hi;

There is a formula

The derivation is on the top of this page:

http://en.wikipedia.org/wiki/Linear_interpolation

It can be derived from what I call the point - point formula of Cartesian geometry. basically, if you want to linearly interpolate between two data points you fit a straight line between those two points and then just plug in.

There is also a method that can do this interpolation. Welcome to the forum!

In mathematics, you don't understand things. You just get used to them.
Some cause happiness wherever they go; others, whenever they go.
If you can not overcome with talent...overcome with effort.

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•  » Formulas
•  » Thiele's Interpolation Formula - Inverse Phenomenon

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The text passage is a discussion thread from a math forum where a user named Reuel is inquiring about Thiele's Interpolation Formula, specifically its derivation and an interesting phenomenon they observed regarding the function's inverse. Another user, bobbym, responds with some insights and suggestions. A third user, kathrina, later joins the thread with a question about the derivation of the linear interpolation formula.
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Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

• Index
•  » Formulas
•  » Thiele's Interpolation Formula - Inverse Phenomenon

## #1 2011-01-29 07:23:55

Reuel
Full Member

Offline

### Thiele's Interpolation Formula - Inverse Phenomenon

Hello!

I am curious and trying to learn more about Thiele's Interpolation Formula, the mechanics of how it works, and understanding, in general, how it is derived, among other things.

That which fascinates me about the formula is an odd inversion of the function within the function itself that seems to appear with any set of three points friendly enough with the function to not divide by zero. For example, if you solve the formula for the three symmetric points

you get the formula

which, on the domain from 0 to 100 creates a pretty graph whose derivatives are 9 when x = 0, 1 when x = 25, and 1/9 when x = 100, simply using the values of x from the points given above.

What's bizarre is that the function has an inverse of itself across the vertical asymptotic line and in the second quadrant which is shifted over by a value equal but negative to the second x value (-25)and up by the second y value (75). The inverse in the second quadrant therefore has derivatives equal to and equidistant to those of the function's curve in the first quadrant, proving they are the same.

How does this work? How can a function contain its own inverse?

My observation of this goings on stems from my attempts at curve fitting which can be found elsewhere in the "Help Me!" portion of the forum, though by now I believe it must be buried several pages back.

Thank you for your input!

## #2 2011-01-29 13:36:08

bobbym
Administrator

Offline

### Re: Thiele's Interpolation Formula - Inverse Phenomenon

In mathematics, you don't understand things. You just get used to them.
Some cause happiness wherever they go; others, whenever they go.
If you can not overcome with talent...overcome with effort.

## #3 2011-01-29 16:32:46

Reuel
Full Member

Offline

### Re: Thiele's Interpolation Formula - Inverse Phenomenon

Yep. You never got back to me. The question I have now is about deriving the formula and, hopefully, learning about the formula itself. I thought it more appropriate, therefore, in the formula forum.

If anyone has a proof/derivation of this formula, I thank ye. I do not have a lot of books on statistics or physics.

## #4 2011-01-29 16:49:52

bobbym
Administrator

Offline

### Re: Thiele's Interpolation Formula - Inverse Phenomenon

Hi Reule;

I am pretty sure that the reason I did not get back to you is that I was unable to find a single smooth curve to join those 2 piecewise functions. Although they look fine graphed they are not one function but 2 very close together. You see interpolation like numerical integration is sensitive to the singularites or discontinuities in the derivative of the function. The two functions have different derivatives at the same point. That is why a fit was not possible in my opinion. I do not know of any rational or polynomial function that have 2 different derivative values at the same point.

In mathematics, you don't understand things. You just get used to them.
Some cause happiness wherever they go; others, whenever they go.
If you can not overcome with talent...overcome with effort.

## #5 2011-01-29 16:54:01

Reuel
Full Member

Offline

### Re: Thiele's Interpolation Formula - Inverse Phenomenon

No, the derivatives would be the same.

Anyway, this question was about deriving the formula itself.

## #6 2011-01-29 17:01:48

bobbym
Administrator

Offline

### Re: Thiele's Interpolation Formula - Inverse Phenomenon

No, the derivatives would be the same.

Yes, in the fit, but a fit was impossible, I think because of the problem with the derivatives being different at the same x value for the two piecewise functions. That is why they could not be joined into one function. You were also asking for a interpolating fit ( exact ). For many discontinous functions that is not possible. Just explaining why there was no reply, I failed in the attempt even though I used thousands of points and 3 computers! Mathematically, I suspect the above reason will not allow the problem to be solved.

Now getting to the Thiele fit that would require reciprocal differences and a continued fraction.

That is how it is done. The p() are the reciprocal differences, very hard to compute by hand. This is clearly a job for a computer.

In mathematics, you don't understand things. You just get used to them.
Some cause happiness wherever they go; others, whenever they go.
If you can not overcome with talent...overcome with effort.

## #7 2011-01-30 10:17:50

Reuel
Full Member

Offline

### Re: Thiele's Interpolation Formula - Inverse Phenomenon

All respect, working in as many "last words" as possible and copying and pasting, so to speak, from Wikipedia isn't helping. I taught myself how to do the formula by hand a year ago when my math teacher refused to teach it to me. What I am looking for is a derivation of the formula.

You know... a proof.

I have contacted a mathematics professor about it from a statistics class. When I obtain the derivation process I shall post it here to benefit others who may also be interested in learning about where the formula comes from.

Thanks...

## #8 2011-01-30 10:24:57

bobbym
Administrator

Offline

### Re: Thiele's Interpolation Formula - Inverse Phenomenon

I have contacted a mathematics professor about it from a statistics class.

That is a good idea because as I understand it Thiele and Fisher ( the statistician ) were coworkers. I have never seen the derivation of it. So I thank you beforehand.

year ago when my math teacher refused to teach it to me.

Maybe he was right? From a Numerical Analyst point of view I am forced to agree. As a practical method it is a hopeless jumble and there are better ways.

In mathematics, you don't understand things. You just get used to them.
Some cause happiness wherever they go; others, whenever they go.
If you can not overcome with talent...overcome with effort.

## #9 2012-03-21 19:17:07

kathrina
Novice

Offline

### Re: Thiele's Interpolation Formula - Inverse Phenomenon

hi can i ask about how does the formula of linear interpolation derived from?

## #10 2012-03-21 19:24:29

kathrina
Novice

Offline

### Re: Thiele's Interpolation Formula - Inverse Phenomenon

help me please i cant understand how does it derived from the only fact i know is that it is connected to the two point slope form

## #11 2012-03-21 19:25:10

bobbym
Administrator

Offline

### Re: Thiele's Interpolation Formula - Inverse Phenomenon

Hi;

There is a formula

The derivation is on the top of this page:

http://en.wikipedia.org/wiki/Linear_interpolation

It can be derived from what I call the point - point formula of Cartesian geometry. basically, if you want to linearly interpolate between two data points you fit a straight line between those two points and then just plug in.

There is also a method that can do this interpolation. Welcome to the forum!

In mathematics, you don't understand things. You just get used to them.
Some cause happiness wherever they go; others, whenever they go.
If you can not overcome with talent...overcome with effort.

• Index
•  » Formulas
•  » Thiele's Interpolation Formula - Inverse Phenomenon

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The text passage is a discussion thread from a math forum where a user named Reuel is inquiring about Thiele's Interpolation Formula, specifically its derivation and an interesting phenomenon they observed regarding the function's inverse. Another user, bobbym, responds with some insights and suggestions. A third user, kathrina, later joins the thread with a question about the derivation of the linear interpolation formula.

TYPE: NO_QUESTION

The passage does not contain a specific mathematical question that requires a numerical or algebraic answer. It is a discussion about a mathematical concept and a request for a derivation, but no direct question or problem to solve is presented.