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Thread Subject:
Parabolic Interpolation

Subject: Parabolic Interpolation

From: Ismail SEZEN

Date: 14 Jan, 2012 20:53:07

Message: 1 of 8

Actualy this question is related mainly about a definition. Is there anyone can describe the definition of "Parabolic Interpolation"? What must we understand when you read the "Parabolic Interpolation" term?

Subject: Parabolic Interpolation

From: Matt J

Date: 14 Jan, 2012 21:08:08

Message: 2 of 8

"Ismail SEZEN" <sezenismail@gmail.com> wrote in message <jespvj$6rm$1@newscl01ah.mathworks.com>...
> Actualy this question is related mainly about a definition. Is there anyone can describe the definition of "Parabolic Interpolation"? What must we understand when you read the "Parabolic Interpolation" term?
===========

This is how I understand it:

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

Subject: Parabolic Interpolation

From: Ismail SEZEN

Date: 14 Jan, 2012 22:01:08

Message: 3 of 8

"Matt J" wrote in message <jesqro$9bu$1@newscl01ah.mathworks.com>...
> This is how I understand it:
> http://en.wikipedia.org/wiki/Successive_parabolic_interpolation

Thank you Matt for the quick answer.
Let me mention that I visited lots of site including the above. I agree with Matt.

Then what do you name a process like below:

1- We've three point. ( x1, x2, x3) and ( f(x1), f(x2), f(x3))
2- I don't know the real function which the values above are taken from. But I know that function would be like parabola. So I choose a second degree polynomial.
3- I'm creating a quadratic function by using this values. [ f(x)=Ax^2+Bx+C ]
4- If I know the quadratic function, I can find the extremum point [Xm and f(Xm) ] by derivative of quadratic function. (Xm=-(B/2A))

Is this Polynomial Interpolation? Curve Fitting? or Lagrange Interpolation? Or Quadratic Interpolation? What is the best term identifies this process?

Excuse me, perhaps this seems a little bit silly (or much more) but I'm really confused about the terms.

Subject: Parabolic Interpolation

From: Bruno Luong

Date: 15 Jan, 2012 08:26:08

Message: 4 of 8

"Ismail SEZEN" <sezenismail@gmail.com> wrote in message <jestv4$i09$1@newscl01ah.mathworks.com>...
> "Matt J" wrote in message <jesqro$9bu$1@newscl01ah.mathworks.com>...
> > This is how I understand it:
> > http://en.wikipedia.org/wiki/Successive_parabolic_interpolation
>
> Thank you Matt for the quick answer.
> Let me mention that I visited lots of site including the above. I agree with Matt.
>
> Then what do you name a process like below:
>
> 1- We've three point. ( x1, x2, x3) and ( f(x1), f(x2), f(x3))
> 2- I don't know the real function which the values above are taken from. But I know that function would be like parabola. So I choose a second degree polynomial.
> 3- I'm creating a quadratic function by using this values. [ f(x)=Ax^2+Bx+C ]
> 4- If I know the quadratic function, I can find the extremum point [Xm and f(Xm) ] by derivative of quadratic function. (Xm=-(B/2A))
>
> Is this Polynomial Interpolation? Curve Fitting? or Lagrange Interpolation? Or Quadratic Interpolation? What is the best term identifies this process?

All four are correct, I would use Quadratic Interpolation.

Bruno

Subject: Parabolic Interpolation

From: Matt J

Date: 15 Jan, 2012 13:25:09

Message: 5 of 8

"Ismail SEZEN" <sezenismail@gmail.com> wrote in message <jestv4$i09$1@newscl01ah.mathworks.com>...
>
> Then what do you name a process like below:
>
> 1- We've three point. ( x1, x2, x3) and ( f(x1), f(x2), f(x3))
> 2- I don't know the real function which the values above are taken from. But I know that function would be like parabola. So I choose a second degree polynomial.
==========

If you don't know the function from which the samples were taken, it would sound to me like this is curve fitting.

Subject: Parabolic Interpolation

From: John D'Errico

Date: 15 Jan, 2012 14:02:08

Message: 6 of 8

"Ismail SEZEN" <sezenismail@gmail.com> wrote in message <jestv4$i09$1@newscl01ah.mathworks.com>...
> "Matt J" wrote in message <jesqro$9bu$1@newscl01ah.mathworks.com>...
> > This is how I understand it:
> > http://en.wikipedia.org/wiki/Successive_parabolic_interpolation
>
> Thank you Matt for the quick answer.
> Let me mention that I visited lots of site including the above. I agree with Matt.
>
> Then what do you name a process like below:
>
> 1- We've three point. ( x1, x2, x3) and ( f(x1), f(x2), f(x3))
> 2- I don't know the real function which the values above are taken from. But I know that function would be like parabola. So I choose a second degree polynomial.
> 3- I'm creating a quadratic function by using this values. [ f(x)=Ax^2+Bx+C ]
> 4- If I know the quadratic function, I can find the extremum point [Xm and f(Xm) ] by derivative of quadratic function. (Xm=-(B/2A))
>
> Is this Polynomial Interpolation? Curve Fitting? or Lagrange Interpolation? Or Quadratic Interpolation? What is the best term identifies this process?
>
> Excuse me, perhaps this seems a little bit silly (or much more) but I'm really confused about the terms.

All of those terms can easily apply here, although some
tend to have subtly different connotations.

Lagrange interpolation does not indicate the order of the
polynomial, so it is perhaps too broad. Furthermore,
Lagrange interpolation usually implies the coefficients
were derived using a specific formula. This does not seem
to be true here.

Curve fitting can to be applied to a broad class of problems
of which this is a subset, but it is perhaps too broad of a
brush to suggest what you are doing here. Thus, one might
use curve fitting to describe a spline model, or a exponential
fit, or any of a variety of schemes.

Polynomial interpolation does describe what you are doing,
but it does not indicate the order of the polynomial. Since
you have exactly 3 points and a quadratic polynomial, why
not be accurate and state what you have done explicitly?

That leaves quadratic interpolation, or perhaps quadratic
polynomial interpolation. It describes exactly what you have
done.

John

Subject: Parabolic Interpolation

From: Matt J

Date: 15 Jan, 2012 14:13:08

Message: 7 of 8

"Ismail SEZEN" <sezenismail@gmail.com> wrote in message <jestv4$i09$1@newscl01ah.mathworks.com>...
>
> Then what do you name a process like below:
>
> 1- We've three point. ( x1, x2, x3) and ( f(x1), f(x2), f(x3))
> 2- I don't know the real function which the values above are taken from. But I know that function would be like parabola. So I choose a second degree polynomial.
> 3- I'm creating a quadratic function by using this values. [ f(x)=Ax^2+Bx+C ]
> 4- If I know the quadratic function, I can find the extremum point [Xm and f(Xm) ] by derivative of quadratic function. (Xm=-(B/2A))
>
> Is this Polynomial Interpolation? Curve Fitting? or Lagrange Interpolation? Or Quadratic Interpolation? What is the best term identifies this process?
================

I amend what I said before. All of these terms apply here, as the others have been saying.

The only footnote is that sometimes the term "quadratic interpolation" refers to an algorithm for iteratively minimizing a 1D function (see the wiki link I gave you) but that doesn't appear to be what you are doing here.

Subject: Parabolic Interpolation

From: John D'Errico

Date: 15 Jan, 2012 14:25:08

Message: 8 of 8

"Matt J" wrote in message <jeumtk$qnk$1@newscl01ah.mathworks.com>...
> "Ismail SEZEN" <sezenismail@gmail.com> wrote in message <jestv4$i09$1@newscl01ah.mathworks.com>...
> >
> > Then what do you name a process like below:
> >
> > 1- We've three point. ( x1, x2, x3) and ( f(x1), f(x2), f(x3))
> > 2- I don't know the real function which the values above are taken from. But I know that function would be like parabola. So I choose a second degree polynomial.
> > 3- I'm creating a quadratic function by using this values. [ f(x)=Ax^2+Bx+C ]
> > 4- If I know the quadratic function, I can find the extremum point [Xm and f(Xm) ] by derivative of quadratic function. (Xm=-(B/2A))
> >
> > Is this Polynomial Interpolation? Curve Fitting? or Lagrange Interpolation? Or Quadratic Interpolation? What is the best term identifies this process?
> ================
>
> I amend what I said before. All of these terms apply here, as the others have been saying.
>
> The only footnote is that sometimes the term "quadratic interpolation" refers to an algorithm for iteratively minimizing a 1D function (see the wiki link I gave you) but that doesn't appear to be what you are doing here.

Perhaps that was more an abuse of the term, not truly
describing what they were doing, i.e., the use of an
interpolant for an optimization problem. Thus the words
"quadratic interpolation" say nothing about the true
problem they are solving - an optimization.

The point is, it is they who are misusing the phrase.

John

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