least squares fit to function

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2 total messages Started by anthony.r.rizzo Tue, 13 Oct 1992 15:16

#486

Author: anthony.r.rizzo
Date: Tue, 13 Oct 1992 15:16

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Some time ago, I asked for net-wisdom on the subject of constrained
least squares.  Many of you responded, letting me know that I was
on track.  First, let me thank all those who responded.  I've since
written my own little program to do constrained least square fits
of n-degree polynomials.  But now I have a slightly off-the-wall
question for you.  Read on. ;-)

Given some explicitely defined function g(x), say, g(x) = sin(x),
can the method of least square errors be applied such that
we derive a second function f(x) = a0 + a1*x + a2*x^2 + ... + an*x^n  
where f(x) is the best least squares fit to g(x)?

I know that this request sounds strange.  But I have a legitimate
application.  The application software that I'm using, ANSYS, 
forces me to input certain quantities as polynomial functions
of temperature.  So I can't get around using a polynomial.
Given this restriction, I at least want to use a best-fit polynomial.

Has anyone ever heard of this being done before?

Tony Rizzo

Re: least squares fit to function

#495

Author: William Skelly
Date: Wed, 14 Oct 1992 02:02

36 lines
2222 bytes

In article <1992Oct13.151606.8435@cbfsb.cb.att.com> rizzo@cbnewsf.cb.att.com (anthony.r.rizzo) writes:
>Some time ago, I asked for net-wisdom on the subject of constrained
>least squares.  Many of you responded, letting me know that I was
>on track.  First, let me thank all those who responded.  I've since
>written my own little program to do constrained least square fits
>of n-degree polynomials.  But now I have a slightly off-the-wall
>question for you.  Read on. ;-)
>
>Given some explicitely defined function g(x), say, g(x) = sin(x),
>can the method of least square errors be applied such that
>we derive a second function f(x) = a0 + a1*x + a2*x^2 + ... + an*x^n  
>where f(x) is the best least squares fit to g(x)?
>
>I know that this request sounds strange.  But I have a legitimate
>application.  The application software that I'm using, ANSYS, 
>forces me to input certain quantities as polynomial functions
>of temperature.  So I can't get around using a polynomial.
>Given this restriction, I at least want to use a best-fit polynomial.
>
>Has anyone ever heard of this being done before?
>
>Tony Rizzo
>

Yes. I think you want to check out Generalised Least-Squares
fitting routines.  I am currently using the one in Numerical 
recipes for which fitting a polynomial is straightforward.
Press et al. recommend their singular value decomposition
over least-squares or weighted least-squares routines.

If you figure out how to use it you might let me know how
I can find the best fit for a spherical variogram using these
routines ;-).  

Cheers,
Chris

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