# polynomial approximation

Started by May 16, 2007
```Hi
This is regarding usage of chebyshev polynomials for function
approximation. I work on this problem during my spare time.

The Chebyshev polynomial coefficients are obtained from unequally
spaced zeroes of the function. But, if we use the same Chebyshev
polynomial to approximate a periodic finction, is there a certain way
to model (a mathematical  equation) the error involved, assuming
infinite precision (or double precision floating point). Eventually, I
would like to translate this to fixed point.

If there are any books that I should refer to, please let me know.

I have got experimental results for various functions using double
precision floating point format but need a way to mdel it.

I apologize if this is the wrong group to post this question

Regards
Nithin

```
```nithin.pal@gmail.com wrote:
> Hi
> This is regarding usage of chebyshev polynomials for function
> approximation. I work on this problem during my spare time.
>
> The Chebyshev polynomial coefficients are obtained from unequally
> spaced zeroes of the function. But, if we use the same Chebyshev
> polynomial to approximate a periodic finction, is there a certain way
> to model (a mathematical  equation) the error involved, assuming
> infinite precision (or double precision floating point). Eventually, I
> would like to translate this to fixed point.
>
> If there are any books that I should refer to, please let me know.
>
> I have got experimental results for various functions using double
> precision floating point format but need a way to mdel it.
>
> I apologize if this is the wrong group to post this question

It may not be the best group, and I'm certainly not the best respondent
to answer it, but I can recommend a book. "An Introduction to the
Approximation of Functions" by Theodore J. Rivlin. Dover reprint ISBN
0-486-64069-8. My copy cost \$3.50; I'm sure it's more now.
http://tinyurl.com/32w6uj

Jerry
--
Engineering is the art of making what you want from things you can get.
&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;
```
```On May 16, 11:38 am, nithin....@gmail.com wrote:
>
> The Chebyshev polynomial coefficients are obtained from unequally
> spaced zeroes of the function. But, if we use the same Chebyshev
> polynomial to approximate a periodic function, is there a certain way
> to model (a mathematical  equation) the error involved, assuming
> infinite precision (or double precision floating point). Eventually, I
> would like to translate this to fixed point.
>
> If there are any books that I should refer to, please let me know.
>
> I have got experimental results for various functions using double
> precision floating point format but need a way to mdel it.
>
> I apologize if this is the wrong group to post this question
>

it's a good group to post this to.  maybe sci.math would be another
appropriate group.

even though i understand how to use Tchebyshev polynomials to
approximate a constant segment (the passband gain in a Tchebyshev Type
1 filter or the stopband in the Tchebyshev Type 2 filter), i don't
know exactly how to use Tchebyshev polynomials to optimally
approximate a given function with finite order polynomial.  for that
problem, i use the Remez exchange algorithm (not to be confused with
Parks-McClellan).

personally, i don't know how to approximate a periodic function with
Tchebyshev polynomials, but if you *do* compute the Fourier series
coefficients of a periodic function and truncate those coefs to a
finite order, you can use those Fourier series coefs to get you a sum
of Tchebyshev polynomial operators with a single cosine as the input.
is that what you want to do?

r b-j

```
```nithin.pal@gmail.com wrote:

> This is regarding usage of chebyshev polynomials for function
> approximation. I work on this problem during my spare time.

My understanding, not having actually done it, is...

One could write the Taylor series expansion of a
function and truncate the series at some point.  For most
functions it will not be optimal over the desired domain
of the function.

One instead writes the function as an expansion of
Chebyshev (sometimes the spelling is different) polynomials,
truncates that series, and then rewrites the truncated
series as an ordinary polynomial.  Those coefficients will
be a little different from the Taylor series, better over
the desired domain, likely a lot worse outside that domain.

Argument reduction is used to reduce the domain needed for
the polynomial expansion using, for example, trigonometric
identities.

-- glen

```
```In article <1179329920.465429.253030@p77g2000hsh.googlegroups.com>,
nithin.pal@gmail.com says...
>
>
>Hi
>This is regarding usage of chebyshev polynomials for function
>approximation. I work on this problem during my spare time.
>
>The Chebyshev polynomial coefficients are obtained from unequally
>spaced zeroes of the function. But, if we use the same Chebyshev
>polynomial to approximate a periodic finction, is there a certain way
>to model (a mathematical  equation) the error involved, assuming
>infinite precision (or double precision floating point). Eventually, I
>would like to translate this to fixed point.
>
>If there are any books that I should refer to, please let me know.
>
>I have got experimental results for various functions using double
>precision floating point format but need a way to mdel it.
>
>I apologize if this is the wrong group to post this question

IIRC, Numerical Recipes has a discussion of this, as well as code.

```