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Re: minimax polynomial determination

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  • Subject: [mg115815] Re: minimax polynomial determination
  • From: Bill Rowe <readnews at>
  • Date: Fri, 21 Jan 2011 04:33:50 -0500 (EST)

On 1/20/11 at 6:30 AM, nbbienia at (Leslaw Bieniasz)

>On Wed, 19 Jan 2011, Bill Rowe wrote:

>>On 1/18/11 at 5:48 AM, nbbienia at (Leslaw Bieniasz)

>>>I need to determine minimax polynomial approximations to a certain
>>>function computed using MATHEMATICA. Unfortunately it is not
>>>possible to calculate exact derivatives of the function. Is there
>>>any way to use the MiniMaxApproximation[] algorithm with
>>>numerically approximated derivatives? I would appreciate an

>>A minmax approximation can be efficiently computed as a Chebyshev
>>series. You don't need to compute the derivative to get the
>>coefficients for a Chebyshev series. All you need do is sample the
>>function at appropriate points. Then the needed coefficients can be
>>computed using a discrete cosine transform.
>>See the applications section of ref/FourierDCT for details of how
>>to sample the function correctly and use FourierDCT to compute the
>>needed coefficients.

>Sorry, I don't grasp that. Minimax polynomial results from the
>minimisation of the norm maximum of the difference between the
>polynomial and a given function. Therefore, some optimisation
>procedure must be used (often the Remez algorithm is used for this

Yes and no. If you want the optimal minimax polynomial, then yes
you will need to use an optimization algorithm. By optimal, I
mean the minimiax polynomial that has the smallest absolute
error for a given polynomial degree. But if you are willing to
settle for a minimax polynomial that is nearly optimal, then the
Chebyshev series will do and can be computed without computing
the derivatives.

See <> for more details.

Also, a very good starting point for the Remez algorithm is a
Chebyshev series.

See <>

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