Re: minimax polynomial determination
- To: mathgroup at smc.vnet.net
- Subject: [mg115815] Re: minimax polynomial determination
- From: Bill Rowe <readnews at sbcglobal.net>
- Date: Fri, 21 Jan 2011 04:33:50 -0500 (EST)
On 1/20/11 at 6:30 AM, nbbienia at cyf-kr.edu.pl (Leslaw Bieniasz) wrote: >On Wed, 19 Jan 2011, Bill Rowe wrote: >>On 1/18/11 at 5:48 AM, nbbienia at cyf-kr.edu.pl (Leslaw Bieniasz) >>wrote: >>>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 >>>example. >>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 >purpose). 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 <http://en.wikipedia.org/wiki/Approximation_theory> for more details. Also, a very good starting point for the Remez algorithm is a Chebyshev series. See <http://en.wikipedia.org/wiki/Remez_algorithm>