Re: minimax polynomial determination

*To*: mathgroup at smc.vnet.net*Subject*: [mg115770] Re: minimax polynomial determination*From*: Leslaw Bieniasz <nbbienia at cyf-kr.edu.pl>*Date*: Thu, 20 Jan 2011 06:30:27 -0500 (EST)*References*: <ih6ee9$3gb$1@smc.vnet.net>

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). It seems that the MiniMaxAlgorithm implemented in MATHEMATICA needs exact derivatives to perform the minimisation. I don't see how the minimax polynomial could be found without minimisation. I can imagine, however, that the minimisation can be performed without using the derivative information, or by using approximate derivatives. So, my question was if one can use somehow the approximate derivatives with the MiniMaxAlgorithm. Leslaw