Dear R-users, I learned today that there exists an interesting topic in numerical analysis names "best polynomial approximation" (BSA). Given a function f the BSA of degree k, say pk, is the polynomial such that pk=arginf sup(|f-pk|) Although given some regularity condition of f, pk is unique, pk IS NOT calculated with least square. A quick google tour show a rich field of research and many algorithms proposed for computing such a task. I was wondered if some of you knows about some R implementations (packages) for computing BSA. Many thanks in advance, Patrizio as usual I apologize for my fragmented English -- +------------------------------------------------- | Patrizio Frederic, PhD | Assistant Professor, | Department of Economics, | University of Modena and Reggio Emilia, | Via Berengario 51, | 41100 Modena, Italy | | tel: +39 059 205 6727 | fax: +39 059 205 6947 | mail: patrizio.frederic at unimore.it +-------------------------------------------------
Hi, My understanding is that Chebyshev polynomials solve the minimax approximation problem. If this correct, what you need is an algorithm to compute Chebyshev polynomial approximation. I have written an R function to do this. See the attached code that contains the function and an example. Is this helpful? I am not sure if there are better algorithms in some R packages. Ravi. -----Original Message----- From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org] On Behalf Of Patrizio Frederic Sent: Monday, May 17, 2010 5:53 PM To: r-help at r-project.org Subject: [R] best polynomial approximation Dear R-users, I learned today that there exists an interesting topic in numerical analysis names "best polynomial approximation" (BSA). Given a function f the BSA of degree k, say pk, is the polynomial such that pk=arginf sup(|f-pk|) Although given some regularity condition of f, pk is unique, pk IS NOT calculated with least square. A quick google tour show a rich field of research and many algorithms proposed for computing such a task. I was wondered if some of you knows about some R implementations (packages) for computing BSA. Many thanks in advance, Patrizio as usual I apologize for my fragmented English -- +------------------------------------------------- | Patrizio Frederic, PhD | Assistant Professor, | Department of Economics, | University of Modena and Reggio Emilia, | Via Berengario 51, | 41100 Modena, Italy | | tel: +39 059 205 6727 | fax: +39 059 205 6947 | mail: patrizio.frederic at unimore.it +------------------------------------------------- ______________________________________________ R-help at r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
I guess you may be looking for the Remez algorithm. AFAIK there is no implementation in one of the R packages. You can find FORTRAN code in the Collected Algorithms of the ACM (no. 604) which probably could be called from R. There appears to exist a discrete, equi-distant(?) version as function 'remez' in the signal package, if that is of any help to you. I have never used it. Regards, Hans Werner P.S.: The Chebyshev polynomials do not compute the "best polynomial approximation", but they provide a nice way to estimate the maximal distance to this best approximating polynomial. Patrizio Frederic wrote:> > Dear R-users, > I learned today that there exists an interesting topic in numerical > analysis names "best polynomial approximation" (BSA). Given a function > f the BSA of degree k, say pk, is the polynomial such that > > pk=arginf sup(|f-pk|) > > Although given some regularity condition of f, pk is unique, pk IS NOT > calculated with least square. A quick google tour show a rich field of > research and many algorithms proposed for computing such a task. > > I was wondered if some of you knows about some R implementations > (packages) for computing BSA. > > Many thanks in advance, > > Patrizio > > as usual I apologize for my fragmented English > > -- > +------------------------------------------------- > | Patrizio Frederic, PhD > | Assistant Professor, > | Department of Economics, > | University of Modena and Reggio Emilia, > | Via Berengario 51, > | 41100 Modena, Italy > | > | tel: +39 059 205 6727 > | fax: +39 059 205 6947 > | mail: patrizio.frederic at unimore.it > +------------------------------------------------- > > ______________________________________________ > R-help at r-project.org mailing list > https://stat.ethz.ch/mailman/listinfo/r-help > PLEASE do read the posting guide > http://www.R-project.org/posting-guide.html > and provide commented, minimal, self-contained, reproducible code. > >-- View this message in context: http://r.789695.n4.nabble.com/best-polynomial-approximation-tp2220439p2221042.html Sent from the R help mailing list archive at Nabble.com.