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Topic:
Fast exponent and logarithm, given initial estimate
Replies:
29
Last Post:
Nov 8, 2004 2:31 AM




Re: Fast exponent and logarithm, given initial estimate
Posted:
Oct 22, 2004 12:41 PM


In article <9215d7ac.0410220624.5902f53b@posting.google.com>, glenlow@pixelglow.com (Glen Low) writes: >> Some advice: try to eliminate the link in your head between "approximation" >> and "taylor expansion" and replace it by "approximation" and "orthogonal >> polynomials" :) Taylor is good in a small neighbourhood around the point >> of interest. Orthogonal polynomials are good in quite a large interval to >> which you can almost always reduce your initial interval to. > >OK. Link eliminated :) > >> > Qn: what tool did you use to generate the minimax polynomial? >> > >> Maple 9.51. The numapprox package, minimax function. The classical algorithm >> associated with minimax approximations is the Remez algorithm (2nd version >> of 1934, I don't have the exact reference at hand). > >Looks like I have to continue to rely on the kindness of strangers :) >, at least until next year when I might be in a position to purchase >one. Or I might have a look at yacas. > >Could you or someone else supply the following minimax polynomials > >* sin(pi*x/2) for x in [0,1] OR in [0,1/2] and [1/2,1], 6 degrees up >to 10e7 or 8 relative error >* log(x) for x in [0,1], either 6 degrees polynomial or 3/2 or 4/1 >rational (Pade?) polynomial with 10e7 or 8 relative error > >I had used <a href="http://www.iancgbell.clara.net/maths/funcs.htm">http://www.iancgbell.clara.net/maths/funcs.htm</a> as a source >for the coefficients but the error terms aren't what I wanted and I >still can't get down to the # of ulps I need. > >Cheers, >Glen Low, Pixelglow Software >www.pixelglow.com
if you have somewhere access to a pc with linux and gcc then you can do anything you want yourself: here is the remezcode: <a href="http://www.netlib.org/cephes/index.html">http://www.netlib.org/cephes/index.html</a> click on remes.tgz if you have access to a mat library then this book will provide you with all information you need: Zbl 0174.20402 Hart, J.F.; Cheney, E.W.; Lawson, C.L.; Maehly, H.J.; Mesztenyi, C.K.; Rice, J.R.; Tha cher, H.G.jun.; Witzgall, C. Computer approximations (English) The SIAM Series in Applied Mathematics. New York etc: John Wiley \& Sons, Inc. X, 343 p. (1968).
hth peter



