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

 Messages: [ Previous | Next ]
 Peter Spellucci Posts: 2,760 Registered: 12/7/04
Re: Fast exponent and logarithm, given initial estimate
Posted: Oct 22, 2004 12:41 PM

glenlow@pixelglow.com (Glen Low) writes:
&gt;&gt; and "taylor expansion" and replace it by "approximation" and "orthogonal
&gt;&gt; polynomials" :-) Taylor is good in a small neighbourhood around the point
&gt;&gt; of interest. Orthogonal polynomials are good in quite a large interval to
&gt;&gt; which you can almost always reduce your initial interval to.
&gt;
&gt;
&gt;&gt; &gt; Qn: what tool did you use to generate the minimax polynomial?
&gt;&gt; &gt;
&gt;&gt; Maple 9.51. The numapprox package, minimax function. The classical algorithm
&gt;&gt; associated with minimax approximations is the Remez algorithm (2nd version
&gt;&gt; of 1934, I don't have the exact reference at hand).
&gt;
&gt;Looks like I have to continue to rely on the kindness of strangers :-)
&gt;, at least until next year when I might be in a position to purchase
&gt;one. Or I might have a look at yacas.
&gt;
&gt;Could you or someone else supply the following minimax polynomials
&gt;
&gt;* sin(pi*x/2) for x in [0,1] OR in [0,1/2] and [1/2,1], 6 degrees up
&gt;to 10e-7 or 8 relative error
&gt;* log(x) for x in [0,1], either 6 degrees polynomial or 3/2 or 4/1
&gt;rational (Pade?) polynomial with 10e-7 or 8 relative error
&gt;
&gt;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
&gt;for the coefficients but the error terms aren't what I wanted and I
&gt;still can't get down to the # of ulps I need.
&gt;
&gt;Cheers,
&gt;Glen Low, Pixelglow Software
&gt;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 remez-code:
<a href="http://www.netlib.org/cephes/index.html">http://www.netlib.org/cephes/index.html</a>
click on remes.tgz
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 \&amp; Sons, Inc. X, 343 p. (1968).

hth
peter

Date Subject Author
10/18/04 Glen Low
10/18/04 Jeremy Watts
10/19/04 Peter Spellucci
10/19/04 Glen Low
10/18/04 bv
10/19/04 Glen Low
10/19/04 George Russell
10/19/04 Glen Low
10/20/04 George Russell
10/20/04 Glen Low
10/21/04 Christer Ericson
10/21/04 Glen Low
10/22/04 Christer Ericson
10/19/04 Martin Brown
10/19/04 Glen Low
10/19/04 Richard Mathar
10/19/04 Glen Low
10/20/04 Gert Van den Eynde
10/20/04 Glen Low
10/20/04 Richard Mathar
10/21/04 Gert Van den Eynde
10/21/04 bv
10/22/04 Glen Low
10/22/04 Peter Spellucci
10/22/04 Glen Low
10/23/04 bv
10/24/04 Gert Van den Eynde
10/25/04 Peter Spellucci
10/20/04 Gert Van den Eynde
11/8/04 Glen Low