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Re: desperately seeking better minimax approximation package
*To*: mathgroup at smc.vnet.net
*Subject*: [mg4194] Re: desperately seeking better minimax approximation package
*From*: Paul Abbott <paul at earwax.pd.uwa.edu.au>
*Date*: Thu, 13 Jun 1996 23:09:36 -0400
*Organization*: University of Western Australia
*Sender*: owner-wri-mathgroup at wolfram.com
John Bunda wrote:
> Specifically, what we'd like to be able to do is approximate
> a continuous function with a even or odd polynomial, such that
> the maximum error is minimized.
What do you need this for? Aren't other types of approximations better
in general (e.g. Pade, Rational functions, Chebyshev polynomial
expansion, ...)?
>For example, sin(x) could be
> approximated as:
> p(x) = c1x + c3*x^3 + c5*x^5 + ...
> where c1, c3, and c5 are chosen to minimize the maximum error
> between p(x) and sin(x). The trouble with the minimax package
> that comes with Mathematica is it requires me to have terms
> with strictly sequential powers - I can't skip the x^2 and x^4
> terms like in the above series.
Perhaps you can use
<<NumericalMath`Approximations`
to do what you want in general? Noting that
Sin[Sqrt[x]]/Sqrt[x] + O[x]^3
2
x x 3
1 - - + --- + O[x]
6 120
you see that you can compute a MiniMaxApproximation for
Sin[Sqrt[x]]/Sqrt[x] over some interval (0 excluded here):
MiniMaxApproximation[Sin[Sqrt[x]]/Sqrt[x],
{x, {0.000001, Pi/2}, 3, 0}];
Then pull out the approximation:
sin[x_] = %[[2,1]];
and turn this into the expansion you want:
x sin[x^2] // Expand
3
0.9999998550999375 x - 0.16666387574085572 x +
5 7
0.00832472373825108 x - 0.00018978623422653169 x
The maximum error occurs for x = Pi/2.
Cheers,
Paul
_________________________________________________________________
Paul Abbott
Department of Physics Phone: +61-9-380-2734
The University of Western Australia Fax: +61-9-380-1014
Nedlands WA 6907 paul at physics.uwa.edu.au
AUSTRALIA http://www.pd.uwa.edu.au/Paul
_________________________________________________________________
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