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MathGroup Archive 1996

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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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