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Re: [vox-tech] Fwd: [OT] How do calculators work?
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Re: [vox-tech] Fwd: [OT] How do calculators work?

Bill Kendrick wrote:
Pete posted this from a non-subscribed address:

Date: Fri, 1 Dec 2006 21:31:14 -0800
From: Peter Salzman <p@cbreak.com>
Subject: [OT] How do calculators work?
To: vox-tech@lists.lugod.org

I've always heard that calculators use truncated Taylor series to
approximate functions like trig and exp functions.

Yet that can't be the whole story:

Taylor approximations require more and more terms for convergence as you
evaluate the series farther and farther away from the point of expansion.

Second, we get into problems with singularities and the radius of
convergence.  The series converges on a complex disk (or a real interval)
that contains no singularities.  That presents a major obstacle for
calculating logarithms.  That's why you always expand log(x + k), rather
than log(x).

So saying that calculators use power series approximations can't be the
whole story.  It's a good zeroth order approxmation to the truth.  What's
the first order correction to the truth?


It appears that CORDIC approximations are an effective solution. [1]

I had heard of them, but had never worked through them myself until
prompted by this question.  I still don't quite follow the theoretical
basis [2]. Tracy Allen's explanation [3] is helpful, but doesn't
address the application of the algorithm to logarithms and square roots.


[1] http://www.jacques-laporte.org/TheSecretOfTheAlgorithms.htm
[2] http://www.jacques-laporte.org/Welther-Unified%20Algorithm.pdf
[3] http://www.emesystems.com/BS2mathC.htm

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