Re: Atan2(x, y) ?? How to implement?? - convergence

From: JEMebius (jemebius_at_xs4all.nl)
Date: 11/22/04 

Date: Mon, 22 Nov 2004 20:55:57 +0100
To: Randy Poe <poespam-trap@yahoo.com>

Convergence is a matter of endurance.

The Taylor series for sin (x) and cos (x) are absolutely convergent for
all values of x.

For large values of x the convergence is not terribly fast, but for all
values of x the convergence is eventually better than
that of any geometric series, however small its ratio.

It looks like you've seen Taylor series for sin(x) and cos(x). What
you've shown are truncated Taylor series, which are therefore
approximations. How good the approximations are depend on how many
terms you include and what value of x you use (these are expansions
near x=0, so the approximation gets worse as x gets far from 0).

Furthermore, these approximations don't converge well for any value of
x, and don't converge at all for most. Pick a value of n and see what
happens at x=0.5 and x = 1.5 for instance.

Most computer implementations of the trig functions are based on what
are called the CORDIC algorithms. Do a search for that and you'll find
plenty of material.

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