From: jsavard@nospam.ath (John Savard)
Newsgroups: sci.math
Subject: Re: CORDIC algo?
Date: Tue, 17 Feb 1998 20:47:45 GMT

marc@aargh.mayn.de (Marc Heuler) wrote:

>Can you explain how the CORDIC algorithm works?  The documents I found on
>the web so far are inconsistent or lack important key information.  When I
>simulate the algo on the computer, I do not get the correct results.

>I'm looking for sin/cos calculation only.  Thanks in advance!

I once looked up the original paper in print.

And I figured out a simpler algorithm (than the one given, using
hyperbolic trig functions) for logs using similar principles.

Anyhow, for sin and cos, the procedure works like this:

Initialize:

X = K
Y = K
theta = 45 degrees

You need a fixed table, containing -

arctan (1.5)
arctan (1.25)
arctan (1.125)
...

continuing for a length corresponding to the precision you want to
achieve.

What you then do is:

Is the target angle less than 45 degrees? 
   Then: Ynew = Y - X/2
         Xnew = X + Y/2
         theta = theta - arctan(1.5)
   Else: Ynew = Y + X/2
         Xnew = X - Y/2
         theta = theta + arctan(1.5)
X = Xnew
Y = Ynew

then repeat, substituting 'theta' for 45 degrees in the test,
and arctan(1.25), arctan(1.125), and so on successively for
arctan(1.5). X/2 and Y/2 become X/4, Y/4 then X/8, Y/8, and so on as
well.

K is chosen so that the square root of X squared plus Y squared comes
out to 1 at the end, so that X is the cosine of the target angle and Y
is the sine of the target angle.

(Actually, you may really have to start by using arctan(2) and Y - X,
X + Y to handle all angles between 0 and 90.)

The idea is that X/2, X/4, etc., can all be obtained via shifting, so
this lets you calculate the trig functions without doing any
multiplies, which were slow on old hardware.

John Savard
==============================================================================

From: Keith Burnett <keith@xylem.demon.co.uk>
Newsgroups: sci.math,uk.education.maths,aus.mathematics
Subject: Re: Algorithm for inverse trigonom. functions
Date: Sat, 10 Oct 1998 18:25:36 +0100

In article <361CBE2A.F9224710@woerle.net>, Klaus =?iso-
8859-1?Q?W=F6rle?= <klaus@woerle.net> writes
>I am looking for an algorithm to calculate the inverse
>trigonometric functions using only sine, cosine and
>+ - * / . A processor system has only the latter and I
>need to have the inverse functions of sine and cosine.
>
>Any idea?
>Thanks a lot for your help!
>
>Klaus
>
>
The CORDIC routine might help, see

ftp://ftp.exeter.edu/pub/peanut/cordic.tex

might help if you can compile a tex/pictex
document. The paper is by Richard Parris,
and I can e-mail you a PDF version if needed.

Below are the references from the paper

References
1. James Kropa, "Calculator Algorithms", Mathematics Magazine, volume 51
(March
1978), pages 106-109.
2. Charles Schelin, "Calculator Function Approximation", The American
Mathematical
Monthly, volume 90 (May 1983), pages 317-325.
3. Xiaobo Hu, Ronald Harber, and Steven Bass, "Expanding the Range of
Con- vergence of
the CORDIC Algorithm", IEEE Transactions on Computers, volume 40
(January 1991),
pages 13-21.

Cheers 
-- 
Keith Burnett