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Square Root
A simple iterative algorithm is used to compute the greatest integer
less than or equal to the square root. Essentially, this is Newton's
linear approximation, computed by finding successive values of the
equation:
x[k]^2 - V
x[k+1] = x[k] - ------------
2 x[k]
...where V is the value for which the square root is being sought. In
essence, what is happening here is that we guess a value for the
square root, then figure out how far off we were by squaring our guess
and subtracting the target. Using this value, we compute a linear
approximation for the error, and adjust the "guess". We keep doing
this until the precision gets low enough that the above equation
yields a quotient of zero. At this point, our last guess is one
greater than the square root we're seeking.
The initial guess is computed by dividing V by 4, which is a heuristic
I have found to be fairly good on average. This also has the
advantage of being very easy to compute efficiently, even for large
values.
So, the resulting algorithm works as follows:
x = V / 4 /* compute initial guess */
loop
t = (x * x) - V /* Compute absolute error */
u = 2 * x /* Adjust by tangent slope */
t = t / u
/* Loop is done if error is zero */
if(t == 0)
break
/* Adjust guess by error term */
x = x - t
end
x = x - 1
The result of the computation is the value of x.
------------------------------------------------------------------
The contents of this file are subject to the Mozilla Public
License Version 1.1 (the "License"); you may not use this file
except in compliance with the License. You may obtain a copy of
the License at http://www.mozilla.org/MPL/
Software distributed under the License is distributed on an "AS
IS" basis, WITHOUT WARRANTY OF ANY KIND, either express or
implied. See the License for the specific language governing
rights and limitations under the License.
The Original Code is the MPI Arbitrary Precision Integer Arithmetic
library.
The Initial Developer of the Original Code is
Michael J. Fromberger <sting@linguist.dartmouth.edu>
Portions created by Michael J. Fromberger are
Copyright (C) 1998, 2000 Michael J. Fromberger. All Rights Reserved.
Contributor(s):
Alternatively, the contents of this file may be used under the
terms of the GNU General Public License Version 2 or later (the
"GPL"), in which case the provisions of the GPL are applicable
instead of those above. If you wish to allow use of your
version of this file only under the terms of the GPL and not to
allow others to use your version of this file under the MPL,
indicate your decision by deleting the provisions above and
replace them with the notice and other provisions required by
the GPL. If you do not delete the provisions above, a recipient
may use your version of this file under either the MPL or the GPL.
$Id$
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