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88 lines
3.1 KiB
Plaintext
88 lines
3.1 KiB
Plaintext
Square Root
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A simple iterative algorithm is used to compute the greatest integer
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less than or equal to the square root. Essentially, this is Newton's
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linear approximation, computed by finding successive values of the
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equation:
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x[k]^2 - V
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x[k+1] = x[k] - ------------
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2 x[k]
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...where V is the value for which the square root is being sought. In
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essence, what is happening here is that we guess a value for the
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square root, then figure out how far off we were by squaring our guess
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and subtracting the target. Using this value, we compute a linear
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approximation for the error, and adjust the "guess". We keep doing
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this until the precision gets low enough that the above equation
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yields a quotient of zero. At this point, our last guess is one
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greater than the square root we're seeking.
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The initial guess is computed by dividing V by 4, which is a heuristic
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I have found to be fairly good on average. This also has the
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advantage of being very easy to compute efficiently, even for large
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values.
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So, the resulting algorithm works as follows:
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x = V / 4 /* compute initial guess */
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loop
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t = (x * x) - V /* Compute absolute error */
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u = 2 * x /* Adjust by tangent slope */
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t = t / u
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/* Loop is done if error is zero */
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if(t == 0)
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break
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/* Adjust guess by error term */
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x = x - t
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end
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x = x - 1
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The result of the computation is the value of x.
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------------------------------------------------------------------
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***** BEGIN LICENSE BLOCK *****
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Version: MPL 1.1/GPL 2.0/LGPL 2.1
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The contents of this file are subject to the Mozilla Public License Version
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1.1 (the "License"); you may not use this file except in compliance with
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the License. You may obtain a copy of the License at
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http://www.mozilla.org/MPL/
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Software distributed under the License is distributed on an "AS IS" basis,
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WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License
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for the specific language governing rights and limitations under the
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License.
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The Original Code is the MPI Arbitrary Precision Integer Arithmetic
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library.
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The Initial Developer of the Original Code is
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Michael J. Fromberger <sting@linguist.dartmouth.edu>
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Portions created by the Initial Developer are Copyright (C) 1998, 2000
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the Initial Developer. All Rights Reserved.
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Contributor(s):
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Alternatively, the contents of this file may be used under the terms of
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either the GNU General Public License Version 2 or later (the "GPL"), or
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the GNU Lesser General Public License Version 2.1 or later (the "LGPL"),
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in which case the provisions of the GPL or the LGPL are applicable instead
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of those above. If you wish to allow use of your version of this file only
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under the terms of either the GPL or the LGPL, and not to allow others to
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use your version of this file under the terms of the MPL, indicate your
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decision by deleting the provisions above and replace them with the notice
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and other provisions required by the GPL or the LGPL. If you do not delete
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the provisions above, a recipient may use your version of this file under
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the terms of any one of the MPL, the GPL or the LGPL.
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***** END LICENSE BLOCK *****
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$Id: sqrt.txt,v 1.2 2005/02/02 22:28:22 gerv%gerv.net Exp $
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