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authorMatt A. Tobin <mattatobin@localhost.localdomain>2018-02-02 04:16:08 -0500
committerMatt A. Tobin <mattatobin@localhost.localdomain>2018-02-02 04:16:08 -0500
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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.
+
+------------------------------------------------------------------
+ This Source Code Form is subject to the terms of the Mozilla Public
+ # License, v. 2.0. If a copy of the MPL was not distributed with this
+ # file, You can obtain one at http://mozilla.org/MPL/2.0/.