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.

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The Original Code is the MPI Arbitrary Precision Integer Arithmetic
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Michael J. Fromberger <sting@linguist.dartmouth.edu>

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Copyright (C) 1998, 2000 Michael J. Fromberger. All Rights Reserved.

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$Id: sqrt.txt,v 1.1 2000/07/14 00:44:37 nelsonb%netscape.com Exp $


