Newton's Method and Square RootsDate: 12/04/98 at 22:34:52 From: hendricks Subject: Newton's Square Root Can you explain how finding square roots by hand relates to Newton's method for approximating the zero of a function? Date: 12/05/98 at 08:22:30 From: Doctor Jerry Subject: Re: Newton's Square Root Hi Hendricks, I don't know anything by the name "Newton's Square Root Theory." I wonder if you can be thinking about the fact that the ancient "divide- and-average" algorithm for approximating sqrt(a) is in fact just Newton's method, applied to the function f(x) = x^2 - a? First, here's the divide-and-average algorithm. Suppose that you wish to calculate the square root of a number A. The divide-and-average algorithm is: 1. Choose a rough approximation G of sqrt(A). 2. Divide A by G and then average the quotient with G, that is, calculate: G* = ((A/G)+G)/2 3. If G* is sufficiently accurate, stop. Otherwise, let G = G* and return to step 2. Here's an example: To calculate the sqrt(2), choose G = 1.5. G* = (2/1.5 + 1.5)/2 = 1.41666666666 G* = (2/1.41666666666+1.41666666666)=1.41421568628 G* = (2/1.41421568628+1.41421568628)=1.41421356238 G* = (2/1.41421356238+1.41421356238)=1.41421356238 The number of correct decimal places more or less doubles with each repetition of step 2. Secondly, Newton's method is a method in calculus for determining a zero of a function. Suppose f has a zero near a; then if we set x_1 = a and define: x_{n+1} = x_n - f (x_n)/f'(x_n), n = 1, 2, 3, ... in many cases the sequence x_1, x_2, ... will converge to the zero near a. It turns out that if f(x) = x^2 - a and we take x_1 = a/2, then Newton's method is the divide-and-average algorithm. - Doctor Jerry, The Math Forum http://mathforum.org/dr.math/ |
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