# rec23 - 6.006 Intro to Algorithms Recitation 23 May 4 2011...

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6.006 Intro to Algorithms Recitation 23 May 4, 2011 Newton’s Method Newton’s method is a method that iteratively computes progressively better approximations to the roots of a real-valued function f ( x ) . Its input is an initial guess x 0 and the function f ( x ) . The method goes as follows: 1. Given x 0 and f ( x ) , initialize n = 0 2. Compute x n +1 = x n - f ( x n ) f 0 ( x n ) 3. Repeat step 2 until f ( x n ) is sufﬁciently close to a root of f ( x ) . An example of Newton’s method in action:

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6.006 Intro to Algorithms Recitation 23 May 4, 2011 Deriving Heron’s Method Heron’s method (or the Babylonian method) is an algorithm that approximates S . We can inter- pret this problem as solving for the roots of the function f ( x ) = x 2 - S . Since p ( S ) is a zero for this problem, we can apply Newton’s method to derive a method to solve for square roots. In this particular case, f ( x n ) = x 2 n - S and f 0 ( x n ) = 2 x n . Plugging this into Newton’s method, we get the iterative step
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## This note was uploaded on 11/11/2011 for the course MATH 180 taught by Professor Byrns during the Spring '11 term at Montgomery College.

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rec23 - 6.006 Intro to Algorithms Recitation 23 May 4 2011...

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