NewtonConvergence - Newton Iteration Consider solution to...

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Newton Iteration Consider solution to the equation z 2 = 2 on the real line. The solution is clearly z = ± . Now consider approximating the solutions of this equation using Newton’s Method. We first set up the function to be set to zero, f ( x ) = x 2 ! 2 = 0 . Then our Newton Function is the following. g ( x ) = x ! x 2 ! 2 2 x The derivative of g ( x ) is the following. ! g ( x ) = 1 2 " 1 x 2 Clearly ! g ( x ) = 0 if and only if x = ± . The graph of ! g ( x ) is given below. Based on the graph of ! g ( x ) and using the Mean Value Theorem, we can analyze exactly what will happen when we apply the Newton Function to any starting value. If we have any positive number 0 < x < 2 , then the Mean Value Theorem says that g ( x ) > 2 . For x > 2 , we have that g ( x ) > 2 as well. We also have that g ( x ) ! 2 < 1 2 x ! 2 . So, for x > 2 we have that g n ( x ) ! < 1 2 " # $ % & ' n x ! 2 . This guarantees that g n ( x ) ! 2 as n !" whether x is less than 2 or greater than 2 as long as x is positive. If x is negative, then g n ( x ) ! " by a similar argument. So, Newton’s ! g ( x ) = 1 2 " 1 x 2
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Method converges for all x except x = 0 .
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This note was uploaded on 07/14/2011 for the course MAC 3473 taught by Professor Block during the Spring '08 term at University of Florida.

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NewtonConvergence - Newton Iteration Consider solution to...

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