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Unformatted text preview: ChemE 109  Numerical and Mathematical Methods in Chemical and Biological Engineering Fall 2007 Solution to Homework Set 3 1. Recall the derivation of the Newton iteration formula discussed in the lecture; it was based on a Taylor series expansion of the function f ( x ) around an estimate of the root x ( m ) , truncated after the linear term. You are asked to derive a more accurate iteration scheme as follows: Start from the Taylor series expansion of f ( x ) around x ( m ) , and truncate it after the quadratic term; derive then a general iteration formula for x ( m +1) , and explain how you would use it. Solution: From Taylor series, we have f ( x root ) = f ( x ) + df dx ( x )( x root x ) + 1 2 d 2 f dx 2 ( x ) ( x root x ) 2 2 + O (( x root x ) 3 = 0 Define a = 1 2 d 2 f dx 2 ( x ) b = df dx ( x ) c = f ( x ) then we have a ( x x ) 2 + b ( x 1 x ) + c = 0 x 1 x = b b 2 4 ac 2 a x 1 = x + b b 2 4 ac 2 a 2. For the nonlinear functions that follow, provide the Newton iteration formula that you would2....
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 Fall '07
 Christofides

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