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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