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Quadratic Newton's Method

  1. Nov 13, 2014 #1
    1. The problem statement, all variables and given/known data
    The Newton iteration formula is based on a Taylor series expansion of the function f(x) around an estimate of the root xn, 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 xn, and truncate it after the quadratic term; derive then a general iteration formula for xn+1, and explain how you would use it.


    2. Relevant equations
    Newton's method equation:
    af2d6f780d8673d64e8cc328ae52631d.png

    Taylor's series expansion with ε=x-x0

    NumberedEquation1.gif

    3. The attempt at a solution
    If you truncate all the terms after the linear term, it becomes a matter of simple rearrangement to isolate xn+1.

    However, when truncating after quadratic term, isolating xn+1 becomes considerably more messy. My question is whether it would be valid to try to isolate xn+1. I have considered using quadratic equation but given the tediousness of this approach I am hoping for a different method.
     
  2. jcsd
  3. Nov 13, 2014 #2
    Sounds like you need to quit dodging the work and go to it.
     
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