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Homework Statement
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.
Homework Equations
Newton's method equation:
Taylor's series expansion with ε=x-x0[/B]
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.