How Does Including the Quadratic Term Affect the Newton Iteration Formula?

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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 x_n, 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_n, and truncate it after the quadratic term; derive then a general iteration formula for x_n+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 x_n+1.

However, when truncating after quadratic term, isolating x_n+1 becomes considerably more messy. My question is whether it would be valid to try to isolate x_n+1. I have considered using quadratic equation but given the tediousness of this approach I am hoping for a different method.

 

[Dr.D]

Sounds like you need to quit dodging the work and go to it.