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

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SUMMARY

The discussion focuses on deriving a more accurate iteration scheme for the Newton iteration formula by truncating the Taylor series expansion of the function f(x) after the quadratic term. The traditional Newton's method, which relies on a linear approximation, is contrasted with this more complex approach that incorporates quadratic terms. Participants emphasize the challenges of isolating xn+1 when using the quadratic term and suggest that while the quadratic equation could be applied, a more efficient method should be sought. The conversation highlights the necessity of understanding Taylor series and the implications of including higher-order terms in iterative methods.

PREREQUISITES
  • Taylor series expansion
  • Newton's method for root finding
  • Quadratic equations
  • Iterative numerical methods
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Students in mathematics or engineering, particularly those studying numerical analysis and iterative methods for solving equations.

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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:
af2d6f780d8673d64e8cc328ae52631d.png


Taylor's series expansion with ε=x-x0[/B]
NumberedEquation1.gif


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

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