Newtons method and convergence

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SUMMARY

Newton's method exhibits quadratic convergence due to its reliance on the derivative of the function being analyzed. Specifically, when applied to functions like x² - 2, the method rapidly approaches the root by squaring the error term at each iteration. This characteristic is rooted in the Taylor series expansion, which demonstrates how the approximation improves significantly as it nears the actual root. Understanding this concept is crucial for effectively utilizing Newton's method in numerical analysis.

PREREQUISITES
  • Understanding of calculus, particularly derivatives and Taylor series.
  • Familiarity with numerical methods and their applications.
  • Basic knowledge of root-finding algorithms.
  • Experience with mathematical software or programming languages for implementing Newton's method.
NEXT STEPS
  • Study the Taylor series expansion and its application in numerical methods.
  • Explore detailed examples of Newton's method applied to various functions.
  • Learn about the geometric interpretation of convergence in root-finding algorithms.
  • Investigate alternative root-finding methods for comparison, such as the bisection method and secant method.
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Students and professionals in mathematics, computer science, and engineering who are interested in numerical methods and optimization techniques.

nobahar
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Hello!
In Newton's method, I am having difficulty understanding why the convergence is quadratic.
I can imagine the closer and closer approximations moving along the x-axis as the base of a right angle triangle, but I cannot see from here how I arrive at the convergence being quadratic.
Any one know of any sources that explain this in a straightforward manner? Or would anyone like to attempt to explain this for me?! It's really bugging me!
Thanks in advance.
 
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Should it be obvious that for, say, x2 - 2, the rate of convergence of the error should be quadratic? The examples I find on the internet are far from obvious, even those that proport to be geometrical interpretations!
Any help would be really appreciated!
Thanks.
 

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