Which Root-Finding Method Should I Use for an Oscillatory Function?

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SUMMARY

The discussion centers on selecting an appropriate root-finding method for an oscillatory function. The user initially employed the Newton-Raphson method but encountered issues with it identifying the first root instead of subsequent roots. They considered the bisection method for its guaranteed convergence within a specified interval, despite its slower performance. The conversation also highlighted Brent's Method as a viable option, suggesting a hybrid approach of using bisection or Brent's method for initial approximation followed by Newton-Raphson for rapid convergence.

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  • Understanding of Newton-Raphson method for root-finding
  • Familiarity with bisection method and its convergence properties
  • Knowledge of Brent's Method and its application in root-finding
  • Basic concepts of oscillatory functions and their behavior
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  • Research the implementation of Brent's Method for root-finding
  • Learn about the advantages and limitations of the bisection method
  • Explore the conditions under which Newton-Raphson can be effectively applied
  • Investigate hybrid root-finding algorithms combining bisection and Newton-Raphson methods
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Mathematicians, engineers, and computer scientists involved in numerical analysis, particularly those working with oscillatory functions and seeking efficient root-finding techniques.

a.mlw.walker
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Hi so I have attached a graph, and I am trying to find the first root of the function. I began using Newton Raphson however due to the oscillatory nature of the curve, sometimes NR would not find the first route but an nth route. I need the first. Therefore what I did was compute the first time the function is known to be negative. At this point the x value and the function value are known. It is also known that x = 0 is greater than 1.
My question then is should I use the bisection method as it is guaranteed to solve for a root lying between x = 0 and f(xn) < 0. However it is slow. The secant method requires two initial values so would that work? Or can I use the Newton Raphson method and set a boundary that it knows the route must be before?

Thanks
 

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If you know the interval that contains the root you are looking for, Brent's Method might be a good choice, or a combination of Brent's method (or bisection) to get close, then NR for rapid convergence.

http://en.wikipedia.org/wiki/Brent's_method
 

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