Finding Real Roots Using Iteration: How to Choose Starting Values

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SUMMARY

The discussion focuses on finding the real roots of the cubic equation x^3 - 6.2x^2 - 11x - 5 = 0 using the technique of iteration, specifically fixed-point iteration. Participants suggest selecting starting values by testing intervals such as [0,1] and [1,2], evaluating the function at the endpoints to identify sign changes, which indicate the presence of a root within that interval. The midpoint of the interval is recommended as a suitable starting point for the iteration process to ensure convergence.

PREREQUISITES
  • Understanding of fixed-point iteration methods
  • Familiarity with cubic equations and their properties
  • Knowledge of evaluating functions and their derivatives
  • Basic concepts of numerical analysis
NEXT STEPS
  • Research fixed-point iteration techniques in numerical analysis
  • Learn how to apply the Intermediate Value Theorem for root finding
  • Explore methods for determining convergence in iterative processes
  • Study the implications of selecting initial values on the convergence of iterative methods
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Mathematicians, engineering students, and anyone interested in numerical methods for solving polynomial equations will benefit from this discussion.

bemigh
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Hey everyone, I'm not sure where I am going wrong here...
I need to find the 3 roots of the following equation:
x^3 - (6.2)x^2-11x-5=0
and i need to find the real roots using the technique of iteration. I understand this technique, however I am not sure which values i should start off testing...(which x1)
Any help would be appreciated...
Steph
 
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this may be misleading but here goes anyway...

do you mena fixed point iteration s.t. you rearrance f(x) to get a convergent sequence? Well pick intervals like [0,1] [1,2] and find the deriavtie at the end points. also find the function at the endpoints
if the the function at one point is negative while hte other is potivie you have a root in that interval and pick maybe the midpoint of the interval since your sequence would converge anyway.
 

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