Newton's Method: Checking that ff'' >0 - Why and What if Not?

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Homework Help Overview

The discussion revolves around Newton's method for finding roots of functions, specifically focusing on the condition that the product of the function and its second derivative, ff'', is greater than zero. Participants are exploring the implications of this condition on the convergence behavior of the method.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster questions the necessity of the condition ff'' >0 and its impact on the effectiveness of Newton's method. Some participants suggest that this condition relates to the monotonic convergence of the sequence towards the root, while others inquire about the consequences if the condition is not satisfied.

Discussion Status

The discussion is active, with participants providing references to external resources and engaging in clarifying the implications of the conditions for convergence. There is an exploration of different convergence rates and the potential for non-convergence under certain circumstances.

Contextual Notes

Participants are considering the conditions under which Newton's method operates effectively, including the implications of having finite derivatives and the behavior of functions near their roots. There is an acknowledgment of the complexity of these conditions and their varying impacts on convergence.

tysonk
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When using Newton's method to find roots, why should we check that ff'' >0 . I can't find an adequate reason for this. Does Newton's method fail otherwise? If so why? Thanks.
 
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You should check out the discussion at http://en.wikipedia.org/wiki/Newton's_method#Analysis The method doesn't necessarily fail, but f f'' >0 is a condition that the sequence monotonically converges to the root. If you read on a bit further into the next section on that page, they explain that f'\neq 0, f'' finite are conditions for quadratic convergence. These latter conditions are much weaker than the f f'' >0 condition.
 
Oh so if that condition is not met, it converges linearly?
 
tysonk said:
Oh so if that condition is not met, it converges linearly?

No, there's no reason to conclude that. If you want to understand the f'\neq 0, f~ f''>0, you might want to consider a few sketches of the behavior of the function to the right of a root. You'll see how that condition leads to the sequence being monotone decreasing.

If the other set of conditions, f'\neq 0, f'' finite, is not met, it's possible that the sequence does not converge at all.
 

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