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

[tysonk]

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.

 

[fzero]

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.

 

[tysonk]

Oh so if that condition is not met, it converges linearly?

 

[fzero]

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.