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.