If I wanted to find the least number of iterations it would take for applying Newton's method on a function, is there a formula that I can use to obtain it? If so, what is that formula?
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1You would need to state what the result is you want to achieve. In general Newton's method will not reach a root in finitely many steps. Outside the basins of quadratic convergence the Newton iteration will mostly behave chaotically, see Newton-fractals. These basins are only large for functions with only a few roots and in low dimension. – Lutz Lehmann May 01 '17 at 17:30
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The best I know is an asymptotic bound: a quadratic rate of convergence. https://en.wikipedia.org/wiki/Newton's_method
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Thank you for doing this. Looks good, and I've learned how to typeset URLs. – avs May 01 '17 at 18:09
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It depends on the geometry of the root. You can assume quadratic convergence as long as it is not a multiple root, ie, $f'(r)\neq0$. With high multiplicities or very small derivatives in the root Newton's method gets stuck, but can be modified to remain quadratic.
Norza
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