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Why use NR if it fails so badly?
Rate of convergence:
- i+1 = i - f(xi) / f '(xi)
Taylor expand f(xi) & f '(xi) to get:
- i+1 = - i2 f ''(xi) / f '(xi) [quadratic!]
Note both f(x) and f '(x) must be evaluated each iteration, plus both must be continuous near root.
Best use of NR is to "polish-up" bisection root.