Newton-Raphson, Cont'd

• 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.

Notes: