Second-order complex Newton root-finding algorithm for determining resonant modes of a nontrivial laser system (Q1324196)

The paper deals with a second-order algorithm obtained by a three-terms Taylor series expansion. There is a misprint in the main equation (10). The correct expression of (10) is given by \[ x_{i+1}= x_ i- {f'(x_ i)\over f''(x_ i)}\pm\sqrt{{[f'(x_ i)]^ 2- 2f(x_ i)f''(x_ i)\over f''(x_ i)^ 2}}.\tag{10} \] The authors don't indicate how the sign \(+\) or \(-\) is taken in (10) in order to obtain a convergent method. An error estimation and some examples are given.