**Remark 1.**

*(a) It follows from condition* (15) *and the estimate:*

$$\begin{aligned} \left| \Omega'(p\_\*)^{-1}[\mathbf{x}, p\_\*; \Omega] \right| &= \left| \Omega'(p\_\*)^{-1}([\mathbf{x}, p\_\*; \Omega] - \Omega'(p\_\*) - \Omega'(p\_\*)) + l \right| \\ &\leq \left| 1 + \left| \Omega'(p\_\*)^{-1}([\mathbf{x}, p\_\*; \Omega] - \Omega'(p\_\*)) \right| \right| \\ &\leq \left| 1 + \gamma |\lambda\_0 - p\_\*| \right| \end{aligned}$$

*and Condition* (14) *can be discarded and M substituted by:*

$$M = M(v) = 1 + \gamma v$$

*or M* = 2, *since υ* ∈ [0, 1*γ* ). *(b) We note that* (2) *does not change if we adopt the conditions of Theorem 1 instead of the stronger ones given in [3]. In practice, for the error bounds, we can consider the computational order of convergence (COC) [10]:*

$$\xi = \frac{\ln \frac{|\delta\_{s+2} - p\_\*|}{|\delta\_{s+1} - p\_\*|}}{\ln \frac{|\delta\_{s+1} - p\_\*|}{|\delta\_s - p\_\*|}}, \quad \text{for each } s = 0, 1, 2, \dots \tag{37}$$

*or the approximated computational order of convergence (ACOC) [10]:*

$$\xi^\* = \frac{\ln \frac{|\delta\_{s+2} - \delta\_{s+1}|}{|\delta\_{s+1} - \delta\_s|}}{\ln \frac{|\delta\_{s+1} - \delta\_s|}{|\delta\_s - \delta\_{s-1}|}}, \quad \text{for each } s = 1, 2, \dots \tag{38}$$

*In practice, we obtain the order of convergence that, avoiding the bounds, involves estimates higher than the first Fréchet derivative.*
