1 Correction to: Calcolo (2016) 53:133–145 https://doi.org/10.1007/s10092-015-0140-5

Note that inequality (9) in the Original Article is incorrect. In fact, by Cauchy–Schwarz Inequality, we have

$$\begin{aligned} \left| F(x_k)^Ty_{k-1}\right| \le \Vert F(x_k)\Vert \Vert y_{k-1}\Vert ,\quad \mathrm{and}\quad \left| F(x_k)^Td_{k-1}\right| \le \Vert F(x_k)\Vert \Vert d_{k-1}\Vert , \end{aligned}$$

which implies that

$$\begin{aligned}&|\frac{F(x_k)^Ty_{k-1}}{d_{k-1}^Ty_{k-1}}|\Vert d_{k-1}\Vert +|\frac{F(x_k)^Td_{k-1}}{d_{k-1}^Ty_{k-1}}|\Vert y_{k-1}\Vert \\&\quad \le \frac{||F(x_k)||||y_{k-1}||}{|d_{k-1}^Ty_{k-1}|}\Vert d_{k-1}\Vert +\frac{\Vert F(x_k)\Vert \Vert d_{k-1\Vert }}{|d_{k-1}^Ty_{k-1}|}\Vert y_{k-1}\Vert . \end{aligned}$$

By Cauchy–Schwarz Inequality again, we have

$$\begin{aligned} |d_{k-1}^Ty_{k-1}|\le \Vert d_{k-1}\Vert \Vert y_{k-1}\Vert , \end{aligned}$$

which means that

$$\begin{aligned} \frac{||F(x_k)||||y_{k-1}||}{|d_{k-1}^Ty_{k-1}|}\Vert d_{k-1}\Vert +\frac{\Vert F(x_k)\Vert \Vert d_{k-1\Vert }}{|d_{k-1}^Ty_{k-1}|}\Vert y_{k-1}\Vert \ge 2\Vert F(x_k)\Vert . \end{aligned}$$

Then we have inequality (9) in the Original Article is incorrect, that is

$$\begin{aligned} \Vert F(x_k)\Vert +\left| \frac{F(x_k)^Ty_{k-1}}{d_{k-1}^Ty_{k-1}}\right| \Vert d_{k-1}\Vert +\left| \frac{F(x_k)^Td_{k-1}}{d_{k-1}^Ty_{k-1}}\right| \Vert y_{k-1}\Vert \le 3\Vert F(x_k)\Vert \end{aligned}$$

is incorrect. Then Remark 2.1 in the Original Article should be modified in the following way:

Remark 2.1

(2) and (3) in the Original Article imply that

$$\begin{aligned} F(x_k)^Td_k=-\Vert F(x_k)\Vert ^2. \end{aligned}$$

By Cauchy–Schwarz inequality, we have

$$\begin{aligned} \Vert F(x_k)\Vert \le \Vert d_k\Vert , \end{aligned}$$

which means that terminates condition \(\Vert F(x_k)\Vert =0\) in Algorithm 2.1 in the Original Article can be implied by \(\Vert d_k\Vert =0\). Therefore, the terminates condition \(\Vert F(x_k)\Vert =0\) in Algorithm 2.1 in the Original Article can be replaced by \(\Vert d_k\Vert =0\).