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Published in:
01-06-2022 | Correction
Correction to: A self-adaptive three-term conjugate gradient method for monotone nonlinear equations with convex constraints
Published in: Calcolo | Issue 2/2022
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Excerpt
Note that inequality (9) in the Original Article is incorrect. In fact, by Cauchy–Schwarz Inequality, we have
$$\begin{aligned} \left| F(x_k)^Ty_{k-1}\right| \le \Vert F(x_k)\Vert \Vert y_{k-1}\Vert ,\quad \mathrm{and}\quad \left| F(x_k)^Td_{k-1}\right| \le \Vert F(x_k)\Vert \Vert d_{k-1}\Vert , \end{aligned}$$
$$\begin{aligned}&|\frac{F(x_k)^Ty_{k-1}}{d_{k-1}^Ty_{k-1}}|\Vert d_{k-1}\Vert +|\frac{F(x_k)^Td_{k-1}}{d_{k-1}^Ty_{k-1}}|\Vert y_{k-1}\Vert \\&\quad \le \frac{||F(x_k)||||y_{k-1}||}{|d_{k-1}^Ty_{k-1}|}\Vert d_{k-1}\Vert +\frac{\Vert F(x_k)\Vert \Vert d_{k-1\Vert }}{|d_{k-1}^Ty_{k-1}|}\Vert y_{k-1}\Vert . \end{aligned}$$
$$\begin{aligned} |d_{k-1}^Ty_{k-1}|\le \Vert d_{k-1}\Vert \Vert y_{k-1}\Vert , \end{aligned}$$
$$\begin{aligned} \frac{||F(x_k)||||y_{k-1}||}{|d_{k-1}^Ty_{k-1}|}\Vert d_{k-1}\Vert +\frac{\Vert F(x_k)\Vert \Vert d_{k-1\Vert }}{|d_{k-1}^Ty_{k-1}|}\Vert y_{k-1}\Vert \ge 2\Vert F(x_k)\Vert . \end{aligned}$$
$$\begin{aligned} \Vert F(x_k)\Vert +\left| \frac{F(x_k)^Ty_{k-1}}{d_{k-1}^Ty_{k-1}}\right| \Vert d_{k-1}\Vert +\left| \frac{F(x_k)^Td_{k-1}}{d_{k-1}^Ty_{k-1}}\right| \Vert y_{k-1}\Vert \le 3\Vert F(x_k)\Vert \end{aligned}$$