Optimal parameter for the SOR-like iteration method for solving the system of absolute value equations

The SOR-like iteration method for solving the absolute value equations~(AVE) of finding a vector $x$ such that $Ax - |x| - b = 0$ with $\nu = \|A^{-1}\|_2 < 1$ is investigated. The convergence conditions of the SOR-like iteration method proposed by Ke and Ma ([{\em Appl. Math. Comput.}, 311:195--202, 2017]) are revisited and a new proof is given, which exhibits some insights in determining the convergent region and the optimal iteration parameter. Along this line, the optimal parameter which minimizes $\|T_\nu(\omega)\|_2$ with $$T_\nu(\omega) = \left(\begin{array}{cc} |1-\omega| & \omega^2\nu \\ |1-\omega| & |1-\omega| +\omega^2\nu \end{array}\right)$$ and the approximate optimal parameter which minimizes $\eta_{\nu}(\omega) =\max\{|1-\omega|,\nu\omega^2\}$ are explored. The optimal and approximate optimal parameters are iteration-independent and the bigger value of $\nu$ is, the smaller convergent region of the iteration parameter $\omega$ is. Numerical results are presented to demonstrate that the SOR-like iteration method with the optimal parameter is superior to that with the approximate optimal parameter proposed by Guo, Wu and Li ([{\em Appl. Math. Lett.}, 97:107--113, 2019]). In some situation, the SOR-like itration method with the optimal parameter performs better, in terms of CPU time, than the generalized Newton method (Mangasarian, [{\em Optim. Lett.}, 3:101--108, 2009]) for solving the AVE.Comment: 23 pages, 7 figures, 7 table

The Optimization and Mathematical Modeling of Quality Attributes of Parboiled Rice Using a Response Surface Method

The response surface methodology was used to optimize the hydrothermal processing conditions based on the rice quality parameters of the Rong Youhua Zhan rice variety (Indica). The effect of soaking temperature (29.77, 40, 55, 70, and 80.23°C), soaking time (67.55, 90, 120, 150, and 170.45 min), and steaming time (1.59, 5, 10, 15, and 18.41 min), each tested at five levels, on percentage of head rice yield (HRY), hardness, cooking time, lightness, and color were determined, with R2 values of 0.96, 0.94, 0.90, 0.88, and 0.94, respectively. HRY, hardness, cooking time, and color increased with process severity while lightness decreased, although HRY decreased after reaching a maximum. The predicted optimum soaking temperature, soaking time, and steaming time were 69.88°C, 150 min, and 6.73 min, respectively, and the predicted HRY, hardness, cooking time, lightness, and color under these conditions were 73.43%, 29.95 N, 32.14 min, 83.03 min, and 12.24 min, respectively, with a composite desirability of 0.9658. The parboiling industry could use the findings of the current study to obtain the desired quality of parboiled rice. This manuscript will be helpful for researchers working on commercializing parboiled rice processes in China as well as in other countries