This paper presents a fourth-order nonlinear conjugate gradient method in equality constrained optimization. The idea is to transform the constrained problem into unconstrained type through the Lagrange multipliers scheme. Using four terms of Taylor series development, we approximate the transformed function (augmented Lagrange function). Lastly, we employ the new fourth-order nonlinear conjugate gradient method in equality constrained optimization to solve the optimization problem. We present the algorithm in steps and some properties of the gradients are proved, using classical results. Also, the convergence analysis has been proved under classical and known assumptions. Furthermore, we present the obtained numerical results and compare them to some existing results. The analysis of results confirms that the new method is accurate.
The development of any conjugate gradient method could be viewed from the perspective of approximating an objective function ) (x f by a functional ) (x F noting that the properties of the functional can be used to characterize the method. Using the functional ) (x F , the new conjugate gradient method was developed and used to solve many nonlinear optimization problems with high efficiency and accuracy. The numeric analysis of its stability and convergence becomes imperative in order to establish the reliability of the method and satisfy the yearnings of its increasing users. In this paper, we present the stability and convergence analysis of the new conjugate gradient method.
This paper investigates the effect of increasing levels of atmospheric carbon dioxide (CO2) on rainwater. The design of this research includes the collection and analysis of recorded partial pressures of carbon dioxide (pCO2) at six air-space control stations in Nigeria. The already established equations for the chemistry of water constitute the theoretical framework of this investigation. These equations resolve into a mathematical model which connects the pCO2 and the activity index of hydrogen ions (pH) in rainwater. A cubic polynomial, which represents the predictive framework of this study, fits the average pCO2, while the model generates the corresponding pH. The obtained results show that the increasing levels of CO2 contribute to climate change and the proportionate decrease of pH in rainwater. An extrapolated result reveals that the acidity of rainwater will increase from 5.3% in 2000 to 93.7% by 2050.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.