Applications of higher-order optimal Newton secant iterative methods in ocean acidification and investigation of long-run implications of emissions on alkalinity of seawater

The Newton secant method is a third-order iterative nonlinear solver. It requires two function and one first derivative evaluations. However, it is not optimal as it does not satisfy the Kung-Traub conjecture. In this work, we derive an optimal fourth-order Newton secant method with the same number of function evaluations using weight functions and we show that it is a member of the King family of fourth-order methods. We also obtain an eighth-order optimal Newton-secant method. We prove the local convergence of the methods. We apply the methods to solve a fourth-order polynomial arising in ocean acidifications and study their dynamics. We use the data of CO2 available from the National Oceanic and Atmospheric Administration from 1959 to 2012 and calculate the pH of the oceans for these years. Finally we further investigate the long-run implications of CO2 emissions on alkalinity of seawater using fully modified ordinary least squares (FMOLS) and dynamic OLS (DOLS). Our findings reveal that a one-percent increase in CO2 emissions will lead to a reduction in seawater alkalinity of 0.85 percent in the long run.