Newton-like method global

The time step, At, is used to switch the method from being a relaxation method to a global Newton method. When the time step is small, e.g., if = 0.1, then the changes in the independent variables are small. The method performs like a damped Newton-Raphson method, where the steps are small but in the direction of the solution and without any oscillation. When the value of At is large, i.e., At = 1000, the method performs like a Newton-Raphson method. The value of At at each column trial determines the speed and stability of the method, The units of the time step are the same as the flows to and from the column. The calculation sequence of the Ketchum method is as follows ... 

Equilibrium compositions of systems of chemical reactions or systems of enzyme-catalyzed reactions can only be calculated by iterative methods, like the Newton-Raphson method, and so computer programs are required. These computer programs involve matrix operations for going back and forth between conservation matrices and stoichiometric number matrices. A more global view of biochemical equilibria can be obtained by specifying steady-state concentrations of coenzymes. These are referred to as calculations at the third level to distinguish them from the first level (chemical thermodynamic calculations in terms of species) and the second level (biochemical thermodynamic calculations at specified pH in terms of reactants). 

Other optimization paths are possible. One can think that a covering of the exponential product e b niay be better achieved by a simultaneous n-dimensional optimization process like Newton-Raphson procedure [58]. In fact, however, the overlap as described in equation (4.19) is an irregular function of the p,e parameter pairs, and possesses multiple maxima and minima with respect to the scale factors variation. A method with a scanning or sweep feature among a given parameter range will be perhaps superior in this case than a global search procedure, and so was adopted. 

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