Newton-Raphson solution

The set of equations to be solved is now (19.22), (19.23), and (19.25). These are marked with boxes above. The Newton-Raphson method requires the complete set of partial derivatives of all of these equations. Taken in order, these are, for the charge balance (19.22) 

With the new set of guesses the process is repeated iteratively until successive estimations of all concentrations stop changing significantly. 

Activity coefficient corrections are treated just as with the rote manual method already discussed. The first time through, all activity coefficients could be set to 1.0 (or some reasonable estimate). Concentrations of species calculated this way are then used to estimate a better set of activity coefficients in each successive iteration. 

The above procedure has been coded in FORTRAN as the program EQBRM and a copy suitable for personal computers is included in this book as Appendix E, along with an example showing the proper format for input data. For different applications it is necessary to choose among the available methods of estimating activity coefficients. For example, the Debye-Huckel equation can often be used for dilute systems such as rivers and groundwater, but concentrated brines will require the Pitzer equations or measured coefficients if they are available. For this reason, a subroutine should be written to calculate activity coefficients for your application. 

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Carry out the first two iterations of the Newton-Raphson solution of the polynomial Eq. (1-10). 

In the SR method, temperatures are the dominant variables and are found by a Newton-Raphson solution of the stage energy balances. Compositions do not have as great an influence in calculating the temperatures as do heat effects or latent heats of vaporization. The component flow rates are found by the tridiagonal matrix method. These are summed to get the total rates, hence the name sum rates. 

Independent variable in a Newton-Raphson solution, Sec. 4.2.5 only. 

Equation set 12.28 is solved along with Equations 12.24,12.25, 12.26, and 12.27. As in the method described for two-product columns, the calculations alternate between a Newton-Raphson solution of equation set 12.28 and an iterative solution of the rest of the equations as described above. 

Table 3.2 Newton-Raphson solution to finding the local minima of Eq. (3.16).

Volumetric Relations. The fluid-filled porosity is not a primary unknown because it is determined by the Pkjk=l,...,Np. In the iterative Newton-Raphson solution scheme, values of 6/ are updated based on new values of the Pjt s at the end of each iteration. 

Vector of independent variables of stage/ for a Newton-Raphson solution in a Naphtali-Sandholm or nonequilibrium method, Sec. 4.2.9 and 4.2.13. 

Equation set 12.19 is equivalent to Equation set 12.18 since N and B are implicit functions of b, N, and K. The independent variables are N and B, and the dependent variables are and K. The calculations alternate between a Newton-Raphson solution of Equation set 12.19 and an iterative solution of Equation set 12.17 as described previously. The algorithm is outlined as follows ... 

Liquid phase compositions and phase ratios are calculated by Newton-Raphson iteration for given K values obtained from LILIK. K values are corrected by a linearly accelerated iteration over the phase compositions until a solution is obtained or until it is determined that calculations are too near the plait point for resolution. 

An alternative, and closely related, approach is the augmented Hessian method [25]. The basic idea is to interpolate between the steepest descent method far from the minimum, and the Newton-Raphson method close to the minimum. This is done by adding to the Hessian a constant shift matrix which depends on the magnitude of the gradient. Far from the solution the gradient is large and, consequently, so is the shift d. One... 

Errors are proportional to At for small At. When the trapezoid rule is used with the finite difference method for solving partial differential equations, it is called the Crank-Nicolson method. The implicit methods are stable for any step size but do require the solution of a set of nonlinear equations, which must be solved iteratively. The set of equations can be solved using the successive substitution method or Newton-Raphson method. See Ref. 36 for an application to dynamic distillation problems. 

Simultaneous solution by the Newton-Raphson method yields x = 0.9121, y = 0.6328. Accordingly, the fractional compositions are ... 

Equation 13-39 is a cubic equation in terms of the larger aspect ratio R2. It can be solved by a numerical method, using the Newton-Raphson method (Appendix D) with a suitable guess value for R2. Alternatively, a trigonometric solution may be used. The algorithm for computing R2 with the trigonometric solution is as follows ... 

Newton-Raphson procedure, 272, 273 Nickel-gold, solid solution (Au6Ni4), enthalpy of formation, 144 solution (Au-Ni), enthalpy of formation, 143... 

This equation must be solved for yn +l. The Newton-Raphson method can be used, and if convergence is not achieved within a few iterations, the time step can be reduced and the step repeated. In actuality, the higher-order backward-difference Gear methods are used in DASSL [Ascher, U. M., and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia (1998) and Brenan, K. E., S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, North Holland Elsevier (1989)]. 

NLP methods provide first and second derivatives. The KKT conditions require first derivatives to define stationary points, so accurate first derivatives are essential to determine locally optimal solutions for differentiable NLPs. Moreover, Newton-Raphson methods that are applied to the KKT conditions, as well as the task of checking second-order KKT conditions, necessarily require second-... 

Of such schemes, two of the most robust and powerful are Newton s method for solving an equation with one unknown variable, and Newton-Raphson iteration, which treats systems of equations in more than one unknown. I will briefly describe these methods here before I approach the solution of chemical problems. Further details can be found in a number of texts on numerical analysis, such as Carnahan et al. (1969). 

In this section we consider how Newton-Raphson iteration can be applied to solve the governing equations listed in Section 4.1. There are three steps to setting up the iteration (1) reducing the complexity of the problem by reserving the equations that can be solved linearly, (2) computing the residuals, and (3) calculating the Jacobian matrix. Because reserving the equations with linear solutions reduces the number of basis entries carried in the iteration, the solution technique described here is known as the reduced basis method. ... 

The computing time required to evaluate Equation 4.19 in a Newton-Raphson iteration increases with the cube of the number of equations considered (Dongarra et al., 1979). The numerical solution to Equations 4.3 1.6, therefore, can be found most rapidly by reserving from the iteration any of these equations that can be solved linearly. There are four cases in which equations can be reserved ... 

At each step in the Newton-Raphson iteration, we evaluate the residual functions and Jacobian matrix. We then calculate a correction vector as the solution to the matrix equation... 

The solution, performed at each step in the Newton-Raphson iteration, is accomplished by setting Equation 10.5 equal to Equation 10.6. We write a residual... 

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