Newton-Raphson iterative method

The NONLIN module is responsible for intializing the concentration vector, C(t), for l i NRCT. Here NRCT is the number of reactants. If there are no equilibrium reactions, then C i) is set to IC i), the initial concentration vector, for 1 < f < NRCT. If equilibrium reactions do exist, then the type (2) equations (with derivatives set to zero) and the Type (1) and Type (3) equations are all solved simultaneously for the equilibrium concentrations of all reactants. Because the equilibrium equations are generally nonlinear, the Newton-Raphson iteration method is used to solve these equations. Also, since there is no symbol manipulation capability in the current version of CRAMS, numerical differentiation is used to calculate the required partial derivatives. That is, the rate expressions cannot at this time be automatically differentiated by analytical methods. A three point differentiation formula is used 27) ... 

The Newton-Raphson iteration method is terminated when either the absolute values of the residuals and the differences between successive iterates are less than a specified tolerance, or when the specified maximum number of iterations is... 

Equations (15-58), (15-59), and (15-60) are solved simultaneously by the Newton-Raphson iterative method in which successive sets of the output variables are produced until the values of the M, E, and H functions are driven to within some tolerance of zero. During the iterations, nonzero values of the functions are called discrepancies or errors. Let the functions and output variables be grouped by stage in order from top to bottom. As will be shown, this is done to produce a block tridiagonal structure for the Jacobian matrix of partial derivatives so that the Thomas algorithm can be applied. Let... 

Step 6. Both theoretical transmittance spectra (obtained in Step 4 and Step 5) are compared with the experimental transmittance and a refined spectrum of attenuation coeflBcients k, is obtained (Steps 5 and 6 use the Newton-Raphson iterative method of numerical solutions of non-Hnear and transcendental equations [47]). 

The Newton-Raphson Iteration method was outlined In sections 2.5.3 (FEM) and 2.6.2 (BEM). The notation introduced in those sections will be used here. 

Fig. 3.18 Residual history of the Newton-Raphson iteration method. For increasing Wagner numbers, the end-residual decreases.

Computational method and estimation of parameters. The system of three differential equations which pre-sents the design model is nonlinear and subject to boundary conditions. For solving numerically the model equations the method of orthogonal collocation was used (90, 91). As collocation functions the so-called shifted Legendre polymonials were applied. As a rule the collocation was done for 5 inner points. The lumped equations were solved by means of the Newton-Raphson iteration method. 

The orbital exponents (, are nonlinear parameters, and the optimization is not trivial. In general, Cj are complex valued and their real and imaginary parts must be individually optimized. For this purpose, the Newton-Raphson iteration method has been used. 

With a suitable first estimate xq, the Newton-Raphson iteration method will in most cases converge very quickly. When applied to programming, the correction I Xj+i —Xi I can be compared with a requirement made for accuracy s x +i—Xj > continue iteration... 

The kinetic equations such as Equation 7.37, Equation 7.38, Equation 7.40, and Equation 7.41 are known as transcendental equations, whose direct solution cannot be obtained. Such a kinetic equation is generally solved by the use of approximation techniques such as Newton-Raphson iterative method and nonlinear least-squares method. But, these methods have limitations of a different nature. For instance, the nonlinear least-squares method, which is most commonly used in such kinetic studies, tends to provide less reliable values of calculated kinetic parameters with increase in the number of such parameters. 

These systems are solved by a step-limited Newton-Raphson iteration, which, because of its second-order convergence characteristic, avoids the problem of "creeping" often encountered with first-order methods (Law and Bailey, 1967) ... 

For nonquadratic but monotonic surfaces, the Newton-Raphson minimization method can be applied near a minimum in an iterative way [24]. 

Geochemists, however, seem to have reached a consensus (e.g., Karpov and Kaz min, 1972 Morel and Morgan, 1972 Crerar, 1975 Reed, 1982 Wolery, 1983) that Newton-Raphson iteration is the most powerful and reliable approach, especially in systems where mass is distributed over minerals as well as dissolved species. In this chapter, we consider the special difficulties posed by the nonlinear forms of the governing equations and discuss how the Newton-Raphson method can be used in geochemical modeling to solve the equations rapidly and reliably. 

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). 

The multidimensional counterpart to Newton s method is Newton-Raphson iteration. A mathematics professor once complained to me, with apparent sincerity, that he could visualize surfaces in no more than twelve dimensions. My perspective on hyperspace is less incisive, as perhaps is the reader s, so we will consider first a system of two nonlinear equations / = a and g = b with unknowns, v and y. 

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. ... 

Fig. 4.4. Comparison of the computing effort, expressed in thousands of floating point operations (Aflop), required to factor the Jacobian matrix for a 20-component system (Nc = 20) during a Newton-Raphson iteration. For a technique that carries a nonlinear variable for each chemical component and each mineral in the system (top line), the computing effort increases as the number of minerals increases. For the reduced basis method (bottom line), however, less computing effort is required as the number of minerals increases.

The set of Eqs. (38)-(40), together with appropriate boundary conditions, was solved numerically using a combination of a finite difference method and a Newton-Raphson iteration [47]. 

For the simulation of the reactor behaviour the system of ordinary differential equations was integrated by means of a Runge-Kutta-Merson method with variable step length, whereas the nonlinear algebraic equations were solved by a Newton-Raphson iteration. 

Eqn (29) may be included into the Newton-Raphson iteration as the n-th equation to determine all the intermediate as well as the final pressure. This however, requires subsequent derivation of the extra row in the Jacobian by the differentiation of eqn (29) with respect to the vector x of all pressures. This leads to fairly involved algebraic expression, so the quickest and safest method of calculating the choked flow conditions in the line segment is by a simple single variable optimization of y from eqn (27) with respect to the final pressure. The vector x is conputed frcm eqn (24) by the straight forward Newton-Raphson iteration for each step in the single variable hillclimbing. 

The code is capable of a variety of iterative treatments of nonlinear problems, including Newton-Raphson iteration and incremental load methods. 

A number of iterative methods exist, as described in Appendix L. TK Solver uses a modified Newton-Raphson iterative procedure (see Sec. L.2), which is satis-fectory for a wide variety of problems. 

Several programs have been written specifically for a very restricted class of equilibrium only problems. The Pit Method of Sillen and Warnquist has been widely used to solve for equilibrium constants in inorganic systems that have one or more simultaneous reversible reactions. DeLand uses goal-seeking routines to facilitate the matching of data, but free energy data for all reactants is required. Bos and Meershoek 24) have written a PL/1 program which uses the Newton-Raphson iteration to compute equilibrium constants in complex systems. 

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