Newton’s iteration method

Our solution method of choice is Newton s iterative method, see Section 1.2. Newton s method requires us to evaluate the Jacobian matrix Df of... 

Figure D.l Newton s iterative method to locate the zero of a function y = / (lo) = 0 through a tangent construction (see text).

According to the equilibria constants and the Pitzer ion-interaction parameters, the solubilities of the quaternary system at 298.15 K have been calculated though the Newton s Iteration Method to solve the non-linearity simultaneous equations system, and shown in Table 9. 

Newton s iterative method considers a first order Taylor series approximation to a function near a root of the function. Halley s method extends this to a consideration of second order terms as in ... 

To solve the set of non-linear transcendental equations the Newton Raphson s iterative method is used. First,... 

In both classes, it is advisable to immediately perform a Newton s iteration to exploit the efficiency of this kind of method as soon as the Jacobian matrix has been either evaluated or updated. 

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... 

A difficulty with the energy conserving method (6), in general, is the solution of the corresponding nonlinear equations [6]. Here, however, using the initial iterate (q + A p , p ) for (q +i, p +i), even for large values of a we did not observe any difficulties with the convergence of Newton s method. 

Some formulas, such as equation 98 or the van der Waals equation, are not readily linearized. In these cases a nonlinear regression technique, usually computational in nature, must be appHed. For such nonlinear equations it is necessary to use an iterative or trial-and-error computational procedure to obtain roots to the set of resultant equations (96). Most of these techniques are well developed and include methods such as successive substitution (97,98), variations of Newton s rule (99—101), and continuation methods (96,102). 

Molecular dynamics, in contrast to MC simulations, is a typical model in which hydrodynamic effects are incorporated in the behavior of polymer solutions and may be properly accounted for. In the so-called nonequilibrium molecular dynamics method [54], Newton s equations of a (classical) many-particle problem are iteratively solved whereby quantities of both macroscopic and microscopic interest are expressed in terms of the configurational quantities such as the space coordinates or velocities of all particles. In addition, shear flow may be imposed by the homogeneous shear flow algorithm of Evans [56]. 

In this chapter the new difference schemes are constructed for the quasilin-ear heat conduction equation and equations of gas dynamics with placing a special emphasis on iterative methods available for solving nonlinear difference equations. Among other things, the convergence of Newton s method is established for implicit schemes of gas dynamics. 

It is worth noting here that Newton s method is quite applicable for solving problem (35) in addition to the well-established method of iterations. 

Numerical calculations for 7 = 5/3 (a = 1.5) showed that the itera.-tions within the framework of Newton s method converge even if the steps r are so large that the shock wave runs over two-three intervals of the grid W/j in one step r. Of course, such a large step is impossible from a computational point of view in connection with accuracy losses. Thus, the restrictions imposed on the step r are stipulated by the desired accuracy rather than by convergence of iterations. 

After that, Newton s method of iterations applies equally well to either of these groups independently. By analogy with the isotermic case the first group of equations is to be solved with a prescribed temperature, while the second one needs the assigned values of rj and v. The essence of the matter in the last case is that the origin of the heat conduction equation is stipulated by the available sources of a dynamical nature. 

With knowledge of v and jy the mth iteration T is recovered from the equations of the second group by Newton s method. 

Steady-state solutions are found by iterative solution of the nonlinear residual equations R(a,P) = 0 using Newton s methods, as described elsewhere (28). Contributions to the Jacobian matrix are formed explicitly in terms of the finite element coefficients for the interface shape and the field variables. Special matrix software (31) is used for Gaussian elimination of the linear equation sets which result at each Newton iteration. This software accounts for the special "arrow structure of the Jacobian matrix and computes an LU-decomposition of the matrix so that qu2usi-Newton iteration schemes can be used for additional savings. 

Solution of Equation 8.65 yields the optimum value for the step-size. The solution can be readily obtained by Newton s method within 3 or 4 iterations using jLta as a starting value. This optimal step-size policy was found to yield very good results. The only problem that it has is that one needs to store the values of state and sensitivity equations at each iteration. For high dimensional systems this is not advisable. 

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). 

Fig. 4.1. Newton s method for solving a nonlinear equation with one unknown variable. The solution, or root, is the value of x at which the residual function R(x) crosses zero. In (a), given an initial guess. vl0,), projecting the tangent to the residual curve to zero gives an improved guess v( l ). By repeating this operation (b), the iteration approaches the root.

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