Newton-Raphson numerical procedure

Newton-Raphson s procedure and the least-squares minimization process were involved in numerical calculations [5]. The basic... 

The computational procedure can now be explained with reference to Fig. 19. Starting from points Pt and P2, Eqs. (134) and (135) hold true along the c+ characteristic curve and Eqs. (136) and (137) hold true along the c characteristic curve. At the intersection P3 both sets of equations apply and hence they may be solved simultaneously to yield p and W for the new point. To determine the conditions at the boundary, Eq. (135) is applied with the downstream boundary condition, and Eq. (137) is applied with the upstream boundary condition. It goes without saying that in the numerical procedure Eqs. (135) and (137) will be replaced by finite difference equations. The Newton-Raphson method is recommended by Streeter and Wylie (S6) for solving the nonlinear simultaneous equations. In the specified-time-... 

We shall not discuss all the numerous energy minimisation procedures which have been worked out and described in the literature but choose only the two most important techniques for detailed discussion the steepest descent process and the Newton-Raphson procedure. A combination of these two techniques gives satisfactory results in almost all cases of practical interest. Other procedures are described elsewhere (1, 2). For energy minimisation the use of Cartesian atomic coordinates is more favourable than that of internal coordinates, since for an arbitrary molecule it is much more convenient to derive all independent and dependent internal coordinates (on which the potential energy depends) from an easily obtainable set of independent Cartesian coordinates, than to evaluate the dependent internal coordinates from a set of independent ones. Furthermore for our purposes the use of Cartesian coordinates is also advantageous for the calculation of vibrational frequencies (Section 3.3.). The disadvantage, that the potential energy is related to Cartesian coordinates in a more complex fashion than to internals, is less serious. 

The Newton-Raphson procedure was used to find e satisfying F(e) = 0. Iterations began at high conversion and the derivative dF/de was found by numerical differentiation. Convergence was obtained in 5 iterations, with 10 critical point evaluations, in about 10 seconds. The computer used was the University of Calgary Honeywell HIS-Multics system. 

The numerical solution of the system (4.86), by a procedure of the Newton-Raphson type with two variables, requires the calculation of the derivatives dtjds and dt /5Wc . The results we obtained for a square lattice are similar to those by Yonezawa and Odagaki180 for a cubic lattice. The most striking feature is the existence, at low concentration, of a gap in the density of states,179 which isolates the zero energy on which a 3 peak builds up. Thus the HCPA produces a forbidden region of energy for the transport the gap and the 3 peak disappear at a critical concentration, analogous to the percolation threshold of the mean-field of resistances. 

The equilibrium configuration of the surface region comprising n layers is determined by solving simultaneously the 4n equations obtained by equating to zero the partial derivatives of AU with respect to each of the variables. The equations so obtained are nonlinear and are solved by an iterative Newton-Raphson procedure (12), which necessitates calculating the second partial derivatives of AU with respect to all possible pairs of variables. A Bendix G15D computer was used for all numerical computations—i.e., evaluation of the various lattice sums, calculation of the derivatives of AU, and solution of the linearized forms in the Newton-Raphson treatment. 

Often the melting point and the heat of fusion at the melting point are used as estimates of T and A Hi. It should be noted that the latter equation is nonlinear, since y- on the right-hand side is a function of x . Hence the determination of x calls for an iterative numerical procedure, such as the Newton-Raphson or the secant methods. 

Equation (13-14) is solved iteratively for V/F, followed by the calculation of values o(x,anAy, from Eqs. (13-12) and (13-13) and L from the total mole balance. Any one of a number of numerical root-finding procedures such as the Newton-Raphson, secant, false-position, or bisection method can be used to solve Eq. (13-14). Values of K, are constants if they are independent of liquid and vapor compositions. Then the resulting calculations are straightforward. Otherwise, the K, values must be periodically updated for composition effects, perhaps... 

The resulting equations describing mass and heat transport are highly non-linear algebraic equations which can be solved numerically using a common procedure such as the Newton-Raphson technique. 

With all the necessary ingredients in place, the task is now to derive a reliable force field. In an automated refinement, the first step is to define in machine-readable form what constitutes a good force field. Following that, the parameters are varied, randomly or systematically (15,42). For each new parameter set, the entire data set is recalculated, to yield the quality of the new force field. The best force field so far is retained and used as the basis for new trial parameter sets. The task is a standard one in nonlinear numerical optimization many efficient procedures exist for selection of the optimum search direction (43). Only one recipe will be covered here, a combination of Newton-Raphson and Simplex methods that has been successfully employed in several recent parameterization efforts (11,19,20,28,44). 

The system of model equations has been solved numerically The activity profile has been computed from eq.(20), by the explicit finite difference scheme with adjustable step length. For given activity distribution the system of model equations has been solved using the implicit finite difference scheme. This leads to a system of nonlinear algebraic equations which has been solved by the Newton-Raphson procedure. 

This method, which is also called the Newton-Raphson method, is an iterative procedure for obtaining a numerical solution to an algebraic equation. An iterative procedure is one that is repeated until the desired degree of accuracy is attained. The procedure is illustrated in Fig. 4.9. We assume that we have an equation written in the form... 

The estimation of surface coverage (eq.7.113) and reaction rate (eq.7.110) should be performed numerically. For instance, the value of surface coverage, could be solved using the Newton-Raphson procedure. An analytical expression (although not strictly correct) can be obtained for the region of medium coverage when it is usually assumed, that (W o-l, hence similar to eq. (7.90)... 

The system of equations (7.189) and (7.190) cannot be solved analytically (except for z=l). The estimation of reaction rates and comparison with experimental data should be done by minimization of the sum of residual squares, while the value of surface coverage from balance equations should be solved numerically using, for instance, the Newton-Raphson procedure. 

These values of f can then be used as a new Iq vector for the next application of Eq. (4.24). This multidimensional Newton-Raphson procedure, which involves the solution of a large number of coupled linear equations, is then repeated until the At values are sufficiently small (convergence). Given the set of f J amplitudes, Eq. (4.16) can then be used to compute E. Although the first applications of the coupled cluster method to quantum chemistry did employ this Newton Raphson scheme, the numerical problems involved... 

Solutions of Equation 12 or equivalent closed forms for < in terms of r/vg are possible by successive numerical approximations, such as the Newton-Raphson method for determining the complex root of F(VT) - r iw)/vo(ico) = 0. This procedure has been used successfully, as discussed by Suggett, but has the disadvantages that an initial estimate of is necessary and that conditions for rapid Or acceptable convergence to the true value are not easily... 

Although this equation is a solvable quadratic (owing to the second order reaction), we will solve it nonetheless by numerical means to demonstrate the general procedure of the implicit integration method. The iteration scheme for the Newton-Raphson procedure is (Appendix A)... 

As it may be hard (or even impossible) to compute the derivatives of F with respect to the manipulated variables, approximations are normally provided for both H and VF in Equations 8.25 and 8.26. (The use of numerical procedures based on variational principles was very popular in the past [ 161 ]. In order to solve variational problems numerically, standard Newton-Raphson procedures are generally used to solve the resulting two-boimdary value problem that is associated with the variational formulation. For this reason, optimization of dynamic problems based on variational principles is also included in this set of SQP-related numerical techniques.)... 

For general case, the IAS equations must be solved numerically and this is quite effectively done with standard numerical tools, such as the Newton-Raphson method for the solution of algebraic equations and the quadrature method for the evaluation of integral. We shall develop below a procedure and then an algorithm for solving the equilibria problem when the gas phase conditions (P, yj are given... 

When the pore half width is greater than the inflexion pore half width d3/ai2, there are two local minima and these must be found by solving numerically eq. (6.10-13). This is done with an interative procedure, such as the Newton-Raphson method. The initial guess for the position at which the minimum occurs is done as follows. Since the minima will occur at a distance about 0.858 Gi2 from the slab surface (see eq. 6.10-8), the initial guess for z in searching for the local minimum is... 

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