Newton-Raphson methods characteristics

Pseudo-Newton-Raphson methods have traditionally been the preferred algorithms with ab initio wave function. The interpolation methods tend to have a somewhat poor convergence characteristic, requiring many function and gradient evaluations, and have consequently primarily been used in connection with semi-empirical and force field methods. 

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-... 

Application to Simultaneous Phase and Chemical Equilibrium. The single-stage process with simultaneous phase and chemical equilibrium is another application of the inside-out concept where the Newton-Raphson method has been employed in a judicious way in the inside loop. There would appear to be no reaction parameter having characteristics that make it suitable as an outside loop iteration variable in the spirit of the inside-out concept. On the other hand, the chemical equilibrium relationships are simple in form, and do not introduce new thermophysical properties that depend in a complicated way on other variables. Thus it makes sense to include them in the inside loop, and to introduce the reaction extents as a new set of inside loop variables. 

The simultaneous solution uses the Newton-Raphson method, which is based on linearizing the model equations. Two characteristics are inherent in this method. Since the equations are highly nonlinear, the success of linearization usually requires good starting values. On the other hand, as the solution is approached, the linearized equations become progressively more accurate and convergence is accelerated. 

Table 4-7 Convergence characteristics exhibited by the Newton-Raphson method (procedure 1) in the solution of the stripper problem, Example 4-3...

Table 8-8 Convergence characteristics of procedures 1 and 2 for Example 8-1 by use of the N(r + 2) Newton-Raphson method...

Table 10-4 Convergence characteristics of the N Newton-Raphson method for Example 10-1...

Example 10-1 was also solved by use of the N Newton-Raphson method.8 The convergence characteristics of this method for this example are shown in Table 10-4. Again as in Chap. 4, the corrections AT were reduced by the factors 1/2, 1/4, 1/8,. .., until the 7 s so obtained were within the range of the curve fits. A comparison of the computer times given in Tables 10-2 and 10-4 shows that the 0 method is several times faster than the N Newton-Raphson method. 

Three well-known numerical methods for solving multivariable problems are presented as well as their convergence characteristics. The methods considered are direct iteration, the Newton-Raphson method, and Broyden s method. 

The Newton-Raphson method for the solution of n equations in n unknowns takes the form given by Eq. (15-3). In the application of this method, it is recommended that the convergence characteristics be checked by solving a wide variety for examples. The use of different initial sets of values for the variables should also be investigated. Also, if only positive roots of the functions are desired, provisions should be made for an alternative selection of variables for the next trial when one or more negative values are computed by an intermediate trial. 

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) ... 

After the Broyden correction for the independent variables has been computed, Broyden proposed that the inverse of the jacobian matrix of the Newton-Raphson equations be updated by use of Householder s formula. Herein lies the difficulty with Broyden s method. For Newton-Raphson formulations such as the Almost Band Algorithm for problems involving highly nonideal solutions, the corresponding jacobian matrices are exceedingly sparse, and the inverse of a sparse matrix is not necessarily sparse. The sparse characteristic of these jacobian matrices makes the application of Broyden s method (wherein the inverse of the jacobian matrix is updated by use of Householder s formula) impractical. 

Equation (7.18) represents a set of N nonlinear algebraic equations, which must be solved by a trial and error method such as Newton-Raphson or successive substitution for y . j (Appendix A). This is the characteristic difference between the implicit and explicit types of solution the easier explicit method allows sequential solution one at a time, while the implicit method requires simultaneous solutions of sets of equations hence, an iterative solution at a given time... 

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