Newton-Raphson procedure

Such step-limiting is often helpful because the direction of correction provided by the Newton-Raphson procedure, that is, the relative magnitudes of the elements of the vector J G, is very frequently more reliable than the magnitude of the correction (Naphtali, 1964). In application, t is initially set to 1, and remains at this value as long as the Newton-Raphson correotions serve to decrease the norm (magnitude) of G, that is, for... 

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

Newton-Raphson Procedure This variant chooses ak = fixf where/ = df/dx and geometrically consists of replacing the graph of... 

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. 

With the help of the steepest descent process usually final absolute d E/3 x(--values of about 0.1 kcal mole-1 A-1 may be reached. By means of the Newton-Raphson procedure these values may in most cases easily be lowered to about 10-6 kcal mole-1 A-1. Only with a few rigid polycyclic systems and with some small molecules were we able to ap-... 

Like Newton s method, the Newton-Raphson procedure has just a few steps. Given an estimate of the root to a system of equations, we calculate the residual for each equation. We check to see if each residual is negligibly small. If not, we calculate the Jacobian matrix and solve the linear Equation 4.19 for the correction vector. We update the estimated root with the correction vector,... 

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 study described above for the water-gas shift reaction employed computational methods that could be used for other synthesis gas operations. The critical point calculation procedure of Heidemann and Khalil (14) proved to be adaptable to the mixtures involved. In the case of one reaction, it was possible to find conditions under which a critical mixture was at chemical reaction equilibrium by using a one dimensional Newton-Raphson procedures along the critical line defined by varying reaction extents. In the case of more than one independent chemical reaction, a Newton-Raphson procedure in the several reaction extents would be a candidate as an approach to satisfying the several equilibrium constant equations, (25). 

Approximate TS structures were located based on the pathways shown in Fig. 31 using the SEAM search algorithm. However, for associative interchange, this leads to an inconsistency in that in order to have different connectivities in reactant and product states, there are only six explicit M-0 bonds while the TS should have seven. Consequently, the seventh ligand is explicitly connected and the structure reoptimized using a simple Newton-Raphson procedure. For vanadium, the SEAM structure is sufficiently good for this procedure to locate a true first-order saddle point (Fig. 32, left) (73). 

A more robust method is tire Newton-Raphson procedure. In Eq. (2.26), we expressed the full force-field energy as a multidimensional Taylor expansion in arbitrary coordinates. If we rewrite this expression in matrix notation, and truncate at second order, we have... 

A very important method is the Newton-Raphson procedure ... 

The iteration counter k and the argument x(l) refers to the macroiterations made in the Newton-Raphson procedure, and they are obviously constant within the context of this section. Let us drop them for convenience. Also, let us explicitly assume that the Jacobian is in fact a positive definite Hessian, and that f(x< >) is a gradient The equation to be solved is thus rewritten in the form... 

The choice of optimization scheme in practical applications is usually made by considering the convergence rate versus the time needed for one iteration. It seems today that the best convergence is achieved using a properly implemented Newton-Raphson procedure, at least towards the end of the calculation. One full iteration is, on the other hand, more time-consuming in second order methods, than it is in more approximative schemes. It is therefore not easy to make the appropriate choice of optimization method, and different research groups have different opinions on the optimal choice. We shall discuss some of the more commonly implemented methods later. 

Before analyzing the energy expression (3 25) with respect to the variational parameters, let us briefly review the multidimensional Newton-Raphson procedure. Assume that the energy is a function of a set of parameters pj, which we arrange as a column vector, p. We now make a Taylor expansion of the energy E = E(p) around a point p0, which we arbitrarily can put equal to zero ... 

Computational aspects on the Newton-Raphson procedure When constructing methods to solve the system of linear equations (4 22) one should be aware of the dimension of the problem. It is not unusual to have Cl expansion comprising 104 - 106 terms, and orbital spaces with more than two hundred orbitals. In such calculations it is obviously not possible to explicitly construct the Hessian matrix. Instead we must look for iterative algorithms... 

When these conditions are not fulfilled the Newton-Raphson procedure may converge only slowly or even diverge. It is then necessary to introduce manual procedures, which drive the solution to the local minimum. A number of such methods have been devised, which in many cases work well. They are based on... 

Thus the Hessian will become singular if we include rotations between the active orbitals. Redundant parameters must not be included in the Newton-Raphson procedure.They are trivial to exclude for the examples given above, but in more general cases a redundant variable may occur as a linear combination of S and T and it might be difficult to exclude them. One of the advantages of the CASSCF method is that all parameters except those given above are non-redundant. 

An approximation, which has been used with some success, is to neglect the coupling term altogether. This leads to the unfolded two-step Newton-Raphson procedure, where the equations to be solved are ... 

The most widely used methods fall into two general categories (1) steepest descent and related methods such as conjugate gradient, which use first derivatives, and (2) Newton-Raphson procedures, which additionally use second derivatives. 

A new accurate fitting of the calculated PES was realised by a Newton-Raphson procedure adopting the same potential form as used by Buck et al. [65]. The agreement between the MO-VB ab initio potential and the Buck potential, determined by direct fitting of the experimental data, is very satisfactory. 

Several authors have recently reported on the use of Newton-Raphson procedures as applied to the solution of the equation describing a complex fractionator for the separation of multicomponent mixtures. However, it appears that the flexibility to handle various types of electrolyte problems, or to handle... 

Calculate all of the independent functions and their norm. Get a new set of temperatures T/b by solving the independent equations during one pass through the Newton-Raphson procedure. 

Note that there are no other side calculations between passes through the Newton-Raphson procedure. Total flow rates and duties are calculated after a solution of the column is found. 

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. 

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