Newton-Raphson scheme

The steepest-descent method does converge towards the expected solution but convergence is slow in the vicinity of the minimum. In order to scale variations, we can use a second-order method. The most straightforward method consists in applying the Newton-Raphson scheme to the gradient vector of the function/to be minimized. Since the gradient is zero at the minimum we can use the updating scheme... 

The Newton-Raphson scheme prescribes the updating formula... 

Write the appropriate driver routine to integrate the equations using either the DASSL [46] (DAE) or VODE (ODE) [49] software. You will need to iteratively solve for the laboratory-referenced molar fluxes N], N2 using a multidimensional root-finding routine, such as MNEWT / FDJAC [319], a Newton-Raphson scheme. 

In addition to further correlating the ground state of a single molecule, the SCVB procedure can also be used to describe its excited states. However, a minimization procedure based on a first-order approach tends not to give good convergence in such cases. Instead, we have adopted a stabilized Newton-Raphson scheme, as in the usual SC approach, but we use an approximate expression for the second derivative that requires only density matrices up to third order [12]. The resulting procedure has been shown to be quite stable. 

Equation (48) leads to a system of M non-linear equations. The solutions can be obtained by using the Newton-Raphson scheme. In a number of synthetic numerical examples the solution was found to be very sensitive to noise and to the extent to which the plume is dissipated, especially for the gradual release scenario but not in the catastrophic release scenarios even in the presence of moderate measurement errors. The noise level used was 5% and 20 and 50% for moderate and high levels, respectively. 

The solutions for bound states are given by Equations (55) and (56), which are used to construct the wave function R and its derivative R. In order to find the energy eigenvalues a guess value eo is proposed at the outset of the procedure, to subsequently be improved upon within a Newton-Raphson scheme until a required precision is attained... 

In the framework of the Newton-Raphson scheme, the generalised displacements Ui+i of the next iteration are obtained by... 

The finite element grid adopted has 8000 elements (200 x 40 rectangular elements). To solve the linear systems resulting from the iterative Newton-Raphson scheme, a parallelized conjugated... 

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... 

This represents an problem in which C nonlinear algebraic equations are to be solved for C unknowns T and y. In general, these equations must be solved by trial, and the standard method of attack is the Newton-Raphson scheme [9]. However, in particular problems, we hope to find alternative algorithms, for the Newton-Raphson method is computationally expensive and slow to converge. 

A comparison with Eq. A. 10 shows we were using the Newton-Raphson scheme by the artificial device of division by x. 

New values of g, and can be obtained by using a Newton-Raphson scheme. The residuals can be expressed as ... 

The underlying solution technique used In this paper Is the Newton-Raphson scheme, refined by Houpert and Hamrock [3], which Incorporates lubricant compressibility, the Roelands pressure-viscosity relationship, an Improved elastic calculation and variable mesh spacing. 

Several effective methods for direct localization of the transition state points have been evolved which do not require a calculation of the whole PES [35-37]. The Newton-Raphson scheme [38] is a standard method for determining any critical points, however, it converges toward the saddle point only in a region sufficiently close to it. One of the best known methods is the Mclver and Komornicki [39] minimization of the norm of the gradient Sg. 

The simplest numerical method for a detailed geometrically and material nonlinear (GMN) analysis is the Newton-Raphson scheme (Crisfield 1979 Bathe 1995), which can be found in three forms (i) the full Newton-Raphson, which is the most accurate, but also the most time consuming, since the tangent stiffness of the structure has to be calculated and factorized within each iteration in the solution procedure (ii) the modified Newton-Raphson, which differs from the full Newton-Raphson in that the calculation and the factorization of the tangent stiffness matrix take place only in some iterations within each step, thus requiring in most cases a larger number of iterations per step but... 

Siegbahn PEM, Heiberg A, Roos BO, Levy B. Comparison of the super-CI and the Newton-Raphson scheme in the complete active space SCF method. Phys Scr. 1980 21 323. 

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