Multidimensional Newtons Method

Consider some point (oq, bo.) within the region of definition of the functions F,G. and suppose that the functions can be represented by an multidimensional Taylor series about this point. Truncating the series after 

For the one-dimensional case, dFjda can usually be estimated using values of F determined at previous guesses. Thus, 

For two- and higher-dimensional solutions, it is probably best to estimate the first partial derivatives by a formula such as 

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There is no natural way to generalize the one-dimensional bisection method to solve this multidimensional problem. But it is possible to generalize Newton s method to this situation. The one-dimensional Newton method was derived using a Taylor expansion, and the multidimensional problem can be approached in the same way. The result involves a 3/V x 3/V matrix of derivatives, J, with elements 7y = dg, /dxj. Note that the elements of this matrix are the second partial derivatives of the function we are really interested in, E(x). Newton s method defines a series of iterates by... 

By the multidimensional Newton-Raphson method the error in the material-balance equation and the computed derivative of the errors with respect to the concentration of components (Jacobian of Y with respect to X) can be used to compute an improved guess for the concentration of components ... 

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... 

The BzzMinimizationQuasiNewton class is designed to solve unconstrained multidimensional minimization problems using the quasi-Newton method as the principal algorithm. If the algorithm does not converge even though... 

The procedure is stopped if all the components of the estimate of the Newton method, d , are reasonably small. As is the case with multidimensional optimization, checking whether a norm of the vector d is lower than an assigned value is not sufficient in itself In fact, it is advisable to adopt a relative measure ... 

B.3 SOLUTION OF NONLINEAR SIMULTANEOUS EOUATIONS (MULTIDIMENSIONAL NEWTON-RAPHSON METHOD)... 

The multidimensional counterpart to Newton s method is Newton-Raphson iteration. A mathematics professor once complained to me, with apparent sincerity, that he could visualize surfaces in no more than twelve dimensions. My perspective on hyperspace is less incisive, as perhaps is the reader s, so we will consider first a system of two nonlinear equations / = a and g = b with unknowns, v and y. 

At this point it may seem as though we can conclude our discussion of optimization methods since we have defined an approach (Newton s method) that will rapidly converge to optimal solutions of multidimensional problems. Unfortunately, Newton s method simply cannot be applied to the DFT problem we set ourselves at the beginning of this section To apply Newton s method to minimize the total energy of a set of atoms in a supercell, E(x), requires calculating the matrix of second derivatives of the form SP E/dxi dxj. Unfortunately, it is very difficult to directly evaluate second derivatives of energy within plane-wave DFT, and most codes do not attempt to perform these calculations. The problem here is not just that Newton s method is numerically inefficient—it just is not practically feasible to evaluate the functions we need to use this method. As a result, we have to look for other approaches to minimize E(x). We will briefly discuss the two numerical methods that are most commonly used for this problem quasi-Newton and... 

We did not define a stopping criterion for the multidimensional version of Newton s method. How would you define such a criterion ... 

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... 

In this section we will discuss the specific mathematical techniques used to estimate chemical equilibria using the sequential approach, which is the foundation for all versions of the FREZCHEM model, except for versions 2 and 10 (see above). The techniques used to solve (find the roots of) the equilibrium relations can be grouped into three classes simple one-dimensional (1-D) techniques, Brents method for more complex 1-D cases, and the Newton-Raphson technique that is used for both 1-D and multidimensional cases. 

Recent years have seen the extensive application of computer simulation techniques to the study of condensed phases of matter. The two techniques of major importance are the Monte Carlo method and the method of molecular dynamics. Monte Carlo methods are ways of evaluating the partition function of a many-particle system through sampling the multidimensional integral that defines it, and can be used only for the study of equilibrium quantities such as thermodynamic properties and average local structure. Molecular dynamics methods solve Newton s classical equations of motion for a system of particles placed in a box with periodic boundary conditions, and can be used to study both equilibrium and nonequilibrium properties such as time correlation functions. 

One of the simplest alternatives for multidimensional root finding is the Newton-Raphson method. For the sake of simplicity we introduce it by... 

The Newton-Raphson method and other algorithms for multidimensional root-finding can be found in C- — - in the section Multidimensional Root-Finding of the GNU Scientific Library (GSL) http //www.gnu.org/software/gsl. 

For the application of the Newton-Raphson method detailed above we must define the functions (as shown in (6.9)), their derivatives (i.e., the Jacobian matrix) and the vector of initial estimates that corresponds to the bulk concentrations of the different species for fc = 0 and, subsequently, to the concentrations of the previous timestep fc — 1. We also have to provide the tolerance criterion together with a maximmn number of iterations such that a convergence error is obtained if the accuracy required is not reached after a reasonable munber of cycles. Different criteria can be imposed in a multidimensional problem ... 

Many different methods attempt to minimize merit functions when the Newton estimate is unsatisfactory. However, there is an additional difference between this approach and multidimensional optimization problems. 

Another systematic way to construct CG models from detailed atomistic simulations is the Newton inversion method [97]. In this method, the structural information extracted from atomistic simulations is used to determine effective potentials for a CG model of the system. Suppose the effective potentials in the CG model are determined by a set of parameters A, where i runs from 1 to the number of parameters in the potential. The set of target properties that are known from atomistic simulations is represented by Aj], where j changes fi om 1 to the number of target properties. By means of the Newton inversion method, a set of nonlinear multidimensional equation between /I, and computed average properties Aj) is solved iteratively. At each iteration of the Newton inversion, the effect of different potential parameters on different averages can be calculated by the following formula [97] ... 

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