Product-difference algorithm

The parameters a and can be obtained from the moments by evaluating Hankel determinants or working out product-difference algorithms. The projected density of states n( ) is then given by... 

This system of 2 M nonlinear equations is ill-conditioned for large M, but can be efficiently solved using the product-difference (PD) algorithm introduced by McGraw (1997). Thus, given the set of 2 M moments on the left-hand side of Eq. (107), the PD algorithm returns wm and lm for m — 1., M. The closed microscopic transport equation for the moments can then be written for k — 0,..., 2 M— 1 as... 

This led to the production of different algorithms, all reported in this volume and all based on Peter s GRID force field. It is noteworthy that GRID-MIF are cur-... 

The procedure can be iterated, considering now the Volterra integro-differential equation (3.44) for x(/), then for 2(0. and so on. The implications of Eq. (3.54) have been discussed in more detail in Chapter III, with comments on product-difference (PD) algorithms. 

The product-difference (PD) algorithm was developed by Gordon (1968) and is based on the theory of continued fractions of Stieltjes. The first step is to construct a matrix P with... 

Chapter 3 provides an introduction to Gaussian quadrature and the moment-inversion algorithms used in quadrature-based moment methods (QBMM). In this chapter, the product-difference (PD) and Wheeler algorithms employed for the classical univariate quadrature method of moments (QMOM) are discussed, together with the brute-force, tensor-product, and conditional QMOM developed for multivariate problems. The chapter concludes with a discussion of the extended quadrature method of moments (EQMOM) and the direct quadrature method of moments (DQMOM). 

Although a great step forward, these methods had a common problem in the restricted formulation of the scoring function. The monomer frequency algorithms are difficult to extend to more than two objectives as they are order dependent and therefore are unable to compare different solutions without great effort. For the stochastic, product based algorithms, a decision has to be made regarding how important each objective is for example does diversity... 

Optimization—that is, minimization of the energy gradient—of a reactant, reaction intermediate (if existing), or product is usually straightforward and the only caveat is to make sure that its corresponding Hessian is positive definite. However, characterization and optimization of a transition structure constitute more difficult tasks and different algorithms [16,17] have been developed to this end. The located transition structure must fulfill the following four requirements, also known as Mclver-Komornicki conditions [18] ... 

The nature of product-based library design lends itself naturally to the application of heuristic search methods such as simulated annealing and genetic algorithms. Several groups have published applications in the latter area, which has been recently reviewed. While all methods differ somewhat in their technical implementations of the different algorithms, by far the... 

Compute the weights and abscissas from the given moments by use of the product-difference (PD)-algorithm. 

Matrix in Golub-Welsch algorithm for which the eigensys-tem are the Gauss-Lobatto quadrature points and weights Product difference matrix of order (2Nq- -l)(2Nq- -l) in PD algorithm... 

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