Recursive method

AUGMENTED SPACE RECURSION METHOD FOR THE CALCULATION OF ELECTRONIC STRUCTURE OF RANDOM ALLOYS... 

In this section, we construct the GF for a finite chain with an applied held (Davison et al 1997), by using the CF elements of the recursion method (Haydock 1980) and thereby build the GF atom-by-atom, in a similar way as the causal-surface GF approach (Pendry et al 1991). 

When the function to be fitted to data does not depend linearly on the parameters, recursive methods must be used. A slightly modified version of the Newton-Raphson method (Chapter 3) will be used (Hamilton, 1964). Let jc be the vector of the n unknowns Xj and y — f(x) the m-vector of observable functions y, = /(jc). The analytical form of the functions f(x) may be the same or not. Let the vector / represent the m observations / of these functions. A vector jc is sought which minimizes the scalar c2 such that... 

As discussed, an alternative to direct diagonalization is by recursion. The recursive diagonalization approach has several attractive features, including more favorable scaling laws, which make it ideally suited for large eigenproblems. For example, some applications of linear-scaling recursive methods in... 

The filtering, namely the construction of energy local bases, can also be carried out using the Lanczos recursion or similar recursive methods. However, filtered vectors at E/ can only be obtained using the Green filter ... 

For time-dependent Hamiltonians, one can reformulate the problem with the (t, l ) scheme,227 in which the time is treated as an extra degree of freedom. Thus, the techniques developed for stationary problems can be applied in a straightforward manner. Applications of recursive methods to laser-driven dynamics have been reported by several authors.99,228-230... 

The aforementioned applications of recursive methods in reaction dynamics do not involve diagonalization explicitly. In some quantum mechanical formulations of reactive scattering problems, however, diagonalization of sub-Hamiltonian matrices is needed. Recursive diagonalizers for Hermitian and real-symmetric matrices described earlier in this chapter have been used by several authors.73,81... 

The relative merit of the two recursive methods in computing both the bound-state and the resonance spectra have been examined and discussed by several of authors,41,42,80,247,314-318 and the consensus is that their efficiency... 

Treacy, M.M.J., Newsam, J.M., and Deem, M.W. (1991) A general recursion method for calculating diffracted intensities from crystals containing planar faults. Proc. R. Soc. Lond., A443, 499-520. 

The proposed technique is numerically "robust", and its results are comparable to those obtained through a recursive method based on the Kalman filter ( L). It should be noted that because the present technique utilizes all of the information simultaneously, the results have been compared to those of the optimal smoother estimates in (1 ), which are "better" than the true filtered estimates. 

At this point all the vibronic eigenstates can be easily evaluated numerically by means of the recursion method or alternatively the renormalization method. 

While the supercell approach works well for localized systems, it is typically necessary to consider a very large supercell. This results in a plane-wave basis replicating not only the relevant electronic states but also vacuum regions imposed by the supercell. A much more efficient method to implement for investigating the electronic structures of localized systems is to use real space methods such as the recursion methods [27] and the moments methods [28], These methods do not require symmetry and their cost grows linearly with the number of inequivalent atoms being considered. For these reasons, real space methods are very useful for a description of the electronic properties of complex systems, for which the usual k-space methods are either inapplicable or extremely costly. 

In considering fluids, a variety of approaches have been used to compute entropy and free energy a free-volume method,111-112 thermodynamic perturbation theory,113-116 thermodynamic integration,117-121 umbrella sampling,122-124 and a Monte Carlo recursion method.125-126 The entropy of association of two protein molecules in water has also been computed.127... 

Computation of the Free Energy of Liquid Water by the Monte Carlo Recursion Method. 

A very simple, and fast convergent, recursive method may be devised from this formula, leaving the rotation sine as a constant and finding, at each iteration step, the optimal tangent value, which will lead to a new sine value for the next iteration. Therefore, calling... 

Key words Impurity exciton, electroabsorption spectra, molecular static method, vibrational density of states, recursion method... 

The effect of charged nickel ions on the vibrational spectra of ZnO crystals was investigated by the recursion method using the shell model. The frequencies of locahzed vibrations of different S5mimetries were calculated with allowance for the atomic displacements in the vicinity of the charged impurities. These data on defect vibrations were applied to the interpretation... 

It is well known that orthogonal polynomials originate from the theory of continued fractions (see, for instance, the books 5, 7, 30, and 31 and references therein). We briefly review now the definitions and the basic properties of these polynomials and their connection with continued fractions, with the main purpose of emphasizing the aspects which are at the basis of the recursion methods. 

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