Direct iteration

An iterative directed-metallating approach to 4,5-substituted indoles starting from gramine was documented <05T6886>. Treatment of gramine 173 with tert-butyllithium and trimethylsilylmethylazide followed by Boc protection gave 4-aminoindole 174. Directed lithiation by the carbamate followed by treatment with DMF gave indole 175. 

Once the basis determinants are selected the problem is to find the lowest eigenvectors of the resulting matrix representation of the Hamiltonian. This is usually done via an iterative direct diagonalization procedure that minimizes the norm of the error vector d, for the state i)... 

Fig. 4 Experimental order parameters, S, of NH bond vectors in GB3 derived from iterative Direct Interpretation of Dipolar Couplings (DIDC) using all six sets of RDCs. The red line marks the order parameters derived from relaxation. Filled symbols represent residues for which the fully anisotropic model was required to get a satisfactory fit to the data, while, for open symbols, the isotropic internal motion model was able to fit the RDCs to within the experimental noise. (Reprinted with permission from [41])...

The molar flux is obtained from Equation 7.128, whereas the enhancement factor Ea is iteratively calculated from Equations 7.121 and 7.127. If the gas film resistance is negligible, l/fcoA is set to zero in Equations 7.127 and 7.128. In this case (fcoA oc), Ea is iterated directly from Equation 7.121. 

We will apply this theorem now to construct Newton s method by an appropriate choice of B. Later, when dealing with discretized differential equations we will also apply the fixed point iteration directly to solve nonlinear systems. 

The level set method is represented by an implicit algorithm. In practice implicit means that we do not iterate directly on the nodes defining the front. We rather define a function from which we can extract information about the front. When evolving the function, we implicitly move the front. This implies that, even though the front may be complicated, the function itself is not necessarily complicated. This is attractive for real-world data sets, which are often noisy and abrupt. 

For large systems, the solution to the Newton equations must be obtained by an iterative, direct algorithm. From (10.8.14), we note that the required linear transformation of an antisymmetric trial matrix b may be expressed as the antisymmetric part of the matrix... 

For liquid-liquid separations, the basic Newton-Raphson iteration for a is converged for equilibrium ratios (K ) determined at the previous composition estimate. (It helps, and costs very little, to converge this iteration quite tightly.) Then, using new compositions from this converged inner iteration loop, new values for equilibrium ratios are obtained. This procedure is applied directly for the first three iterations of composition. If convergence has not occurred after three iterations, the mole fractions of all components in both phases are accelerated linearly with the deviation function... 

For the iteration algorithm (5) the optimal estimations (6) are directly used by a second back loop to block B (long dashed line in Fig. 1). 

In many cases, the methods used to solve identification problems are based on an iterative minimization of some performance criterion measuring the dissimilarity between the experimental and the synthetic data (generated by the current estimate of the direct model). In our case, direct quantitative comparison of two Bscan images at the pixels level is a very difficult task and involves the solution of a very difficult optimization problem, which can be also ill-behaved. Moreover, it would lead to a tremendous amount of computational burden. Segmented Bscan images may be used as concentrated representations of the useful... 

The same procedure is done for the y-direction. The combination of the two new matrices Ax and Ay then represents the result of the first iteration ... 

With XRD applied to bulk materials, a detailed structural analysis of atomic positions is rather straightforward and routine for structures that can be quite complex (see chapter B 1.9) direct methods in many cases give good results in a single step, while the resulting atomic positions may be refined by iterative fitting procedures based on simulation of the diffraction process. 

For large Cl calculations, the frill matrix is not fonned and stored in the computer s memory or on disk rather, direct CF methods [ ] identify and compute non-zero and inunediately add up contributions to the sum jCj. Iterative methods [, in which approximate values for the Cj coefficients are refined tlirough sequential application of to the preceding estimate of the vector, are employed to solve... 

Csaszar P and Pulay P 1984 Geometry optimization by direct inversion in the iterative subspace J. Moi. Struct. (Theochem) 114 31... 

Ionova I V and Carter E A 1995 Crbital-based direct inversion in the iterative subspace for the generalized valence bond method J. Chem. Phys. 102 1251... 

Hamilton T P and Pulay P 1986 Direct Inversion In the Iterative subspace (DNS) optimization of open-shell, excited-state and small multiconfiguratlonal SCF wavefunctlons J. Chem. Phys. 84 5728... 

In the fir.st iteration proces.s, the class values of the atoms of the structure. show information already known (the degree of the nodes). Hence Morgan takes the neighboring atoms into account. He considers the environment of an atom by summing class values of all directly adjacent atoms. This process results in a new class value called the extended connectivity (EC) value of the atom. The new EC value expresses indirectly the neighborhood of the adjacent atoms in a second sphere (Figure 2-43). 

The PEOE method basically consists of only three loops, one running over all the atoms of a molecule, a second loop through each iteration, and a third loop running over all the directly bonded neighbors of an atom. The cycle is usually stopped... 

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