Nest iteration method

Finally, it should be noted that the multigrid method can be used as either an iterative process or as a direct solver (the so-called full multigrid or nested iteration method ). 

For the solution of Equation 10.25 the inverse of matrix A is computed by iterative techniques as opposed to direct methods often employed for matrices of low order. Since matrix A is normally very large, its inverse is more economically found by an iterative method. Many iterative methods have been published such as successive over-relaxation (SOR) and its variants, the strongly implicit procedure (SIP) and its variants, Orthomin and its variants (Stone, 1968), nested factorization (Appleyard and Chesire, 1983) and iterative D4 with minimization (Tan and Let-keman. 1982) to name a few. 

Noise allocation followed by scalar quantization and Huffman coding. In this method, no explicit bit allocation is performed. Instead, an amount of allowed noise equal to the estimated masked threshold is calculated for each scalefactor band. The scalefactors are used to perform a coloration of the quantization noise (i.e. they modify the quantization step size for all values within a scalefactor band) and are not the result of a normalization procedure. The quantized values are coded using Huffman coding. The whole process is normally controlled by one or more nested iteration loops. The technique is known as analysis-by-synthesis quantization control. It was first introduced for OCF [Brandenburg, 1987], PXFM [Johnston, 1989b] and ASPEC [Brandenburg et al., 1991], In a practical application, the following computation steps are performed in an iterative fashion ... 

Thus, the approximate Newton search direction in TN methods is obtained by allowing a nonzero residual norm rk = rj = H p + gA at each step. The size of this residual is monitored systematically according to the progress made. This formulation leads to a doubly nested iteration structure for every outer Newton iteration k (associated with xk) there corresponds an inner loop for pk p p 1,. . . . ... 

We believe that a better way to solve the set of Eqs. 11.5.1-11.5.10 is to solve all the equations simultaneously using Newton s method (Algorithm C.2). This approach avoids the nested iterations of the foregoing procedure and keeps both thermodynamic property... 

FIG. 12 Schematic representation of the iterative scheme. We have two nested iterative loops the time loop (1) for updating the density fields p, and within each time iteration an iterative loop (2) for updating the external potential fields U. We start with an initial guess for the external potential fields. We use Eq. (17) to generate a set of unique density fields. The cohesive chemical potential E [relation (3)] can be calculated from the density fields by Eq. (8). The total chemical potential /x(4) can now be found from Eq. 6. We update the density fields (5) [by using the old and updated fields in Eq. (23)] and accept the density fields if the condition (26) is satisfied. If this is not the case, the external potential fields are updated by a steepest descent method. 

The various procedures that address side chain generation as an energy-driven conformational search can be grouped as systematic, combinatorial, and stochastic methods. The CONGEN procedure " includes five different tree-based implementations of the first two approaches. The first methods, named ALL, generates all possible conformations by a series of nested iterations over every side chain, and hence is limited to 7 or 8 residues. The second method, named FIRST, uses... 

As previously commented, the standard method for solving equations is Newton s method. But this requires the calculation of a Jacobian matrix at each iteration. Even assuming that accurate derivatives can be calculated, this is frequently the most time-consuming activity for some problems, especially if nested nonlinear procedures are used. On the other hand, we can also consider the class of quasi-Newton methods where the Jacobian is approximated based on differences in x and/(x), obtained from previous iterations. Here, the motivation is to avoid evaluation of the Jacobian matrix. 

Increment the coxmter p p + and repeat the nested cycle of equations (29) and (30) until convergence. This iterative procedure is related to the normal method of solving the nonlinear PB equation for screening charge densities. Indeed, the DHH analysis can be regarded as... 

We also have studied briefly the effects of increasing the depths of loop nests on the slicing time. This is interesting since loops will necessitate conventional fixed-point iterations in the RD data dependency analysis used by PDG-based slicing, whereas our SLV-based method uses a different mechanism to handle loops. 

For many years the standard preconditioner has been the method of nested factorisation due to [8]. The iteration scheme of choice, based on well known approaches to nonlinear optimisation, has been the orthomin method of [153]. Only recently have there been challenges to nested factorisation from multigrid methods and domain decomposition methods. Deeper analysis may lead to improvements on basic orthomin. (See [134] for a complete overview and [150] for a review of multigrid.)... 

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