Bookkeeping parameter

We introduce into (4.1) a small bookkeeping parameter A that will finally be put equal to unity. Thus we get the auxiliary differential equation... 

First, we have an abundancy of relaxation methods. Recall from chapter 3 that such a method is characterized by the subsequent variation of only a few parameters at a time, and that they demand an efficient bookkeeping of which entries of r to update, when the current subset of elements of d are varied. Of this class, the simplest methods in use are the Gauss-Seidel family of methods. Essentially only one element at a time gets updated. Let us simplify by using an algorithmic notation, where the iteration counter is dropped, and we use the replacement operator = instead of equalities ... 

Similar equations have been proposed by Swain et al. to correlate all nucleophilic displacements on carbon. However, the number of parameters has grown to four, and the usefulness of such a system of bookkeeping is open to question. ... 

The main advantage of the far formula (16) is that it allows for a product decomposition into the particle components and can be evaluated in linear time with respect to the number of particles with constant cutoffs. To draw advantage from this, some additional bookkeeping is necessary, see [17]. For MMM2D error estimates exist, which allow the tuning of the method this is important since the evaluation is split up into the near and far formula, leading to a highly non-uniform error distribution (see Fig. 1). With optimized parameters, the method can be tuned to scale like... 

One of the important parameters in this form of bookkeeping is the size of the cell V. There is a trade off between cycle time and accuracy. This procedure only includes interactions between particles in nearest-neighbor cells. If u / is less than or equal to the interaction range, then important interactions will be missed. On the other hand, if v is large so that on the average a large fraction of the N particles occupy each cell, the procedure becomes as inefficient as if no bookkeeping were used. The optimum value of v has been located in an... 

A few other bookkeeping sections in the code convert the input solution array from an x-y labeled two index matrix into a single column array if needed on lines 454 through 459 and a final loop converts the single column solution array into a j,i two index matrix of solution values on lines 486 through 488. The type of solution desired may be specified as a table of values in order to pass additional parameters to the PDE solver. If a table is specified for the typsola parameter, two additional parameters, rx and ry, can be extracted from the input list as seen on line 444. For a direct sparse matrix solution with the SPM parameter these are not used. For the COE and SOR methods, this feature can be used to specify a spectral radius value with only the rx parameter used. For the ADI method these two values can be used to specify wx and wy parameters. In all cases, default values of these parameters are available, if values are not input to the various functions. 

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