Replicated algorithms

As shown in the Level column of Figure 10, the various injected volumes can be designated as individual component calibration levels. By doing this, experiments 10-27 can also be used to help validate multilevel, multi-replicate calibration algorithms on a chromatographic data system. 

In practice the equation of motion is solved first without considering the constraint force and in the next step the constraint forces are obtained by correcting the positions such that the molecule conserves its minimum structure, i.e., such that the constraints are fiilfilled. For small molecules direct inversion is possible, for large molecules iterative procedures are applied (60). This means that each constraint is corrected after the other until a certain convergence is reached. This algorithm is called Shake (65). Another important aspect of simulations concerns periodic boundary conditions. A virtual replication of the central box at each of its planes is carried out in order to avoid surface effects. A detailed description can be found in the excellent textbooks of Allen and Tildesley (60) and Frenkel and Smit (61). 

Using the BH algorithm to estimate the effects, we note that column AB has the same signs as column C, column AC has those of column B, column BC those of column A and column ABC those of I. Hence the linear combination of observations in column A, 1a, can be used to estimate not only the main effect of A but also the BC interaction (1a = A + BC 1 a = A — BC if we select C = —AB as generator). Two (or more) effects that share this type of relationship are termed aliases". As a consequence, aliasing" is a direct result of fractional replication. In many practical situations, it will be possible to select a fraction of the experiments so that the main effect and the low-order interactions that are of interest become aliased (confounded) only with high-order interactions, which are probably negligible. 

The algorithm used is attributed to J. B. J. Read. For many manipulations on large matrices it is only practical for use with a fairly large computer. The data are arranged in two matrices by sample i and nuclide j one matrix, V, contains the amount of each nuclide in each sample the other matrix, E, contains the variances of these numbers, as estimated from counting statistics, agreement between replicate analyses, and known analytical errors. It is also possible to add an arbitrary term Fik to each variance to account for random effects between samples not considered in the model this is usually done in terms of an additional fractional error. Zeroes are inserted for missing data in cases in which not all nuclides were measured in every sample. 

If the exact algorithms described above are not feasible for a given data set (the limitation is usually number of taxa), then various heuristic approaches can be tried. The heuristics used should be described in sufficient detail that they can be replicated, and so that alternative searches can be attempted. It is also worthwhile to discuss the number of alternative solutions examined, to give a sense of the thoroughness of the search. 

Standard Curve Inject each calibration standard in triplicate. Normal instrument linearity extends to 25 ng/mL. If nonlinear calibration capability is not available, limit the working calibration curve to <25 ng/mL. Use the calibration algorithms provided in the instrument software. Recheck calibration periodically (<15 samples) by running a 25- or 50-ng/ mL calibration standard interspersed with samples. If recheck differs from calibration by >10%, recalibrate the instrument. The instrumental detection limit (DL) and quantitation limit (QL), in picograms, may be based on 7 to 10 replicates of the Sample Preparation Blank and calculated as follows ... 

To date the most efficient parallel SCF algorithms have been based on replication within each processor of several 0 N ) matrices, limiting the maximum calculation size and forcing an unacceptably low ratio of processors to memory. This restriction led to increased activity in the development of distributed-data schemes, some of which we now consider. 

Smith has also described the implementation of MD on parallel machines, with particular reference to hypercube computers. In work designed to introduce a novice to the main aspects of parallel computing in MD, Smith described three particular algorithms replicated data (RD), systolic loop (SLS-G), and parallelized link cells (PLC), all of which have good load balancing. The performance characteristics of each algorithm and the factors affecting... 

Windemuth and Schulten described a further variant of the systolic loop algorithm, the replicated systolic loop for the efficient calculation of macromolecular force fields on the Connection Machine (e.g., CM-1, CM-2, CM-5). The full force field was separated into bond interactions and nonbonding interactions only the latter were implemented on the Connection Machine, and parallelized by the replicated systolic loop algorithm. The former, less computationally intensive tasks were performed by an existing, conventional MD code on the front end. 

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