Iterative definition

Such an iterative definition is identical with the outer iteration process used to compute TV in a critical system ... 

To decide which molecule to add at each iteration requires the dissimilarity values between each molecule remaining in the database and those already placed into the subset to be calculated. Again, this can be achieved in several ways. Snarey et al. investigated two conunon definitions, MaxSum and MaxMin. If there are m molecules in the subset then... 

Fig. 8,20 First five iterations of an SDCA system starting from a 4-neighbor Euclidean lattice seeded with a single non-zero site at the center. The global transition rule F consists of totalistic-value and restricted totalistic topology rules C — (26,69648,32904)[3 3[ (see text for rule definitions). Solid sites have <7=1.

In another important class of cases, the matrix A is positive definite. When this is so, both the Gauss-Seidel iteration and block relaxation converge, but the Jacobi iteration may or may not. 

Generally speaking, the investigator carries out such simulations to show whether the postulated scheme matches the data. To carry out the simulations, the researcher fixes the rate constants of some steps at their established values and evaluates the ones that are unknown by iterative adjustment, the criterion being the quality of the match. Obviously, one is not likely to produce a definitive answer if too many steps have rate constants that are allowed to be adjusted at will. One may also be tempted to adjust one (more ) of the supposedly established rate constants. This action is perilous, without independent cause for suspicion. In that event, one should very seriously consider replicating the original experiments that defined that step. Not to do so invites the construction of an unsound house of cards. ... 

For convenience of presentation, model building can be divided into four phases (1) problem definition and formulation, (2) preliminary and detailed analysis, (3) evaluation, and (4) interpretation application. Keep in mind that model building is an iterative procedure. Figure 2.2 summarizes the activities to be carried out,... 

Difficulty 3 can be ameliorated by using (properly) finite difference approximation as substitutes for derivatives. To overcome difficulty 4, two classes of methods exist to modify the pure Newton s method so that it is guaranteed to converge to a local minimum from an arbitrary starting point. The first of these, called trust region methods, minimize the quadratic approximation, Equation (6.10), within an elliptical region, whose size is adjusted so that the objective improves at each iteration see Section 6.3.2. The second class, line search methods, modifies the pure Newton s method in two ways (1) instead of taking a step size of one, a line search is used and (2) if the Hessian matrix H(x ) is not positive-definite, it is replaced by a positive-definite matrix that is close to H(x ). This is motivated by the easily verified fact that, if H(x ) is positive-definite, the Newton direction... 

If the BFGS algorithm is applied to a positive-definite quadratic function of n variables and the line search is exact, it will minimize the function in at most n iterations (Dennis and Schnabel, 1996, Chapter 9). This is also true for some other updating formulas. For nonquadratic functions, a good BFGS code usually requires more iterations than a comparable Newton implementation and may not be as accurate. Each BFGS iteration is generally faster, however, because second derivatives are not required and the system of linear equations (6.15) need not be solved. 

Solving a QP with a positive-definite Hessian is fairly easy. Several good algorithms all converge in a finite number of iterations see Section 8.3. However, the Hessian of the QP presented in (8.69), (8.70), and (8.73) is V2L (x,X), and this matrix need not be positive-definite, even if (x, X) is an optimal point. In addition, to compute V2L, one must compute second derivatives of all problem functions. 

Most NLP solvers evaluate the first-order optimality conditions and declare optimality when a feasible solution meets these conditions to within a specified tolerance. Problems that reach what appear to be optimal solutions in a practical sense but require many additional iterations to actually declare optimality may be sped up by increasing the optimality or feasibility tolerances. See Equations (8.31a) and (8.31b) for definitions of these tolerances. Conversely, problems that terminate at points near optimality may often reach improved solutions by decreasing the optimality or feasibility tolerances if derivative accuracy is high enough. 

Glossary This is an initial set of definitions of terms used to define the problem or requirements. The glossary is developed in parallel with a type model, either for the system specification or of the domain itself. The glossary will be maintained through the iterations of the system specifications. 

On the contrary, the definition of the collision process, Eq. (429), is such that through a sequence of such events, the perturbation caused by the external force may be propagated at long distances. For instance, in the diagram of Fig. 21, corresponding to a typical term of the iterative solution of Eq. (428), the T operators are not localized around the B-particle. This allows long-range hydrodynamical effects. 

In the vicinity of the minimum, the H should be positive-definite. This may not be the case everywhere in which case there is a small but real danger of iterating towards a saddle instead of the minimum. It is therefore highly advisable, especially when the data scatter about the best-fit straight line, plane, or hyper-plane, to use the best possible initial estimate. Most commonly, one of the linear estimates (Section 5.1) will be good enough. 

Multiple alignments of repeats are constructed in an iterative manner. The initial alignment is based on definitions from determined protein structures or else from the literature. In the initial database search step, a profile constructed from the multiple alignment is compared with a sequence database. Top scoring sequences are considered using complementary approaches such as PSI-BLAST and FASTA to provide the two thresholds minimum E value and minimum number of repeats per protein required. After one or two iterations, the final alignment and the thresholds are stored in the SMART database to allow the detection of repeats in any sequence. 

Validation is the process of proving that a method is acceptable for its intended purpose. It is important to note that it is the method not the results that is validated. The most important aspect of any analytical method is the quality of the data it ultimately produces. The development and validation of a new analytical method may therefore be an iterative process. Results of validation studies may indicate that a change in the procedure is necessary, which may then require revalidation. Before a method is routinely used, it must be validated. There are a number of criteria for validating an analytical method, as different performance characteristics will require different validation criteria. Therefore, it is necessary to understand what the general definitions and schemes mean in the case of the validation of CE methods (Table 1). Validation in CE has been reviewed in references 1 and 2. The validation of calibrations for analytical separation techniques in general has been outlined in reference 3. The approach to the validation of CE method is similar to that employed for HPLC methods. Individual differences will be discussed under each validation characteristic. 

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