Criterion of convergence

The convergence of calculation can be indicated by evaluating the residual of Eq (35) at each node point during iteration. As a criterion of convergence, the maximum residual should be less than a preassigned value, say 10 . ... 

In step 3, a criterion of convergence may be introduced to terminate the iterations. Two other points should be mentioned instead of taking the correct Singles-CI matrix, we may resort to a simpler one, omitting single bi-electronic integrals and using only Fock-matrix elements as ... 

Another drawback was a criterion of convergence by GA. It is intrinsically difficult or impossible to verify the goal of optimization by GA. In fact fluctuation of maximum value and that of dispersion of catalyst composition are main indexes for the convergence. Instead of GA, therefore, more straightforward method was applied. It is so-called "grid search" (GS) all the activities of all possible combination of catalyst component were predicted by ANN. The global optimum on the ANN was found rapidly with the assistance of GS (13. 23. 39. 40. 42- 44). 

A starting value for x +j will be guessed (and once the integrations have begun a good value of this vector will be its value at the last timestep, Xt) and equation (2.78) can be evaluated repeatedly until some criterion of convergence is satisfied. 

In general, it is found that as q increases, one is compelled to use smaller values of A in (9.77) and large numbers of iterations to get a convergent result. As a criterion of convergence, we can choose the quantity... 

Both estimates, Eq. (15) and Eq.(18), must agree if the method has truly achieved uniform sampling. Therefore, in addition to the condition / < /thr the agreement of both estimates can be viewed as another criterion of convergence. After both methods agree, either one can be used. In this work we have chosen to present the estimates obtained from integration of the forces as these curves have a smoother appearance. 

By employing successive substitution of the tear variables and the criterion of Eq. (13-83), convergence was achieved slowly, but without oscillation, in 23 iterations. Computed products are. 

Note that the change in energy between the current and next points is not an explicit criterion for convergence. It is reflected in the tests of the size of the next step, since small steps near a minimum will usually result in small changes in the energy. 

This criterion allows the computation of the radius of convergence p by means of convergent series. The numerical evaluation requires an estimate of the remainder terms of the series r(x). Relations (10) and (12) provide the inequalities... 

Regarding accuracy, the finite difference approximations for the radial derivatives converge O(Ar ). The approximation for the axial derivative converges 0(Az), but the stability criterion forces Az to decrease at least as fast as Ar. Thus, the entire computation should converge O(Ar ). The proof of convergence requires that the computations be repeated for a series of successively smaller grid sizes. 

Compare P° ° with the actual total pressure given, P. ff it is sufficiently close to P (perhaps using a relative convergence criterion of 10 ), the guess T is correct. The vapor composition can then be calculated fi om Eq. (4.5). 

These results can be contrasted with results obtained using a less stringent equilibrium criterion of = 0.0005 S cm-1 min-1 as shown in the open circle data in Fig. 6. A large discrepancy is observed in the middle- o2 region. A difference between the measured values of Po2 by the sensors at the top and bottom of the cell of more than one order of magnitude in p0 was found under these conditions. In general, the p0 difference between the two sensors is slow to converge due to the slow equilibrium kinetics and consequently, both conductivity and log p0 difference criteria are needed to ensure that equilibrium is reached. The open circle data in Fig. 6 are similar to results in other ferrite perovskite oxides and reflect nonequilibrium behavior.14 17... 

Most applications of ECD spectroscopy in supramolecular chemistry, which can be counted to hundreds, involve proofs of existence and determination of properties (chirality) of potentially chiral supramolecular structures. The convergence of the ECD amplitude is taken as a criterion of pure enantiomers. The loss of structure upon the change of physico-chemical properties can be followed by the decrease of CD signal. Another concept uses chromophores [24] for testing the three dimensional structure of supramolecular systems. If the isolated chromophore is not chiral and induced circular dichroism (ICD) is observed in the system, we can sometimes deduce the structure of the chromophoric environment (see section 8.4.2). Ab initio calculations open another possibility to employ ECD into supramolecular chemistry (Section 8.1.3). 

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