Convergent ordering

The first result, for At = 0.5, shows a negative result for a(t ax) due to the very large value for At. For smaller values of At, the calculated values for a tmax) are physically realistic and the errors decrease by roughly a factor of 4 as the time step decreases by a factor of 4. Thus, the error is proportional to At. Euler s method is said to converge order At, denoted 0(At). 

Convergence order At for Euler s method is based on more than the empirical observation in Example 2.4. The order of convergence springs directly from the way in which the derivatives in Equations (2.11) are calculated. The simplest approximation of a first derivative is... 

Bragg-Williams treatment of convergent ordering in solid solutions... 

We will not carry this analysis further, since in writing the correction function on the form e(z,y) we have not explicitly included the dependence on derivatives of f. However, we note that this section provides reason for a classification of defect correction schemes by a new concept, called convergence order. A procedure which guarantees the asymptotic inequalities (within some disk)... 

Suppose you want to solve the equation x = e z on a pocket calculator. Suggest a simple method, and verify that it works by working out its asymptotic convergence properties What is the convergence order If this is a first-order procedure, what is the convergence rate A d= lim 001j Show that the NR method, applied as f(x) = x - e x = 0, takes the form... 

In practice, the asymptotic convergence order in not so important. For a large system it takes so many iterations to reach the asymptotic behaviour, that convergence to machine accuracy has occurred much earlier. Of prime importance is instead to have a good enough approximation to the true Hessian. 

Figure 15. Calculated cooling paths over 13 decades of cooling rates showing dependence of the low-temperature site occupancies of FeMnSi04 on cooling rate. The room-temperature site occupancy, or degree of non-convergent order, is a direct measure of the cooling rate of the sample.

Parthasarathy HB, Schall JD, Graybiel AM (1992) Distributed but convergent ordering of corticostriatal projections analysis of the frontal eye field and the supplementary eye field in the macaque monkey. J. 

This is a study of the non-convergent ordering of Mg and Ni in synthetic olivines by means of the neutron powder diffraction and X-ray absorption spectroscopy (EXAFS). The implications of the intracristalline ordering to olivine crystal chemistry and their application to the consideration of thermodynamic relations, geothermometry, geo-speedometry, and minor or trace element distribution control is discussed in detail. 

In the previous chapter, we discussed the growth of error in numerical methods for differential equations. We saw that if the time interval is fixed, the error obeys the power law relationship with stepsize that is predicted by the convergence theory. We also saw that this did not contradict the exponential growth in the error with time (when the stepsize is fixed). The latter issue casts doubt on the reUance on the convergence order as a means for assessing the suitability of an integrator for molecular dynamics. 

The implication of backward error analysis is that the convergence order is still directly relevant in molecular simulations, even when the accuracy of trajectories (described by the convergence theorem) cannot be verified. We can think of the modified energy surface as a rippled version of the original. The order determining the allowed magnitude of the fiuctuation. 

In practice all the methods presented here are convergent in the strong sense, but their strong convergence orders may be low, typically r = 1/2. 

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