Criteria for convergence

B. Klahn, W.A. Bingel, The convergence of the Rayleigh-Ritz method in quantum chemistry. I. The criteria for convergence, Theor. Chim. Acta 44 (1977) 9-26. 

However, the criteria for convergence of a seqnence and of a series are different. A convergent seqnence may not generate a convergent series. A seqnence formed by partial sums of a convergent series is convergent. 

Note that for the shape of curve chosen on Fig. 1.4, jc, is nearer to the root x than either q or X2- This, of course, will not always be the case with all functions. Discussion of criteria for convergence will be given in the next section. 

Yon may hke to reflect on our criteria for good disconnections (see for example frame 76). Two of them could be called simply guides to help us find convergent syntheses. Which ones ... 

We specified tight SCF convergence criteria for all jobs (SCF=Tight). Here are the results of our calculations, as well as the computed and observed energy differences ... 

The convergent approach to dendrimer synthesis has evolved considerably over the past dozen years. With the benefit of hindsight, it is clear that the key criteria for a successful convergent synthesis include ... 

In the calculation, we have examined the effect of some parameter on total energy and stmcture of surface and crystal. These parameters include plane wave cutoff energy, k-point and SCF convergence criteria, and the geometry optimization convergence criteria for the geometry optimization task. 

Suppose f(t) is known from experimental data, then for each function p (t) the left-hand sidecan be calculated. This procedure can be repeated N -f- 1 times,givingN H- 1 results, usable to calculate the coefficients a, Uj,... a . Important criteria for the choice of the function p (t) are the fast convergence and the accuracy of the method. 

The cumulative probability functions listed in Table 2.1.6 can be used to give some measure of correlation between the various criteria for orbital size. Once a particular value for P is chosen, it can be equated to each of the expresssions in turn and the resulting transcendental equation solved numerically. The p values thus obtained for P = 0.50, 0.90, 0.95, and 0.99 are tabulated in Table 2.1.7. The significant quantites are the size ratios, with the Is p value as standard, which provide a rational scale of relative orbital size based on any adopted probability criterion. As the prescribed P value approaches unity, the size ratios gradually decrease in magnitude and orbitals in the same shell tend to converge to a similar size. 

The Optimal Tearing Criterion. Various criteria for tearing have been suggested from time to time as the most suitable to minimize the total amount of calculation required for convergence. These include ... 

P/ and qj = n. Kandiner and Brinkley (6) assumed values for the derived species, n , and solved Equation (11) for the moles of components, nj. Equations (10) were solved for n and convergence criteria checked. For convergence, components need be selected as those species present in the highest concentration at equilibrium. When this is not the case, Equation (11) frequently gives negative values for nj. 

O Neill, T.J., Tebbutt, J.S., Challis, R.E. 2001. Convergence criteria for scattering models of ultrasonic wave propagation in suspensions of particles. IEEE Trans. Ultrasonics, Ferro-electrics and Frequency Control 48, 419-424. 

Even if computer applications verify the convergence of the approximations as the number m increases and the set

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Donald B. Boyd (Eli Lilly and Company), David W. Smith, James J. P. Stewart (U.S. Air Force Academy), and Erich Wimmer (Cray Research) Importance of Criteria for Self-Consistent Field Convergence and Geometry Optimization in AMI, MNDO, and MINDO/3 Molecular Orbital Calculations. ... 

For converging sections that are flood loaded and have a full cross section (i.e., hoppers), use the same design criteria used for mass flow bins discussed above. 

Many other factors besides perceptual illusions could also affect the accuracy of retrospective self-reports. On the other hand, research has identified the conditions that increase the likelihood of accurate self-report data. And the ultimate reason why survey studies of alcohol use have advanced knowledge is that scientists have looked to converging evidence within studies (Do the different sources of data that I ve collected in this study lead me to the same, reasonedsle conclusions ) and replication across studies (Do different researchers doing the same study under different conditions get similar results ) as the main criteria for confidence in their results. 

Although it has been contended that further controlled research is needed to diagnose a withdrawal syndrome in human marijuana users (Smith 2002), criteria for cannabinoid withdrawal have been proposed (Budney et al. 2003). Moreover, converging lines of evidence from retrospective, outpatient, and inpatient stud-... 

The final step is to check for convergence by one of two criteria either the variation of the V/s and T/s from the previous iteration is within a given tolerance, or Equations 13.2 through 13.5 are satisfied within tolerance. 

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