Convergence, monotonic

When computing a molecular energy, or other energy requiring high levels of complexity, we need some sweet assurance that the calculations will converge monotonically to the "correct result." This assurance is provided by the variational theorem, which says that the Rayleigh56 ratio ... 

It is shown by Bellman that for any N the x,y plane is divided into two decision regions just as in the case of infinite N. It is also found that as N increases the decision regions converge monotonically to those for A = 00, and indeed that there is an Ao such that they are identical for A > No-... 

As regards the ground state of a given symmetry, the fundamental theoretical tool is the existence of the energy lower bound. This fact allows Rayleigh-Ritz variational calculations to converge monotonically upon variation of the size of the basis set and/or of linear or nonlinear variational parameters. 

In an interval I, for one equation f(x)=0 with solution X = 5 e I, the Newton-Raphson iteration converges monotonic (at least after the first iteration step) and quadratic to x = when f (x) and f (x) are non-zero and continuous for all x e I [ 78], [119]. This is equivalent with saying that the function f(x) must be convex and have a unique solution in the interval. Figure 3.14- gives a visualization of these properties. 

Fig. 3.142 In one dimension the Newton iteration converges monotonic and quadratic to the solution when the function f(x) has continuous non-zero first and second derivatives.

If Eq is calculated from (3.184) using the same matrix P as was used to form F, then Eq will be an upper bound to the true energy at any stage of the iteration and will usually converge monotonically from above to the converged result. If one adds the nuclear-nuclear repulsion to the electronic energy Eq one obtains the total energy E ... 

The great attraction of Feller s formula, besides its simplicity, is that no empiricism of any kind is involved. It has been used extensively by Dunning and co-workers.For applications to properties other than energies, which may not converge monotonically, Martin and Taylor propose the following generalization of the Feller formula ... 

FIGURE 3.2 Possible results of increasing the order of Moller-Plesset calculations. The circles show monotonic convergence. The squares show oscillating convergence. The triangles show a diverging series. 

However, this convergence is not monotonic. Sometimes, the smallest calculation gives a very accurate result for a given property. There are four sources of error in ah initio calculations ... 

We consider penalized operator equations approximating variational inequalities. For equations with strongly monotonous operators we construct an iterative method, prove convergence of solutions, and obtain error estimates. 

A convergence justification of the nonlinear terms 5 p rrf )ij to the term 5 p m )ij can be done by monotonicity arguments. The details are omitted here. 

The restrictions of the definition (5.127) are the same as before It gives correct results only for monotonically evolving functions w/(f) and i)if(t) should fastly enough approach its steady-state value m/(oo) for convergence of the integral in (5.127). 

Let us now consider the errors in the CC3 S( — 2) Cauchy moment (the static polarizability). From the monotonic convergence of the CCSD doubly augmented basis-set calculations of S( — 2) in Table 1 it appears that the difference between the 5Z and 6Z results should give a good estimate of the CCSD basis-set error at the 6Z level. CC3/d-aug-pV5Z 5 ( —2) Cauchy moment is 2.670 a.u. Using the... 

The function BSE(/, n) therefore increases monotonically with n and asymptotes to the true standard error associated with error estimate has converged, which is not subject to the extremes of numerical uncertainty associated with the tail of a correlation function. Furthermore, the blockaveraging analysis directly includes all trajectory information (all frames). 

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