Convergence bounds

Handy, C. R., Bessis, Sigismondi, G., and Morley, T. D., (1988) Rapidly Converging Bounds for the Ground State Energy of Hydrogenic Atoms in Superstrong Magnetic Fields , Phys. Rev. Lett. 60, 253. 

The proof of this theorem is in Ref. [42]. Obviously, the critical quantity in the convergence bound is the distance y of the training sample from the decision boundary. 

We have encountered oscillating and random behavior in the convergence of open-shell transition metal compounds, but have never tried to determine if the random values were bounded. A Lorenz attractor behavior has been observed in a hypervalent system. Which type of nonlinear behavior is observed depends on several factors the SCF equations themselves, the constants in those equations, and the initial guess. 

Using the term weakly compact we mean only that every bounded sequence contains a weakly converging subsequence. The same is related to the term - weakly compact. 

Proof Let pm G be a minimizing sequence. It is bounded in and hence the convergence (2.135) can be assumed. For every m, the solution of the following variational inequality can be found ... 

This method requires solution of sets of linear equations until the functions are zero to some tolerance or the changes of the solution between iterations is small enough. Convergence is guaranteed provided the norm of the matrix A is bounded, F(x) is Bounded for the initial guess, and the second derivative of F(x) with respect to all variables is bounded. See Refs. 106 and 155. 

The development of an SC procedure involves a number of important decisions (1) What variables should be used (2) What equations should be used (3) How should variables be ordered (4) How should equations be ordered (5) How should flexibility in specifications be provided (6) Which derivatives of physical properties should be retained (7) How should equations be linearized (8) If Newton or quasi-Newton hnearization techniques are employed, how should the Jacobian be updated (9) Should corrections to unknowns that are computed at each iteration be modified to dampen or accelerate the solution or be kept within certain bounds (10) What convergence criterion should be applied ... 

While this bound is not a particularly strong one and convergence is generally faster in practice [goles90], it does clearly point out the important fact that transient times are linearly bounded by the lattice size n. Notice that this is not true of more general classes of matrices, even those of the preceding section that are both symmetric and integer-valued. Equation 5.140 shows that the transient time depends on both A and 26 — Al if both A and b are arbitrary (save, perhaps, for A s symmetry), there is of course no particular reason to expect t to be linearly... 

But the convergence rate depends on the parameter w. However, there are analytical estimates for ui and the convergence rate when the subsidiary information on the spectral bounds of the operator D A + A+) is available, but their determination is some problem in itself. Just for this reason the parameter w is so chosen as to minimize the total number of iterations. In dealing with numerous similar problems this approach is more convenient and effective. 

It turns out that the convergence of the aforementioned methods become worse on account of widely varying bounds of the spectra of difference operators. 

The procedure is repeated until convergence is reached. Since we are interested in bound states where ej <0, no problem of divergenee or eusps eonditions is raised. But the method ean be adapted to more general situations by introdueing a translation of the energy origin in Eq. 21. 

Thus, Equation 13.36 can be used to accelerate the convergence and is known as the Wegstein method11. If q = 0 in Equation 13.36, the method becomes direct substitution. If q < 0, acceleration of the solution occurs. Bounds are normally set for the value of q to prevent unstable behavior. 

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