Convergence radius

This scheme is concurrent for both the density and p2 at every iteration step, one first tests whether A is less than the maximum value allowed according to Eq. (192), then calculates p2 according to Eq. (191), and propagates the density to the next step. It is important to know that no extra density normalization effort is needed because the density is always normalized by choosing the value for p2 according to Eq. (191). Niunerical tests show that the steepest-descent scheme still has an instability problem and the convergence radius for A is quite small. 

Second case. Next we consider the case where the condition (4. 5) is satisfied. Here we make the curve F as yet unspecified. According to (4.17) we obtain as a lower bound of the convergence radius of the series (2. 5)... 

Else, the sequence y an is unbounded. In that case, s (x) diverges everywhere except for x = x0, and one says that the convergence radius is R = 0. 

Now assume a non-zero convergence radius. One can then prove that ... 

Another common example arises in the expansion of the interaction energy of two atoms as a function of the inverse interatomic distance 1/Rab- This turns out to be a Taylor series which differ from the correct energy by a real contribution with zero Taylor expansion. The difference may be attributed to so-called exchange repulsion. In this case, it is nowadays known that die Taylor series has a zero convergence radius, so that the energy expression constitutes an example of an asymptotic series (to be defined in a moment) which is non-convergent for all Rab-... 

This series cannot be used to evaluate E(l), since the convergence radius is only This is due to two branch points, at A = i. The simple replacement of A with /i, which we define as... 

Find the Taylor series for log(x) expanded around x0 = 1. Determine the convergence radius. Explain which feature of the function is responsible for the limited convergence radius. Then show that a better way of calculating the logarithm is provided by... 

It is expected that a suitable choice of g will improve the convergence of the power series as the singularity of tj/ may be cancelled by the zero of g. In most cases, the Pade approximation /jg provides with a better approximation than the corresponding power series truncated at any order q, especially when /c is comparable to (or even greater than) the convergence radius of the power series [Eq. (342)] (Takeshi, 1999). 

The latest expansions are power series in e and their convergence depend on the singularities of the analytic function u = u(e, ), which are at e = 0.6627434- . This is the convergence radius of the given series (see Wintner, 1941). 

In the case of the exchange reaction the effective Planck s constant is heff = 1.77 X 10 and is thus smaller than This fact is in agreement with the apparent speedup of fhe convergence of fhe CRP values, see Figure 5.13b, in comparison wifh the case. Finally, the convergence is very fast and pronounced for the case of fhe heavy (hypofhefical) atoms, see Figure 5.13c, for which fteff = 6.9 x 10 which is much smaller than the estimated convergence radius. 

The second-order energy per atom has the same asymptotic behaviour as in the periodic or linear 2A + 1 chain it diverges logarithmically. Therefore, perturbation theory seems to be of very little value in this context The reason is the shrinking of the convergence radius R of the series expansion of the eigenvalue as k approaches the Fermi value. 

In any case, perturbation theory cannot provide an estimate of the asymptotic value of the Peierls distortion because the convergence radius of the expansion is rapidly shrinking to zero as N grows large. 

The convergence radius of the substitution method is similar to the one required for stability in an explicit method and is related to the product of the integration step and the maximum eigenvalue of the Jacobian. 

Figure 12.10 shows an example of a one-dimensional function and its associated gradient norm. It is clear that a gradient norm minimization will only locate one of the two stationary points if started near x = 1 or x = 9. Most other starting points will converge on the shallow part of the function near x = 5. The often very small convergence radius makes gradient norm minimizations impractical for routine use. 

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