Cubic convergence

Maximizing the absolute value of the kurtosis of the components of y (Eq. 18.12), one of the columns of the separating matrix W is found, and so one independent component at a time is identified. The other columns are estimated subsequently. The algorithm has a cubic convergence and typically convergence by 20 iterations. From Eqs. (18.13), (18.14), and (18.15), the output matrix of independent components y can be written as... 

Comparing Tables 11.3 and 11.4, we find that, in terms of macro iterations, the Newton method converges faster than the quasi-Newton method. The quasi-Newton method works better than the Newton method in the first few iterations, but the local cubic convergence of the Newton method then takes over and ensures that this method gives the smallest number of macro iterations. However, since each Newton iteration is an order of magnitude mote expensive than each quasi-Newton or Davidson iteration, the quasi-Newton and Davidson methods are far more cost-effective. 

Compare E — Ecnv. lIRnll and C — Ccnvil for = 1 and = 2 in the Newton optimization of Exercise 11.11.5. Do the errors exhibit the expected cubic convergence Compare with the errors of the Davidson method. 

All three error measures in Table IIS.11.3 show cubic convergence. Comparing with the Davidson method in Table 1 IS. 11.2, we find that, for this particular system, one Newton iteration corresponds to three Davidson iterations. 

In the next section we derive the Taylor expansion of the coupled cluster cubic response function in its frequency arguments and the equations for the required expansions of the cluster amplitude and Lagrangian multiplier responses. For the experimentally important isotropic averages 7, 7i and yx we give explicit expressions for the A and higher-order coefficients in terms of the coefficients of the Taylor series. In Sec. 4 we present an application of the developed approach to the second hyperpolarizability of the methane molecule. We test the convergence of the hyperpolarizabilities with respect to the order of the expansion and investigate the sensitivity of the coefficients to basis sets and correlation treatment. The results are compared with dispersion coefficients derived by least square fits to experimental hyperpolarizability data or to pointwise calculated hyperpolarizabilities of other ab inito studies. 

Determine the relative rates of convergence for (1) Newton s method, (2) a finite difference Newton method, (3) quasi-Newton method, (4) quadratic interpolation, and (5) cubic interpolation, in minimizing the following functions ... 

In the exercises for Chapter 2 we suggested calculations for several materials, including Pt in the cubic and fee crystal structures and ScAl in the CsCl structure. Repeat these calculations, this time developing numerical evidence that your results are well converged in terms of sampling k space and energy cutoff. 

A better alternative is to use the difference structure factor AF in the summations. The electrostatic properties of the procrystal are rapidly convergent and can therefore be easily evaluated in direct space. Stewart (1991) describes a series of model calculations on the diatomic molecules N2, CO, and SiO, placed in cubic crystal lattices and assigned realistic mean-square amplitudes of vibration. He reports that for an error tolerance level of 1%, (sin 0/2)max = 1-1.1 A-1 is adequate for the deformation electrostatic potential, 1.5 A-1 for the electric field, and 2.0 A 1 for the deformation density and the deformation electric field gradient (which both have Fourier coefficients proportional to H°). 

In looking at the polynomial fit we see that the convergence is primarily linear since the coefficient of the linear term is much larger than those of the cubic and higher terms. 

Detailed description of the domains of convergence of hyper geometric series in terms of amoeba of the discriminant of the polynomial has been given recently in Passare and Tsikh (2004). The discriminant A(a) is an irreducible polynomial with integer coefficients in terms of the coefficients , of polynomial (54) that vanishes if this polynomial has multiple roots. For instance, for cubic polynomial the discriminant is... 

The following case study demonstrates the convergence behavior for the LH mechanism (50) with irreversible first stage (i.e. r i = 0). In this case the kinetic polynomial (51) always has (structurally unstable with respect to feasibility) zero root whereas three other roots could be found from the cubic equation... 

To obtain the anharmonic terms in the potential, on the other hand, the choice of coordinates is important 130,131). The reason is that the anharmonic terms can only be obtained from a perturbation expansion on the harmonic results, and the convergence of this expansion differs considerably from one set of coordinates to another. In addition it is usually necessary to assume that some of the anharmonic interaction terms are zero and this is true only for certain classes of internal coordinates. For example, one can define an angle bend in HjO either by a rectilinear displacement of the hydrogen atoms or by a curvilinear displacement. At the harmonic level there is no difference between the two, but one can see that a rectilinear displacement introduces some stretching of the OH bonds whereas the curvilinear displacement does not. The curvilinear coordinate follows more closely the bottom of the potential well (Fig. 12) than the linear displacement and this manifests itself in rather small cubic stretch-bend interaction constants whereas these constants are larger for rectilinear coordinates. A final and important point about the choice of curvilinear coordinates is that they are geometrically defined (i.e. independent of nuclear masses) so that the resulting force constants do not depend on isotopic species. At the anharmonic level this is not true for rectilinear coordinates as it has been shown that the imposition of the Eckart conditions, that the internal coordinates shall introduce no overall translation or rotation of the body, forces them to have a small isotopic dependence 132). 

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