Iterative characteristics

However, each of these forms possesses a spurious root and has other characteristics (maxima or minima) that often give rise to convergence problems with common iterative-solution techniques. 

These systems are solved by a step-limited Newton-Raphson iteration, which, because of its second-order convergence characteristic, avoids the problem of "creeping" often encountered with first-order methods (Law and Bailey, 1967) ... 

In the highly nonlinear equilibrium situations characteristic of liquid separations, the use of priori initial estimates of phase compositions that are not very close to the true compositions of these phases can lead to divergence of iterative computations or to spurious convergence upon feed composition. 

For an industrial application it is necessary to separate the response of a real crack from artifacts, and to derive information about the geometry and the location of the crack. For this purpose we have developed a filter which is sensitive to the characteristic features of a signal caused by a crack and amplifies it, whereas signals without these typical features are suppressed. In Fig. 5.1 first results obtained with such an iterative filter algorithm are shown. 

This example illustrates the simplified approach to film blowing. Unfortunately in practice the situation is more complex in that the film thickness is influenced by draw-down, relaxation of induced stresses/strains and melt flow phenomena such as die swell. In fact the situation is similar to that described for blow moulding (see below) and the type of analysis outlined in that section could be used to allow for the effects of die swell. However, since the most practical problems in film blowing require iterative type solutions involving melt flow characteristics, volume flow rates, swell ratios, etc the study of these is delayed until Chapter 5 where a more rigorous approach to polymer flow has been adopted. 

Ri seems to be related to solute size, but it is obtained by an iterative curve-fitting procedure. The parameters M and D are characteristic of the solvent. For water Af = 0. A solvophobicity parameter Sp is then defined for the solvent having value M... 

If the ecjuations have the proper characteristics, the iterative process will eventually converge. Commonly used convergence criteria are of two types ... 

The methods of simple and of inverse iteration apply to arbitrary matrices, but many steps may be required to obtain sufficiently good convergence. It is, therefore, desirable to replace A, if possible, by a matrix that is similar (having the same roots) but having as many zeros as are reasonably obtainable in order that each step of the iteration require as few computations as possible. At the extreme, the characteristic polynomial itself could be obtained, but this is not necessarily advisable. The nature of the disadvantage can perhaps be made understandable from the following observation in the case of a full matrix, having no null elements, the n roots are functions of the n2 elements. They are also functions of the n coefficients of the characteristic equation, and cannot be expressed as functions of a smaller number of variables. It is to be expected, therefore, that they... 

Because of the convenient mathematical characteristics of the x -value (it is additive), it is also used to monitor the fit of a model to experimental data in this application the fitted model Y - ABS(/(x,. ..)) replaces the expected probability increment ACP (see Eq. 1.7) and the measured value y, replaces the observed frequency. Comparisons are only carried out between successive iterations of the optimization routine (e.g. a simplex-program), so that critical X -values need not be used. For example, a mixed logarithmic/exponential function Y=Al LOG(A2 + EXP(X - A3)) is to be fitted to the data tabulated below do the proposed sets of coefficients improve the fit The conclusion is that the new coefficients are indeed better. The y-column shows the values actually measured, while the T-columns give the model estimates for the coefficients A1,A2, and A3. The x -columns are calculated as (y- Y) h- Y. The fact that the sums over these terms, 4.783,2.616, and 0.307 decrease for successive approximations means that the coefficient set 6.499... yields a better approximation than either the initial or the first proposed set. If the x sum, e.g., 0.307,... 

One of the drawbacks of the first iteration, however, is that computation of energy quantities, e.g. orbital and total energies, requires to evaluate the integrals occurring in Eq. 3 on the basis of the ( )il )(p)- Unfortunately, the transcendental functions in terms of which the (]>il Hp) are expressed at the end of the first iteration do not lead to closed form expressions for these integrals and a numerical procedure is therefore needed. This constitutes a barrier to carry out further iterations to improve the orbitals by approaching the HE limit. A compromise has been proposed between a fully numerical scheme and the simple first iteration approach based on the fact that at the end of each iteration the < )j(k)(p) s entail the main qualitative characteristics of the exact solution and most... 

As previously described, Eq. 6 contains two constants characteristic of the system and the sample, feo and S, which can be determined by two chromatographic mns differing only in tc. These two values allow to calculate log fe using Eq. 4. However, because there is no empirical solution, values of log few and S have to be computed by iteration. Such procedures are included in several commercially available LC software packages, such as Drylab (Rheodyne, CA, USA), Chromsword (Merck, Darmstadt, Germany), ACD/LC simulator (Advanced Chemical Development, Toronto, Canada) or Osiris (Datalys, Grenoble, Erance). This approach was comprehensively described and successfully applied for accurate log P determination of several solutes with diverse chemical structures [9, 12, 43, 50]. 

In order to improve the convergence characteristics and robustness of the Gauss-Newton method, Levenberg in 1944 and later Marquardt (1963) proposed to modify the normal equations by adding a small positive number, y2, to the diagonal elements of A. Namely, at each iteration the increment in the parameter vector is obtained by solving the following equation... 

The solution of these dynamic nonlinear differential equations is considerably more complex than the previous systems considered. In particular, stable solution methods are based on physically realistic multiphase flow functions that have the following properties relative permeability functions are non-negative, monotoni-cally increasing with their respective saturation, and are zero at vanishing saturations, and capillary pressure is monotonically increasing with respect to the saturation of the non-wetting phase. It is necessary that any iterative scheme for estimating the multiphase flow functions retain these characteristics at each step. 

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