Iteration procedure

In vapor-liquid equilibria, it is relatively easy to start the iteration because assumption of ideal behavior (Raoult s law) provides a reasonable zeroth approximation. By contrast, there is no obvious corresponding method to start the iteration calculation for liquid-liquid equilibria. Further, when two liquid phases are present, we must calculate for each component activity coefficients in two phases since these are often strongly nonlinear functions of compositions, liquid-liquid equilibrium calculations are highly sensitive to small changes in composition. In vapor-liquid equilibria at modest pressures, this sensitivity is lower because vapor-phase fugacity coefficients are usually close to unity and only weak functions of composition. For liquid-liquid equilibria, it is therefore more difficult to construct a numerical iteration procedure that converges both rapidly and consistently. 

The special estimates used in BLIPS, which are essentially pure phases of the "solvent components" (98%, with 2% of the other solvent) are chosen to avoid these problems in this iterative procedure. 

Bubble-point and dew-point pressures are calculated using a first-order iteration procedure described by the footnote to Equation (7-25). 

We will see that superseding the functional fi(p ) in the form of Gibbs measure (4) ensures the linearity of equation (1), simplifies the iteration procedure, and naturally provides the support of any expected feature in the image. The price for this is, that the a priori information is introduced in more biased, but quite natural form. 

To search for the forms of potentials we are considering here simple mechanical models. Two of them, namely cluster support algorithm (CSA) and plane support algorithm (PSA), were described in details in [6]. Providing the experiments with simulated and experimental data, it was shown that the iteration procedure yields the sweeping of the structures which are not volumetric-like or surface-like, correspondingly. While the number of required projections for the reconstruction is reduced by 10 -100 times, the quality of reconstruction estimated quantitatively remained quite comparative (sometimes even with less artefacts) with that result obtained by classic Computer Tomography (CT). 

A new one-dimensional mierowave imaging approaeh based on suecessive reeonstruetion of dielectrie interfaees is described. The reconstruction is obtained using the complex reflection coefficient data collected over some standard waveguide band. The problem is considered in terms of the optical path length to ensure better convergence of the iterative procedure. Then, the reverse coordinate transformation to the final profile is applied. The method is valid for highly contrasted discontinuous profiles and shows low sensitivity to the practical measurement error. Some numerical examples are presented. 

Despite the complexity of these expressions, it is possible to hrvert transport coefficients to obtain infomiation about the mtemiolecular potential by an iterative procedure [111] that converges rapidly, provided that the initial guess for V(r) has the right well depth. 

The electron density, pj, of the embedded cluster/adsorbate atoms is calculated using quantum chemistry methods (HF, PT, multireference SCF, or Cl). The initial step in this iterative procedure sets to zero,... 

The con vergen ce of. SCK in teraction s is n o( always succcssfu I. In the simplest iteration procedure, iterations proceed without the aid of cither an external con vergen cc accelerator or an cxtrapola-tor. This often leads to slow convergence. 

When fitting two structures, the aim is to find the relative orientations of the two molecules in which the RMSD is a minimum. Many methods have been devised to perform this seemingly irmocuous calculation. Some algorithms, such as that described by Ferro and Hermans [Ferro and Hermans 1977] use an iterative procedure in which the one molecule is moved relative to the other, gradually reducing the RMSD. Other methods locate the best fit directly, such as Kabsch s algorithm [Kabsch 1978]. 

To avoid imposition of unrealistic exit boundary conditions in flow models Taylor et al. (1985) developed a method called traction boundary conditions . In this method starting from an initial guess, outflow condition is updated in an iterative procedure which ensures its consistency with the flow regime immediately upstream. This method is successfully applied to solve a number of turbulent flow problems. 

As mentioned in Chapter 2, the numerical solution of the systems of algebraic equations is based on the general categories of direct or iterative procedures. In the finite element modelling of polymer processing problems the most frequently used methods are the direet methods. 

A common iterative procedure is to solve the problem of interest by repeated calculations that do not initially give the correct answer but get closer to it as the calculation is repeated, perhaps many times. The approximate solution is said to converge on the correct solution. Although no human would be willing to repeat an iterative calculation thousands of times to converge on the right answer, the computer does, and, because of its speed, it often arrives at the answer in a reasonable amount of time. 

Having filled in all the elements of the F matr ix, we use an iterative diagonaliza-tion procedure to obtain the eigenvalues by the Jacobi method (Chapter 6) or its equivalent. Initially, the requisite electron densities are not known. They must be given arbitrary values at the start, usually taken from a Huckel calculation. Electron densities are improved as the iterations proceed. Note that the entire diagonalization is carried out many times in a typical problem, and that many iterative matrix multiplications are carried out in each diagonalization. Jensen (1999) refers to an iterative procedure that contains an iterative procedure within it as a macroiteration. The term is descriptive and we shall use it from time to time. 

The Q-equilibrate method is applicable to the widest range of chemical systems. It is based on atomic electronegativities only. An iterative procedure is used to adjust the charges until all charges are consistent with the electronegativities of the atoms. This is perhaps the most often used of these methods. 

The solution oftheRHForUHFequationsisan iterative procedure with two principal issues. First is the question of what to use for an initial guess and second, whether the solutions will converge quickly or at all. The initial guess affects the convergence also, as an exact guess would immediately converge. 

Fix the parameter c. To linearize the left-hand side of (2.166), for arbitrary x G we construct the following iterative procedure for... 

---