Iterative computations

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. 

We have the makings of an iterative computer method. Start by assuming values for the matr ix elements and calculate electron densities (charge densities and bond orders). Modify the matr ix elements according to the results of the electron density calculations, rediagonalize using the new matrix elements to get new densities, and so on. When the results of one iteration are not different from those of the last by more than some specified small amount, the results are self-consistent. 

Conjugate Gradient methods compute the conjugate directions hj by iterative computation involving the gradient gj without... 

By employing successive substitution of the tear variables and the criterion of Eq. (13-83), convergence was achieved slowly, but without oscillation, in 23 iterations. Computed products are. 

The holistic thermodynamic approach based on material (charge, concentration and electron) balances is a firm and valuable tool for a choice of the best a priori conditions of chemical analyses performed in electrolytic systems. Such an approach has been already presented in a series of papers issued in recent years, see [1-4] and references cited therein. In this communication, the approach will be exemplified with electrolytic systems, with special emphasis put on the complex systems where all particular types (acid-base, redox, complexation and precipitation) of chemical equilibria occur in parallel and/or sequentially. All attainable physicochemical knowledge can be involved in calculations and none simplifying assumptions are needed. All analytical prescriptions can be followed. The approach enables all possible (from thermodynamic viewpoint) reactions to be included and all effects resulting from activation barrier(s) and incomplete set of equilibrium data presumed can be tested. The problems involved are presented on some examples of analytical systems considered lately, concerning potentiometric titrations in complex titrand + titrant systems. All calculations were done with use of iterative computer programs MATLAB and DELPHI. 

To find the best a priori conditions of analysis, the equilibrium analysis, based on material balances and all physicochemical knowledge involved with an electrolytic system, has been done with use of iterative computer programs. The effects resulting from (a) a buffer chosen, (b) its concentration and (c) complexing properties, (d) pH value established were considered in simulated and experimental titrations. Further effects tested were tolerances in (e) volumes of titrants added in aliquots, (f) pre-assumed pH values on precision and accuracy of concentration measured from intersection of two segments obtained in such titrations. 

T vo main streams of computational techniques branch out fiom this point. These are referred to as ab initio and semiempirical calculations. In both ab initio and semiempirical treatments, mathematical formulations of the wave functions which describe hydrogen-like orbitals are used. Examples of wave functions that are commonly used are Slater-type orbitals (abbreviated STO) and Gaussian-type orbitals (GTO). There are additional variations which are designated by additions to the abbreviations. Both ab initio and semiempirical calculations treat the linear combination of orbitals by iterative computations that establish a self-consistent electrical field (SCF) and minimize the energy of the system. The minimum-energy combination is taken to describe the molecule. 

Employing an iterative computer program, in conjunction with the above equations, Katz et al. examined a wide range of values for (k) and (Vmw) nd calculated the volume change on mixing of a series of methanol/water mixtures having assumed volume fractions of methanol. The results for each selected values of (k) and (Vmw) and each volume fraction of methanol were compared with experimentally determined values of (vj) and the specific values of (k) and (Vmw) that gave the minimum error... 

Newton Iteration Compute Xi by Newton iterations on the reduced KKT system. 

Linear models with respect to the parameters represent the simplest case of parameter estimation from a computational point of view because there is no need for iterative computations. Unfortunately, the majority of process models encountered in chemical engineering practice are nonlinear. Linear regression has received considerable attention due to its significance as a tool in a variety of disciplines. Hence, there is a plethora of books on the subject (e.g., Draper and Smith, 1998 Freund and Minton, 1979 Hocking, 1996 Montgomery and Peck, 1992 Seber, 1977). The majority of these books has been written by statisticians. 

Prior work on the use of critical point data to estimate binary interaction parameters employed the minimization of a summation of squared differences between experimental and calculated critical temperature and/or pressure (Equation 14.39). During that minimization the EoS uses the current parameter estimates in order to compute the critical pressure and/or the critical temperature. However, the initial estimates are often away from the optimum and as a consequence, such iterative computations are difficult to converge and the overall computational requirements are significant. 

Furthermore, in order to avoid any iterative computations for each critical point, we use the experimental measurements of the state variables instead of their unknown true values. In the above equations r, and r2 are residuals which can be easily calculated for any equation of state. 

The corresponding liquid-phase chemistry can be used to promote ion formation by appropriate choice of solvent and pH, salt addition to form M.Na+ or M.NH4+, and postcolumn addition of reagents. The primary applications of ESI-MS are in the biopolymer field. The phenomenon of routine multiple charging is exclusive to electrospray, which makes it a very valuable technique in the fine chemical and biochemical field, because mass spectrometers can analyse high-molecular-mass samples without any need to extend their mass range, and without any loss of sensitivity. However, with ESI, molecules are not always produced with a distribution of charge states [137], Nevertheless, this phenomenon somehow complicates the determination of the true mass of the unknown. With conventional low-resolution mass spectrometers, the true mass of the macromolecule is determined by an indirect and iterative computational method. 

We are now in a position that allows the easy, non-iterative computation of the species distribution curves of any acid. The program EDTA.m performs the computations for edta, a 6-protic acid ... 

The values of A and B are determined by an iterative computer technique using equations (23) and (27). As a first step, an arbitrary equation of state is used to generate 3 V/3T and to calculate Cp, v and The equation of state for water (31)... 

Relaxation methods involve iteratively seeking a convergent solution to the Laplace equation. In the present case, for instance, if we rewrite the coefficient matrix A = I + E, where the latter matrix consists of elements that are all small compared to 1, the matrix Laplace equation takes the form = EU + b. One begins the calculation with values U = b [or, equivalently, U = 0] and iteratively computes successive values The calculation terminates when a specified limit of accuracy is achieved. One such measure involves calculating the proportional differences ... 

An appreciable progress in the fundamentals for characterization of branched polymers has been achieved to date by different investigators93-95. Therefore, at present, the physico-chemical behavior of branched polymers in solution as well as in bulk can be elucidated to a great extent on the molecular basis. Kurata et al. proposed a useful, iterative computer method for estimating the degree of branching... 

The numerical solutions of QCE equations are obtained by an iterative computer algorithm. Given the ab initio quantities to construct the qt (initially for a guessed set of cluster populations) and the specific pressure P0 and temperature T0 of a desired state, the program finds the numerical volumes (and associated cluster population distributions) that satisfy (13.68) for the desired Pq. This matching condition may be written in the form of a polynomial equation... 

Until the advent of computers, multicomponent distillation problems were solved manually by making tray-by-tray calculations of heat and material balances and vapor-liquid equilibria. Even a partially complete solution of such a problem required a week or more of steady work with a mechanical desk calculator. The alternatives were approximate methods such as those mentioned in Sections 13.7 and 13.8 and pseudobinary analysis. Approximate methods still are used to provide feed data to iterative computer procedures or to provide results for exploratory studies. 

The hydroxy a-amino acids l-serine and l-threonine, used as models for the 2-amino-2-deoxy glyconic acids, have been complexed with Ni(II) at 37 °C in aqueous solutions of 0.15M potassium nitrate. Values for the stability constants were obtained from iso-pH titration data which were collected by alternate, small, incremental additions of metal ion and potassium hydroxide being made such that the pH of the solution remained nearly constant. The data were consistent with the predominance of MLn species, along with additional protonated and hydrolyzed complexes. There was no evidence for the involvement of the hydroxyl group in chelation. By the same iterative computations the complexes formed between borate and mannitol have been analyzed, and the stability constants have been calculated. Complexes with mannitohborate stoichiometries of I.T, 1 2, 1 3, and 2 1 were proposed. 

This equation is appropriate for iterative computer calculation. Simplifications have been proposed by Thodos.s... 

A drawback of the gamut-constraint method is that it may fail to find an estimate of the illuminant. This may happen if the resulting intersected convex hull A4n is the empty set. Therefore, care must be taken not to produce an empty intersection. There are several ways to address this problem. One possibility would be to iteratively compute the intersection by considering all of the vertices of the observed gamut in turn. If, as a result of the intersection, the intersected hull should become empty, the vertex is discarded and we continue with the last nonempty hull. Another possibility would be to increase the size of the two convex hulls that are about to be intersected. If the intersection should become empty, the size of both hulls is increased such that the intersection is nonempty. A simple implementation would be to scale each of the two convex hulls by a certain amount. If the intersection is still empty, we again increase the size of both hulls by a small amount. 

The least-squares methods may generally be applied using one of two possible approaches. One of them requires a computation of the function gradients whilst the other does not. Gradient methods use iterative computation of corrections to consecutive approximate solutions g[Pg.265]

Fig. 15.7 Iterative computational scheme used by Mewes et al. [Reprinted by permission from D. Mewes, S. Luther, and K. Riest, Simultaneous Calculation of Roll Deformation and Polymer Flow in the Calendering Process, Int. Polym. Process., 17, 339-346 (2002).]...

The modeling of electrochemical processes has evolved over the past 50 years to the point where complex problems involving multiple reactions, temperature variations, and physical property variations can be treated. Essentially all contemporary models require iterative computer techniques to simulate system behavior. 

Computer programs for simulation of proton NMR spectra are available. If accurate measurements of chemical shifts and coupling constants for all of the protons can be obtained, the simulated spectrum will be congruent with the actual spectrum. In many cases, at least some of the spin systems will be first order. If not, reasonable estimates of shifts and coupling constants may be made, and the iterative computer program will adjust the values until the simulation matches the actual spectrum—assuming, of course, that the identification is valid. 

The statistical properties of PLS2 are still not well understood and may not even be optimal for many calibration problems. The solution produced by PLS2 is dependent on how its iterative computations are initialized. A usual practice is to initialize, PLS2 with the column from Y with the greatest correlation to X. Initialization with other columns of Y produces different results. 

In the 1960s, the start of application of computers to the practice of marine research gave a pulse to the development of numerical diagnostic hydrodynamic models [33]. In them, the SLE (or the integral stream function) field is calculated from the three-dimensional density field in the equation of potential vorticity balance over the entire water column from the surface to the bottom. The iterative computational procedure is repeated until a stationary condition of the SLE (or the integral stream function) is reached at the specified fixed density field. Then, from equations of momentum balance, horizontal components of the current vector are obtained, while the continuity equation provides the calculations of the vertical component. The advantage of this approach is related to the absence of the problem of the choice of the zero surface and to the account for the coupled effect of the baroclinicity of... 

This procedure was necessary because unique solutions were not obtained when 25 values of cx and c2 as well as 17 values of dx and d2 were derived by an iterative computer-procedure. Swain s own doubts about the results of this treatment have been emphasised by Streitwieser (1956a), who pointed out that values of do not reflect sensitivity to solvent nucleophilicity (Swain used cx/c2 to characterize reactivity). 

The flash calculation illustrated by Example 10.3 for a system obeying Raoult s law was solved by a very simple trial procedure. This was possible becau K-values (K( = yjxi) could be calculated from knowledge of T and P alone When the K-values depend not only on T and P but also on the phase compos1 tions, their calculation is inherently more difficult. Moreover, since the pha compositions are not initially known, they are most conveniently found by iterative computation scheme. 

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