Processes iterative

Equations (7-8) and (7-9) are then used to calculate the compositions, which are normalized and used in the thermodynamic subroutines to find new equilibrium ratios,. These values are then used in the next Newton-Raphson iteration. The iterative process continues until the magnitude of the objective function 1g is less than a convergence criterion, e. If initial estimates of x, y, and a are not provided externally (for instance from previous calculations of the same separation under slightly different conditions), they are taken to be... 

No a priori information about the unknown profile is used in this algorithm, and the initial profile to start the iterative process is chosen as (z) = 1. Moreover, the solution of the forward problem at each iteration can be obtained with the use of the scattering matrices concept [8] instead of a numerical solution of the Riccati equation (4). This allows to perform reconstruction in a few seconds of a microcomputer time. The whole algorithm can be summarized as follows ... 

These quartic equations are solved in an iterative maimer and, as such, are susceptible to convergence difficulties. In any such iterative process, it is important to start with an approximation reasonably close to the final result. In CC theory, this is often achieved by neglecting all of tlie temis tliat are nonlinear in the t amplitudes (because the ts are assumed to be less than unity in magnitude) and ignoring factors that couple different doubly-excited CSFs (i.e. the sum over i, f, m and n ). This gives t amplitudes that are equal to the... 

A general work flow scheme - an iterative process - for the generation of conformations is shown in Figure 2-103. 

Let us define knowledge as the perception of the logical relations among the structures of the information. One thing we have to bear in mind is that any systematic treatment of information needs some previous knowledge. Therefore, research, in the long run, is always an iterative process, as depicted on Figure 4-1. 

In this particular case, the calculations are completely symmetrical up to Eqs. (8-30). Evei ything we have said for a we can also say for p. At self-consistency, a = p so we can substitute a for p at any point in the iterative process, knowing that as we approach self-consistency for one, we approach the same self-consistent value for the other. 

Understanding how the force field was originally parameterized will aid in knowing how to create new parameters consistent with that force field. The original parameterization of a force field is, in essence, a massive curve fit of many parameters from different compounds in order to obtain the lowest standard deviation between computed and experimental results for the entire set of molecules. In some simple cases, this is done by using the average of the values from the experimental results. More often, this is a very complex iterative process. 

The designer usually wants to specify stream flow rates or parameters in the process, but these may not be directly accessible. For example, the desired separation may be known for a distiUation tower, but the simulation program requires the specification of the number of trays. It is left up to the designer to choose the number of trays that lead to the desired separation. In the example of the purge stream/ reactor impurity, a controller module may be used to adjust the purge rate to achieve the desired reactor impurity. This further complicates the iteration process. 

Recalciilate the , and continue this iterative procedure until it converges to a fixed value for X, yi- This sum is appropriate to the pressure P for which the calculations have been made. Unless the sum is unity, the pressure is adjusted and the iteration process is repeated. Systematic adjustment of pressure P continues until X, yi = 1- The pressure and vapor compositions so found are the equilibrium values for the given temperature and hquid-phase composition as predicted by the equation of state. 

The choice M = 0.250 is to prevent negative coefficients and thus an unstable iterative process. 

The analysis of a risk—that is, its estimation—leads to the assessment of that risk and the decision-making processes of selecting the appropriate level of risk reduction. In most studies this is an iterative process of risk analysis and risk assessment until the risk is reduced to some specified level. The subjec t of acceptable or tolerable levels of risk that coiild be applied to decision making on risks is a complex subject which will not oe addressed in this section. 

The goal of measurement selection is to identify a set of measurements that, when interpreted, will lead to unique values for the model parameters, insensitive to uncertainties in the measurements. This is an iterative process where ... 

Structure calculation algorithms in general assume that the experimental list of restraints is completely free of errors. This is usually true only in the final stages of a structure calculation, when all errors (e.g., in the assignment of chemical shifts or NOEs) have been identified, often in a laborious iterative process. Many effects can produce inconsistent or incorrect restraints, e.g., artifact peaks, imprecise peak positions, and insufficient error bounds to correct for spin diffusion. 

In his early survey of computer experiments in materials science , Beeler (1970), in the book chapter already cited, divides such experiments into four categories. One is the Monte Carlo approach. The second is the dynamic approach (today usually named molecular dynamics), in which a finite system of N particles (usually atoms) is treated by setting up 3A equations of motion which are coupled through an assumed two-body potential, and the set of 3A differential equations is then solved numerically on a computer to give the space trajectories and velocities of all particles as function of successive time steps. The third is what Beeler called the variational approach, used to establish equilibrium configurations of atoms in (for instance) a crystal dislocation and also to establish what happens to the atoms when the defect moves each atom is moved in turn, one at a time, in a self-consistent iterative process, until the total energy of the system is minimised. The fourth category of computer experiment is what Beeler called a pattern development... 

UOTEN is adjusted downward in speed and UlTEN is adjusted upward in speed in an iterative process until the minimum wind speed, UC, that will entrain the plume into a building s cavity is found. The critical wind speed is then adjusted to the anemometer height, using the reverse of the power law above, as follows ... 

The techniques of design verification identified in clause 4.4.7 can be used to verify that the design output meets the design input requirements. However, design verification is often an iterative process. As features are determined, their compliance with the require-... 

These messages indicate that the SCF calculation, which is an iterative process, failed to converge. The predicted energy should accordingly be ignored. ... 

The Isodensity PCM (IPCM) model defines the cavity as an isodensity surface of the molecule. This isodensity is determined by an iterative process in which an SCF cycle is performed and converged using the current isodensity cavity. The resultant wavefunction is then used to compute an updated isodensity surface, and the cycle is repeated until the cavity shape no longer changes upon completion of the SCF. 

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

It may happen that many steps are needed before this iteration process converges, and the repeated numerical solution of Eqs. III.21 and III.18 becomes then a very tedious affair. In such a case, it is usually better to try to plot the approximate eigenvalue E(rj) as a function of the scale factor rj, particularly since one can use the value of the derivative BE/Brj, too. The linear system (Eq. III. 19) may be written in matrix form HC = EC and from this and the normalization condition Ct C = 1 follows... 

In order to solve Eq. III.49, one can try to use the formula E k+D = f E k), which leads to a first-order iteration procedure. Starting from a trial value Z (0), one obtains a series E 1), E 2), E 3),. . . which may be convergent or divergent. In both cases, one can go over to a second-order iteration process, which is most easily derived by solving the equation F(E) — 0 by means of Newton-Raphson s formula... 

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