Method simple iteration

There are various ways to obtain the solutions to this problem. The most straightforward method is to solve the full problem by first computing the Lagrange multipliers from the time-differentiated constraint equations and then using the values obtained to solve the equations of motion [7,8,37]. This method, however, is not computationally cheap because it requires a matrix inversion at every iteration. In practice, therefore, the problem is solved by a simple iterative scheme to satisfy the constraints. This scheme is called SHAKE [6,14] (see Section V.B). Note that the computational advantage has to be balanced against the additional work required to solve the constraint equations. This approach allows a modest increase in speed by a factor of 2 or 3 if all bonds are constrained. 

The simple iteration scheme. By formally setting n = 1 in formula (29) the preceding is referred to as the simple iteration method... 

By making n iterations of the simple iteration method we find that... 

Recall that it is fairly common to write the iteration number k over the sought function y within the frameworks of iterative methods available for difference equations. The same procedure works in the simple iteration scheme (SIS) which has been designed for problem (37) ... 

Tj, 2/(7i -b 72), thereby justifying estimate (17) and the convergence of the minimal residual method with the same rate as occurred before for the simple iteration method with the exact values 71 and... 

During the course of MRM the same procedures (13 ) and (14) are workable with increased volume of calculations in connection with formula (14) for as compared with the simple iteration method. 

No doubt, several conclusions can be drawn from such reasoning. First, the method being employed above converges in the space Ha with the same rate as the simple iteration method although it occurs in one of the subordinate norms. Second, the minimal residual method converges in the space Ha, that is, in a more stronger norm. 

The method involves a simple iteration on only one variable, pH. Simple interval-halving convergence (see Chap. 4) can be used very effectively. The titration curves can be easily converted into simple functions to include in the computer program. For example, straight-line sections can be used to interpolate between data points. 

Because the function given by Eq. (4) neither obliterates nor strongly suppresses the high Fourier frequencies in the data, we would expect a linear method to perform relatively well. A simple iterative approach based on the direct method of Section I of Chapter 3 does, in fact, prove effective. 

We have developed a simple, iterative synthetic method for the preparation of hydroxypropyl derivatives of phenolic and aliphatic alcohols which allows complete definition and control of the degree of chain extension in the products. This methodology has been applied to the preparation of a series of lignin model compounds having hydroxypropyl chain extension degrees of 1-... 

In what follows problem (37) will be treated as a model one in the further comparison of various methods in a step-by-step fashion in line with established priorities and answering real needs. We concentrate primarily on the total number of the iterations required in the simple iteration method (34)-(34/) and the method with optimal set of Chebyshev s parameters (14), (29). 

There are iterative methods (e.g., Jacobi, Gauss-Seidel, Newton) whose purpose is simply to provide solutions for the steady-state equations, others (e.g., Euler and its improved versions) aim to give trajectories. Cycling will be felt as a disagreeable iteration artifact in the first case, as an indication of a probably cyclic trajectory in the second case. The relation between the behavior in a simple iteration method (e.g., Jacobi) and the real trajectory is interesting, if not simple. Consider, for instance, a simple negative loop comprising three inhibitory elements ... 

In an admirably compact note in 1965 Nesbet used a very simple idea to generate a powerful method for the calculation of the individual eigenvalues and eigenvectors of very large matrices, which is particularly well-behaved if the lowest root is required. The method is iterative and uses the following technique ... 

V. 17. Consider a rigid package restrained by four symmetrically disposed tension tiedowns. A requirement of the simplified method is to predict upper bound values of tiedown force and hence, by reaction, forces on the package attachment and the conveyance. This method is apphcable only to statically determinate systems, and simple iterative assumptions are made on the system behaviour to derive upper bound forces. 

The simplest organization of the iteration process is simple iteration, when the found composition xd or xb from the previous iteration is used to determine composition xb or x for the following iteration at the direct or indirect separation, respectively. Besides simple iteration, one can also use other more complicated but more reliable and faster methods. 

Eor multivariate calibration in analytical chemistry, the partial least squares (PLS) method [19], is very efficient. Here, the relations between a set of predictors and a set (not just one) of response variables are modeled. In multicomponent calibration the known concentrations of / components in n calibration samples are collected to constitute the response matrix Y (n rows, / columns). Digitization of the spectra of calibration samples using p wavelengths yields the predictor matrix X (n rows, p columns). The relations between X and Y are modeled by latent variables for both data sets. These latent variables (PLS components) are constructed to exhaust maximal variance (information) within both data sets on the one hand and to be maximally correlated for the purpose of good prediction on the other hand. From the computational viewpoint, solutions are obtained by a simple iterative procedure. Having established the model for calibration samples. comp>o-nent concentrations for future mixtures can be predicted from their spectra. A survey of multi-component regression is contained in [20],... 

In cases where we can predict, by intuition or on the grounds of experimental data, which constituents will prevail in the state of equilibrium, we may apply the simple iteration method with advantage The following usually is a sufficient condition... 

Bose-Einstein condensates. Following the exciting progress with cold atoms the experimental challenge became to prepare ultracold molecules. Unfortunately the laser-cooling techniques applied to atoms are not directly transferable to molecules, since these methods rely upon cyclic absorption and spontaneous emission within an almost pure two-level system. The multilevel structure of the electronic states of molecules and the manifold of allowed transitions does not allow for simple iterative excitation with one single laser line. 

Note that the exit concentration of reactant A is measured at the end of the tubular reactor where B enters. Generally, only the entrance concentrations are known. For countercurrent processes, one may use a simple iterative method, assuming a certain exit concentration of A,... 

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