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. 

These considerations throw further light on the simple iteration for finding roots of a matrix A. For... 

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) ... 

It seems clear that in solving the system (72) the number of the iterations within the framework of the explicit scheme with optimal set of Chebyshev s parameters or of the simple iteration scheme is proportional to 1/x/ or l/ 7, thus causing an enormous growth as 0. 

The two-layer simple iteration scheme permits us to find the first iteration... 

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. 

PLS has been introduced in the chemometrics literature as an algorithm with the claim that it finds simultaneously important and related components of X and of Y. Hence the alternative explanation of the acronym PLS Projection to Latent Structure. The PLS factors can loosely be seen as modified principal components. The deviation from the PCA factors is needed to improve the correlation at the cost of some decrease in the variance of the factors. The PLS algorithm effectively mixes two PCA computations, one for X and one for Y, using the NIPALS algorithm. It is assumed that X and Y have been column-centred as usual. The basic NIPALS algorithm can best be demonstrated as an easy way to calculate the singular vectors of a matrix, viz. via the simple iterative sequence (see Section 31.4.1) ... 

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. 

A system of linear equations as in Eq. (1) and (2) is employed. Rather than the value B of the bonding indicators in each actinide metal, AB variations are calculated with respect to the configuration of a reference state. The reference state configuration is inspired by the Engel-Brewer correlations, amply used for transition metals and alloys It is seen that the system of equations contains one equation less than the number of unknowns, so that only a range of the Ah solutions can he determined. However, this range can be shown, by a simple iterative procedure, to be limited. 

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 order to estimate the transcendental number e, we will expand the exponential function ex in a power series using a simple iterative procedure starting from its definition Eq. (25) together with Eq. (12). As a prelude, we first find the power series expansion of the geometric series y — 1/(1 + x), iterating the equivalent expression ... 

The fiber is suspended in the liquid, which means that due to small time scales given by the pure viscous nature of the flow, the hydrodynamic force and torque on the particle are approximately zero [26,51]. Numerically, this means that the velocity and traction fields on the particle are unknown, which differs from the previous examples where the velocity field was fixed and the integral equations were reduced to a system of linear equations in which velocities or tractions were unknown, depending on the boundary conditions of the problem. Although computationally expensive, direct integral formulations are an effective way to find the velocity and traction fields for suspended particles using a simple iterative procedure. Here, the initial tractions are assumed and then corrected, until the hydrodynamic force and torque are zero. 

This methodology has been used for the preparation of L-threitol (4) (Scheme 9.9) and erythritol derivatives.93 This simple iterative process has been used for the simple alditols,93 94 deoxyalditols,95 and aldoses93 and for all of the L-hexoses.96,97... 

Eqs. (11.42) and (11.43) are very convenient for design calculations when the mass flow rate of condensate is specified and the required temperature difference is to be determined. However, when the condensation rate is not specified, the solution of Eq. (11.42) requires an iterative procedure since the Reynolds number cannot be calculated a priori. A simple iterative approach is described in Example 11.3. In the laminar regime, if the condensation rate is not known and the temperature difference is specified, iteration can be avoided by using Eq. (11.21) instead of Eq. (11.43). 

For example, when solving for yk and P, we do not have values necessary for calculation of the d>k, and when solving for xk and T, we can evaluate neither the P 1 nor the yk. Simple iterative procedures, described in the following paragraphs, allow efficient solution of each of the four types of problem. 

Once the first amino acid is fixed to the column, reagents are added simply by passing solutions down the column. Any excess or by-products are washed off. Finally, the product is released bypassing a solution of CF3CO2H down the column. The simplicity and reliability of this type of simple iterative process, with two steps per cycle, has made automated peptide synthesis common laboratory practice. 

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