Iteration/iterative algorithms

For the iteration algorithm (5) the optimal estimations (6) are directly used by a second back loop to block B (long dashed line in Fig. 1). 

Equation (2.106) gives rise to an implicit scheme except for 0 = 0. The application of implicit schemes for transient problems yields a set of simultaneous equations for the field unknown at the new time level n + 1. As can be seen from Equation (2.111) some of the terms in the coefficient matrix should also be evaluated at the new time level. Therefore application of the described scheme requires the use of iterative algorithms. Various techniques for enhancing the speed of convergence in these algorithms can be found in the literature (Pittman, 1989). 

The described scheme can also be incorporated into iterative algorithms and used to solve steady-state flow problems (Zienkiewicz and Wu, 1991). 

Statistical and algebraic methods, too, can be classed as either rugged or not they are rugged when algorithms are chosen that on repetition of the experiment do not get derailed by the random analytical error inherent in every measurement,i° 433 is, when similar coefficients are found for the mathematical model, and equivalent conclusions are drawn. Obviously, the choice of the fitted model plays a pivotal role. If a model is to be fitted by means of an iterative algorithm, the initial guess for the coefficients should not be too critical. In a simple calculation a combination of numbers and truncation errors might lead to a division by zero and crash the computer. If the data evaluation scheme is such that errors of this type could occur, the validation plan must make provisions to test this aspect. 

Although the overall cost of the conjugate gradient algorithm may be higher than that of some of the iterative algorithms described in Section III.B, the algorithm allows us easily to restrict the spectral and temporal stmcture of optimal pulses and enables us to incorporate the exact form of the laser-molecule interactions. 

This follows from the orthogonality of the eigenvectors v, and V2. We have preferred the residual matrix because this approach is used in iterative algorithms for the calculation of eigenvectors, as is explained in Section 31.4. 

In real practice, the location m and the variance have to be estimated from real data. An iterative algorithm, similar to the one used in Chapter 10 for the robust covariance estimation, is used to calculate the trust function. The main advantage of using this algorithm is that the convergence is warranted. 

As expected in an iterative algorithm, we start from an initial guess for the parameters. This parameter vector is subsequently improved by the addition of an appropriate parameter shift vector 8p, resulting in a better, but probably still not perfect, fit. From this new parameter vector the process is repeated until the optimum is reached. 

The Singular Value Decomposition, SVD, has superseded earlier algorithms that perform Factor Analysis, e.g. the NIPALS or vector iteration algorithms. SVD is one of the most stable, robust and powerful algorithms existing in the world of numerical computing. It is clearly the only algorithm that should be used for any calculation in the realm of Factor Analysis. 

It is possible to use an iterative algorithm to determine the exact positions of the minima. Again, in such a program the rate constants can be fitted individually, irrespective of the others. 

As we have seen with the previous iterative refinement and ITTFA, convergence generally is very sluggish. Even with moderately complex systems, it is often too slow to be useful. There are alternative, non-iterative methods that compare favourably with the above iterative algorithms. 

This, however, means that both the y-data and the scores have to be multiplied by the appropriate weights /wi and then the classical OLS-based procedure can be applied. Practically, starting values for the weights have to be determined, and they are updated using an iterative algorithm. 

Equations (4.15) and (4.16) give the iteration algorithm for reguessing the two new values each time through the loop. Four partial derivatives must be ealeulated, either analytically or numerically, at each iteration step. 

The iterative algorithm is (Wang and Henke, Hydro. Proc., Vol. 45, 1966, page 155)... 

Here we synthesize the concepts of the last four sections, (i) CSE, (ii) reconstruction, (iii) purification, and (iv) a contracted power method, to obtain an iterative algorithm for the direct calculation of the 2-RDM. 

The rest of the exchange and correlation effects will be taken into account to the first two orders of PT by the total interelectron interaction [13-19], The electron density is determined by an iteration algorithm [11, 14], In the first iteration we... 

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