Point-successive

While the first point was addressed directly in Protocol 7.1, we will address the other points successively in the next paragraphs. 

With all of the point-successive methods, only one array in the computer is needed to store both o k) and 6 k+1) because the element o k) is never again needed once o k+1) is computed. This contrasts with the point-simultaneous methods, where, for each n, the elements of 6 k) must be available corresponding to the full range over which [s] m is finite. 

We have also learned that the point-successive methods need not demand more computer memory than point-simultaneous methods, and that the seemingly inherent asymmetry of the point-successive methods can be overcome. Furthermore, the linear methods described in this section are... 

For this work, the spectrometer function s(x) was determined by the method outlined in Section II.G.3 of Chapter 2. In digitizing the data, a sample density was chosen to accommodate about 70 samples taken across the full width at half maximum of s(x). A 25-point cubic polynomial smoothing filter was used in the deconvolution procedure to control high-frequency noise. Instead of the convolution in Eq. (13), the point-successive modification described in Section III.C.2 of Chapter 3 was employed. In Eq. (24) of Chapter 3, we replaced k with the expression... 

III.C.3) and the use of special sampling methods to gain the rapid-convergence advantage of point-successive iteration without inducing asymmetry (Section III.C.2). 

There are two different procedures by which the iteration may be carried out. One may substitute each improved coefficient, along with all the others, into the succeeding equation. This is known as the point successive procedure. Alternatively, one may determine all the unknown coefficients in all the equations using only the set of coefficients determined from the previous iteration. This is known as the point simultaneous procedure. Undesirable convergence properties of the point simultaneous procedure motivated its abandonment early in the research, however. The former procedure converges almost always when the tolerance is not chosen exceedingly small. The tolerance determines when the iteration is stopped. The iterations continue until the maximum change in any coefficient over its... 

There is an important disadvantage stemming from variable separation and use of the FFT, however. One is limited to the use of the point simultaneous procedure in the determination of the coefficients. That is, all of the coefficients must be computed in an iteration before they can be resubstituted back in the iterative equations. With the point successive method, the improved coefficient determined by one equation is substituted in the succeeding equation to render additional improvement. With the test data treated in this work, the point simultaneous procedure converged much more slowly than the point successive method. 

For a fixed gas rate Qg, if the liquid rate is increased sufficiently high, the criterion of Equation 62 will be violated and the bubble train will break at the smallest constriction along the train. As the liquid rate is increased higher and higher, the train will break at other points (successively higher Rk s) with the result that the number of continuous trains will increase, but their lengths will decrease. 

There are essentially six types of procedures to solve constrained nonlinear optimization problems. The three methods considered more successful are the successive LP, the successive quadratic programming, and the generalized reduced gradient method. These methods use different strategies but the same information to move from a starting point to the optimum, the first partial derivatives of the economic model, and constraints evaluated at the current point. Successive LP is used in a number of solvers including MINOS. Successive quadratic programming is the method of... 

If the elements present in the soUd have an incomplete strongly localized shell, the spectra show special features. This is the case, for example, for the rare earths, and we shall discuss this point successively for the emission and for the absorption processes. 

By applying Aitken s method to a linearly convergent sequence obtained from fixed point (successive substitution) iteration, we can accelerate the convergence to quadratie order. This procedure is known as the Steffenson s method, which leads to Steffenson s algorithm as follows. 

Davison ET (1985) Amitriptyhne-induced torsade de pointes. Successful therapy with atrial pacing. J Electrocardiol 18 299-301... 

For the analysis and interpretation of the experimental results, in the first instance the software should perform a pre-processing step of raw data, extracting the required number of complete cycles and, optionally, the sequence of evenly-spaced interpolated points. Successively, the software gives forth the relevant Fourier analysis. The adopted numerical procedure is based on the above-mentioned discrete Fourier transformation (DFT) algorithm (i.e., the summation over all points in Eq. (31)). As shown in the previous paragraph, the algorithm allows any number n and any particular values of periods NA / n (frequencies,/ ) to be selected, among all the possible discrete values, for the analysis of observed data. 

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