Search variables

In principle, one could solve equations (la), (2a) and (3a) for K, a and. Unfortunately, and [n] are often highly correlated and it is recommended that only one of these data be used per standard. A practical procedure is to estimate using narrow MWD polystyrene standards leaving two unknowns, K and a. To illustrate the method, suppose Mjj and M data are available for the single broad MWD standard. Dividing equation (2a) by (la) eliminates a and one is left with a single-variable search for 3. 

Once 3 is known a direct calculation using either equation (la) or (2a) may be done. A similar single-variable search procedure may... 

A single-variable search for 3 results when equation (6) for i=l... 

A single-variable search for D2 is followed by a direct calculation of Dx using equation (lOc) for at least two broad MWD standards. A direct calculation using equations (lOa) or (lOb) pro-... 

If all of the variables appeared in all of the equations, then the use of optimum seeking methods for the direction of the calculations would be impractical because the search dimensionality would become excessive. However, the opposite is true for these applications. The system of equations is sparse only a few variables are present in each equation. This requires that only a few variables need to be search variables with the rest being state variables. Search variables must be chosen carefully. Generally, the most constrained variables should be chosen as search variables, and the least constrained variables chosen as state variables. The opposite choice will often drive the highly constrained variables into the infeasible region causing computational difficulties. Also for the applications illustrated in this paper, minor equilibrium species should not be chosen as search variables. 

Loy (11) has published a procedure based upon the method of Balke and Hamielec which uses a more efficient Iterative, single-variable search algorithm which relies upon the fact that the dispersity (M /it) is a function of C2 only. The computer program Incorporating this much faster algorithm converges to the optimum C2 within 36 iterations. 

Malawer and Montana (14) have developed an efficient iterative, sequential single variable search algorithm which relies upon a polydisperse standard with known Mn and My values. A direct graphical proof of the algorithm was presented. 

Equations (12)-(15) provide fgur independent equations relating the parameters r)> a/R and s- Thus a unique solution is possible. However, m may be difficult to measure for systems with relativefy fast adsorption. In that case, the problem reduces to a single variable search to best fit the uptake curve using equations (12)-(14). 

Laplace transform variable Search direction Step response coefficient Time... 

There is one unknown (D2) in Equation (7.51), which can be determined by a single-variable search technique. Once D2 is known, the intercept of the calibration curve, D, is determined by ... 

The critical value of SD is that for which the second factor becomes exactly zero for n N. It can be readily calculated by a one-variable search, for any assumed values of the parameters. 

If xi and X2 are varied one at a time, then the method is known as a univariate search and is the same as carrying out successive line searches. If the step length is determined so as to find the minimum with respect to the variable searched, then the calculation steps toward the optimum, as shown in Figure 1.15a. This method is simple to implement, but can be very slow to converge. Other direct methods include pattern searches such as the factorial designs used in statistical design of experiments (see, for example, Montgomery, 2001), the EVOP method (Box, 1957) and the sequential simplex method (Spendley et ah, 1962). 

Knowing the values of the first partial derivatives of the economic model and constraint equations at a feasible point, the reduced gradient can be computed by Eq. (20). Then the reduced gradient line is used to locate the optimum by a single variable search on a, the parameter of the reduced gradient line in Eq. (21). 

Optimization using the alternating variable search method... 

If the product cannot be disassembled and reassembled, the technique to use is paired comparisons. The concept is to select pairs of good and bad units and compare them, using whatever visual, mechanical, electrical, chemical, etc., comparisons are possible, recording whatever differences are noticed. Do this for several pairs, continuing until a pattern of differences becomes evident. In many cases, a half-dozen paired comparisons is enough to detect repeatable differences. The units chosen for this test should be selected at random to establish statistical confidence in the results. If the number of differences detected is more than four, then use of variables search is indicated. For four or fewer, a full factorial analysis can be done. 

Nondimensionalization of the equations, which is usually desirable, can lead in this case to a computational complication. Many authors have used the length L to nondimensionalize axial position so that the dimensionless length varies from 0 to 1. In that case L enters the problem formulation explicitly, and both the length and initial stress must be adjusted on each iteration, requiring a two-variable search. Convergence of the iterations is still rapid, in part because Equation 7.41 gives an excellent initial approximation for L, but clearly such a formulation should be avoided. 

As discussed above, differences between spectra of cultured tumour cell lines may be subtle and difficult to discriminate from normal biological variability. Searching for diagnostic features in this relatively simple system may be compared to searching for the proverbial needle in a haystack, and multivariate pattern recognition methods are required. With this in mind, the task of identifying spectroscopic differences between sections of intact tissue becomes a daunting one. Not only must one find the subtle... 

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