Binary Searches

A binary search is a robust and easily implemented method for finding a root of a single equation, F a) = 0. It is necessary to know bounds, amm < a < dmax, within which the root exists. If F a j ) and Fia af) differ in sign, there will be an odd number of roots within the bounds and a binary search wiU... 

Simple evaluation of a single CSTR using a binary search... 

Binary search to findWAin Wmin= 6250 lower bound, kg/hr Wmax =100000 upper bound FOR 1 = 1 TO 24... 

This example found the reactor throughput that would give the required annual capacity. For prescribed values of the design variables T and V, there is only one answer. The program uses a binary search to find that answer, but another root-finder could have been used instead. Newton s method (see Appendix 4) will save about a factor of 4 in computation time. 

The golden section search is the optimization analog of a binary search. It is used for functions of a single variable, F a). It is faster than a random search, but the difference in computing time will be trivial unless the objective function is extremely hard to evaluate. 

A more rigorous solution based on marching ahead in enthalpy according to Equation 7.42 is given in the next 16 lines of code. The temperature is found from the enthalpy using a binary search. The code is specific to the initial conditions of this problem. Results are very similar to those for marching temperature directly. 

If the mixture has gelled, the program proceeds to calculate P(Fa° ) and P(Fg° ) using a binary search method (lines 2510-2770). This method is more convenient that the earlier approach of Bauer and Budde (10) who used Newton s method, since derivatives of the functions are not required. The program also calculates the probability generating functions used to calculate sol fractions and the two crosslink densities (lines 2800-3150). Finally, the sol fraction and crosslink densities are calculated and printed out (lines 3160-3340). The program then asks for a new percents of reaction for the A and B groups. To quit enter a percent reaction for A of >100. 

Two successful automated procedures have been given. In each case one must first find the peak, and a binary search with doubled limits on each reversal is the fastest search method if the peak position is unknown. One is then on either the upper or the lower branch of either the substrate or the epilayer parabola (it is in fact a very shallow and distorted parabola). Fewster first showed that the branch conld be detected by driving down in nntil the intensity is about halved from that of the peak, then driving down in. If the intensity rises one is on the upper branch of the parabola, if it falls one is on the lower branch. Fewster then alternated steps in and steps in nntil the peak was fonnd. The latter is determined by the intensity falling instead of rising on beginning the step. 

The task is then to find the 7 that fits the requirements. This can easily be done using a binary search, and the function EE FAC is provided for this. 

Cheng and Holland, 1999). Sequential bifurcation is related to binary search (Wan et al., 2004) which searches a sorted array by repeatedly dividing the search interval in half, beginning with an interval covering the whole array. First, we give an outline of the sequential bifurcation procedure (Section 2.1). Second, we present the assumptions and notation of sequential bifurcation (Section 2.2), and, third, we illustrate the procedure through our case study (Section 2.3). 

We now describe a binary-search-like method that efficiently finds a pair of values PjTPj-i among Pq,.. ., Pm satisfying pj-i — Pj > Starting with the entire interval [1, m], the search is repeatedly narrowed down to an arbitrary interval [a, 5]. At each stage, the middle value is computed (approximately)... 

C BINARY SEARCH METHOD AND THE MINIMUM REFLUX RATIO... 

The tandem stream approach is also important in true flow-based titrations [17,21], The number and length of carrier, titrant and titrand plugs can be efficiently modified in real time according to a concentration-orientated feed-back mechanism relying on an extrapolative algorithm for the determination of the titration endpoint. This approach was called binary search. 

M. Korn, L.F.B.P. Gouveia, E. Oliveira, B.F. Reis, Binary search in flow titration employing photometric end-point detection, Anal. Chim. Acta 313 (1995) 177. 

More recently, flow titrations have been implemented in multi-commuted flow systems. With these systems, the exact amounts of the solutions involved can be modified in real time according to a concentration-oriented feedback mechanism. The flow titration really mimics a true titration because an analytical curve is not needed. A flow system exploiting a binary search to define the end point of an acid—base titration [324] is a good example of this approach. Stream directing solenoid valves were used to modify the sample and titrant volumes after every measurement via a feedback mechanism. Samples with concentrations within a range of two orders of magnitude could be titrated without modifying the flow manifold. 

Use the binary-search technique to find a zero of a function. 

This subroutine uses a binary search to close the material balance. 

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