The Bisection Algorithm

Given a continuous function fix), defined in the interval [a, h] with fia) and fib) being of opposite sign, then there exists a value p ia p b), for which fip) = 0. There may be more than one value of p. 

The method calls for a repeated halving of the subinterval [a, h] and at each stop, locating the half region containing p. 

To begin the process of repeated halving, we make a first iteration to find p (call this pf and let Pi be the midpoint of [a, b] that is. 

Some convergence tests could be used to stop the iteration. Given a tolerance e 0, we generate Pi, P2, , p until one of the following conditions is met 

Difficulties may arise using any of these stopping criteria, especially the criterion (A.la). If there is no knowledge beforehand regarding the function / or the approximate location of the exact root p, Eq. A.lh is the recommended stopping criterion. 

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The roots x of the nonlinear equation G x) = 0 can be obtained from a standard numerical protocol, such as the bisection algorithm. Hence... 

Stability of the system is examined at the minimum, maximum, and midpoints of the range of search of K. . That half of the interval in which the change from negative to positive (stable to unstable system) occurs is retained by the bisection algorithm. This new interval is bisected again and the evaluation of the system stability is repeated, until the convergence criterion, which is < 0.001, is met. 

Compare the usable data range from the bisection based algorithm with that from the graphical method. 

MATLAB has the function fzero which performs this bisection algorithm. The M-file bisec.m (Figure 2.11) uses fzero to calculate the root for this heat capacity example. 

This is an exaiq>le showing how to solve a nonlinear equation by y. calling the fzero function (bisection algorithm) in Matlab. 

In this section two iterative schemes for calculating the density given values for the pressure and temperature are described the bisection or interval-halving method, and the Newton-Raphson technique. These methods and others are described in more detail by Burden et al. (1978), who also give algorithms for these procedures. 

The system is unstable because of the positive real components of the root.s. At the midrange (K = 50) the system is still stable because all the real parts of the roots are negative. The bisection method continues its search in the range 50-100. In a total of 19 evaluations, the algorithm arrives at the critical value of in the range... 

Apart from the help a reference energy provides, we note again that, in particular, the bisection of the configuration space Equation 11.29 assists the construction of a CE such as Equation 11.46 the reduced configuration space helps because the CE must be valid for fewer configurations, and the reduced rank (2 instead of 4) helps because the genetic algorithm in our CE code UNCLE [88] can select the best clusters more easily. 

The false-position method is similar to the bisection method but improves on the iterative algorithm by making use of the magnitudes of the function at the upper and lower position values. The iterative algorithm is ... 

Giese and York (GY) [68] used the branch-free FMM algorithm of Watson et al. [69] and the recursive bisection ideas of Perez-Jorda and Yang (PJY) [70] to create an adaptive FMM for systems of particles composed of point multipoles, as opposed to the trivial case of point charges (monopoles). GY spent most of their effort in... 

The minimum value of the function Qi(r) can be determined numerically using a standard algorithm such as Brendt interval bisection. Assuming that the minimum occurs at r = r2, the value Q2 = Qi(r2) = Q(ci(r2), 2) coincides with the absolute minimum of Q(c, w, r). 

Compared to solveadiabxy. m for the adiabatic CSTR case in Section 3.1, the above MATLAB function solveNadiabxy. m depends on the two extra parameters Kc and yc that were defined following equation (3.9). It uses MATLAB s built-in root finder fzero.m. As explained in Section 3.1, such root-finding algorithms are not very reliable for finding multiple steady states near the borders of the multiplicity region. The reason - as pointed out earlier in Section 1.2 - is geometric the points of intersection of the linear and exponential parts of equations such as (3.16) are very shallow, and their values are very hard to pin down via either a Newton or a bisection method, especially near the bifurcation points. 

How close to the bifurcation limits does your bisection program succeed when the graphics solutions are used as starting points for fzero What are the sizes of the residues in the computed solutions near the bifurcation points Which of the proposed steady-state finders of part (a) or (b) do you prefer Be careful and monitor your hybrid algorithm s effort via clock and etime. 

A hybrid method, bisecting K-means, combines the divisive hierarchical and K-means methods to produce a controlled number of hierarchical document clusters. It has been shown to perform as good as or better than hierarchical methods while retaining the performance of the K-means approach [32]. The process of this method involves bisecting a selected cluster of documents (biggest or poorest quality) into two smaller clusters but optimizing the centroids to obtain new clusters with the best possible quality. An example of an implementation of this type of method is the Oracle Text hierarchical K-means algorithm. 

The idea of safeguard methods consists of combining a method which is convergent for an arbitrary initial interval Iq with a rapidly convergent method. Such a combination may consist of bisection and inverse interpolation. Fig. 6.2 shows the principles of the algorithm each step is discussed in the sequel. 

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