Bisection algorithm

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. 

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

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. 

N.Dyn and D.Levin Smooth Interpolation by bisection algorithms. pp335-337 in Approximation Theory 5 (eds Chui, Schumaker and Ward), 1986... 

J.-L.Merrien A family of Hermite interpolants by bisection algorithms. Numerical Algorithms 2, ppl87-200, 1992... 

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. 

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. 

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

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 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. 

A problem similar to the evaluation of J[p, p ] is encountered in simulations of systems of classical particles (point charges), where the particle-particle interaction also has a quadratic O(N ) scaling. For this problem, three algorithms with linear or near-linear scaling have been introduced recently the Fast Multipole Method (FMM), Tree Codes (TC), and the Recursive Bisection Method (RBM). The success of these three methods has prompted its application to the 7[p, p ] problem. ... 

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. 

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