The Bisection Method: Which Root?

The Bisection Method: Which Root?

Coupled bisection for root ordering

We consider the problem of solving multiple ‘‘coupled’’ root-finding problems at once, in that we can evaluate every function at the same point simultaneously. Using a dynamic programming formulation, we show that a sequential bisection algorithm is a close-to-optimalmethod for finding a rankingwith respect to the zeros of these functions. We show the ranking can be found in linear time, prove ...

Complexity of the bisection method

The bisection method is the consecutive bisection of a triangle by the median of the longest side. In this paper we prove a subexponential asymptotic upper bound for the number of similarity classes of triangles generated on a mesh obtained by iterative bisection, which previously was known only to be finite. The relevant parameter is γ/σ, where γ is the biggest and σ is the smallest angle of t...

Generalized Probabilistic Bisection for Stochastic Root-Finding

We consider numerical schemes for root finding of noisy responses through generalizing the Probabilistic Bisection Algorithm (PBA) to the more practical context where the sampling distribution is unknown and location-dependent. As in standard PBA, we rely on a knowledge state for the approximate posterior of the root location. To implement the corresponding Bayesian updating, we also carry out ...

On Faster Convergence of the Bisection Method

Let AABC be a triangle with vertices A, B, and C. It is "bisected" as follows: choose a/the longest side (say AB) of AABC, let D be the midpoint of AB, then replace AABC by two triangles aA.DC and ADBC. Let Ag| be a given triangle. Bisect Aqj into two triangles A¡¡ and Aj2. Next bisect each Aj(-, /= 1, 2, forming four new triangles A2j-, i = 1,2, 3,4. Continue thus, forming an infinite sequence...