Abstract
The paper summarizes author’s investigations in tuning a multithreaded interval branch-and-prune algorithm for nonlinear systems and presents the developed solver. New results for using the box-consistency enforcing operator and a new variant of the initial exclusion phase are presented. Also, a new heuristic to choose the coordinate for bisection is considered. Extensive numerical experiments are analyzed to provide the satisfying version of the algorithm.
Highlights
We consider the problem of finding all solutions of nonlinear systems of equations, i.e., systems of the form: f (x) = 0, (1)
– a sophisticated heuristic to choose the bisected component [27], – an initial exclusion phase of the algorithm – based on Sobol sequences [28], – an additional test based on quadratic approximation of a single equation and the Hansen’s method [18] to solve quadratic equations with interval coefficients; see [29]
The heuristic we propose is described by Algorithm
Summary
We consider the problem of finding all solutions of nonlinear systems of equations, i.e., systems of the form:. The kinematic chain starts in the point (0, 0) and the effector is supposed to be placed in the point (1, 1) and oriented orthogonally (under the right angle) to the OY axis This problem can be formulated as the following system of equations:. Interval methods (see, e.g., [18, 20, 37]) are a well-known approach to find all solutions of both kinds of systems Their essence is to perform operations on (possibly multidimensional) intervals (so-called boxes in Rn; see Fig. 2) instead of specific numbers (vectors), so that, if a ∈ a and b ∈ b, (a b ∈ a b), i.e., the result of an operation on numbers belongs to the result of operation on intervals, containing the arguments. The solver is targeted at underdetermined problems, yet it could be used for well-determined ones,
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