Hybridization of bisection method for solving non-differentiable nonlinear equations

What is it about?

The bisection method is a well known method for solving non-differentiable nonlinear equations in a closed initial search interval. Sometimes, due to the nature of some problems or applications It is utilized as an auxiliary tool in solving solving of systems of equations, optimization problems, imprecise problems, special functions, dynamic systems, etc. In all instances, a valid zero localization is needed. This need is satisfied either with the validation of Bolzano’s criterion on specific search intervals or by adopting techniques and methods of finding the number of zeros of a function, like the topological degree.

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Why is it important?

The importance of the proposed method is revealed from the motivation of solving effectively real life problems and developing simple, reliable and robust solvers. The proposed, equivalent form of bisection method which is combined with interval bisection steps, empowers the classic bisection scheme as a zero-localization process. The new form is simpler and zero-existence conditions are not required. In addition, no-zero areas are discarded with certainty, resulting in several cases enclosures for more than one zero, exploiting at the same time the efficiency of an interval method and the constant behavior of classic bisection method.

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