Improved Iterative Methods for Solving High Order Polynomial Equations

Many calculations in engineering and scientific computation can summarized to the problem of solving a polynomial equation. Based on Sturm theorem, an adaptive algorithm for real root isolation is shown. This algorithm will firstly find the isolate interval for all the real roots rapidly. And then approximate the real roots by subdividing the isolate intervals and extracting subintervals each of which contains one real root. This method overcomes all the shortcomings of dichotomy method and iterative method. It doesn’t need to compute derivative values, no need to worry about the initial points, and could find all the real roots out parallelly.

On a Generalized Sturm Theorem

Sturm oscillation theorem for second order differential equations was generalized to systems and higher order equations with positive leading coefficient by several authors. What we propose here is a Sturm type oscillation theorem for indefinite systems with Dirichlet boundary conditions of the formwhere p

Branch identification and motion domain analysis of Stephenson type six-bar linkages

A general approach for branch identification and motion domain analysis of Stephenson type six-bar linkages is presented. By applying the Sturm theorem to the input-output polynomial equation, the dead-centre positions of the linkage are first evaluated and classified into two groups in order to discriminate the upper and lower bounds of the motion domains. The circuits of the linkage are then identified by matching the dead centres to the branches, which are attributed in accordance with the case where the input is assigned to a joint within the four-bar chain. Finally, the branches and motion domains of the more complicated case where the input is given through one of the uncoupled joints within the five-bar chain, are identified by mapping the circuits onto the domain of the specified input joint. This approach does not rely on the coupler curve of the constituent four-link mechanism. This is also suitable for computer implementation and can be systematically applied to all types of Stephenson linkages, regardless of the types of joints and the selection of input-output pair.