Computer tools for the construction and analysis of some efficient root-finding simultaneous methods

Using the tools provided by computer algebra system Mathematica, we consider two iterative methods of high efficiency for the simultaneous approximation of simple or multiple (real or complex) zeros of algebraic polynomials. The proposed methods are based on the fourth-order Schr?der-like methods of the first and second kind. We prove that the order of convergence of both basic total-step simultaneous methods is equal to five. Using corrective approximations produced by methods of order two, three and four for finding a single multiple zero, the convergence order is increased from five to six, seven, and eight, respectively. The increased convergence speed is attained with negligible number of additional arithmetic operations, which significantly increases the computational efficiency of the accelerated methods. Convergence properties of the proposed methods are demonstrated by numerical examples and graphics visualization by plotting trajectories of zero approximations. Flows of iterative processes, presented by these trajectories, point to the stability and robustness of the proposed methods.

Symmetric Bowtie Decompositions of the Complete Graph

Given a bowtie decomposition of the complete graph $K_v$ admitting an automorphism group $G$ acting transitively on the vertices of the graph, we give necessary conditions involving the rank of the group and the cycle types of the permutations in $G$. These conditions yield non–existence results for instance when $G$ is the dihedral group of order $2v$, with $v\equiv 1, 9\pmod{12}$, or a group acting transitively on the vertices of $K_9$ and $K_{21}$. Furthermore, we have non–existence for $K_{13}$ when the group $G$ is different from the cyclic group of order $13$ or for $K_{25}$ when the group $G$ is not an abelian group of order $25$. Bowtie decompositions admitting an automorphism group whose action on vertices is sharply transitive, primitive or $1$–rotational, respectively, are also studied. It is shown that if the action of $G$ on the vertices of $K_v$ is sharply transitive, then the existence of a $G$–invariant bowtie decomposition is excluded when $v\equiv 9\pmod{12}$ and is equivalent to the existence of a $G$–invariant Steiner triple system of order $v$. We are always able to exclude existence if the action of $G$ on the vertices of $K_v$ is assumed to be $1$–rotational. If, instead, $G$ is assumed to act primitively then existence can be excluded when $v$ is a prime power satisfying some additional arithmetic constraint.

Computing a Manipulator Regressor Without Acceleration Feedback

SUMMARYA manipulator regressor is an n x l matrix function in the dynamic expression τ = Y r or τ = Wr, which linearizes the robotic dynamics with respect to a properly defined inertia parameter vector ζr є R1. Many modern adaptive controllers require on-line computation of a regressor to estimate the unknown inertia parameters and ensure robustness of the closed-loop system.While the computation of Y is studied by Atkeson, An and Hollerbach1 and Khosla and Kanade,2 the computation of W for a general n–link robot has not been reported in the literature. This paper presents an algorithm to compute W for a general n–link robotic manipulator. The variables used to construct the regressor matrix are directly available from the outward iteration of a Newton-Euler algorithm; some additional arithmetic operations and first-order, low-pass filtering are needed. The identification of unknown inertia parameters is also discussed.