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Browsing by Author "McCarthy, J. Michael"

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    Generalized Linear Product Homotopy Algorithms and the Computation of Reachable Surfaces
    Su, Hai-Jun; McCarthy, J. Michael; Watson, Layne T. (Department of Computer Science, Virginia Polytechnic Institute & State University, 2003)
    In this paper, we apply a homotopy algorithm to the problem of finding points in a moving body that lie on specific algebraic surfaces for a given set of spatial configurations of the body. This problem is a generalization of Burmester's determination of points in a body that lie on a circle for five planar positions. We focus on seven surfaces that we term "reachable" because they correspond to serial chains with two degree-of-freedom positioning structures combined with a three degree-of-freedom spherical wrist. A homotopy algorithm based on generalized linear products is used to provide a convenient estimate of the number of solutions of these polynomial systems. A parallelized version of this algorithm was then used to numerically determine all of the solutions.
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    POLSYS GLP: A Parallel General Linear Product Homotopy Code for Solving Polynomial Systems of Equations
    Su, Hai-Jun; McCarthy, J. Michael; Sosonkina, Masha; Watson, Layne T. (Department of Computer Science, Virginia Polytechnic Institute & State University, 2004)
    Globally convergent, probability-one homotopy methods have proven to be very effective for finding all the isolated solutions to polynomial systems of equations. After many years of development, homotopy path trackers based on probability-one homotopy methods are reliable and fast. Now, theoretical advances reducing the number of homotopy paths that must be tracked, and in the handling of singular solutions, have made probability-one homotopy methods even more practical. POLSYS GLP consists of Fortran 95 modules for nding all isolated solutions of a complex coefficient polynomial system of equations. The package is intended to be used on a distributed memory multiprocessor in conjunction with HOMPACK90 (Algorithm 777), and makes extensive use of Fortran 95 derived data types and MPI to support a general linear product (GLP) polynomial system structure. GLP structure is intermediate between the partitioned linear product structure used by POLSYS PLP (Algorithm 801) and the BKK-based structure used by PHCPACK. The code requires a GLP structure as input, and although nding the optimal GLP structure is a dicult combinatorial problem, generally physical or engineering intuition about a problem yields a very good GLP structure. POLSYS GLP employs a sophisticated power series end game for handling singular solutions, and provides support for problem denition both at a high level and via hand-crafted code. Dierent GLP structures and their corresponding Bezout numbers can be systematically explored before committing to root finding.