A Globally Convergent Parallel Algorithm for Zeros of Polynomial Systems
نویسنده
چکیده
POLYNOMIAL systems of equations frequently arise in solid modelling, robotics, computer vision, chemistry, chemical engineering, and mechanical engineering. Locally convergent iterative methods such as quasi-Newton methods may diverge or fail to find all meaningful solutions of a polynomial system. This paper proposes a parallel homotopy algorithm for polynomial systems of equations that is guaranteed globally convergent (always converges from an arbitrary starting point) with probability one, finds all solutions to the polynomial system, and has a large amount of inherent parallelism. Some mathematical background material on homotopy algorithms and polynomial systems is included. A particular coarse-grained decomposition strategy is given in detail, and the results of some experiments on an Intel iPSC-32 hypercube are presented. Solving nonlinear systems of equations is a central problem in numerical analysis, with enormous significance for science and engineering. A very special case, namely small polynomial systems of equations with many real solutions, occurs frequently enough in solid modelling, robotics, computer vision, chemical equilibrium computations, chemical process design, mechanical engineering, and other areas to justify special algorithms. To put polynomial systems in perspective and for the purpose of discussion here, there are three classes of nonlinear systems of equations: (1) large systems with sparse Jacobian matrices, (2) small transcendental (nonpolynomial) systems with dense Jacobian matrices, and (3) small polynomial systems with dense Jacobian matrices. Sparsity for small problems is not significant, and large systems with dense Jacobian matrices are intractable, so these two classes are not counted. Of course medium sized problems are also of practical interest, but the boundaries between small, medium, and large, change with computer hardware technology and algorithmic development. Depending on algorithmic efficiency, hardware capability, and the significance of sparsity, a medium sized problem is treated like it belongs to one of the above three classes anyway, so there is no need for a “medium” class. Large sparse nonlinear systems of equations, such as equilibrium equations in structural mechanics, have two aspects: highly nonlinear and recursive scalar computations, and large
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Message length effects for solving polynomial systems on a hypercube
Polynomial systems of equations frequently arise in solid modelling, robotics, computer vision, chemistry, chemical engineering, and mechanical engineering. Locally convergent iterative methods such as quasi-Newton methods may diverge or fail to find all meaningful solutions of a polynomial system. Recently a homotopy algorithm has been proposed for polynomial systems that is guaranteed globall...
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