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Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components
, 2001
"... In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities. In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposi ..."
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Cited by 56 (26 self)
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In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities. In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposition of the solution set. In particular, ignoring multiplicities, our algorithms lay out the decomposition of the set of solutions into irreducible components, by finding, at each dimension, generic points on each component. As byproducts, the computation also determines the degree of each component and an upper bound on itsmultiplicity. The bound issharp (i.e., equal to one) for reduced components. The algorithms make essential use of generic projection and interpolation, and can, if desired, describe each irreducible component precisely as the common zeroesof a finite number of polynomials.
Numerical Homotopies to compute generic Points on positive dimensional Algebraic Sets
 Journal of Complexity
, 1999
"... Many applications modeled by polynomial systems have positive dimensional solution components (e.g., the path synthesis problems for fourbar mechanisms) that are challenging to compute numerically by homotopy continuation methods. A procedure of A. Sommese and C. Wampler consists in slicing the com ..."
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Cited by 50 (24 self)
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Many applications modeled by polynomial systems have positive dimensional solution components (e.g., the path synthesis problems for fourbar mechanisms) that are challenging to compute numerically by homotopy continuation methods. A procedure of A. Sommese and C. Wampler consists in slicing the components with linear subspaces in general position to obtain generic points of the components as the isolated solutions of an auxiliary system. Since this requires the solution of a number of larger overdetermined systems, the procedure is computationally expensive and also wasteful because many solution paths diverge. In this article an embedding of the original polynomial system is presented, which leads to a sequence of homotopies, with solution paths leading to generic points of all components as the isolated solutions of an auxiliary system. The new procedure significantly reduces the number of paths to solutions that need to be followed. This approach has been implemented and applied to...
Symmetric Newton Polytopes for Solving Sparse Polynomial Systems
 ADV. APPL. MATH
, 1994
"... The aim of this paper is to compute all isolated solutions to symmetric polynomial systems. Recently, it has been proved that modelling the sparse structure of the system by its Newton polytopes leads to a computational breakthrough in solving the system. In this paper, it will be shown how the Lift ..."
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Cited by 20 (9 self)
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The aim of this paper is to compute all isolated solutions to symmetric polynomial systems. Recently, it has been proved that modelling the sparse structure of the system by its Newton polytopes leads to a computational breakthrough in solving the system. In this paper, it will be shown how the Lifting Algorithm, proposed by Huber and Sturmfels, can be applied to symmetric Newton polytopes. This symmetric version of the Lifting Algorithm enables the efficient construction of the symmetric subdivision, giving rise to a symmetric homotopy, so that only the generating solutions have to be computed. Efficiency is obtained by combination with the product homotopy. Applications illustrate the practical significance of the presented approach.
Balancing the Lifting Values to Improve the Numerical Stability of Polyhedral Homotopy Continuation Methods
, 1997
"... Polyhedral homotopy continuation methods exploit the sparsity of polynomial systems so that the number of solution curves to reach all isolated solutions is optimal for generic systems. The numerical stability of tracing solution curves of polyhedral homotopies is mainly determined by the height of ..."
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Cited by 13 (3 self)
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Polyhedral homotopy continuation methods exploit the sparsity of polynomial systems so that the number of solution curves to reach all isolated solutions is optimal for generic systems. The numerical stability of tracing solution curves of polyhedral homotopies is mainly determined by the height of the powers of the continuation parameter. To reduce this height we propose a procedure that operates as an intermediate stage between the mixedvolume computation and the tracing of solution curves. This procedure computes new lifting values of the support of a polynomial system. These values preserve the structure of the mixedcell configuration obtained from the mixedvolume computation and produce betterbalanced powers of the continuation parameter in the polyhedral homotopies. AMS Subject Classification. 14Q99, 52A39, 52B20, 65H10. Key words and phrases. Polyhedral homotopies, path following, numerical stability, balancing. email: gaota@math.msu.edu y Research was supported in p...
Improving the efficiency of exclusion algorithms
 Advances in Geometry
, 2001
"... Exclusion algorithms are a wellknown tool in the area of interval analysis, see, e.g., [5, 6], for finding all solutions of a system of nonlinear equations. They also have been introduced in [14, 15] from a slightly different viewpoint. In particular, such algorithms seem to be very... ..."
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Cited by 9 (1 self)
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Exclusion algorithms are a wellknown tool in the area of interval analysis, see, e.g., [5, 6], for finding all solutions of a system of nonlinear equations. They also have been introduced in [14, 15] from a slightly different viewpoint. In particular, such algorithms seem to be very...
Five Precision Points Synthesis Of Spatial Rrr Manipulators Using Interval Analysis
 ASME J. Mech. Des
, 2002
"... In this paper, the geometric design problem of seriallink robot manipulators with three revolute (R) joints is solved for the first time using an interval analysis method. In this problem, five spatial positions and orientations are defined and the dimensions of the geometric parameters of the ..."
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Cited by 5 (1 self)
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In this paper, the geometric design problem of seriallink robot manipulators with three revolute (R) joints is solved for the first time using an interval analysis method. In this problem, five spatial positions and orientations are defined and the dimensions of the geometric parameters of the 3R manipulator are computed so that the manipulator will be able to place its endeffector at these prespecified locations. Denavit and Hartenberg parameters and 4x4 homogeneous matrices are used to formulate the problem and obtain the design equations and an interval method is used to search for design solutions within a predetermined domain. At the time of writing this paper, six design solutions within the search domain and an additional twenty solutions outside the domain have been found. KEYWORDS Geometric Design, Robot Manipulators, Interval Analysis
Isotropic coordinates, circularity, and Bezout numbers: planar kinematics from a new perspective
 PROC. ASME DES. ENG. TECH. CONF., AUG. 18–22
, 1996
"... It is commonly recognized that a convenient formulation for problems in planar kinematics is obtained by considering links to be vectors in the complex plane. However, scant attention has been paid to the natural interpretation of complex vectors as isotropic coordinates. These coordinates, often co ..."
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Cited by 4 (2 self)
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It is commonly recognized that a convenient formulation for problems in planar kinematics is obtained by considering links to be vectors in the complex plane. However, scant attention has been paid to the natural interpretation of complex vectors as isotropic coordinates. These coordinates, often considered a special trick for analyzing fourbar motion, are in fact uniquely suited to two new techniques for analyzing polynomial systems: the BKK bound and the productdecomposition bound. From this synergistic viewpoint, a fundamental formulation of planar kinematics is developed and used to prove several new results, mostly concerning the degree and circularity of the motion of planar linkages. Useful for both analysis and synthesis of mechanisms, the approach both simpli es theoretical proofs and facilitates the numerical solution of mechanism problems.
Geometric Design of Spatial RRR Manipulators Using Interval Analysis
"... In this paper, the geometric design problem of seriallink robot manipulators with three revolute (R)joints is solved for the first time using an interval analysis method. In this problem, five spatial positions and orientations are deftned and the dimensions of the geometric parameters of the 3R m ..."
Abstract
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In this paper, the geometric design problem of seriallink robot manipulators with three revolute (R)joints is solved for the first time using an interval analysis method. In this problem, five spatial positions and orientations are deftned and the dimensions of the geometric parameters of the 3R manipulator are computed so that the manipulator will be able to place its endeffector at these prespecified locations. Denavit and Hartenberg parameters and 4x4 homogeneous matrices are used to formulate the problem and obtain the design equations and an interval method is used to search for design solutions within a predetermined domain. At the time of writing this paper, six design solutions within the search domain and an additional twenty solutions outside the domain have been found.
Forward Kinematics Analysis of a 3PRS Parallel Manipulator
"... Abstract—In this article the homotopy continuation method (HCM) to solve the forward kinematic problem of the 3PRS parallel manipulator is used. Since there are many difficulties in solving the system of nonlinear equations in kinematics of manipulators, the numerical solutions like NewtonRaphson ..."
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Abstract—In this article the homotopy continuation method (HCM) to solve the forward kinematic problem of the 3PRS parallel manipulator is used. Since there are many difficulties in solving the system of nonlinear equations in kinematics of manipulators, the numerical solutions like NewtonRaphson are inevitably used. When dealing with any numerical solution, there are two troublesome problems. One is that good initial guesses are not easy to detect and another is related to whether the used method will converge to useful solutions. Results of this paper reveal that the homotopy continuation method can alleviate the drawbacks of traditional numerical techniques. Keywords—Forward kinematics; Homotopy continuation method; Parallel manipulators; Rotation matrix