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28
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...
Uncalibrated Euclidean reconstruction: a review
, 1999
"... This paper provides a review on techniques for computing a threedimensional model of a scene from a single moving camera, with unconstrained motion and unknown parameters. In the classical approach, called autocalibration or selfcalibration, camera motion and parameters are recovered first, using ..."
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Cited by 35 (8 self)
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This paper provides a review on techniques for computing a threedimensional model of a scene from a single moving camera, with unconstrained motion and unknown parameters. In the classical approach, called autocalibration or selfcalibration, camera motion and parameters are recovered first, using rigidity; then structure is easily computed. Recently, new methods based on the idea of stratification have been proposed. They upgrade the projective structure, achievable from correspondences only, to the Euclidean structure, by exploiting all the available constraints.
Symmetric Functions Applied to Decomposing Solution Sets of Polynomial Systems
 SIAM J. Numer. Anal
, 2001
"... Many polynomial systems have solution sets comprising multiple irreducible components, possibly of dierent dimensions. A fundamental problem of numerical algebraic geometry is to decompose such a solution set, using oatingpoint numerical processes, into its components. ..."
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Cited by 35 (21 self)
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Many polynomial systems have solution sets comprising multiple irreducible components, possibly of dierent dimensions. A fundamental problem of numerical algebraic geometry is to decompose such a solution set, using oatingpoint numerical processes, into its components.
Using monodromy to decompose solution sets of polynomial systems into irreducible components
 PROCEEDINGS OF A NATO CONFERENCE, FEBRUARY 25  MARCH 1, 2001, EILAT
, 2001
"... ..."
Symbolicnumeric sparse interpolation of multivariate polynomials
 In Proc. Ninth Rhine Workshop on Computer Algebra (RWCA’04), University of Nijmegen, the Netherlands (2004
, 2006
"... We consider the problem of sparse interpolation of an approximate multivariate blackbox polynomial in floatingpoint arithmetic. That is, both the inputs and outputs of the blackbox polynomial have some error, and all numbers are represented in standard, fixedprecision, floating point arithmetic. ..."
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Cited by 32 (5 self)
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We consider the problem of sparse interpolation of an approximate multivariate blackbox polynomial in floatingpoint arithmetic. That is, both the inputs and outputs of the blackbox polynomial have some error, and all numbers are represented in standard, fixedprecision, floating point arithmetic. By interpolating the black box evaluated at random primitive roots of unity, we give efficient and numerically robust solutions. We note the similarity between the exact BenOr/Tiwari sparse interpolation algorithm and the classical Prony’s method for interpolating a sum of exponential functions, and exploit the generalized eigenvalue reformulation of Prony’s method. We analyze the numerical stability of our algorithms and the sensitivity of the solutions, as well as the expected conditioning achieved through randomization. Finally, we demonstrate the effectiveness of our techniques in practice through numerical experiments and applications. 1.
Numerical Irreducible Decomposition using Projections from Points on the Components
 In Symbolic Computation: Solving Equations in Algebra, Geometry, and Engineering, volume 286 of Contemporary Mathematics
"... To classify positive dimensional solution components of a polynomial system, we construct polynomials interpolating points sampled from each component. In previous work, points on an idimensional component were linearly projected onto a generically chosen (i + 1)dimensional subspace. In this p ..."
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Cited by 19 (13 self)
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To classify positive dimensional solution components of a polynomial system, we construct polynomials interpolating points sampled from each component. In previous work, points on an idimensional component were linearly projected onto a generically chosen (i + 1)dimensional subspace. In this paper, we present two improvements. First, we reduce the dimensionality of the ambient space by determining the linear span of the component and restricting to it. Second, if the dimension of the linear span is greater than i + 1, we use a less generic projection that leads to interpolating polynomials of lower degree, thus reducing the number of samples needed. While this more ecient approach still guarantees  with probability one  the correct determination of the degree of each component, the mere evaluation of an interpolating polynomial no longer certi es the membership of a point to that component. We present an additional numerical test that certi es membership in this new situation. We illustrate the performance of our new approach on some wellknown test systems.
Homotopies for intersecting solution components of polynomial systems
 SIAM J. Numer. Anal
, 2004
"... Abstract. We show how to use numerical continuation to compute the intersection C = A∩B of two algebraic sets A and B, where A, B, and C are numerically represented by witness sets. Enroute to this result, we first show how to find the irreducible decomposition of a system of polynomials restricted ..."
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Cited by 19 (13 self)
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Abstract. We show how to use numerical continuation to compute the intersection C = A∩B of two algebraic sets A and B, where A, B, and C are numerically represented by witness sets. Enroute to this result, we first show how to find the irreducible decomposition of a system of polynomials restricted to an algebraic set. The intersection of components A and B then follows by considering the decomposition of the diagonal system of equations u − v = 0 restricted to {u, v} ∈ A × B. One offshoot of this new approach is that one can solve a large system of equations by finding the solution components of its subsystems and then intersecting these. It also allows one to find the intersection of two components of the two polynomial systems, which is not possible with any previous numerical continuation approach.
Advances in Polynomial Continuation for Solving Problems in Kinematics
, 2004
"... For many mechanical systems, including nearly all robotic manipulators, the set of possible configurations that the links may assume can be described by a system of polynomial equations. Thus, solving such systems is central to many problems in analyzing the motion of a mechanism or in designing a m ..."
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Cited by 16 (8 self)
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For many mechanical systems, including nearly all robotic manipulators, the set of possible configurations that the links may assume can be described by a system of polynomial equations. Thus, solving such systems is central to many problems in analyzing the motion of a mechanism or in designing a mechanism to achieve a desired motion. This paper describes techniques, based on polynomial continuation, for numerically solving such systems. Whereas in the past, these techniques were focused on finding isolated roots, we now address the treatment of systems having higherdimensional solution sets. Special attention is given to cases of exceptional mechanisms, which have a higher degree of freedom of motion than predicted by their mobility. In fact, such mechanisms often have several disjoint assembly modes, and the degree of freedom of motion is not necessarily the same in each mode. Our algorithms identify all such assembly modes, determine their dimension and degree, and give sample points on each.
Numerical Factorization of Multivariate Complex Polynomials
 Theoretical Comput. Sci
, 2003
"... One can consider the problem of factoring multivariate complex polynomials as a special case of the decomposition of a pure dimensional solution set of a polynomial system into irreducible components. The importance and nature of this problem however justify a special treatment. ..."
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Cited by 14 (4 self)
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One can consider the problem of factoring multivariate complex polynomials as a special case of the decomposition of a pure dimensional solution set of a polynomial system into irreducible components. The importance and nature of this problem however justify a special treatment.