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45
Solving Systems of Polynomial Equations
 American Mathematical Society, CBMS Regional Conferences Series, No 97
, 2002
"... Abstract. One of the most classical problems of mathematics is to solve systems of polynomial equations in several unknowns. Today, polynomial models are ubiquitous and widely applied across the sciences. They arise in robotics, coding theory, optimization, mathematical biology, computer vision, gam ..."
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Cited by 145 (10 self)
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Abstract. One of the most classical problems of mathematics is to solve systems of polynomial equations in several unknowns. Today, polynomial models are ubiquitous and widely applied across the sciences. They arise in robotics, coding theory, optimization, mathematical biology, computer vision, game theory, statistics, machine learning, control theory, and numerous other areas. The set of solutions to a system of polynomial equations is an algebraic variety, the basic object of algebraic geometry. The algorithmic study of algebraic varieties is the central theme of computational algebraic geometry. Exciting recent developments in symbolic algebra and numerical software for geometric calculations have revolutionized the field, making formerly inaccessible problems tractable, and providing fertile ground for experimentation and conjecture. The first half of this book furnishes an introduction and represents a snapshot of the state of the art regarding systems of polynomial equations. Afficionados of the wellknown text books by Cox, Little, and O’Shea will find familiar themes in the first five chapters: polynomials in one variable, Gröbner
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 37 (22 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
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Introduction to numerical algebraic geometry
 In Solving Polynomial Equations, Series: Algorithms and Computation in Mathematics
, 2005
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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.
A rankrevealing method with updating, downdating and applications
 SIAM J. Matrix Anal. Appl
"... Abstract. A new rank revealing method is proposed. For a given matrix and a threshold for nearzero singular values, by employing a globally convergent iterative scheme as well as a deflation technique the method calculates approximate singular values below the threshold one by one and returns the a ..."
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Cited by 23 (7 self)
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Abstract. A new rank revealing method is proposed. For a given matrix and a threshold for nearzero singular values, by employing a globally convergent iterative scheme as well as a deflation technique the method calculates approximate singular values below the threshold one by one and returns the approximate rank of the matrix along with an orthonormal basis for the approximate null space. When a row or column is inserted or deleted, algorithms for updating/downdating the approximate rank and null space are straightforward, stable and efficient. Numerical results exhibiting the advantages of our code over existing packages based on twosided orthogonal rankrevealing decompositions are presented. Also presented are applications of the new algorithm in numerical computation of the polynomial GCD as well as identification of nonisolated zeros of polynomial systems.
Numerical Irreducible Decomposition using PHCpack
, 2003
"... Homotopy continuation methods have proven to be reliable and efficient to approximate all isolated solutions of polynomial systems. In this paper we show how we can use this capability as a blackbox device to solve systems which have positive dimensional components of solutions. We indicate how the ..."
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Cited by 21 (14 self)
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Homotopy continuation methods have proven to be reliable and efficient to approximate all isolated solutions of polynomial systems. In this paper we show how we can use this capability as a blackbox device to solve systems which have positive dimensional components of solutions. We indicate how the software package PHCpack can be used in conjunction with Maple and programs written in C. We describe a numerically stable algorithm for decomposing positive dimensional solution sets of polynomial systems into irreducible components.
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 21 (14 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.
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 21 (14 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.
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 18 (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.