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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.
Efficient incremental algorithms for the sparse resultant and the mixed volume
 J. Symbolic Computation
, 1995
"... We propose a new and efficient algorithm for computing the sparse resultant of a system of n + 1 polynomial equations in n unknowns. This algorithm produces a matrix whose entries are coefficients of the given polynomials and is typically smaller than the matrices obtained by previous approaches. Th ..."
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Cited by 51 (7 self)
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We propose a new and efficient algorithm for computing the sparse resultant of a system of n + 1 polynomial equations in n unknowns. This algorithm produces a matrix whose entries are coefficients of the given polynomials and is typically smaller than the matrices obtained by previous approaches. The matrix determinant is a nontrivial multiple of the sparse resultant from which the sparse resultant itself can be recovered. The algorithm is incremental in the sense that successively larger matrices are constructed until one is found with the above properties. For multigraded systems, the new algorithm produces optimal matrices, i.e., expresses the sparse resultant as a single determinant. An implementation of the algorithm is described and experimental results are presented. In addition, we propose an efficient algorithm for computing the mixed volume of n polynomials in n variables. This computation provides an upper bound on the number of common isolated roots. A publicly available implementation of the algorithm is presented and empirical results are reported which suggest that it is the fastest mixed volume code to date.
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...
Sparse Elimination and Applications in Kinematics
, 1994
"... This thesis proposes efficient algorithmic solutions to problems in computational algebra and computational algebraic geometry. Moreover, it considers their application to different areas where algebraic systems describe kinematic and geometric constraints. Given an arbitrary system of nonlinear mul ..."
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Cited by 47 (10 self)
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This thesis proposes efficient algorithmic solutions to problems in computational algebra and computational algebraic geometry. Moreover, it considers their application to different areas where algebraic systems describe kinematic and geometric constraints. Given an arbitrary system of nonlinear multivariate polynomial equations, its resultant serves in eliminating variables and reduces root finding to a linear eigenproblem. Our contribution is to describe the first efficient and general algorithms for computing the sparse resultant. The sparse resultant generalizes the classical homogeneous resultant and exploits the structure of the given polynomials. Its size depends only on the geometry of the input Newton polytopes. The first algorithm uses a subdivision of the Minkowski sum and produces matrix...
Orthogonal Maximal Abelian *Subalgebras of the N×n Matrices and Cyclic NRoots
 Institut for Matematik, U. of Southern Denmark
, 1996
"... It is proved that for n = 5, there is up to isomorphism only one pair of orthogonal maximal abelian subalgebras (MASA's) in the n \Theta nmatrices. The same result holds trivially for n = 2 and n = 3, but de la Harpe, Jones, Munemasa and Watatani have shown that, for every prime number n 7, there ..."
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Cited by 33 (3 self)
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It is proved that for n = 5, there is up to isomorphism only one pair of orthogonal maximal abelian subalgebras (MASA's) in the n \Theta nmatrices. The same result holds trivially for n = 2 and n = 3, but de la Harpe, Jones, Munemasa and Watatani have shown that, for every prime number n 7, there are at least two nonisomorphic pairs of MASA's in the n \Theta n matrices. We draw connections to the research of Backelin, Bjorck and Froberg on cyclic nroots, and use their classification of cyclic 7roots to construct five nonisomorphic pairs of MASA's in the 7 \Theta 7 matrices. 1 1 Introduction Let A and B be two maximal abelian subalgebras (MASA's) of the algebra of complex n \Theta n matrices. A and B are orthogonal in the sense of Popa [16], i.e. A " B = C 1 and of the product of the orthogonal projections EA and EB of M n (C ) onto A and B (with respect to the HilbertSchmidt norm) is equal to the orthogonal projection EA"B of M n (C ) onto C 1. This means that B ae M n (C ...
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.
Polyhedral End Games for Polynomial Continuation
 Numerical Algorithms
, 1998
"... Bernshtein's theorem provides a generically exact upper bound on the number of isolated solutions a sparse polynomial system can have in (C ) n , with C = C n f0g. When a sparse polynomial system has fewer than this number of isolated solutions some face system must have solutions in (C ..."
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Cited by 21 (8 self)
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Bernshtein's theorem provides a generically exact upper bound on the number of isolated solutions a sparse polynomial system can have in (C ) n , with C = C n f0g. When a sparse polynomial system has fewer than this number of isolated solutions some face system must have solutions in (C ) n . In this paper we address the process of recovering a certificate of deficiency from a diverging solution path. This certificate takes the form of a face system along with approximations of its solutions. We apply extrapolation to estimate the cycle number and the face normal. Applications illustrate the practical usefulness of our approach. keywords : homotopy continuation, polynomial systems, Newton polytopes, Bernshtein bound, cycle number. AMS(MOS) Classification : 14Q99, 52A39, 52B20, 65H10. 1 Introduction All isolated complex solutions to polynomial systems can be approximated numerically by homotopy continuation methods. The strategy is to set up a collection of implicitly d...