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25
Introduction to numerical algebraic geometry
 In Solving Polynomial Equations, Series: Algorithms and Computation in Mathematics
, 2005
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On location and approximation of clusters of zeros: Case of embedding dimension one
 Foundations of Computational Mathematics
"... Abstract. Isolated multiple zeroes or clusters of zeroes of analytic maps with several variables are known to be difficult to locate and approximate. This article is in the vein of the αtheory, initiated by M. Shub and S. Smale in the beginning of the eighties. This theory restricts to simple zeroe ..."
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Cited by 9 (2 self)
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Abstract. Isolated multiple zeroes or clusters of zeroes of analytic maps with several variables are known to be difficult to locate and approximate. This article is in the vein of the αtheory, initiated by M. Shub and S. Smale in the beginning of the eighties. This theory restricts to simple zeroes, i.e., where the map has corank zero. In this article we deal with situations where the analytic map has corank one at the multiple isolated zero, which has embedding dimension one in the frame of deformation theory. These situations are the least degenerate ones and therefore most likely to be of practical significance. More generally, we define clusters of embedding dimension one. We provide a criterion for locating such clusters of zeroes and a fast algorithm for approximating them, with quadratic convergence. In case of a cluster with positive diameter our algorithm stops at a distance of the cluster
Evaluation of jacobian matrices for Newton’s method with deflation to approximate isolated singular solutions of polynomial systems
 SNC 2005 Proceedings. International Workshop on SymbolicNumeric Computation
, 2005
"... Abstract. For isolated singular solutions of polynomial systems, we can restore the quadratic convergence of Newton’s method by deflation. The number of deflation stages is bounded by the multiplicity of the root. A preliminary implementation performs well in case only a few deflation stages are nee ..."
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Cited by 9 (2 self)
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Abstract. For isolated singular solutions of polynomial systems, we can restore the quadratic convergence of Newton’s method by deflation. The number of deflation stages is bounded by the multiplicity of the root. A preliminary implementation performs well in case only a few deflation stages are needed, but suffers from expression swell as the number of deflation stages grows. In this paper we describe how a directed acyclic graph of derivative operators guides an efficient evaluation of the Jacobian matrices produced by our deflation algorithm. We illustrate how the symbolicnumeric deflation algorithm can be used within PHCmaple interfacing Maple with PHCpack.
Approximate Bivariate Factorization, a Geometric Viewpoint
, 2007
"... We briefly present and analyze, from a geometric viewpoint, strategies for designing algorithms to factor bivariate approximate polynomials in C[x, y]. Given a composite polynomial, stably squarefree, satisfying a genericity hypothesis, we describe the effect of a perturbation on the roots of its d ..."
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Cited by 9 (1 self)
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We briefly present and analyze, from a geometric viewpoint, strategies for designing algorithms to factor bivariate approximate polynomials in C[x, y]. Given a composite polynomial, stably squarefree, satisfying a genericity hypothesis, we describe the effect of a perturbation on the roots of its discriminant with respect to one variable, and the perturbation of the corresponding monodromy action on a smooth fiber. A novel geometric approach is presented, based on guided projection in the parameter space and continuation method above randomly chosen loops, to reconstruct from a perturbed polynomial a nearby composite polynomial and its irreducible factors. An algorithm and its ingredients are described.
Interfacing with the numerical homotopy algorithms in phcpack
 in PHCpack, Proceedings of ICMS 2006
, 2006
"... Abstract. Our Maple package PHCmaple came to existence in 2004 when it provided a convenient interface to the basic functionality of phc, a program which is a part of PHCpack and implements numeric algorithms for solving polynomial systems using polynomial homotopy continuation. Following the recent ..."
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Cited by 8 (7 self)
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Abstract. Our Maple package PHCmaple came to existence in 2004 when it provided a convenient interface to the basic functionality of phc, a program which is a part of PHCpack and implements numeric algorithms for solving polynomial systems using polynomial homotopy continuation. Following the recent development of PHCpack the package has been extended with functions that deal with singular polynomial systems, in particular, the deflation procedures that guarantee the ability to refine approximations to an isolated solution even if it is multiple. We see PHCmaple as a part of a larger project to integrate a numerical solver in a computer algebra system. 1
A numerical local dimension test for points on the solution set of a system of polynomial equations
 SIAM J. NUMER. ANAL
"... The solution set V of a polynomial system, i.e., the set of common zeroes of a set of multivariate polynomials with complex coefficients, may contain several components, e.g., points, curves, surfaces, etc. Each component has attached to it a number of quantities, one of which is its dimension. Giv ..."
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Cited by 8 (3 self)
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The solution set V of a polynomial system, i.e., the set of common zeroes of a set of multivariate polynomials with complex coefficients, may contain several components, e.g., points, curves, surfaces, etc. Each component has attached to it a number of quantities, one of which is its dimension. Given a numerical approximation to a point p on the set V, this article presents an efficient algorithm to compute the maximum dimension of the irreducible components of V which pass through p, i.e., a local dimension test. Such a test is a crucial element in the homotopybased numerical irreducible decomposition algorithms of Sommese, Verschelde, and Wampler. This article presents computational evidence to illustrate that the use of this new algorithm greatly reduces the cost of socalled “junkpoint filtering, ” previously a significant bottleneck in the computation of a numerical irreducible decomposition. For moderate size examples, this results in well over an order of magnitude improvement in the computation of a numerical irreducible decomposition. As the computation of a numerical irreducible decomposition is a fundamental backbone operation, gains in efficiency in the irreducible decomposition algorithm carry over to the many computations which require this decomposition as an initial step. Another feature of a local dimension test is that one can now compute the irreducible components in a prescribed dimension without first computing the numerical irreducible decomposition of all higher dimensions. For example, one may compute the isolated solutions of a polynomial system without having to carry out the full numerical irreducible decomposition.
Finding all real points of a complex curve
 In Algebra, Geometry and Their Interactions
, 2006
"... An algorithm is given to compute the real points of the irreducible onedimensional complex components of the solution sets of systems of polynomials with real coefficients. The algorithm is based on homotopy continuation and the numerical irreducible decomposition. An extended application is made t ..."
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Cited by 7 (2 self)
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An algorithm is given to compute the real points of the irreducible onedimensional complex components of the solution sets of systems of polynomials with real coefficients. The algorithm is based on homotopy continuation and the numerical irreducible decomposition. An extended application is made to GriffisDuffy platforms, a class of StewartGough platform robots. 2000 Mathematics Subject Classification. Primary 65H10; Secondary 65H20, 14Q99. Key words and phrases. Homotopy continuation, numerical algebraic geometry, real polynomial systems. In this article we give a numerical algorithm to find the real zero and
Higherorder deflation for polynomial systems with isolated singular solutions
 In IMA Volume 146: Algorithms in Algebraic Geometry
, 2008
"... Given an approximation to a multiple isolated solution of a polynomial system of equations, we have provided a symbolicnumeric deflation algorithm to restore the quadratic convergence of Newton’s method. Using firstorder derivatives of the polynomials in the system, our method creates an augmented ..."
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Cited by 7 (1 self)
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Given an approximation to a multiple isolated solution of a polynomial system of equations, we have provided a symbolicnumeric deflation algorithm to restore the quadratic convergence of Newton’s method. Using firstorder derivatives of the polynomials in the system, our method creates an augmented system of equations which has the multiple isolated solution of the original system as a regular root. In this paper we consider two approaches to computing the “multiplicity structure ” at a singular isolated solution. An idea coming from one of them gives rise to our new higherorder deflation method. Using higherorder partial derivatives of the original polynomials, the new algorithm reduces the multiplicity faster than our first method for systems which require several firstorder deflation steps. 2000 Mathematics Subject Classification. Primary 65H10. Secondary 14Q99, 68W30. Key words and phrases. Deflation, isolated singular solutions, Newton’s method, multiplicity, polynomial systems, reconditioning, symbolicnumeric computations.
The closedness subspace method for computing the multiplicity structure of a polynomial system, to appear
 in Interactions of Classical and Numerical Algebraic
, 2009
"... Abstract. The multiplicity structure of a polynomial system at an isolated zero is identified with the dual space consisting of differential functionals vanishing on the entire polynomial ideal. Algorithms have been developed for computing dual spaces as the kernels of Macaulay matrices. These previ ..."
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Cited by 6 (1 self)
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Abstract. The multiplicity structure of a polynomial system at an isolated zero is identified with the dual space consisting of differential functionals vanishing on the entire polynomial ideal. Algorithms have been developed for computing dual spaces as the kernels of Macaulay matrices. These previous algorithms face a formidable obstacle in handling Macaulay matrices whose dimensions grow rapidly when the problem size and the order of the differential functionals increase. This paper presents a new algorithm based on the closedness subspace strategy that substantially reduces the matrix sizes in computing the dual spaces, enabling the computation of multiplicity structures for large problems. Comparisons of timings and memory requirements demonstrate a substantial improvement in computational efficiency. 1.