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Introduction to numerical algebraic geometry
 In Solving Polynomial Equations, Series: Algorithms and Computation in Mathematics
, 2005
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Finding all real points of a complex curve
, 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 ..."
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Cited by 20 (8 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
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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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.
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 16 (3 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.
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 14 (4 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.
Isosingular Sets and Deflation
, 2011
"... This article introduces the concept of isosingular sets, which are irreducible algebraic subsets of the set of solutions to a system of polynomial equations that share a common singularity structure. The definition of these sets depends on deflation, a procedure that uses differentiation to regulari ..."
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Cited by 13 (7 self)
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This article introduces the concept of isosingular sets, which are irreducible algebraic subsets of the set of solutions to a system of polynomial equations that share a common singularity structure. The definition of these sets depends on deflation, a procedure that uses differentiation to regularize solutions. A weak form of deflation has proven useful in regularizing algebraic sets, making them amenable to treatment by the algorithms of numerical algebraic geometry. We introduce a strong form of deflation and define deflation sequences, which are similar to the sequences arising in ThomBoardman singularity theory. We then define isosingular sets in terms of deflation sequences. We also define the isosingular local dimension and examine the properties of isosingular sets. While isosingular sets are of theoretical interest as constructs for describing singularity structures of algebraic sets, they also expand the kinds of algebraic sets that can be investigated with methods from numerical algebraic geometry.
Numerical computation of the genus of an irreducible curve within an algebraic set
, 2007
"... The common zero locus of a set of multivariate polynomials (with complex coefficients) determines an algebraic set. Any algebraic set can be decomposed into a union of irreducible components. Given a one dimensional irreducible component, i.e. a curve, it is useful to understand its invariants. Th ..."
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Cited by 12 (4 self)
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The common zero locus of a set of multivariate polynomials (with complex coefficients) determines an algebraic set. Any algebraic set can be decomposed into a union of irreducible components. Given a one dimensional irreducible component, i.e. a curve, it is useful to understand its invariants. The most important invariants of a curve are the degree, the arithmetic genus and the geometric genus (where the geometric genus denotes the genus of a desingularization of the projective closure of the curve). This article presents a numerical algorithm to compute the geometric genus of any onedimensional irreducible component of an algebraic set.