Numerically Computing Real Points on Algebraic Sets

Hauenstein, Jonathan D.

doi:10.1007/s10440-012-9782-3

Cited by 63 publications

(65 citation statements)

References 49 publications

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“…Much of the theory underlying the ideas of this article was known by Morgan and Sommese in the 1980s [18] and has since been repeated in various forms, for example in [24,8]. The main contribution of this article is in the application of this theory in the setting of regeneration, not in the theory itself.…”

Section: Justificationmentioning

confidence: 99%

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Perturbed homotopies for finding all isolated solutions of polynomial systems

Bates

Davis

Eklund

et al. 2014

Applied Mathematics and Computation

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a b s t r a c tGiven a polynomial system f : C N ! C n , the methods of numerical algebraic geometry produce numerical approximations of the isolated solutions of f ðzÞ ¼ 0, as well as points on any positive-dimensional components of the solution set, Vðf Þ. Some of these methods are guaranteed to find all isolated solutions (nonsingular and singular alike), while others may not find singular solutions. One of the most recent advances in this field is regeneration, an equation-by-equation solver that is often more efficient than other methods. However, the basic form of regeneration will not necessarily find all isolated singular solutions of a polynomial system without the additional cost of using deflation.The aim of this article is two-fold. First, more generally, we consider the use of perturbed homotopies for solving polynomial systems. In particular, we propose first solving a perturbed version of the polynomial system, followed by a parameter homotopy to remove the perturbation. Such perturbed homotopies are sometimes more efficient than regular homotopies, though they can also be less efficient. Second, a useful consequence is that the application of this perturbation to regeneration will yield all isolated solutions, including all singular isolated solutions. This version of regeneration -perturbed regeneration -can decrease the efficiency of regeneration but increases its applicability.

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Section: Justificationmentioning

confidence: 99%

“…A more general statement than Lemma 3.3 is given as an exercise in [7] and a related result for a pure d-dimensional algebraic subset is presented in [8], for d > 0. For the specific setting of the lemma, the proof is trivial:…”

Section: Justificationmentioning

confidence: 99%

Perturbed homotopies for finding all isolated solutions of polynomial systems

Bates

Davis

Eklund

et al. 2014

Applied Mathematics and Computation

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“…Six of them have degree 4 while two have degree 6. Since a computation using the method of Hauenstein in [9] yields a smooth real point on each of the first 6, they are necessarily real radical. That computation also shows that the degree 6 curves have real points, but those points were singular.…”

Section: Real Numerical Irreducible Decompositionmentioning

confidence: 99%

Real Numerical Algebraic Geometry: Finding All Real Solutions of a Polynomial System

Bates¹,

Brake²,

Hao³

et al. 2014

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“…This is the way used in [17,Theorem 5] to generate solution paths leading to real solutions on V using the homotopy continuation method. If V is compact and smooth, and the variables x1, .…”

Section: Definition 1 [1 Notation 24] For An Arbitrary Pointmentioning

confidence: 99%

Verified error bounds for real solutions of positive-dimensional polynomial systems

Yang

Zhi

Zhu

2013

Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation

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In this paper, we propose two algorithms for verifying the existence of real solutions of positive-dimensional polynomial systems. The first one is based on the critical point method and the homotopy continuation method. It targets for verifying the existence of real roots on each connected component of an algebraic variety V ∩ R n defined by polynomial equations. The second one is based on the low-rank moment matrix completion method and aims for verifying the existence of at least one real roots on V ∩ R n . Combined both algorithms with the verification algorithms for zerodimensional polynomial systems, we are able to find verified real solutions of positive-dimensional polynomial systems very efficiently for a large set of examples.

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Numerically Computing Real Points on Algebraic Sets

Cited by 63 publications

References 49 publications

Perturbed homotopies for finding all isolated solutions of polynomial systems

Perturbed homotopies for finding all isolated solutions of polynomial systems

Real Numerical Algebraic Geometry: Finding All Real Solutions of a Polynomial System

Verified error bounds for real solutions of positive-dimensional polynomial systems

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