On the Geometry and Topology of the Solution Variety for Polynomial System Solving

Beltrán, Carlos; Shub, Michael

doi:10.1007/s10208-012-9134-8

Cited by 8 publications

(8 citation statements)

References 22 publications

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“…Our feeling, based on a series of papers on the condition metric: [2], [3], [6], [7], [9], is that a good choice for the homotopy path (f t ) and the initial zero ζ a will induce a lifted path (f t , ζ t ) close to the condition geodesic connecting (f a , ζ a ) to (f b , ζ b ). We are far from accomplishing this task.…”

Section: Thenmentioning

confidence: 99%

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Adaptive step-size selection for homotopy methods to solve polynomial equations

Dedieu

Malajovich

Shub

2012

IMA Journal of Numerical Analysis

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Abstract. Given a C 1 path of systems of homogeneous polynomial equations ft, t ∈ [a, b] and an approximation xa to a zero ζa of the initial system fa, we show how to adaptively choose the step size for a Newton based homotopy method so that we approximate the lifted path (ft, ζt) in the space of (problems, solutions) pairs. The total number of Newton iterations is bounded in terms of the length of the lifted path in the condition metric.

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Section: Thenmentioning

confidence: 99%

“…V,(f,ζ) is the Riemannian metric in V inherited from the usual metric on P(H (d) ) × P(C n+1 ). This condition metric is studied in more details in Beltrán-Dedieu-Malajovich-Shub [2] and [3], Beltrán-Shub [6] and [7], and in Boito-Dedieu [9].…”

Section: Introductionmentioning

confidence: 99%

Adaptive step-size selection for homotopy methods to solve polynomial equations

Dedieu

Malajovich

Shub

2012

IMA Journal of Numerical Analysis

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“…). In [19] we prove that µ is an equivariant Morse function defined in W with a unique orbit of minima given by the orbit B of the pair of (5.2) under the action of the unitary group (U, (h, ζ))…”

Section: Also Properties Of Geodesics As We Learn Them Can Inform The...mentioning

confidence: 96%

The complexity and geometry of numerically solving polynomial systems.

Beltrán¹,

Shub²

2013

Recent Advances in Real Complexity and Computation

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These pages contain a short overview on the state of the art of efficient numerical analysis methods that solve systems of multivariate polynomial equations. We focus on the work of Steve Smale who initiated this research framework, and on the collaboration between Stephen Smale and Michael Shub, which set the foundations of this approach to polynomial system-solving, culminating in the more recent advances of Carlos Beltrán, Luis

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“…We consider here a general example in GL 5 . Let us take X 0 = diag (13,7,3,9,5) and X 1 = U SV * , where S = diag (8,5,2,4,6) and U, V are randomly chosen orthogonal matrices. Notice that choosing one of the endpoints as a positive diagonal matrix does not cause any loss of generality, since we can always apply orthogonal transformations that send a given matrix to a diagonal form.…”

Section: General Matricesmentioning

confidence: 99%

“…The interest of considering the condition metric in certain spaces of matrices or, more generally, in spaces of polynomial systems, comes from recent papers by Shub [19], Beltrán and Shub [3], [4], and Beltrán, Dedieu, Malajovich, and Shub [2], where these authors follow geodesics for the condition metric in certain solution varieties to improve classical complexity bounds for Bézout's theorem. These geodesics are designed to avoid ill-conditioned problems.…”

Section: Introductionmentioning

confidence: 99%

The Condition Metric in the Space of Rectangular Full Rank Matrices

Boito¹,

Dedieu

2010

SIAM J. Matrix Anal. & Appl.

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The condition metric in spaces of polynomial systems has been introduced and studied in a series of papers by Beltrán, Dedieu, Malajovich, and Shub. The interest of this metric comes from the fact that the associated geodesics avoid ill-conditioned problems and are a useful tool to improve classical complexity bounds for Bézout's theorem. The linear case is examined here: using nonsmooth nonconvex analysis techniques, we study the properties of condition geodesics in the space of full rank, real, or complex rectangular matrices. Our main results include an existence theorem for the boundary problem, a differential inclusion for such geodesics based on Clarke's generalized gradients, regularity properties, and a detailed description of a few particular cases: diagonal and unitary matrices. Moreover, we study condition geodesics from a numerical viewpoint, and we develop an effective algorithm that allows us to compute geodesics with given endpoints and helps to illustrate theoretical results and formulate new conjectures.

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On the Geometry and Topology of the Solution Variety for Polynomial System Solving

Cited by 8 publications

References 22 publications

Adaptive step-size selection for homotopy methods to solve polynomial equations

Adaptive step-size selection for homotopy methods to solve polynomial equations

The complexity and geometry of numerically solving polynomial systems.

The Condition Metric in the Space of Rectangular Full Rank Matrices

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