Penalty Function Based Critical Point Approach to Compute Real Witness Solution Points of Polynomial Systems

Wu, Wenyuan; Chen, Changbo; Reid, Greg

doi:10.1007/978-3-319-66320-3_27

Cited by 7 publications

(10 citation statements)

References 26 publications

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“…Intuitively, this provides an approximate solution to the problem (2.1) for large enough β. Indeed, the paper [24] proves the following result justifying such intuitions.…”

Section: Preliminariessupporting

confidence: 54%

“…In this section, we recall some preliminary results that were introduced in [24] for computing witness points of rank-deficient polynomial systems.…”

Section: Preliminariesmentioning

confidence: 99%

“…To overcome this algebraic rank deficiency problem, the paper [24] introduces a penalty function based approach. Instead of solving the optimization problem (2.1), one considers the following unconstrained optimization problem:…”

Section: Preliminariesmentioning

confidence: 99%

“…When J f is rank-deficient, one of the main obstacles is that it is hard to find a tracing direction since the dimension of the nullspace of J f is at least two. Our starting point for solving this problem is a penalty function based method for computing witness points for every connected component of f −1 (0) [24]. We then extend it to a companion curve tracing method.…”

mentioning

confidence: 99%

“…The paper is structured as follows. In Section 2, we recall how to compute at least one witness point for every connected component of an arbitrary real algebraic variety, whose defining system allows to be rank-deficient, via the so-called penalty function method [24]. In Section 3, we illustrate by a simple example the challenge for tracing real algebraic curves defined by rank-deficient systems, with the initial witness points provided by the penalty function method, as well as the main idea of our companion curve method for handling this problem.…”

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confidence: 99%

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A Companion Curve Tracing Method for Rank-deficient Polynomial Systems

Wu,

Chen

2021

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We propose a method for tracing implicit real algebraic curves defined by polynomials with rank-deficient Jacobians. For a given curve f −1 (0), it first utilizes a regularization technique to compute at least one witness point per connected component of the curve. We improve this step by establishing a sufficient condition for testing the emptiness of f −1 (0). We also analyze the convergence rate and carry out an error analysis for refining the witness points. The witness points are obtained by computing the minimum distance of a random point to a smooth manifold embedding the curve while at the same time penalizing the residual of f at the local minima. To trace the curve starting from these witness points, we prove that if one drags the random point along a trajectory inside a tubular neighborhood of the embedded manifold of the curve, the projection of the trajectory on the manifold is unique and can be computed by numerical continuation. We then show how to choose such a trajectory to approximate the curve by computing eigenvectors of certain matrices. Effectiveness of the method is illustrated by examples.

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“…Intuitively, this provides an approximate solution to the problem (2.1) for large enough β. Indeed, the paper [24] proves the following result justifying such intuitions.…”

Section: Preliminariessupporting

confidence: 54%

“…In this section, we recall some preliminary results that were introduced in [24] for computing witness points of rank-deficient polynomial systems.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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