Multiplicity hunting and approximating multiple roots of polynomial systems

Giusti, M; Yakoubsohn, Jean-Claude

doi:10.1090/conm/604/12070

Cited by 16 publications

(28 citation statements)

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“…And further, we also compare our method with the other four deflation methods [9] on the following four small systems. 5 1 } at (0, 0, −1) with µ = 18. The result (see also in [9]) is below, where method A is in [2,12], method B is in [8], method C is in [5], method D is in [9], method E is our method VDSS.…”

Section: Experiments and Resultsmentioning

confidence: 99%

“…5 1 } at (0, 0, −1) with µ = 18. The result (see also in [9]) is below, where method A is in [2,12], method B is in [8], method C is in [5], method D is in [9], method E is our method VDSS. In Table 3, we denote P oly the number of the polynomials of the final deflation system and V ar the number of the variables in the final deflation system.…”

Section: Experiments and Resultsmentioning

confidence: 99%

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A new deflation method for verifying the isolated singular zeros of polynomial systems

Cheng

Dou

Wen

2020

Journal of Computational and Applied Mathematics

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In this paper, we develop a new deflation technique for refining or verifying the isolated singular zeros of polynomial systems. Starting from a polynomial system with an isolated singular zero, by computing the derivatives of the input polynomials directly or the linear combinations of the related polynomials, we construct a new system, which can be used to refine or verify the isolated singular zero of the input system. In order to preserve the accuracy in numerical computation as much as possible, new variables are introduced to represent the coefficients of the linear combinations of the related polynomials. To our knowledge, it is the first time that considering the deflation problem of polynomial systems from the perspective of the linear combination. Some acceleration strategies are proposed to reduce the scale of the final system. We also give some further analysis of the tolerances we use, which can help us have a better understanding of our method. The experiments show that our method is effective and efficient. Especially, it works well for zeros with high multiplicities of large systems. It also works for isolated singular zeros of non-polynomial systems.

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Section: Experiments and Resultsmentioning

confidence: 99%

Section: Experiments and Resultsmentioning

confidence: 99%

A new deflation method for verifying the isolated singular zeros of polynomial systems

Cheng

Dou

Wen

2020

Journal of Computational and Applied Mathematics

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“…This construction improves the constructions in [18,5] since no new variables are added. It also improves the constructions presented in [14] and the "kerneling" method of [11] by adding a smaller number of equations at each deflation step. Note that, in [11], there are smart preprocessing and postprocessing steps which could be utilized in combination with our method.…”

Section: Deflation Using First Differentialsmentioning

confidence: 99%

“…It also improves the constructions presented in [14] and the "kerneling" method of [11] by adding a smaller number of equations at each deflation step. Note that, in [11], there are smart preprocessing and postprocessing steps which could be utilized in combination with our method. In the preprocessor, one adds directly partial derivatives of polynomials which are zero at the root.…”

Section: Deflation Using First Differentialsmentioning

confidence: 99%

On deflation and multiplicity structure

Hauenstein

Mourrain

Szántó

2017

Journal of Symbolic Computation

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International audienceThis paper presents two new constructions related to singular solutions of polynomial systems. The first is a new deflation method for an isolated singular root. This construction uses a single linear differential form defined from the Jacobian matrix of the input, and defines the deflated system by applying this differential form to the original system. The advantages of this new deflation is that it does not introduce new variables and the increase in the number of equations is linear in each iteration instead of the quadratic increase of previous methods. The second construction gives the coefficients of the so-called inverse system or dual basis, which defines the multiplicity structure at the singular root. We present a system of equations in the original variables plus a relatively small number of new variables that completely deflates the root in one step. We show that the isolated simple solutions of this new system correspond to roots of the original system with given multiplicity structure up to a given order. Both constructions are " exact " in that they permit one to treat all conjugate roots simultaneously and can be used in certification procedures for singular roots and their multiplicity structure with respect to an exact rational polynomial system

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“…The proof of the non-quantified quadratic convergence [24,Theorem 3.16] of Algorithm 1 in [24] has also been simplified. There are other approaches to compute isolated multiple zeros or zero clusters, e.g., corrected Newton methods [33,5,6,7,14,15,34,35,29], deflation techniques [32,48,31,20,4,21,22,45,3,36,24,27,11,26,18,16]. We refer to [10,16] for excellent introductions of previous works on approximating multiple zeros.…”

Section: Introductionmentioning

confidence: 99%

On isolation of simple multiple zeros and clusters of zeros of polynomial systems

Hao

Jiang

et al. 2019

Math. Comp.

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Given a polynomial system f associated with a simple multiple zero x of multiplicity µ, we give a computable lower bound on the minimal distance between the simple multiple zero x and other zeros of f . If x is only given with limited accuracy, we propose a numerical criterion that f is certified to have µ zeros (counting multiplicities) in a small ball around x. Furthermore, for simple double zeros and simple triple zeros whose Jacobian is of normalized form, we define modified Newton iterations and prove the quantified quadratic convergence when the starting point is close to the exact simple multiple zero. For simple multiple zeros of arbitrary multiplicity whose Jacobian matrix may not have a normalized form, we perform unitary transformations and modified Newton iterations, and prove its non-quantified quadratic convergence and its quantified convergence for simple triple zeros.

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Multiplicity hunting and approximating multiple roots of polynomial systems

Cited by 16 publications

References 0 publications

A new deflation method for verifying the isolated singular zeros of polynomial systems

A new deflation method for verifying the isolated singular zeros of polynomial systems

On deflation and multiplicity structure

On isolation of simple multiple zeros and clusters of zeros of polynomial systems

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