2021
DOI: 10.48550/arxiv.2105.08472
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Yet another eigenvalue algorithm for solving polynomial systems

Abstract: In latest years, several advancements have been made in symbolic-numerical eigenvalue techniques for solving polynomial systems. In this article, we add to this list by reducing the task to an eigenvalue problem in a considerably faster and simpler way than in previous methods. This results in an algorithm which solves systems with isolated solutions reliably and efficiently, outperforming homotopy methods in overdetermined cases. We provide an implementation in the proof-ofconcept Julia package EigenvalueSolv… Show more

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Cited by 3 publications
(5 citation statements)
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“…, 6, which characterizes when the three conics {h i = 0} ⊂ P 2 intersect. It has 21894 terms and can be computed as a 6 × 6 determinant, see [6,Chapter 3,§2]. Plugging in the coefficients, i.e.…”
Section: Resultantsmentioning
confidence: 99%
See 1 more Smart Citation
“…, 6, which characterizes when the three conics {h i = 0} ⊂ P 2 intersect. It has 21894 terms and can be computed as a 6 × 6 determinant, see [6,Chapter 3,§2]. Plugging in the coefficients, i.e.…”
Section: Resultantsmentioning
confidence: 99%
“…This can be done using any numerical method for solving polynomial systems. Recent eigenvalue methods are described in [2]. In case of many variables, it is favorable to use the polyhedral homotopies introduced in [13].…”
Section: Contour Integration and Homotopy Continuationmentioning
confidence: 99%
“…Nevertheless, there are efforts to overcome this obstacle using a variant called border basis, e.g., [75]. The same phenomena appear in the resultant computations [80], where there also recent efforts for improvements [13]. Numerical solvers are almost exclusively iterative algorithms that exploit a variant of Newton operator and they perform their computations in floating point arithmetic, e.g., [10,89,96].…”
Section: Methods For Solving Polynomial Systemsmentioning
confidence: 99%
“…Numerical solvers are almost exclusively iterative algorithms that exploit a variant of Newton operator and they perform their computations in floating point arithmetic, e.g., [10,89,96]. There are also approaches based on numerical linear algebra techniques, mainly on eigenvalue computations e.g., [13,17]. The most prominent representatives are the Homotopy Continuation (HC) algorithms [7,9,21,45,96,42].…”
Section: Methods For Solving Polynomial Systemsmentioning
confidence: 99%
“…This can be done using any numerical method for solving polynomial systems. Recent eigenvalue methods are described in [2]. In case of many variables, it is favorable to use the polyhedral homotopies introduced in [13].…”
Section: Contour Integration and Homotopy Continuationmentioning
confidence: 99%