Yet another eigenvalue algorithm for solving polynomial systems

Bender, Matías R.; Telen, Simon

doi:10.48550/arxiv.2105.08472

Cited by 3 publications

(5 citation statements)

References 32 publications

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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%

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Contour Integration for Eigenvector Nonlinearities

Claes,

Meerbergen,

Telen

2023

SIAM J. Matrix Anal. Appl.

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Solving polynomial eigenvalue problems with eigenvector nonlinearities (PEPv) is an interesting computational challenge, outside the reach of the well-developed methods for nonlinear eigenvalue problems. We present a natural generalization of these methods which leads to a contour integration approach for computing all eigenvalues of a PEPv in a compact region of the complex plane. Our methods can be used to solve any suitably generic system of polynomial or rational function equations.

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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%

Contour Integration for Eigenvector Nonlinearities

Claes,

Meerbergen,

Telen

2023

SIAM J. Matrix Anal. Appl.

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“…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%

GPU-Based Homotopy Continuation for Minimal Problems in Computer Vision

Chien¹,

Fan²,

Abdelfattah³

et al. 2021

Preprint

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Systems of polynomial equations arise frequently in computer vision, especially in multiview geometry problems. Traditional methods for solving these systems typically aim to eliminate variables to reach a univariate polynomial, e.g., a tenth-order polynomial for 5-point pose estimation, using clever manipulations, or more generally using Grobner basis, resultants, and elimination templates, leading to successful algorithms for multiview geometry and other problems. However, these methods do not work when the problem is complex and when they do, they face efficiency and stability issues. Homotopy Continuation (HC) can solve more complex problems without the stability issues, and with guarantees of a global solution, but they are known to be slow. In this paper we show that HC can be parallelized on a GPU, showing significant speedups up to 26 times on polynomial benchmarks. We also show that GPU-HC can be generically applied to a range of computer vision problems, including 4view triangulation and trifocal pose estimation with unknown focal length, which cannot be solved with elimination template but they can be efficiently solved with HC. GPU-HC opens the door to easy formulation and solution of a range of computer vision problems.

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Section: Contour Integration and Homotopy Continuationmentioning

confidence: 99%