A Blackbox Polynomial System Solver on Parallel Shared Memory Computers

Verschelde, Jan

doi:10.1007/978-3-319-99639-4_25

Cited by 4 publications

(3 citation statements)

References 19 publications

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“…The start system g(x) = 0 in a homotopy to solve f (x) = 0 was obtained by running the plain blackbox solver (the extended version is described in [40]) on 12 cores tracking 11,016 is less than two minutes. For reproducibility, the seed in the random number generators was 7131.…”

Section: One Fourfold Root Of Cyclic 9-rootsmentioning

confidence: 99%

Locating the Closest Singularity in a Polynomial Homotopy

Verschelde¹,

Viswanathan²

2022

Preprint

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A polynomial homotopy is a family of polynomial systems, where the systems in the family depend on one parameter. If for one parameter we know a regular solution, then what is the nearest value of the parameter for which the solution in the polynomial homotopy is singular? Applying the ratio theorem of Fabry on the solution paths defined by the homotopy, extrapolation methods can accurately locate the nearest singularity. Once the radius of convergence is known, then via a transformation of the continuation parameter, the series expansions of the solution curves will have convergence radius equal to one. To compute all coefficients of the series we propose the quaternion Fourier transform.

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Section: One Fourfold Root Of Cyclic 9-rootsmentioning

confidence: 99%

Locating the Closest Singularity in a Polynomial Homotopy

Verschelde¹,

Viswanathan²

2022

Preprint

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“…Consider the following code snippet. This numerical output is the essence of the blackbox solver for positive dimensional solution sets [Ver18].…”

Section: A Series Expansion For the Solution Starts Its Development Amentioning

confidence: 99%

“…At the time of writing, this paper is based on version 0.9.5 of phcpy, whereas version 0.1.5 was current at the time of [Ver14]. An example of these changes is that the software described in [SVW03] was recently parallelized for phcpy [Ver18].…”

Section: Introductionmentioning

confidence: 99%

Solving Polynomial Systems with phcpy

Otto¹,

Forbes²,

Verschelde³

2019

Proceedings of the Python in Science Conference

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The solutions of a system of polynomials in several variables are often needed, e.g.: in the design of mechanical systems, and in phase-space analyses of nonlinear biological dynamics. Reliable, accurate, and comprehensive numerical solutions are available through PHCpack, a FOSS package for solving polynomial systems with homotopy continuation.This paper explores new developments in phcpy, a scripting interface for PHCpack, over the past five years. For instance, phcpy is now available online through a JupyterHub server featuring Python2, Python3, and SageMath kernels. As small systems are solved in real-time by phcpy, they are suitable for interactive exploration through the notebook interface. Meanwhile, phcpy supports GPU parallelization, improving the speed and quality of solutions to much larger polynomial systems. From various model design and analysis problems in STEM, certain classes of polynomial system frequently arise, to which phcpy is well-suited.

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Robust Numerical Tracking of One Path of a Polynomial Homotopy on Parallel Shared Memory Computers

Telen

Barel

Verschelde

2020

Lecture Notes in Computer Science

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We consider the problem of tracking one solution path defined by a polynomial homotopy on a parallel shared memory computer. Our robust path tracker applies Newton's method on power series to locate the closest singular parameter value. On top of that, it computes singular values of the Hessians of the polynomials in the homotopy to estimate the distance to the nearest different path. Together, these estimates are used to compute an appropriate adaptive step size. For ndimensional problems, the cost overhead of our robust path tracker is O(n), compared to the commonly used predictor-corrector methods. This cost overhead can be reduced by a multithreaded program on a parallel shared memory computer.

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A Blackbox Polynomial System Solver on Parallel Shared Memory Computers

Cited by 4 publications

References 19 publications

Locating the Closest Singularity in a Polynomial Homotopy

Locating the Closest Singularity in a Polynomial Homotopy

Solving Polynomial Systems with phcpy

Robust Numerical Tracking of One Path of a Polynomial Homotopy on Parallel Shared Memory Computers

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