Numerical Instability of Resultant Methods for Multidimensional Rootfinding
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Numerical instability.pdf
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1.18 MB
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Author(s)
Noferini, Vanni
Townsend, Alex John
Date Issued
March 2016
Journal
SIAM Journal on Numerical Analysis
Citation
Noferini, Vanni, and Alex Townsend. “Numerical Instability of Resultant Methods for Multidimensional Rootfinding.” SIAM J. Numer. Anal. 54, no. 2 (January 2016): 719–743. © 2016, Society for Industrial and Applied Mathematics.
Version
Final published version
Abstract
Hidden-variable resultant methods are a class of algorithms for solving multidimensional polynomial rootfinding problems. In two dimensions, when significant care is taken, they are competitive practical rootfinders. However, in higher dimensions they are known to miss zeros, calculate roots to low precision, and introduce spurious solutions. We show that the hidden variable resultant method based on the Cayley (Dixon or Bézout) matrix is inherently and spectacularly numerically unstable by a factor that grows exponentially with the dimension. We also show that the Sylvester matrix for solving bivariate polynomial systems can square the condition number of the problem. In other words, two popular hidden variable resultant methods are numerically unstable, and this mathematically explains the difficulties that are frequently reported by practitioners. Regardless of how the constructed polynomial eigenvalue problem is solved, severe numerical difficulties will be present. Along the way, we prove that the Cayley resultant is a generalization of Cramer's rule for solving linear systems and generalize Clenshaw's algorithm to an evaluation scheme for polynomials expressed in a degree-graded polynomial basis.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1137/15m1022513
