Are resultant methods numerically unstable for multidimensional rootfinding?

Noferini, Vanni and Townsend, Alex (2015) Are resultant methods numerically unstable for multidimensional rootfinding? [MIMS Preprint]

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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) resultant is inherently and spectacularly numerically unstable by a factor that grows exponentially with the dimension. We also show that the Sylvester resultant 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. 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.

Item Type:	MIMS Preprint
Uncontrolled Keywords:	resultants, rootfinding, conditioning, multivariate polynomials, Cayley, Sylvester
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 13 Commutative rings and algebras MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User:	Dr V Noferini
Date Deposited:	06 Jul 2015
Last Modified:	08 Nov 2017 18:18
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/2332

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