Newton's method with deflation for isolated singularities of polynomial systems

Authors:
Anton Leykin;Jan Verschelde;Ailing Zhao
Affiliations:
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL;Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL;Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL
Venue:
Theoretical Computer Science
Year:
2006

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Abstract

We present a modification of Newton's method to restore quadratic convergence for isolated singular solutions of polynomial systems. Our method is symbolic-numeric: we produce a new polynomial system which has the original multiple solution as a regular root. Using standard bases, a tool for the symbolic computation of multiplicities, we show that the number of deflation stages is bounded by the multiplicity of the isolated root. Our implementation performs well on a large class of applications.