Higher-Order Deflation for Polynomial Systems With Isolated Singular Solutions

  • Anton Leykin
  • Jan Verschelde
  • Ailing Zhao
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 146)


Given an approximation to a multiple isolated solution of a system of polynomial equations, we provided a symbolic-numeric deflation algorithm to restore the quadratic convergence of Newton’s method. Using first-order derivatives of the polynomials in the system, our first-order deflation method creates an augmented system that has the multiple isolated solution of the original system as a regular solution.

In this paper we consider two approaches to computing the “multiplicity structure” at a singular isolated solution. An idea coming from one of them gives rise to our new higher-order deflation method. Using higher-order partial derivatives of the original polynomials, the new algorithm reduces the multiplicity faster than our first method for systems which require several first-order deflation steps. In particular: the number of higher-order deflation steps is bounded by the number of variables.

Key words

Deflation isolated singular solutions Newton’s method multiplicity polynomial systems reconditioning symbolic-numeric computations 

AMS(MOS) subject classifications

Primary 65H10 Secondary 14Q99 68W30 


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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Anton Leykin
    • 1
  • Jan Verschelde
    • 2
  • Ailing Zhao
    • 2
  1. 1.Institute for Mathematics and its ApplicationsUniversity of MinnesotaMinneapolisUSA
  2. 2.Department of Mathematics, Statistics, and Computer ScienceUniversity of Illinois at ChicagoChicagoUSA

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