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The Condition Number for Nonlinear Problems

  • Lenore Blum
  • Felipe Cucker
  • Michael Shub
  • Steve Smale

Abstract

The goal of this chapter is to describe a measure of condition for problems that search for a solution of nonlinear systems of equations. Eventually a condition number μ is defined as a bound on the infinitesimal error of a solution caused by an infinitesimal error in the defining system of equations. With our emphasis on polynomial systems, we impose a norm on the space of such systems that reflects an important computational invariant, the distance between the zeros. To avoid the distortion caused by very large zeros, the analysis and metrics are defined in a projective space setting. The result is a unitarily invariant theory.

Keywords

Condition Number Tangent Space Unitary Transformation Polynomial System Real Vector Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Lenore Blum
    • 1
    • 2
  • Felipe Cucker
    • 2
    • 3
  • Michael Shub
    • 4
  • Steve Smale
    • 2
  1. 1.International Computer Science InstituteBerkeleyUSA
  2. 2.Department of MathematicsCity University of Hong KongKowloonHong Kong
  3. 3.Universitat Pompeu FabraBarcelonaSpain
  4. 4.IBM T.J. Watson Research CenterYorktown HeightsUSA

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