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Condition Numbers and the Loss of Precision of Linear Equations

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

Abstract

The condition number of an invertible real or complex n x n matrix A is defined as
$$ \kappa (A) = \left\| A \right\|\;\left\| {{{A}^{{ - 1}}}} \right\|, $$
where ‖A‖ is the operator norm
$$ \left\| A \right\| = \mathop{{\sup }}\limits_{{x \ne 0}} \frac{{\left\| {Ax} \right\|}}{{\left\| x \right\|}} $$
and ℝ n or ℂ n is given the usual inner product. The condition number measures the relative error in the solution of the system of linear equations
$$ {\text{Ax = v}}{\text{.}} $$

Keywords

Condition Number Unit Sphere Linear Case Average Loss Conjugate Gradient Algorithm 
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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