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

Manifold 

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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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