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Condition pp 223-232 | Cite as

Average Analysis of the RCC Condition Number

  • Peter Bürgisser
  • Felipe Cucker
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 349)

Abstract

In Chap.  11 we considered the primal–dual pair ( SP)–( SD) of linear programming optimization problems and analyzed two algorithms that in case both problems are feasible, return optimizers x and y for them, respectively. Recall that here \(A \in \mathbb {R}^{m\times n}\), \(b \in \mathbb {R}^{m}\), \(c \in \mathbb {R}^{n}\), nm≥1, and d=(A,b,c).

To analyze these algorithms we introduced the condition number Open image in new window and the main results in the previous chapter bound the cost of these algorithms by and respectively. Furthermore, this task can be done with finite precision and the result is correct as long as the machine epsilon satisfies This means that the number of digits or bits necessary to perform the computation is bounded by Open image in new window .

The goal of this chapter, following a line of thought well established in our development, is to eliminate Open image in new window from these bounds via a probabilistic analysis. We do so for Gaussian triples d, i.e., we assume that the entries of A, b, and c are i.i.d. random variables with standard normal distribution.

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Peter Bürgisser
    • 1
  • Felipe Cucker
    • 2
  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany
  2. 2.Department of MathematicsCity University of Hong KongHong KongHong Kong SAR

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