Jahresbericht der Deutschen Mathematiker-Vereinigung

, Volume 117, Issue 3, pp 225–229 | Cite as

The Unavoidable Condition… A Report on the Book

Peter Bürgisser, Felipe Cucker: “Condition. The Geometry of Numerical Algorithms”. Grundlehren der mathematischen Wissenschaften 349, Springer 2013, 554 pp.
Book Review
  • 55 Downloads

References

The average study of condition with Gaussian and non Gaussian distributions:

  1. 1.
    Azaïs, J.-M., Wschebor, M.: Level Sets and Extrema of Random Processes and Fields. Wiley, New York (2009) CrossRefGoogle Scholar

The first elegant response at the Smale’s 17th problem which is detailed in the book… but to have the complete story, see also in the bibliography of the book, the papers of Beltrán-Pardo of years 2007, 2008 and 2009:

  1. 2.
    Beltrán, C., Pardo, L.M.: Fast linear homotopy to find approximate zeros of polynomial systems. Found. Comput. Math. 11(1), 95–129 (2011) MathSciNetCrossRefGoogle Scholar

Smoothed analysis of homotopy for polynomial systems:

  1. 3.
    Bürgisser, P., Cucker, F.: On a problem posed by Steve Smale. Ann. Math. 174(3), 1785–1836 (2011) CrossRefGoogle Scholar

An concise view of condition:

  1. 4.
    Dedieu, J.-P.: Condition operators, condition numbers and condition number theorem for the generalized eigenvalue problem. Linear Algebra Appl. 263, 1–24 (1997) MathSciNetCrossRefGoogle Scholar

A pioneer of condition:

  1. 5.
    Gastinel, N.: Linear Numerical Analysis. Academic Press, San Diego (1970) Google Scholar

A first notion of condition for linear programming:

  1. 6.
    Renegar, J.: Incorporating condition measures into the complexity theory of linear programming. SIAM J. Optim. 5(3), 506–524 (1995) MathSciNetCrossRefGoogle Scholar

The five precursor papers for condition at the dawn of the twenty-first century:

  1. 7.
    Shub, M., Smale, S.: Complexity of Bézout’s theorem I: geometric aspects. J. Am. Math. Soc., 459–501 (1993) Google Scholar
  2. 8.
    Shub, M., Smale, S.: Complexity of Bézout’s theorem II: volumes and probabilities. Computational Algebraic Geometry, 267–285 (1993) Google Scholar
  3. 9.
    Shub, M., Smale, S.: Complexity of Bézout’s theorem III: condition number and packing. J. Complex. 9(1), 4–14 (1993) MathSciNetCrossRefGoogle Scholar
  4. 10.
    Shub, M., Smale, S.: Complexity of Bézout’s theorem IV: probability of success; extensions. SIAM J. Numer. Anal. 33(1), 128–148 (1996) MathSciNetCrossRefGoogle Scholar
  5. 11.
    Shub, M., Smale, S.: Complexity of Bézout’s theorem V: polynomial time. Theor. Comput. Sci. 133(1), 141–164 (1994) MathSciNetCrossRefGoogle Scholar

… Without forgetting the bound of number steps of linear homotopy to get approximate zero with respect the integral of square of condition:

  1. 12.
    Shub, M.: Complexity of Bézout’s theorem VI: Geodesics in the condition (number) metric. Found. Comput. Math. 9(2), 171–178 (2009) MathSciNetCrossRefGoogle Scholar

The existence of a homotopy path bounded by the log of condition:

  1. 13.
    Beltrán, C., Shub, M.: Complexity of Bézout’s theorem VII: distance estimates in the condition (number) metric. Found. Comput. Math. 9(2), 179–195 (2009) MathSciNetCrossRefGoogle Scholar

The pioneer paper of smoothed analysis. last version… but the better!

  1. 14.
    Spielman, D.A., Teng, S.-H.: Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time. http://arxiv.org/pdf/cs/0111050.pdf

Copyright information

© Deutsche Mathematiker-Vereinigung and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.ToulouseFrance

Personalised recommendations