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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.

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References

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

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

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

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Smoothed analysis of homotopy for polynomial systems:

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

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An concise view of condition:

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

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A pioneer of condition:

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

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A first notion of condition for linear programming:

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

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

  1. Shub, M., Smale, S.: Complexity of Bézout’s theorem I: geometric aspects. J. Am. Math. Soc., 459–501 (1993)

  2. Shub, M., Smale, S.: Complexity of Bézout’s theorem II: volumes and probabilities. Computational Algebraic Geometry, 267–285 (1993)

  3. Shub, M., Smale, S.: Complexity of Bézout’s theorem III: condition number and packing. J. Complex. 9(1), 4–14 (1993)

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  4. Shub, M., Smale, S.: Complexity of Bézout’s theorem IV: probability of success; extensions. SIAM J. Numer. Anal. 33(1), 128–148 (1996)

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  5. Shub, M., Smale, S.: Complexity of Bézout’s theorem V: polynomial time. Theor. Comput. Sci. 133(1), 141–164 (1994)

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

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

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

  1. 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)

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

  1. 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

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Authors and Affiliations

  1. Toulouse, France

    Jean-Claude Yakoubsohn

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  1. Jean-Claude Yakoubsohn
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Correspondence to Jean-Claude Yakoubsohn.

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Yakoubsohn, JC. The Unavoidable Condition… A Report on the Book. Jahresber. Dtsch. Math. Ver. 117, 225–229 (2015). https://doi.org/10.1365/s13291-015-0117-y

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