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Michael Shub

  1. Smale's Fundamental Theorem of Algebra reconsidered.

    Втр, 04/03/2012 - 15:01 — SomNambul
    Authors: Michael Shub, Diego Armentano
    Subjects: Numerical Analysis
    Abstract

    In his 1981 Fundamental Theorem of Algebra paper Steve Smale initiated the
    complexity theory of finding a solution of polynomial equations of one complex
    variable by a variant of Newton's method. In this paper we reconsider his
    algorithm in the light of work done in the intervening years. Our main theorem
    raises more problems than it solves.

  2. Adaptative Step Size Selection for Homotopy Methods to Solve Polynomial Equations.

    Ср, 04/13/2011 - 15:01 — SomNambul
    Authors: Gregorio Malajovich, Jean-Pierre Dedieu, Michael Shub
    Subjects: Numerical Analysis
    Abstract

    Given a C^1 path of systems of homogeneous polynomial equations f_t, t in
    [a,b] and an approximation x_a to a zero zeta_a of the initial system f_a, we
    show how to adaptively choose the step size for a Newton based homotopy method
    so that we approximate the lifted path (f_t,zeta_t) in the space of (problems,
    solutions) pairs.

    The total number of Newton iterations is bounded in terms of the length of
    the lifted path in the condition metric.

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