On a problem posed by Steve Smale
Peter Bürgisser,Felipe Cucker +1 more
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In this article, a linear homotopy algorithm was proposed to solve the 17th problem of the 21st century, which asks for the existence of a deterministic algorithm computing an approximate solution of a system of n complex polynomials in n unknowns in time polynomial, on the average, in the size N of the input system.Abstract:
The 17th of the problems proposed by Steve Smale for the 21st century asks for the existence of a deterministic algorithm computing an approximate solution of a system of n complex polynomials in n unknowns in time polynomial, on the average, in the size N of the input system. A partial solution to this problem was given by Carlos Beltr an and Luis Miguel Pardo who exhibited a randomized algorithm doing so. In this paper we further extend this result in several directions. Firstly, we exhibit a linear homotopy algorithm that eciently implements a nonconstructive idea ofread more
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Condition : The Geometry of Numerical Algorithms
Peter Brgisser,Felipe Cucker +1 more
TL;DR: This book puts special emphasis on the probabilistic analysis of numerical algorithms via the analysis of the corresponding condition, and contains a condition-based course on linear programming that fills a gap between the current elementary expositions of the subject based on the simplex method and those focusing on convex programming.
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A Deterministic Algorithm to Compute Approximate Roots of Polynomial Systems in Polynomial Average Time
TL;DR: A deterministic algorithm that computes an approximate root of n complex polynomial equations in n unknowns in average Polynomial time with respect to the size of the input, in the Blum–Shub–Smale model with square root is described.
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Robust Certified Numerical Homotopy Tracking
Carlos Beltrán,Anton Leykin +1 more
TL;DR: In this paper, a path-following algorithm for finding approximate zero of polynomial equations in the Turing machine model has been proposed, and the total bit complexity is linear in the length of the path in the condition metric, and polynomially in the logarithm of the maximum of the condition number along the path.
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What is numerical algebraic geometry
TL;DR: This article provides a short introduction to numerical algebraic geometry with the subsequent articles in this special issue considering three current research topics: solving structured systems, certifying the results of numerical computations, and performing algebraic computations numerically via Macaulay dual spaces.
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Certified numerical homotopy tracking
Carlos Beltrán,Anton Leykin +1 more
TL;DR: In this article, a regular homotopy path connecting an approximate zero of the start system to an approximate 0 of the target system is tracked using the alpha theory of Smale.
References
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Mathematical problems for the next century
TL;DR: Arnabels invitation is inspired in part by Hilbert's list of 1900 (see e.g. [Browder, 1976]) and I have used that list to help design this essay.
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PRIMES is in P
TL;DR: In this paper, an unconditional deterministic polynomial-time algorithm that determines whether an input number is prime or composite is presented. But the algorithm is not deterministic in the sense that
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