Felipe Cucker
Fast Computation of Zeros of Polynomial Systems with Bounded Degree under Finite-precision.
Пнд, 05/07/2012 - 15:01 — SomNambulAuthors: Felipe Cucker, Vera Roshchina, Irenee Briquel, Javier Pena
Subjects: Numerical Analysis
AbstractA solution for Smale's 17th problem, for the case of systems with bounded
degree was recently given. This solution, an algorithm computing approximate
zeros of complex polynomial systems in average polynomial time, assumed
infinite precision. In this paper we describe a finite-precision version of
this algorithm. Our main result shows that this version works within the same
time bounds and requires a precision which, on the average, amounts to a
polynomial amount of bits in the mantissa of the intervening floating-point
numbers.A Numerical Algorithm for Zero Counting. III: Randomization and Condition.
Пнд, 07/12/2010 - 15:01 — SomNambulAuthors: Felipe Cucker, Teresa Krick, Gregorio Malajovich, Mario Wschebor
Subjects: Numerical Analysis
AbstractIn a recent paper (Cucker, Krick, Malajovich and Wschebor, A Numerical
Algorithm for Zero Counting. I: Complexity and accuracy, J. Compl.,24:582-605,
2008) we analyzed a numerical algorithm for computing the number of real zeros
of a polynomial system. The analysis relied on a condition number kappa(f) for
the input system f. In this paper, we look at kappa(f) as a random variable
derived from imposing a probability measure on the space of polynomial systems
and give bounds for both the tail P{kappa(f) > a} and the expected value E(log
kappa(f)).Smoothed Analysis of Moore-Penrose Inversion.
Пт, 02/26/2010 - 16:01 — SomNambulAuthors: Peter Buergisser, Felipe Cucker
Subjects: Numerical Analysis
AbstractWe perform a smoothed analysis of the condition number of rectangular
matrices. We prove that, asymptotically, the expected value of this condition
number depends only of the elongation of the matrix, and not on the center and
variance of the underlying probability distribution.A Numerical Algorithm for Zero Counting. II: Distance to Ill-posedness and Smoothed Analysis.
Чт, 09/24/2009 - 12:35 — SomNambulAuthors: Felipe Cucker, Teresa Krick, Gregorio Malajovich, Mario Wschebor
Subjects: Numerical Analysis
AbstractWe show a Condition Number Theorem for the condition number of zero counting
for real polynomial systems. That is, we show that this condition number equals
the inverse of the normalized distance to the set of ill-posed systems (i.e.,
those having multiple real zeros). As a consequence, a smoothed analysis of
this condition number follows.A numerical algorithm for zero counting II: Randomization and Condition.
Ср, 09/23/2009 - 13:19 — SomNambulAuthors: Felipe Cucker, Teresa Krick, Gregorio Malajovich, Mario Wschebor
Subjects: Numerical Analysis
AbstractThis paper was witdrawn by the authors.
A numerical algorithm for zero counting II: Randomization and Condition.
Ср, 09/23/2009 - 13:19 — SomNambulAuthors: Felipe Cucker, Teresa Krick, Gregorio Malajovich, Mario Wschebor
Subjects: Numerical Analysis
AbstractThis paper was witdrawn by the authors.
On a Problem Posed by Steve Smale.
Пнд, 09/14/2009 - 09:01 — SomNambulAuthors: Peter Buergisser, Felipe Cucker
Subjects: Numerical Analysis
AbstractThe 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 Beltran 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 efficiently implements a non-constructive idea of Mike Shub.- Читать далее
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