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Felipe Cucker

Publications and source records attributed to Felipe Cucker.

34 records · Page 2Linked to original sources
arXiv2012

Fast Computation of Zeros of Polynomial Systems with Bounded Degree under Finite-precision

A 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.

math.NA↗
arXiv2011

On a Problem Posed by Steve Smale

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 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. This algorithm is then used in a randomized algorithm, call it LV, a la Beltran-Pardo. Secondly, we perform a smoothed analysis (in the sense of Spielman and Teng) of algorithm LV and prove that its smoothed complexity is polynomial in the input size and $σ^{-1}$, where $σ$ controls the size of of the random perturbation of the input systems. Thirdly, we perform a condition-based analysis of LV. That is, we give a bound, for each system $f$, of the expected running time of LV with input $f$. In addition to its dependence on $N$ this bound also depends on the condition of $f$. Fourthly, and to conclude, we return to Smale's 17th problem as originally formulated for deterministic algorithms. We exhibit such an algorithm and show that its average complexity is $N^{O(\log\log N)}$. This is nearly a solution to Smale's 17th problem.

math.NA↗
arXiv2011

A Numerical Algorithm for Zero Counting. III: Randomization and Condition

In 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)).

math.NA↗
arXiv2010

Coverage processes on spheres and condition numbers for linear programming

This paper has two agendas. Firstly, we exhibit new results for coverage processes. Let $p(n,m,α)$ be the probability that $n$ spherical caps of angular radius $α$ in $S^m$ do not cover the whole sphere $S^m$. We give an exact formula for $p(n,m,α)$ in the case $α\in[π/2,π]$ and an upper bound for $p(n,m,α)$ in the case $α\in [0,π/2]$ which tends to $p(n,m,π/2)$ when $α\toπ/2$. In the case $α\in[0,π/2]$ this yields upper bounds for the expected number of spherical caps of radius $α$ that are needed to cover $S^m$. Secondly, we study the condition number ${\mathscr{C}}(A)$ of the linear programming feasibility problem $\exists x\in\mathbb{R}^{m+1}Ax\le0,x\ne0$ where $A\in\mathbb{R}^{n\times(m+1)}$ is randomly chosen according to the standard normal distribution. We exactly determine the distribution of ${\mathscr{C}}(A)$ conditioned to $A$ being feasible and provide an upper bound on the distribution function in the infeasible case. Using these results, we show that $\mathbf{E}(\ln{\mathscr{C}}(A))\le2\ln(m+1)+3.31$ for all $n>m$, the sharpest bound for this expectancy as of today. Both agendas are related through a result which translates between coverage and condition.

math.PR↗
arXiv2010

Smoothed Analysis of Moore-Penrose Inversion

We 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.

math.NA↗
arXiv2009

Adversarial Smoothed Analysis

The purpose of this note is to extend the results on uniform smoothed analysis of condition numbers from \cite{BuCuLo:07} to the case where the perturbation follows a radially symmetric probability distribution. In particular, we will show that the bounds derived in \cite{BuCuLo:07} still hold in the case of distributions whose density has a singularity at the center of the perturbation, which we call {\em adversarial}.

math.NA↗
arXiv2008

Componentwise condition numbers of random sparse matrices

We prove an O(log n) bound for the expected value of the logarithm of the componentwise (and, a fortiori, the mixed) condition number of a random sparse n x n matrix. As a consequence, small bounds on the average loss of accuracy for triangular linear systems follow.

math.NA↗
arXiv2008

A Numerical Algorithm for Zero Counting. I: Complexity and Accuracy

We describe an algorithm to count the number of distinct real zeros of a polynomial (square) system f. The algorithm performs O(n D kappa(f)) iterations where n is the number of polynomials (as well as the dimension of the ambient space), D is a bound on the polynomials' degree, and kappa(f) is a condition number for the system. Each iteration uses an exponential number of operations. The algorithm uses finite-precision arithmetic and a polynomial bound for the precision required to ensure the returned output is correct is exhibited. This bound is a major feature of our algorithm since it is in contrast with the exponential precision required by the existing (symbolic) algorithms for counting real zeros. The algorithm parallelizes well in the sense that each iteration can be computed in parallel polynomial time with an exponential number of processors.

cs.CC↗
arXiv2007

Flocking in noisy environments

We consider a perturbed version of the dynamics of a flock introduced by Cucker and Smale ("Emergent behaviour in flocks") and prove, under similar conditions, that nearly-alignment (a concept that is precised in the text) is achieved with a certain probability, bounded from below.

math.DS↗
arXiv2006

The probability that a small perturbation of a numerical analysis problem is difficult

We prove a general theorem providing smoothed analysis estimates for conic condition numbers of problems of numerical analysis. Our probability estimates depend only on geometric invariants of the corresponding sets of ill-posed inputs. Several applications to linear and polynomial equation solving show that the estimates obtained in this way are easy to derive and quite accurate. The main theorem is based on a volume estimate of ε-tubular neighborhoods around a real algebraic subvariety of a sphere, intersected with a disk of radius σ. Besides εand σ, this bound depends only the dimension of the sphere and on the degree of the defining equations.

math.NA↗
arXiv2006

Smoothed analysis of complex conic condition numbers

Smoothed analysis of complexity bounds and condition numbers has been done, so far, on a case by case basis. In this paper we consider a reasonably large class of condition numbers for problems over the complex numbers and we obtain smoothed analysis estimates for elements in this class depending only on geometric invariants of the corresponding sets of ill-posed inputs. These estimates are for a version of smoothed analysis proposed in this paper which, to the best of our knowledge, appears to be new. Several applications to linear and polynomial equation solving show that estimates obtained in this way are easy to derive and quite accurate.

math.NA↗
arXiv2003

Counting complexity classes for numeric computations II: algebraic and semialgebraic sets

We define counting classes #P_R and #P_C in the Blum-Shub-Smale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over R, or of systems of polynomial equalities over C, respectively, turn out to be natural complete problems in these classes. We investigate to what extent the new counting classes capture the complexity of computing basic topological invariants of semialgebraic sets (over R) and algebraic sets (over C). We prove that the problem of computing the (modified) Euler characteristic of semialgebraic sets is FP_R^{#P_R}-complete, and that the problem of computing the geometric degree of complex algebraic sets is FP_C^{#P_C}-complete. We also define new counting complexity classes in the classical Turing model via taking Boolean parts of the classes above, and show that the problems to compute the Euler characteristic and the geometric degree of (semi)algebraic sets given by integer polynomials are complete in these classes. We complement the results in the Turing model by proving, for all k in N, the FPSPACE-hardness of the problem of computing the k-th Betti number of the zet of real zeros of a given integer polynomial. This holds with respect to the singular homology as well as for the Borel-Moore homology.

cs.CC↗
arXiv2003

On Tail Decay and Moment Estimates of a Condition Number for Random Linear Conic Systems

In this paper we study the distribution tails and the moments of a condition number which arises in the study of homogeneous systems of linear inequalities. We consider the case where this system is defined by a Gaussian random matrix and characterise the exact decay rates of the distribution tails, improve the existing moment estimates, and prove various limit theorems for large scale systems. Our results are of complexity theoretic interest, because interior-point methods and relaxation methods for the solution of systems of linear inequalities have running times that are bounded in terms of the logarithm and the square of the condition number respectively.

math.NA↗