A numerical algorithm for zero counting. II: Distance to ill-posedness and smoothed analysis

Cucker, Felipe; Krick, Teresa; Malajovich, Gregorio; Wschebor, Mario

doi:10.1007/s11784-009-0127-4

Cited by 20 publications

(56 citation statements)

References 30 publications

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“…We observe that in contrast with the complexity analyses (of condition numbers closely related to κ aff ) in the literature (see, e.g., [2,3,[8][9][10][11]), the bounds in eorem 6.7 depend on E x ∈[−a, a] n (κ aff (f , x) n ) and not on max x ∈[−a, a] κ aff (f , x) n . Whereas the former has finite expectation (over f ), the la er has not.…”

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confidence: 71%

“…Although our approach follows the condition-based ideas of, e.g., [1,8,9,12,18], the complexity analysis in this paper would have been impossible without the continuous amortization technique developed in the exact numerical context [4,5]. We hope that this merging of techniques will start a fruitful exchange of ideas between different approaches to continuous computation.…”

Section: Introductionmentioning

confidence: 99%

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Plantinga-Vegter Algorithm takes Average Polynomial Time

Cucker

Ergür

Tonelli-Cueto

2019

Proceedings of the 2019 International Symposium on Symbolic and Algebraic Computation

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We exhibit a condition-based analysis of the adaptive subdivision algorithm due to Plantinga and Vegter. e first complexity analysis of the PV Algorithm is due to Burr, Gao and Tsigaridas who proved a O 2 τ d 4 log d worst-case cost bound for degree d plane curves with maximum coefficient bit-size τ .is exponential bound, it was observed, is in stark contrast with the good performance of the algorithm in practice. More in line with this performance, we show that, with respect to a broad family of measures, the expected time complexity of the PV Algorithm is bounded by O(d 7 ) for real, degree d, plane curves. We also exhibit a smoothed analysis of the PV Algorithm that yields similar complexity estimates. To obtain these results we combine robust probabilistic techniques coming from geometric functional analysis with condition numbers and the continuous amortization paradigm introduced by Burr, Krahmer and Yap. We hope this will motivate a fruitful exchange of ideas between the different approaches to numerical computation. CCS CONCEPTS•Mathematics of computing → Computations on polynomials; Interval arithmetic; • eory of computation → Design and analysis of algorithms; Computational geometry; KEYWORDS computational algebraic geometry, numerical methods, adaptive subdivision methods, isotopy of curves, complexity ACKNOWLEDGMENTSWe thank Michael Burr for useful discussions.

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confidence: 71%

Section: Introductionmentioning

confidence: 99%

Plantinga-Vegter Algorithm takes Average Polynomial Time

Cucker

Ergür

Tonelli-Cueto

2019

Proceedings of the 2019 International Symposium on Symbolic and Algebraic Computation

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“…where we used (28) for the second inequality. If ( ∞ ) 0 for all ∈ G, then ∞ ∈ S and d S (x, S) d S (x, ∞ ), hence (27) and we are done. So suppose that ( ∞ ) < 0 for some ∈ G and let s > 0 be the smallest real number such that ( s ) = 0 for some ∈ G. By construction, the set H ≔ { ∈ G | ( s ) = 0} is nonempty and element of G \ H is positive at s .…”

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confidence: 99%

Computing the Homology of Basic Semialgebraic Sets in Weak Exponential Time

Bürgisser

Cucker

Lairez

2018

J. ACM

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We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of basic semialgebraic sets which works in weak exponential time. That is, out of a set of exponentially small measure in the space of data, the cost of the algorithm is exponential in the size of the data. All algorithms previously proposed for this problem have a complexity which is doubly exponential (and this is so for almost all data). INTRODUCTIONSemialgebraic sets (that is, subsets of Euclidean spaces defined by polynomial equations and inequalities with real coefficients) come in a wide variety of shapes and this raises the problem of describing a given specimen, from the most primitive features, such as emptiness, dimension, or number of connected components, to finer ones, such as roadmaps, Euler-Poincaré characteristic, Betti numbers, or torsion coefficients.The Cylindrical Algebraic Decomposition (CAD) introduced by Collins [23] and Wüthrich [63] in the 1970's provided algorithms to compute these features that worked within time (sD) 2 O(n) where s is the number of defining equations, D a bound on their degree and n the dimension of the ambient space. Subsequently, a substantial effort was devoted to design algorithms for these problems with single exponential algebraic complexity bounds [4, and references therein], that is, bounds of the form (sD) n O(1) . Such algorithms have been found for deciding emptiness [6,37,49], for counting connected components [7,19,20,38,39], computing the dimension [9,41], the Euler-Poincaré characteristic [2], the first few Betti numbers [3], the top few Betti numbers [5] and roadmaps (embedded curves with certain topological properties) [11,21,50].As of today, however, no single exponential algorithm is known for the computation of the whole sequence of the homology groups (Betti numbers and torsion coefficients). For complex smooth projective varieties, Scheiblechner [52] has been able to provide an algorithm computing the Betti numbers (but not the torsion coefficients) in single exponential time relying on the algebraic De Rham cohomology. The same author provided a lower bound for this problem (assuming integer coefficients) in [51], where the problem is shown to be PS -hard.

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“…Note that the inclusion of M makes definition scale-invariant. This condition number is similar to the definition ofκ(f ) of Cucker et al [4] for the problem of computing isolated roots of polynomial systems R n → R n . Our definition is somewhat simpler than theirs, however, because of the assumption we have imposed that the domain of interest is [0, 1] n+1 rather than all of R n+1 .…”

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confidence: 75%

“…Our definition is somewhat simpler than theirs, however, because of the assumption we have imposed that the domain of interest is [0, 1] n+1 rather than all of R n+1 . This simplifying assumption obviates the need for introducing projective space and scaling of coefficients as in [4].…”

mentioning

confidence: 99%

A Condition Number Analysis of an Algorithm for Solving a System of Polynomial Equations with One Degree of Freedom

Srijuntongsiri¹,

Vavasis²

2011

SIAM J. Sci. Comput.

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This article considers the problem of solving a system of n real polynomial equations in n + 1 variables. We propose an algorithm based on Newton's method and subdivision for this problem. Our algorithm is intended only for nondegenerate cases, in which case the solution is a 1-dimensional curve. Our first main contribution is a definition of a condition number measuring reciprocal distance to degeneracy that can distinguish poor and well-conditioned instances of this problem. (Degenerate problems would be infinitely ill conditioned in our framework.) Our second contribution, which is the main novelty of our algorithm, is an analysis showing that its running time is bounded in terms of the condition number of the problem instance as well as n and the polynomial degrees. Introduction.We consider the problem of finding all zeros of a polynomial function f : [0, 1] n+1 → R n . The zero-set of such a function will, in the generic case, be a 1-dimensional algebraic set. Our algorithm is enumerative in nature and therefore is feasible only in the case of small values of n. We refer to this problem as the single-degree-of-freedom polynomial system (SDPS) problem.Perhaps the most common application of the SDPS problem is finding the intersection of two rational or polynomial surfaces, the so-called surface/surface intersection (SSI) problem. In this case, one is given two polynomials p 1 , p 2 both mapping R 2 to R 3 that each parametrizes a surface. The problem is to find their intersection, i.e., all points (s, t, u, v) in [0, 1] 4 such that p 1 (s, t) − p 2 (u, v) = 0. Other applications arise in robotics and motion planning. A final application is global optimization, in which one finds all the local minimizers by constructing a network of paths that connect local minimizers (see, e.g., [15]).Our proposed algorithm is a hybrid between subdivision and iterative methods. This hybrid idea has been used to solve SSI by Koparkar [14] and to solve line/surface intersection by Toth [24]. The approach is to subdivide the domain recursively, discard the subdomains found to contain no solutions, and invoke an iterative method to locate a solution once it is certain that the iterative method converges. The convergence tests used by Toth and Koparkar are both based on contraction mapping and evaluating ranges of functions.

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A numerical algorithm for zero counting. II: Distance to ill-posedness and smoothed analysis

Cited by 20 publications

References 30 publications

Plantinga-Vegter Algorithm takes Average Polynomial Time

Plantinga-Vegter Algorithm takes Average Polynomial Time

Computing the Homology of Basic Semialgebraic Sets in Weak Exponential Time

A Condition Number Analysis of an Algorithm for Solving a System of Polynomial Equations with One Degree of Freedom

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