## A Numerical Algorithm for Zero Counting. I: Complexity and Accuracy (2008)

Citations: | 3 - 2 self |

### BibTeX

@MISC{Cucker08anumerical,

author = {Felipe Cucker and Teresa Krick and Gregorio Malajovich and Mario Wschebor},

title = {A Numerical Algorithm for Zero Counting. I: Complexity and Accuracy},

year = {2008}

}

### OpenURL

### Abstract

We describe an algorithm to count the number of distinct real zeros of a polynomial (square) system f. The algorithm performs O(log(nDκ(f))) iterations (grid refinements) 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 κ(f) is a condition number for the system. Each iteration uses an exponential number of operations. The algorithm uses finite-precision arithmetic and a major feature in our results is a bound for the precision required to ensure the returned output is correct which is polynomial in n and D and logarithmic in κ(f). The algorithm parallelizes well in the sense that each iteration can be computed in parallel time polynomial in n, logD and log(κ(f)).

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Citation Context ...f‖ κ(f) = max min µnorm(f, x), . x∈Sn ‖f(x)‖∞ Remark 2.1. The quantity κ(f) is closely related to other condition numbers for similar problems. A version of the quantity µnorm(f, ζ) was introduced in =-=[21, 22, 23]-=- (see also [2, Chapter 12]) for a complex polynomial system f and a zero ζ of f in the complex unit sphere S n C ⊂ Cn+1 . The normalized condition number of such a system f was then defined to be µnor... |

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Citation Context ... n x ↦→ Nf (x) = exp x where exp x is the exponential map at x, ( −Df(x) −1 |TxS nf(x) ) exp x h = cos(‖h‖)x + sin(‖h‖) ‖h‖ h. Furthermore, the standard invariants of α-theory, introduced by Smale in =-=[24]-=-, can be defined as: ∥ β(f, x) = ∥Df(x) −1 |TxS nf(x) ∥ , Df(x) γ(f, x) = sup ∥ −1 |TxS nDk 1/(k−1) f(x) |(TxS n ) k , k! ∥ Remark 4.1. k≥2 α(f, x) = β(f, x)γ(f, x). (i) It is easy to see that β(f, x)... |

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Citation Context ...f‖ κ(f) = max min µnorm(f, x), . x∈Sn ‖f(x)‖∞ Remark 2.1. The quantity κ(f) is closely related to other condition numbers for similar problems. A version of the quantity µnorm(f, ζ) was introduced in =-=[21, 22, 23]-=- (see also [2, Chapter 12]) for a complex polynomial system f and a zero ζ of f in the complex unit sphere S n C ⊂ Cn+1 . The normalized condition number of such a system f was then defined to be µnor... |

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Citation Context ...f‖ κ(f) = max min µnorm(f, x), . x∈Sn ‖f(x)‖∞ Remark 2.1. The quantity κ(f) is closely related to other condition numbers for similar problems. A version of the quantity µnorm(f, ζ) was introduced in =-=[21, 22, 23]-=- (see also [2, Chapter 12]) for a complex polynomial system f and a zero ζ of f in the complex unit sphere S n C ⊂ Cn+1 . The normalized condition number of such a system f was then defined to be µnor... |

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Citation Context ...-posed inputs) with a better behavior viz the accumulation of round-off errors? For the problem of deciding the existence of (or computing) a zero of a polynomial system such algorithms were given in =-=[8, 6, 18]-=-. The goal of this article is to describe and analyze a numerical algorithm for zero counting. We will do so by developping appropriate versions of the tools used in [8, 6]. Let d1, . . . , dn ∈ N and... |

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Citation Context ...-posed inputs) with a better behavior viz the accumulation of round-off errors? For the problem of deciding the existence of (or computing) a zero of a polynomial system such algorithms were given in =-=[8, 6, 18]-=-. The goal of this article is to describe and analyze a numerical algorithm for zero counting. We will do so by developping appropriate versions of the tools used in [8, 6]. Let d1, . . . , dn ∈ N and... |

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Citation Context .... Algorithms counting connected components (and hence, in the zero-dimensional case, solutions) based on this method can be found in [14, 16], and in the straight-line program model of computation in =-=[1]-=-. These algorithms parallelise well in the sense that one can devise versions of them working in parallel polynomial time when an exponential number of processors is available. The #PR-completeness of... |

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Citation Context ...thod by Grigoriev and Vorobjov [13, 12] which uses exponential time. Algorithms counting connected components (and hence, in the zero-dimensional case, solutions) based on this method can be found in =-=[14, 16]-=-, and in the straight-line program model of computation in [1]. These algorithms parallelise well in the sense that one can devise versions of them working in parallel polynomial time when an exponent... |

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Citation Context ... 2 2 n 2(n+1)D κ(f) α∗ grid points to decide whether they are in A(f). These can be done independently. Then, we need to compute the number of connected components of Gη. This can be done (see, e.g., =-=[15]-=-) in parallel time O(ln(|Vη|)) 2 where |Vη| denotes the number of vertices of Gη and therefore, in parallel time at most O(n 2 (ln(nDκ(f)) 2 + ln(α∗) 2 )). Since this is the dominant step in the compu... |

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Citation Context ...em 1.1. One bounds how small u needs to be to guarantee a correct answer. Such a bound, needless to say, also depends on the condition of the data d. Examples of this type of analysis can be found in =-=[4, 6, 7, 8]-=-. In each of these references a condition number for the problem at hand occurs in the error analysis. We note that the one in [6] is essentially our κ(f). The rest of the paper is organized as follow... |

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Citation Context ...t years considerable attention was put on the complexity of counting problems over the reals. The counting complexity class #P R was introduced [20] and completeness results for #P R were established =-=[3]-=- for natural geometric problems notably, for the computation of the Euler characteristic of semialgebraic sets. As one could expect, the “basic” #P Rcomplete problem consists of counting the real zero... |

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Citation Context ... of zeros near points x with α(f, x) small enough. (iii) The Newton iteration presented above is not the iteration known as ‘projective Newton’. There is an alpha theory for that method, available in =-=[19]-=-. Here we use slight modifications of the quantities α, β and γ, more adapted to our purposes. We set β(f, x) := µnorm(f, x) ‖f(x)‖∞ ‖f‖ γ(f, x) := D3/2 2 µnorm(f, x) α(f, x) := β(f, x)γ(f, x). The de... |

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Citation Context ...thod by Grigoriev and Vorobjov [13, 12] which uses exponential time. Algorithms counting connected components (and hence, in the zero-dimensional case, solutions) based on this method can be found in =-=[14, 16]-=-, and in the straight-line program model of computation in [1]. These algorithms parallelise well in the sense that one can devise versions of them working in parallel polynomial time when an exponent... |

8 |
Counting problems over the reals
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Citation Context ...omial in n, log D and log(κ(f)). 1 Introduction In recent years considerable attention was put on the complexity of counting problems over the reals. The counting complexity class #P R was introduced =-=[20]-=- and completeness results for #P R were established [3] for natural geometric problems notably, for the computation of the Euler characteristic of semialgebraic sets. As one could expect, the “basic” ... |

7 |
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Citation Context ...em 1.1. One bounds how small u needs to be to guarantee a correct answer. Such a bound, needless to say, also depends on the condition of the data d. Examples of this type of analysis can be found in =-=[4, 6, 7, 8]-=-. In each of these references a condition number for the problem at hand occurs in the error analysis. We note that the one in [6] is essentially our κ(f). The rest of the paper is organized as follow... |

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Citation Context ...each point z in Bf (x) the Newton sequence starting at z converges to ζ. 4.3 Background material Theorem 4.3 is a consequence of the following two results, which are restatements of results proved in =-=[10]-=-. While [10] deals with Newton iteration on arbitrary complete real analytic Riemannian manifolds, here we reword them in terms of Newton iteration on the unit sphere S n (Example 1 in [10]). The γ-Th... |

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(Show Context)
Citation Context ...-posed inputs) with a better behavior viz the accumulation of round-off errors? For the problem of deciding the existence of (or computing) a zero of a polynomial system such algorithms were given in =-=[8, 6, 18]-=-. The goal of this article is to describe and analyze a numerical algorithm for zero counting. We will do so by developping appropriate versions of the tools used in [8, 6]. Let d1, . . . , dn ∈ N and... |

1 |
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Citation Context ... on quantifier elimination for the theory of the reals. Its complexity is hyperexponential. Algorithms with improved complexity (doubly exponential) were devised in the 70s by Collins [5] and Wütrich =-=[27]-=-. A breakthrough was reached a decade ∗ Partially supported by City University SRG grant 7002106. † Partially supported by grants UBACyT X112/06-09, CONICET PIP 2461/00 and ANPCyT 33671/05. ‡ Partiall... |