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 ( log ( n D κ ( 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...

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Bibliographic Details
Published in:Journal of Complexity Vol. 24; no. 5; pp. 582 - 605
Main Authors: Cucker, Felipe, Krick, Teresa, Malajovich, Gregorio, Wschebor, Mario
Format: Journal Article
Language:English
Published: Elsevier Inc 01.10.2008
Subjects:
Counting algorithms
Finite precision
Polynomial systems
Polynomial systems
Finite precision
Counting algorithms
ISSN:0885-064X, 1090-2708
Online Access:Get full text
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Summary:We describe an algorithm to count the number of distinct real zeros of a polynomial (square) system f . The algorithm performs O ( log ( n D κ ( 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 of our results is a bound for the precision required to ensure that 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 polynomial time in n , log D and log ( κ ( f ) ) .
ISSN:0885-064X
1090-2708
DOI:10.1016/j.jco.2008.03.001