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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| Vydáno v: | Journal of Complexity Ročník 24; číslo 5; s. 582 - 605 |
|---|---|
| Hlavní autoři: | , , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Elsevier Inc
01.10.2008
|
| Témata: | |
| ISSN: | 0885-064X, 1090-2708 |
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| Abstract | 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
)
)
. |
|---|---|
| AbstractList | 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
)
)
. |
| Author | Malajovich, Gregorio Krick, Teresa Wschebor, Mario Cucker, Felipe |
| Author_xml | – sequence: 1 givenname: Felipe surname: Cucker fullname: Cucker, Felipe email: macucker@cityu.edu.hk organization: Department of Mathematics, City University of Hong Kong, Hong Kong – sequence: 2 givenname: Teresa surname: Krick fullname: Krick, Teresa email: krick@dm.uba.ar organization: Departamento de Matemática, Univ. de Buenos Aires, Argentina – sequence: 3 givenname: Gregorio surname: Malajovich fullname: Malajovich, Gregorio email: gregorio@ufrj.br organization: Depto. de Matemática Aplicada, Univ. Federal do Rio de Janeiro, Brazil – sequence: 4 givenname: Mario surname: Wschebor fullname: Wschebor, Mario email: wschebor@cmat.edu.uy organization: Centro de Matemática, Universidad de la República, Uruguay |
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| Cites_doi | 10.1006/jcom.1993.1002 10.1016/S0304-3975(98)00190-X 10.1016/j.jco.2005.10.001 10.1016/S0747-7171(88)80006-3 10.1016/0304-3975(94)00065-4 10.1093/imanum/23.3.395 10.1007/BF01202001 10.1016/j.jco.2004.10.001 10.1006/jcom.1999.0503 10.1145/300515.300519 10.1016/S0747-7171(88)80005-1 10.1016/j.jco.2005.11.001 10.1145/79147.214077 10.1137/S1052623401386794 10.1137/0733008 |
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| Keywords | Polynomial systems Finite precision Counting algorithms |
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| References | Bank, Giusti, Heintz, Pardo (b1) 2005; 21 Cucker, Zhou (b9) 2007 Higham (b17) 1996 Grigoriev, Vorobjov (b13) 1988; 5 Malajovich (b19) 1994; 133 Heintz, Roy, Solerno (b16) 1994 Shub, Smale (b22) 1993; 9 Weyl (b26) 1932 Cucker (b6) 1999; 15 Meer (b20) 2000; 242 Shub, Smale (b21) 1993; 6 Bürgisser, Cucker (b3) 2006; 22 Grigoriev, Vorobjov (b14) 1992; 2 Smale (b24) 1986 Han, Wagner (b15) 1990; 37 Dedieu, Priouret, Malajovich (b10) 2003; 23 Shub, Smale (b23) 1996; 33 Golub, Van Loan (b11) 1996 Cucker, Smale (b8) 1999; 46 Grigoriev (b12) 1988; 5 Blum, Cucker, Shub, Smale (b2) 1998 Wüthrich (b27) 1976; vol. 43 Cheung, Cucker (b4) 2006; 22 Li (b18) 2003; vol. 11 Collins (b5) 1975; vol. 33 Cucker, Peña (b7) 2002; 12 Tarski (b25) 1951 Collins (10.1016/j.jco.2008.03.001_b5) 1975; vol. 33 Malajovich (10.1016/j.jco.2008.03.001_b19) 1994; 133 Shub (10.1016/j.jco.2008.03.001_b22) 1993; 9 Heintz (10.1016/j.jco.2008.03.001_b16) 1994 Wüthrich (10.1016/j.jco.2008.03.001_b27) 1976; vol. 43 Weyl (10.1016/j.jco.2008.03.001_b26) 1932 Shub (10.1016/j.jco.2008.03.001_b23) 1996; 33 Li (10.1016/j.jco.2008.03.001_b18) 2003; vol. 11 Cheung (10.1016/j.jco.2008.03.001_b4) 2006; 22 Cucker (10.1016/j.jco.2008.03.001_b6) 1999; 15 Shub (10.1016/j.jco.2008.03.001_b21) 1993; 6 Grigoriev (10.1016/j.jco.2008.03.001_b12) 1988; 5 Blum (10.1016/j.jco.2008.03.001_b2) 1998 Han (10.1016/j.jco.2008.03.001_b15) 1990; 37 Tarski (10.1016/j.jco.2008.03.001_b25) 1951 Higham (10.1016/j.jco.2008.03.001_b17) 1996 Bürgisser (10.1016/j.jco.2008.03.001_b3) 2006; 22 Meer (10.1016/j.jco.2008.03.001_b20) 2000; 242 Dedieu (10.1016/j.jco.2008.03.001_b10) 2003; 23 Smale (10.1016/j.jco.2008.03.001_b24) 1986 Grigoriev (10.1016/j.jco.2008.03.001_b13) 1988; 5 Golub (10.1016/j.jco.2008.03.001_b11) 1996 Cucker (10.1016/j.jco.2008.03.001_b7) 2002; 12 Cucker (10.1016/j.jco.2008.03.001_b8) 1999; 46 Bank (10.1016/j.jco.2008.03.001_b1) 2005; 21 Cucker (10.1016/j.jco.2008.03.001_b9) 2007 Grigoriev (10.1016/j.jco.2008.03.001_b14) 1992; 2 |
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| Snippet | 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
)
)
)... |
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| Title | A numerical algorithm for zero counting, I: Complexity and accuracy |
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