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...
Uložené v:
| Vydané v: | Journal of Complexity Ročník 24; číslo 5; s. 582 - 605 |
|---|---|
| Hlavní autori: | , , , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Elsevier Inc
01.10.2008
|
| Predmet: | |
| ISSN: | 0885-064X, 1090-2708 |
| On-line prístup: | Získať plný text |
| Tagy: |
Pridať tag
Žiadne tagy, Buďte prvý, kto otaguje tento záznam!
|
| Shrnutí: | 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 |