Algorithms for solving linear systems over cyclotomic fields
We consider the problem of solving a linear system A x = b over a cyclotomic field. Cyclotomic fields are special in that we can easily find a prime p for which the minimal polynomial m ( z ) for the field factors into a product of distinct linear factors. This makes it possible to develop fast modu...
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| Vydáno v: | Journal of symbolic computation Ročník 45; číslo 9; s. 902 - 917 |
|---|---|
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| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Elsevier Ltd
01.09.2010
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| Témata: | |
| ISSN: | 0747-7171, 1095-855X |
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| Abstract | We consider the problem of solving a linear system
A
x
=
b
over a cyclotomic field. Cyclotomic fields are special in that we can easily find a prime
p
for which the minimal polynomial
m
(
z
)
for the field factors into a product of distinct linear factors. This makes it possible to develop fast modular algorithms.
We give two output sensitive modular algorithms, one using multiple primes and Chinese remaindering, the other using linear
p
-adic lifting. Both use rational reconstruction to recover the rational coefficients in the solution vector. We also give a third algorithm which computes the solutions as ratios of two determinants modulo
m
(
z
)
using Chinese remaindering only. Because this representation is
d
=
deg
m
(
z
)
times more compact in general, we can compute it the fastest.
We have implemented the algorithms in Maple. Our benchmarks show that the third method is fastest on random inputs, but on real inputs arising from problems in computational group theory, the first two methods are faster because the solutions have small rational coefficients. |
|---|---|
| AbstractList | We consider the problem of solving a linear system
A
x
=
b
over a cyclotomic field. Cyclotomic fields are special in that we can easily find a prime
p
for which the minimal polynomial
m
(
z
)
for the field factors into a product of distinct linear factors. This makes it possible to develop fast modular algorithms.
We give two output sensitive modular algorithms, one using multiple primes and Chinese remaindering, the other using linear
p
-adic lifting. Both use rational reconstruction to recover the rational coefficients in the solution vector. We also give a third algorithm which computes the solutions as ratios of two determinants modulo
m
(
z
)
using Chinese remaindering only. Because this representation is
d
=
deg
m
(
z
)
times more compact in general, we can compute it the fastest.
We have implemented the algorithms in Maple. Our benchmarks show that the third method is fastest on random inputs, but on real inputs arising from problems in computational group theory, the first two methods are faster because the solutions have small rational coefficients. |
| Author | Chen, Liang Monagan, Michael |
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| Cites_doi | 10.1006/jsco.1995.1051 10.1090/S0002-9904-1946-08538-9 10.1016/S0747-7171(86)80021-9 10.1007/BF01459082 10.1016/j.jco.2005.04.002 |
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| Issue | 9 |
| Keywords | Linear systems Cyclotomic polynomials Modular algorithms Cyclotomic fields |
| Language | English |
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Complexity doi: 10.1016/j.jco.2005.04.002 |
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| Snippet | We consider the problem of solving a linear system
A
x
=
b
over a cyclotomic field. Cyclotomic fields are special in that we can easily find a prime
p
for... |
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| SubjectTerms | Cyclotomic fields Cyclotomic polynomials Linear systems Modular algorithms |
| Title | Algorithms for solving linear systems over cyclotomic fields |
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