On the complexity of inverting integer and polynomial matrices
An algorithm is presented that probabilistically computes the exact inverse of a nonsingular n × n integer matrix A using ( n 3 ( log | | A | | + log κ ( A ) ) ) 1 + o ( 1 ) bit operations. Here, | | A | | = max i j | A i j | denotes the largest entry in absolute value, κ ( A ) : = n | | A - 1 | | |...
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| Published in: | Computational complexity Vol. 24; no. 4; pp. 777 - 821 |
|---|---|
| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
Cham
Springer International Publishing
01.12.2015
|
| Subjects: | |
| ISSN: | 1016-3328, 1420-8954 |
| Online Access: | Get full text |
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| Summary: | An algorithm is presented that probabilistically computes the exact inverse of a nonsingular
n
×
n
integer matrix
A
using
(
n
3
(
log
|
|
A
|
|
+
log
κ
(
A
)
)
)
1
+
o
(
1
)
bit operations. Here,
|
|
A
|
|
=
max
i
j
|
A
i
j
|
denotes the largest entry in absolute value,
κ
(
A
)
:
=
n
|
|
A
-
1
|
|
|
|
A
|
|
is the condition number of the input matrix, and the “+
o
(1)” in the exponent indicates a missing factor
c
1
(
log
n
)
c
2
(
loglog
|
|
A
|
|
)
c
3
for positive real constants
c
1
,
c
2
,
c
3
. A variation of the algorithm is presented for polynomial matrices that computes the inverse of a nonsingular
n
×
n
matrix whose entries are polynomials of degree
d
over a field using
(
n
3
d
)
1
+
o
(
1
)
field operations. Both algorithms are randomized of the Las Vegas type: failure may be reported with probability at most 1/2, and if failure is not reported, then the output is certified to be correct in the same running time bound. |
|---|---|
| ISSN: | 1016-3328 1420-8954 |
| DOI: | 10.1007/s00037-015-0106-7 |