Computation of the Smith normal form of polynomial matrices
Computation
of
the
Smith
normal Gilles
Laboratoire
LMC-lMAG,
We describe
46 Av. F. Viallet,
matrices.
and
a new algorithm
of the Smith This
normal
algorithm
form
and pre- and post-multipliers
istic
polynomial
time.
putation
reduces
over
the
field
good
worst-case
the normal in determin-
Noticing
that
to a linear
of the
of polynomial
computes
complexity
a rigourous
methods
trices
and
The a
bound,,
related
This
paper
establishes
for
the
be
computed
Smith
Smith
normal in
normal
ideal
We will
in
also
well
known
some
with
using
Euclid’s
input
from
problems
the
a principal
over
within
Hermite
Hermite
point
of view
to be solved
when
the
Kannan
of matrices
and
Bachem
mite
and
Smith
mial
time.
The
peated
have
done,
been the
arithmetic faster
entries,
computed
the in
is computed
for
matrix.
first
order
Their
bounds during
which
by several
by Chou the
authors
[5, 19, 9, 1’2]. More
algorithms
have
been
the
its
and
trices,
mod-
root
We
is
granted provided that the copies are not made or distributed for diract commercial advantage, the ACM copyright notice and tha title of the publication and ite date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or spacific permission. ACM4SSAC ‘93-7/93/Kiev, Ukraine @1993 ACM 0.89791 -60~2/93/0007/0209< ..$l” .50
tipliers)
to copy
without
fee all or part of this material
directly
nomial itly
new
is in ~, time the
there
multipliers bounded
lengths
of the randomly
where
problems.
matrix
Smith
degree
3 that
Q[a]
7 is the
class
We obtain
a good-conditionning matrix
in
without
as not
normal
such
manner.
form result
matrix. a way
being
variables.
TO and mul-
of sequential
the
on ma-
REDUCTION
(the
with
form
form
in many
The
has,
Smith
in a deterministic
section
cho-
matrix.
of the the
input
the input
characterized
of a large
over
input
exist
a certain
to
can a con-
“good -conditionning”
were
computable in
of the As
polynomially
coefficients
shown
the multipliers
randomized
leading
30RM
computing
gularize
209
not
show
have
is to “pre-condition”
it with
In
is given
computation
say a “conditioning”
the
by [11].
time.
coefficient
key idea
of Hermite,
were will
SMITH
Permission
and
of a polynomial
They
in [8]
are
Unfortunately
i.e.
it
trans-
computation
polynomial that
probability,
of the
and
entries
of this
diagonal.
authors
cost
of the
shown
matrix,
form
in
theorem,
algorithm
The
form
have
dimensions
repetition
cost
Smith
by multiplying
high
the
whose
Their
polynomial appeared
was done
form.
randomized
they
form
equivalence
probabilistic
is the
The
form,
Hermite
recently,
given
the
sen constant
computausing
form in
(or
breakthrough
Vegas
probability,
form.
matrix
re-
appearing
improved in
Her-
last
the Smith
Smith
first form
remainder
multipliers
time
normal
the
normal
Chinese
divisor
polynomial
such that if S is the Smith form of A,
V
high
matrices.
polynousing
with
sequence,
to
Frumkin,
[6, 14] that
be
of the integers
and then
U1O determinant asymptotically
shown
can
of the
of digits
[3] changing were
integer
diagonalization
calculations
Collins tions
have
forms
triangularizations
on the number the
with
the
a Las
be obtain
in
case
paper
Hermite
be computed. In
compute
encountered
Hermite
And
Smith
to
polynomials.
common
direct
in-
sufficient
to those
into
[13].
the
on the
U and
of the
are
have
based
= S. The
that
[7, 18] but
they
Kannan
are not
greatest
A first
ma-
polynomials
of those
similar
a matrix
for computing
as an
forms
not
this
is a triangularnormal
but
UAV
matrix.
form
[10]:
methods
polynomial
the
they
coefficients are
for
for
of
but
algorithm.
by
algorithm
forms)
The
the domain input
normal form
Those
a theoretical
can
defined
of the
matrix.
remain
matrices
given
be applied
a polynomial
bringing
time did
time.
computed
the
post-multipliers
polync)mial
a diagonalization
form:
of the
and
of polynomial
is generally
deal
intermediate ization
form
it is entirely
consists
pm-
deterministic
form
domain,
and
that
the
problems
computing
determinant
degrees
calculations,
when
was
Introduction
also the
bound
method
1
can
in the
of modulo in [16].
bound
correctly
problem
we obtain
matrices
study
developed
These volved
the com-
algebra
coefficients,
polynomial
F38031 Grenoble C6dex.
has been
for the compu-
form
of
Villard
Abstract tation
form
poly-
by explicWe trian-
that
at each
step,
the
diagonal
corresponding Furthermore, recent
aa a “big
gcd”,
shown
that
Q[~]
the
over
authors
the
Q . Using
the
idea
of the
Smith
by triangularizing
show
They
have
form
over
matrix
in section
over
a big
Q[z]
4 that
may
matrix
Kannan’s
algorithm
[13]
the
Hermite
normal
steps,
step
i the
form,
Algorithm Input: At”)
Previous
for
Hermite.
Put
form.
process
It works
in n – 1
(i + 1) x (i + 1) principal
and
the
matrix
minor
is denoted
by Ati).
KHNF := A,
the
n x n matrix.
(z’ + 1) x (i+
triangular
Q .
Reduce
1) principal
H
At
step
the
first
minor
in upper
form. off-diagonal
Output:
on
results
is an elimination
forifromlton–1
also
over
after
principal
2
algorithm
to compute
is in Hermite
an efficient
Hermite
form
[15]
Hermite
of a big
we will
of the
be computed
developed
triangularization
their
computation
Kannan’s
The
the
Marlin
subresultants.
computation
to
and
with
By viewing
have
on generalized
reduces
the
bounds.
2.1
the
form.
Lombardi
complexity
based
is exactly
Smith
we use can be combined
of Labhalla,
good
method
we obtain
of the
the idea
method
to obtain
coefficient
coefficient
entries
of the
(i+
1) x (i+
1)
minor. :=
A(n-l).
■
i unimodular i +
row
1 rows
1 ,. ... i, become
operations
only.
zero by left
The
are performed
entries
multiplying
Ai+l,j,
j
=
by the Bezout’s
matrices: will
We put
restrict
ourselves
matrices,
with
no
but
great
matrices.
the
this
section
[7, 18] concerning
the
matrix of Q[x],
and
The
degrees
absolute
values)
of the
spectively
bounded
is called
unimodular
element
of
column
and
recall
basic
results
normal
forms
and
Smith
the
n whose
main and
of the
d and
/3.
entries
algorithms
the
singular
for
coefficient entries
form.
Q[*]”
the
‘n
A non
is in
non
form: Let
the
singular
UA
h;
are hl,l
‘n
square form
diagonal
ma-
if it
are monic,
is
in each
entry
are of
= hj,l
formed and
is left
equivalent mal
with
common
the jirst
of the Hermite
divisor
of A; H
of A
J i = 2, . . .,n.
UAV
Let s;
= S, denote
the i x i minors normal i=2
form >. ... n.
S
matrix
S which
U
V
and
the greatest
of A;
the
diagonal
of A
are
S1 =
is in Smith
nor-
unimodular. common
divisor
entries s!
and
of all
of the Smith si
=
and r = unimodular
off-diagonal
of the
entry
diagonal
PAj,j row
haa degree
entry
in its
+ qAi+l,j. operations
strictly
lower
column.
mite
1 ([13])
normal
singular
Algorithm
form
matrix
(and over
KHNF
jinds
the multiplier)
Q[x]
the
Her-
of a square
in polynomial
non
time.
ing
the
unicit
polynomial
Hermite
form
y of the
tiplier
form,
is unique:
from
time
can one
can
=
HA-1.
U
A, in a polynomial
system
over
method
has
view
and
Q
seems the
also It
unique
computFrom
that
is possible
the
the
mul-
to
build
of operations,
from
to be costly
next
[10].
a linear
is (H, U).
solution
developed
for
in
deduce
number
whose
been
algorithm
be found
a parallel
in sequential.
This
point We will
of use
result.
of all
i columns
normalform
h~,$ = h~,i/h~_l,i_l
to a unique
form:
each
that
instead,
U unimodular.
the greatest
entries
A of Q[z]nxn
matrix
, 4+l,j), to perform
zero
Smith normal form. A non singular square matrix S of Q[z]~’~ is in Smith normal form if it is diagonal, its diagonal entries are monic, each divides the next. Every non singular matrtx A of Q[x]nxn is (III)
(IV)
than
Theorem
rnatr~z H which is in He~i~e
= H,
denote
i x i minors
the diagonal
gcd(Aj,j
‘
)
their
lengths
is a non
normal
entries
preceding
equ~v~lent to a Un@e normal
=
it is easy
so that
are
of Q[z]n
singular
Hermite
its diagonal
entries
Every
(II)
r
Then
of A are re-
A matrix
determinant
with
degree,
(I)
the
rectangular
A different
triangular,
lower
generalized
some
!l Aj,j/r
(
to
if its
normal
H
upper
be
P –Ai+l,j/r
in-
of Q .
Hermite trix
by
singular
we
Hermite
computation.
non
could
A of dimension
polynomials (log
square
approach
difficulties
In
of an input
to
S~/s~_l,
2.2 We
Subresultants present
in
Lombardi
and
Hermite
form
solution
over
appropriate
this
Q , but
for
the computation
the
210
that
operations
coefficients
[10],
avoid
the
The
main
idea
Sylvester
computation Hermite
Sylvester
only, of the
method
they
of the if the
the As in
Q[z]
use of the the
Hermite
over
system.
subresultants [17]
[15].
computation
the
by row
section
Marlin
to generalize
from
for
then
Labhalla, reduce
to a linear cost
matrix
gtd.
the
system
of finding
the
of the
method
matrix
and
of polynomial form.
the last
polynomial
of they
Indeed,
is
of the gcd to
we know
is triangularized non zero row
gives
The
method
matrix
first
A a matrix
bound
on the
(UA
= H).
degrees
A(d)
If d is a bound take
6 =
cl,
...,
It provides by the
n(d+
the
3 ([11])
dimension
n with
entries
role
Then
of the
of the Hermite
6 is a
multiplier
Sylvester
of the entries
the
canonical
a natural
elements
less than Li
of the
en the
Q[z]n.
Let
Theorem
where
in Q,
U
d. then
the
is obtained
form
and
basis
of
of the
the
Q-vector
whose
space
entries
have
This
de-
. . ..e~.
row-vectors
zd+6en,6,e
of A.
with
A($)
entries
7r) .
n). .,en). is the
n(6
in Q whose
“ “ “ ~~d-’~n,
2 ([15])
form
of A followed
by row
In section form
In
in
of
. . . . ~n]
-the
the
base B(6))
of A(t)
using
by the reduction
4 below of the
consists
method
we extend with
Herrnite are
matrix
iterations
Smith
this
row
theorem,
gorithm
Hermite
for
two
matrix.
tries
of the
iterations
Smith
normal
form
opera-
With
are the
entries 3.4),
high
for
The
The
H’,
form.
number
unit
lower
are chosen
the
Smith
simple on
con-
Hermite.
to compute
do not we
form
will
we
algorithm
good
a root
of
how
to
show
consequently time
the
gcd
columns
refer
to
the
will
give
to compute
characterizations
entm’es
diagonal
all i,
A
entries
of the
of A is equal
(II)
of the Hermite
and the
of the Hermite
Smith
form
of the i x i minors
new
have
if
and
formed
form only
with
if
the i
to the gcd of all i x i minors
method
for
Smith
entries
that
en-
viously,
Hermite,
([11],
at random
matrix
Hermite
normal
form
of A’.
Hermite
normal
form
of ‘Al.
form
of the
next
also con-
doing the
some
diagonal
Smith
in this
section.
is then
Herstage
form.
section,
on the generalized
entries
Notice
a
subthat
it
form:
as seen
pre-
directly
obtained
by
by
column
operations
3.4).
entries
be an
conditioning
. . . . an)
of Qn–l
is replaced
columns
B
good
given B{
:=
are those method
main
form but
that
triangular
off-diagonal
1 Let
A
in Q .
Al
the
Smith form,
bssed in
on
Smith
lemma
Definition
whose
focus
the
the
to ensure of the
algorithm,
the
for
the
The
the
of B
triangular
idea
is studied
to
reducing
([1 1], on
main
ants,
diagonal
matrix
are computed
the
computing
a triangular operations
corresponding suffices
algorithms
row operations.
we propose
column
result
the only
computing
We give
input
that
perform
method in
con-
new
seen
form
(az,~s,
A2 :=
by the
n x n of
such linear
B
that:
non is
singular a (n
–
if the first
combination
ma1)-uple column
of the
other
by
:=Bl+~zBz+~sBs+...+~~B~,
then
A2 if it is diagonal the
The
simple
form~ of A.
:= AC.
whose
us
entries
sists
al-
the
the
A’
In fact,
normal
form.
the
chosen
of this of
of
although
Ccmsequently
application
form
randomized
a randomly
say
Smith
gives
and
is the
a way
(that
polynomial
form.
3
Algorithm RSNF Input: A, n x n matrix.
of Q
Smith
of r,
time.
4 below
matrices
are
triz.
Output:
the
a root
theorems
provide
3 and
1 Let
mite
normal
aa 0(n3)
probability,
form,
of the
not
of the diagonal
of A
for the
Smith.
in pre-conditionning it by
a second
of H’,
C :=
of
complexity
matrices
such
(IV)
first
computations
[14].
suffice.
consists
Hermite
lemma transpose
the
y bounded
A by multiplying
stant
variables
form.
and on its transpose
of [11]
matrix
– 1)/2
and per-
operations
algorithm.
is theoretical
in practice
does
and Smith
com-
of the o~-diagonal
column
to compute
in iterating
on the
by
does not form
previous
Remark
of the usual
of
bounded
of A.
Randomized
The
two
sections
Smith
We
2.3
of Q[z]”x”
entries
computes
time
the
a deterministic
operations.
a triangularization
computation
of
compute
+ 1) x
row-vectors
. . . . L,,
‘The row-vectors (written
by a triangulari.zation only,
in n(n
in polynomial
polynomial
theorem
the
entries
of the
if C
RSNF
pre-conditionning
. . :;t::;”;n’t;~;~:6;;;”
tions
be a matrix r
that
multipliers
sequence
d +6:
6 + 1) matrix
normal
such
algorithm
The
module
are the
Theorem
A
degrees
is a polynomial
O(n3d)
of A, we can
form
bssis
of Q[z]n
=(zd+Jel,
be the
puted
There
degree
matrix.
Let the
A($).
formed
B(5)
to the input
entries
on the degrees
(n – l)d.
Let
in associating
with
plays
by triangularizing
grees
consists A(JJ
entries
construction
or Failed.
of C can
in [11].
gcd
equal
■
be chosen
is detailed
The
in a subset
to the gcd of all
The tries
211
gcd~
of
the
Smith
normal Gilles
Laboratoire
LMC-lMAG,
We describe
46 Av. F. Viallet,
matrices.
and
a new algorithm
of the Smith This
normal
algorithm
form
and pre- and post-multipliers
istic
polynomial
time.
putation
reduces
over
the
field
good
worst-case
the normal in determin-
Noticing
that
to a linear
of the
of polynomial
computes
complexity
a rigourous
methods
trices
and
The a
bound,,
related
This
paper
establishes
for
the
be
computed
Smith
Smith
normal in
normal
ideal
We will
in
also
well
known
some
with
using
Euclid’s
input
from
problems
the
a principal
over
within
Hermite
Hermite
point
of view
to be solved
when
the
Kannan
of matrices
and
Bachem
mite
and
Smith
mial
time.
The
peated
have
done,
been the
arithmetic faster
entries,
computed
the in
is computed
for
matrix.
first
order
Their
bounds during
which
by several
by Chou the
authors
[5, 19, 9, 1’2]. More
algorithms
have
been
the
its
and
trices,
mod-
root
We
is
granted provided that the copies are not made or distributed for diract commercial advantage, the ACM copyright notice and tha title of the publication and ite date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or spacific permission. ACM4SSAC ‘93-7/93/Kiev, Ukraine @1993 ACM 0.89791 -60~2/93/0007/0209< ..$l” .50
tipliers)
to copy
without
fee all or part of this material
directly
nomial itly
new
is in ~, time the
there
multipliers bounded
lengths
of the randomly
where
problems.
matrix
Smith
degree
3 that
Q[a]
7 is the
class
We obtain
a good-conditionning matrix
in
without
as not
normal
such
manner.
form result
matrix. a way
being
variables.
TO and mul-
of sequential
the
on ma-
REDUCTION
(the
with
form
form
in many
The
has,
Smith
in a deterministic
section
cho-
matrix.
of the the
input
the input
characterized
of a large
over
input
exist
a certain
to
can a con-
“good -conditionning”
were
computable in
of the As
polynomially
coefficients
shown
the multipliers
randomized
leading
30RM
computing
gularize
209
not
show
have
is to “pre-condition”
it with
In
is given
computation
say a “conditioning”
the
by [11].
time.
coefficient
key idea
of Hermite,
were will
SMITH
Permission
and
of a polynomial
They
in [8]
are
Unfortunately
i.e.
it
trans-
computation
polynomial that
probability,
of the
and
entries
of this
diagonal.
authors
cost
of the
shown
matrix,
form
in
theorem,
algorithm
The
form
have
dimensions
repetition
cost
Smith
by multiplying
high
the
whose
Their
polynomial appeared
was done
form.
randomized
they
form
equivalence
probabilistic
is the
The
form,
Hermite
recently,
given
the
sen constant
computausing
form in
(or
breakthrough
Vegas
probability,
form.
matrix
re-
appearing
improved in
Her-
last
the Smith
Smith
first form
remainder
multipliers
time
normal
the
normal
Chinese
divisor
polynomial
such that if S is the Smith form of A,
V
high
matrices.
polynousing
with
sequence,
to
Frumkin,
[6, 14] that
be
of the integers
and then
U1O determinant asymptotically
shown
can
of the
of digits
[3] changing were
integer
diagonalization
calculations
Collins tions
have
forms
triangularizations
on the number the
with
the
a Las
be obtain
in
case
paper
Hermite
be computed. In
compute
encountered
Hermite
And
Smith
to
polynomials.
common
direct
in-
sufficient
to those
into
[13].
the
on the
U and
of the
are
have
based
= S. The
that
[7, 18] but
they
Kannan
are not
greatest
A first
ma-
polynomials
of those
similar
a matrix
for computing
as an
forms
not
this
is a triangularnormal
but
UAV
matrix.
form
[10]:
methods
polynomial
the
they
coefficients are
for
for
of
but
algorithm.
by
algorithm
forms)
The
the domain input
normal form
Those
a theoretical
can
defined
of the
matrix.
remain
matrices
given
be applied
a polynomial
bringing
time did
time.
computed
the
post-multipliers
polync)mial
a diagonalization
form:
of the
and
of polynomial
is generally
deal
intermediate ization
form
it is entirely
consists
pm-
deterministic
form
domain,
and
that
the
problems
computing
determinant
degrees
calculations,
when
was
Introduction
also the
bound
method
1
can
in the
of modulo in [16].
bound
correctly
problem
we obtain
matrices
study
developed
These volved
the com-
algebra
coefficients,
polynomial
F38031 Grenoble C6dex.
has been
for the compu-
form
of
Villard
Abstract tation
form
poly-
by explicWe trian-
that
at each
step,
the
diagonal
corresponding Furthermore, recent
aa a “big
gcd”,
shown
that
Q[~]
the
over
authors
the
Q . Using
the
idea
of the
Smith
by triangularizing
show
They
have
form
over
matrix
in section
over
a big
Q[z]
4 that
may
matrix
Kannan’s
algorithm
[13]
the
Hermite
normal
steps,
step
i the
form,
Algorithm Input: At”)
Previous
for
Hermite.
Put
form.
process
It works
in n – 1
(i + 1) x (i + 1) principal
and
the
matrix
minor
is denoted
by Ati).
KHNF := A,
the
n x n matrix.
(z’ + 1) x (i+
triangular
Q .
Reduce
1) principal
H
At
step
the
first
minor
in upper
form. off-diagonal
Output:
on
results
is an elimination
forifromlton–1
also
over
after
principal
2
algorithm
to compute
is in Hermite
an efficient
Hermite
form
[15]
Hermite
of a big
we will
of the
be computed
developed
triangularization
their
computation
Kannan’s
The
the
Marlin
subresultants.
computation
to
and
with
By viewing
have
on generalized
reduces
the
bounds.
2.1
the
form.
Lombardi
complexity
based
is exactly
Smith
we use can be combined
of Labhalla,
good
method
we obtain
of the
the idea
method
to obtain
coefficient
coefficient
entries
of the
(i+
1) x (i+
1)
minor. :=
A(n-l).
■
i unimodular i +
row
1 rows
1 ,. ... i, become
operations
only.
zero by left
The
are performed
entries
multiplying
Ai+l,j,
j
=
by the Bezout’s
matrices: will
We put
restrict
ourselves
matrices,
with
no
but
great
matrices.
the
this
section
[7, 18] concerning
the
matrix of Q[x],
and
The
degrees
absolute
values)
of the
spectively
bounded
is called
unimodular
element
of
column
and
recall
basic
results
normal
forms
and
Smith
the
n whose
main and
of the
d and
/3.
entries
algorithms
the
singular
for
coefficient entries
form.
Q[*]”
the
‘n
A non
is in
non
form: Let
the
singular
UA
h;
are hl,l
‘n
square form
diagonal
ma-
if it
are monic,
is
in each
entry
are of
= hj,l
formed and
is left
equivalent mal
with
common
the jirst
of the Hermite
divisor
of A; H
of A
J i = 2, . . .,n.
UAV
Let s;
= S, denote
the i x i minors normal i=2
form >. ... n.
S
matrix
S which
U
V
and
the greatest
of A;
the
diagonal
of A
are
S1 =
is in Smith
nor-
unimodular. common
divisor
entries s!
and
of all
of the Smith si
=
and r = unimodular
off-diagonal
of the
entry
diagonal
PAj,j row
haa degree
entry
in its
+ qAi+l,j. operations
strictly
lower
column.
mite
1 ([13])
normal
singular
Algorithm
form
matrix
(and over
KHNF
jinds
the multiplier)
Q[x]
the
Her-
of a square
in polynomial
non
time.
ing
the
unicit
polynomial
Hermite
form
y of the
tiplier
form,
is unique:
from
time
can one
can
=
HA-1.
U
A, in a polynomial
system
over
method
has
view
and
Q
seems the
also It
unique
computFrom
that
is possible
the
the
mul-
to
build
of operations,
from
to be costly
next
[10].
a linear
is (H, U).
solution
developed
for
in
deduce
number
whose
been
algorithm
be found
a parallel
in sequential.
This
point We will
of use
result.
of all
i columns
normalform
h~,$ = h~,i/h~_l,i_l
to a unique
form:
each
that
instead,
U unimodular.
the greatest
entries
A of Q[z]nxn
matrix
, 4+l,j), to perform
zero
Smith normal form. A non singular square matrix S of Q[z]~’~ is in Smith normal form if it is diagonal, its diagonal entries are monic, each divides the next. Every non singular matrtx A of Q[x]nxn is (III)
(IV)
than
Theorem
rnatr~z H which is in He~i~e
= H,
denote
i x i minors
the diagonal
gcd(Aj,j
‘
)
their
lengths
is a non
normal
entries
preceding
equ~v~lent to a Un@e normal
=
it is easy
so that
are
of Q[z]n
singular
Hermite
its diagonal
entries
Every
(II)
r
Then
of A are re-
A matrix
determinant
with
degree,
(I)
the
rectangular
A different
triangular,
lower
generalized
some
!l Aj,j/r
(
to
if its
normal
H
upper
be
P –Ai+l,j/r
in-
of Q .
Hermite trix
by
singular
we
Hermite
computation.
non
could
A of dimension
polynomials (log
square
approach
difficulties
In
of an input
to
S~/s~_l,
2.2 We
Subresultants present
in
Lombardi
and
Hermite
form
solution
over
appropriate
this
Q , but
for
the computation
the
210
that
operations
coefficients
[10],
avoid
the
The
main
idea
Sylvester
computation Hermite
Sylvester
only, of the
method
they
of the if the
the As in
Q[z]
use of the the
Hermite
over
system.
subresultants [17]
[15].
computation
the
by row
section
Marlin
to generalize
from
for
then
Labhalla, reduce
to a linear cost
matrix
gtd.
the
system
of finding
the
of the
method
matrix
and
of polynomial form.
the last
polynomial
of they
Indeed,
is
of the gcd to
we know
is triangularized non zero row
gives
The
method
matrix
first
A a matrix
bound
on the
(UA
= H).
degrees
A(d)
If d is a bound take
6 =
cl,
...,
It provides by the
n(d+
the
3 ([11])
dimension
n with
entries
role
Then
of the
of the Hermite
6 is a
multiplier
Sylvester
of the entries
the
canonical
a natural
elements
less than Li
of the
en the
Q[z]n.
Let
Theorem
where
in Q,
U
d. then
the
is obtained
form
and
basis
of
of the
the
Q-vector
whose
space
entries
have
This
de-
. . ..e~.
row-vectors
zd+6en,6,e
of A.
with
A($)
entries
7r) .
n). .,en). is the
n(6
in Q whose
“ “ “ ~~d-’~n,
2 ([15])
form
of A followed
by row
In section form
In
in
of
. . . . ~n]
-the
the
base B(6))
of A(t)
using
by the reduction
4 below of the
consists
method
we extend with
Herrnite are
matrix
iterations
Smith
this
row
theorem,
gorithm
Hermite
for
two
matrix.
tries
of the
iterations
Smith
normal
form
opera-
With
are the
entries 3.4),
high
for
The
The
H’,
form.
number
unit
lower
are chosen
the
Smith
simple on
con-
Hermite.
to compute
do not we
form
will
we
algorithm
good
a root
of
how
to
show
consequently time
the
gcd
columns
refer
to
the
will
give
to compute
characterizations
entm’es
diagonal
all i,
A
entries
of the
of A is equal
(II)
of the Hermite
and the
of the Hermite
Smith
form
of the i x i minors
new
have
if
and
formed
form only
with
if
the i
to the gcd of all i x i minors
method
for
Smith
entries
that
en-
viously,
Hermite,
([11],
at random
matrix
Hermite
normal
form
of A’.
Hermite
normal
form
of ‘Al.
form
of the
next
also con-
doing the
some
diagonal
Smith
in this
section.
is then
Herstage
form.
section,
on the generalized
entries
Notice
a
subthat
it
form:
as seen
pre-
directly
obtained
by
by
column
operations
3.4).
entries
be an
conditioning
. . . . an)
of Qn–l
is replaced
columns
B
good
given B{
:=
are those method
main
form but
that
triangular
off-diagonal
1 Let
A
in Q .
Al
the
Smith form,
bssed in
on
Smith
lemma
Definition
whose
focus
the
the
to ensure of the
algorithm,
the
for
the
The
the
of B
triangular
idea
is studied
to
reducing
([1 1], on
main
ants,
diagonal
matrix
are computed
the
computing
a triangular operations
corresponding suffices
algorithms
row operations.
we propose
column
result
the only
computing
We give
input
that
perform
method in
con-
new
seen
form
(az,~s,
A2 :=
by the
n x n of
such linear
B
that:
non is
singular a (n
–
if the first
combination
ma1)-uple column
of the
other
by
:=Bl+~zBz+~sBs+...+~~B~,
then
A2 if it is diagonal the
The
simple
form~ of A.
:= AC.
whose
us
entries
sists
al-
the
the
A’
In fact,
normal
form.
the
chosen
of this of
of
although
Ccmsequently
application
form
randomized
a randomly
say
Smith
gives
and
is the
a way
(that
polynomial
form.
3
Algorithm RSNF Input: A, n x n matrix.
of Q
Smith
of r,
time.
4 below
matrices
are
triz.
Output:
the
a root
theorems
provide
3 and
1 Let
mite
normal
aa 0(n3)
probability,
form,
of the
not
of the diagonal
of A
for the
Smith.
in pre-conditionning it by
a second
of H’,
C :=
of
complexity
matrices
such
(IV)
first
computations
[14].
suffice.
consists
Hermite
lemma transpose
the
y bounded
A by multiplying
stant
variables
form.
and on its transpose
of [11]
matrix
– 1)/2
and per-
operations
algorithm.
is theoretical
in practice
does
and Smith
com-
of the o~-diagonal
column
to compute
in iterating
on the
by
does not form
previous
Remark
of the usual
of
bounded
of A.
Randomized
The
two
sections
Smith
We
2.3
of Q[z]”x”
entries
computes
time
the
a deterministic
operations.
a triangularization
computation
of
compute
+ 1) x
row-vectors
. . . . L,,
‘The row-vectors (written
by a triangulari.zation only,
in n(n
in polynomial
polynomial
theorem
the
entries
of the
if C
RSNF
pre-conditionning
. . :;t::;”;n’t;~;~:6;;;”
tions
be a matrix r
that
multipliers
sequence
d +6:
6 + 1) matrix
normal
such
algorithm
The
module
are the
Theorem
A
degrees
is a polynomial
O(n3d)
of A, we can
form
bssis
of Q[z]n
=(zd+Jel,
be the
puted
There
degree
matrix.
Let the
A($).
formed
B(5)
to the input
entries
on the degrees
(n – l)d.
Let
in associating
with
plays
by triangularizing
grees
consists A(JJ
entries
construction
or Failed.
of C can
in [11].
gcd
equal
■
be chosen
is detailed
The
in a subset
to the gcd of all
The tries
211
gcd~
