A note on polynomial arithmetic analogue of Halton sequences
A Note on Polynomial Arithmetic Analogue of Halton Sequences SHU TEZUKA
and TAKESHI
IBM Research,
In this article, sequences the
respect
generator
of the
for
practical
sheds
on
a theoretical
sequences
first
Categories
and
General
by Faure
Subject
Key
of these
finite
in
between
the
Theory Phrases:
from
that
for
can be characterized
Fb. This of low
is equal
result
in terms
of
to
provides
discrepancy
Pascal’s
Halton
1 < h < k,
us with
sequences
triangle
and
useful
and
also
low-discrepancy
of his sequences.
G. 1.4 [Numerical
Algorithm,
We show
of the matrix
class
Methodologies]:
Words
fields.
sequences
elements of this
for the analysis
Descriptors:
and
over
the (i, J) element
connection
1.6 [Computing
Terms:
Additional
coordinate
implementation
explored
Differentiation;
hth
arithmetic
b ~, . . . . b~ are distinct
information light
(O, k )-sequences in a prime power base b > h obtained
To be precise,
i, j > 1, and
Laboratory
to polynomial
matrix triangle.
TOKUYAMA
Research
we investigate
with
the Pascal’s
where
Tokyo
Analysis]: Simulation
Discrepancy,
Faure
Quadrature
and
Numerical
and Modeling
sequences,
Halton
sequences,
Pascal’s
triangle
1. INTRODUCTION The notion of discrepancy, which plays an important role in Quasi-Monte in [0, 1]*, Carlo methods, is defined as follows: for N points X., Xl, . . . , X~_l k > 1, and a subinterval J = ll~.l [0, ZLt), where O < u, < 1 for 1< i < k, we define the discrepancy as A(J; D$
where volume
A(J;
Authors’
address:
not made of the
email:
to copy without
publication
and
Tokyo
{tezukaj
its
date
Research of this
commercial
appear,
Machinery.
– V(J)
,
Laboratory,
1623-14
V(J)
the
is
J. We use
Shimotsuruma,
Yamoto,
ttoku}@trlvm.vnet.ibm.com.
fee all or part
for direct
for Computing
and
material
is granted
advantage,
the ACM
notice
is given
To copy otherwise,
that
provided copyright
copying
that notice
the copies
is by permission
or to republish,
requires
are
and the title of the
a fee and\or
permission.
@ 1994 ACM ACM
Research,
or distributed
Association specific
IBM
N) N
of n, O < n < N with X. E J; is extended over all subintervals
is the number and the supremum
242, Japan;
Permission
SUp J
N)
of J;
Kanagawa
) =
1049-3301/94/0700-0279
Transactions
on Modeling
$03.50 and
Computer
Simulation,
Vol
4, No.
3, July
1994,
Pages
279-284.
280
.
the
term
[O, Ilh,
S. Tezuka
and T. Tokuyama
low-discrepancy
such that
sequence
for all
N
>
denote
to
a sequence
1, the discrepancy < C~(log
D~)
XO,
of the first
Halton sequences with and showed that these
respect to polynomial arithmetic sequences constitute a new class
In
this
article,
obtained
from
we this
concentrate class
is
N)k\N,
depending only on the dimension k. [Tezuka 1993], we proposed an analogous
sequences
. . . . in
N points
where Ch is a constant In a previous article
sequences.
Xl,
on the
optimal
of low-discrepancy
version
of
over finite fields of low-discrepancy case,
sequences,
i.e.,
(O, k )-
and
show
that the generator matrix for these sequences can be characterized by using Pascal’s triangle. This result alleviates the effort needed for practical implementation of this explicit theoretical quences.
class of low-discrepancy relationship between
sequences and also provides these sequences and Faure
an se-
2. OVERVIEW Before
introducing
following
the
definition
of (t,
k )-sequences
in base
b, we need
the
notions:
Definition
A b-ary
2.1.
box is an interval
of the form
k
E=
~
[afib-~k,(ak
+
l) b-~’),
h=l
with
integers
Definition
points
b~
= bt-m.
Now,
we define
Definition
(t, k )-sequences Let
2.3.
of points set consisting
point
the
t,
O < ah < bd’
in base
for 1 < h < k.
b.
be an integer.
t >0
b is a sequence
base >
integers
Let O < t < m be integers. A (t,m, k )-net is a point set of l]k such that A( E; b ~) = b’ for every b-ary box E with
2.2.
in [0,
V(E)
m
> 0 and
d~
A (t,k )-sequence,
in [0, 1] k such that of the [X~]~ with
XO, Xl,
for all integers jbm
s
n
0
presented a general let k > 1 and b >2
construction principle for (t, k)and B = {O, 1, . . . . b – 1}.Accord-
we define
(1) a commutative (2)
bisections large j;
ACM
Transactions
*JIB
ring +
on Modeling
R R
with
for
and
identity
and card(R)
j = 1, 2,...,
Computer
Slmulatlon,
with
Vol
~j(0)
4, No
= b; = O for
3, July
1994,
all
sufficiently
Polynomial Analogue (3)
of Halton Sequences
Ak,:R - B for h = 1, 2,..., k and i = 1, 2,..., h < k and all sufficiently large i; and
bisections
for 1
and TAKESHI
IBM Research,
In this article, sequences the
respect
generator
of the
for
practical
sheds
on
a theoretical
sequences
first
Categories
and
General
by Faure
Subject
Key
of these
finite
in
between
the
Theory Phrases:
from
that
for
can be characterized
Fb. This of low
is equal
result
in terms
of
to
provides
discrepancy
Pascal’s
Halton
1 < h < k,
us with
sequences
triangle
and
useful
and
also
low-discrepancy
of his sequences.
G. 1.4 [Numerical
Algorithm,
We show
of the matrix
class
Methodologies]:
Words
fields.
sequences
elements of this
for the analysis
Descriptors:
and
over
the (i, J) element
connection
1.6 [Computing
Terms:
Additional
coordinate
implementation
explored
Differentiation;
hth
arithmetic
b ~, . . . . b~ are distinct
information light
(O, k )-sequences in a prime power base b > h obtained
To be precise,
i, j > 1, and
Laboratory
to polynomial
matrix triangle.
TOKUYAMA
Research
we investigate
with
the Pascal’s
where
Tokyo
Analysis]: Simulation
Discrepancy,
Faure
Quadrature
and
Numerical
and Modeling
sequences,
Halton
sequences,
Pascal’s
triangle
1. INTRODUCTION The notion of discrepancy, which plays an important role in Quasi-Monte in [0, 1]*, Carlo methods, is defined as follows: for N points X., Xl, . . . , X~_l k > 1, and a subinterval J = ll~.l [0, ZLt), where O < u, < 1 for 1< i < k, we define the discrepancy as A(J; D$
where volume
A(J;
Authors’
address:
not made of the
email:
to copy without
publication
and
Tokyo
{tezukaj
its
date
Research of this
commercial
appear,
Machinery.
– V(J)
,
Laboratory,
1623-14
V(J)
the
is
J. We use
Shimotsuruma,
Yamoto,
ttoku}@trlvm.vnet.ibm.com.
fee all or part
for direct
for Computing
and
material
is granted
advantage,
the ACM
notice
is given
To copy otherwise,
that
provided copyright
copying
that notice
the copies
is by permission
or to republish,
requires
are
and the title of the
a fee and\or
permission.
@ 1994 ACM ACM
Research,
or distributed
Association specific
IBM
N) N
of n, O < n < N with X. E J; is extended over all subintervals
is the number and the supremum
242, Japan;
Permission
SUp J
N)
of J;
Kanagawa
) =
1049-3301/94/0700-0279
Transactions
on Modeling
$03.50 and
Computer
Simulation,
Vol
4, No.
3, July
1994,
Pages
279-284.
280
.
the
term
[O, Ilh,
S. Tezuka
and T. Tokuyama
low-discrepancy
such that
sequence
for all
N
>
denote
to
a sequence
1, the discrepancy < C~(log
D~)
XO,
of the first
Halton sequences with and showed that these
respect to polynomial arithmetic sequences constitute a new class
In
this
article,
obtained
from
we this
concentrate class
is
N)k\N,
depending only on the dimension k. [Tezuka 1993], we proposed an analogous
sequences
. . . . in
N points
where Ch is a constant In a previous article
sequences.
Xl,
on the
optimal
of low-discrepancy
version
of
over finite fields of low-discrepancy case,
sequences,
i.e.,
(O, k )-
and
show
that the generator matrix for these sequences can be characterized by using Pascal’s triangle. This result alleviates the effort needed for practical implementation of this explicit theoretical quences.
class of low-discrepancy relationship between
sequences and also provides these sequences and Faure
an se-
2. OVERVIEW Before
introducing
following
the
definition
of (t,
k )-sequences
in base
b, we need
the
notions:
Definition
A b-ary
2.1.
box is an interval
of the form
k
E=
~
[afib-~k,(ak
+
l) b-~’),
h=l
with
integers
Definition
points
b~
= bt-m.
Now,
we define
Definition
(t, k )-sequences Let
2.3.
of points set consisting
point
the
t,
O < ah < bd’
in base
for 1 < h < k.
b.
be an integer.
t >0
b is a sequence
base >
integers
Let O < t < m be integers. A (t,m, k )-net is a point set of l]k such that A( E; b ~) = b’ for every b-ary box E with
2.2.
in [0,
V(E)
m
> 0 and
d~
A (t,k )-sequence,
in [0, 1] k such that of the [X~]~ with
XO, Xl,
for all integers jbm
s
n
0
presented a general let k > 1 and b >2
construction principle for (t, k)and B = {O, 1, . . . . b – 1}.Accord-
we define
(1) a commutative (2)
bisections large j;
ACM
Transactions
*JIB
ring +
on Modeling
R R
with
for
and
identity
and card(R)
j = 1, 2,...,
Computer
Slmulatlon,
with
Vol
~j(0)
4, No
= b; = O for
3, July
1994,
all
sufficiently
Polynomial Analogue (3)
of Halton Sequences
Ak,:R - B for h = 1, 2,..., k and i = 1, 2,..., h < k and all sufficiently large i; and
bisections
for 1
