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Quadratic congruential pseudorandom numbers: Distribution of lagged pairs
JOURNAL OF COMPUTATIONALAND APPLIED MATHEMATICS
EISEVIER
Journal of Computational
and Applied Mathematics
79 (1997) 75-85
Quadratic congruential pseudorandom numbers: Distribution of lagged pairs Jiirgen Eichenauer-Herrmann Fuchbereich Muthematik, Technische Hochschule Darmstadt, SchloJgartenstraJe Received
7, D-64289 Darmstadt, Germany
10 August 1995
Abstract The present paper deals with the quadratic congruential method with power of two modulus for generating uniform pseudorandom numbers. Statistical independence properties of the generated sequences (~,),~a are studied based on the distribution of lagged pairs (x,,x,+z). Upper and lower bounds for the discrepancy of the corresponding point sets in the unit square are established. Keywords: Uniform pseudorandom dence; Lagged pairs; Discrepancy;
numbers; Quadratic congruential Exponential sums
method; Power of two modulus;
Statistical
indepen-
AMS classjfication: 65C10, 1 lK45
1. Introduction Nonlinear congruential methods of generating uniform pseudorandom numbers in the interval [0, 1) have been studied intensively during the last years. Reviews of the rapid development of this progressively increasing field of research can be found in the survey articles [3,4,6,17,18,20-221 and in the excellent monograph of Niederreiter [ 191. The earliest nonlinear congruential approach is the quadratic congruential method proposed by Knuth [ 13, p. 251, which recently received considerable attention [ 1,2,5,7-l 11. The present paper concentrates on the particularly important case of a power of two modulus m = 2”’ with some integer u > 5. Let Z, = (0,1, . . . , n - 1} for integers n>l. For parameters a, b,c E Z, a quadratic congruential sequence (Y,,),,&~ of elements of Z, is defined recursively by yn+l-ay~+by,+c(modm),
n20.
of quadratic congruential pseudorandom numbers in the interval [0,1) is obtained by x, = y,/m for n 20. The sequences (x,,),~o and (yn)naO are purely periodic with the
A sequence (x, In30
0377-0427/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved PII SO377-0427(96)00158-6
76
J. Eichenauer-Herrmann I Journal of Computational and Applied Mathematics 79 (1997)
75-85
maximum possible period length m if and only if a E 0 (mod 2), b G a + 1 (mod 4), and c E 1 (mod 2) (cf. [13, p. 341). The results in [lo] on the distribution of pairs of successive quadratic congruential pseudorandom numbers indicate that a particularly interesting case is obtained if the parameters satisfy the more restrictive conditions a E 2 (mod 4), b E 3 (mod 4) and c = 1 (mod 2). From now on, it is always assumed that these conditions are satisfied. Statistical independence properties of the generated sequences are very important for their usability in stochastic simulations. They can be analysed based on the discrepancy of s-tuples of pseudorandom numbers. For N arbitrary points to, tl, . . . , tN_, E [0, 1 >”the discrepancy is defined by DN(f0, tl, **., G-l 1 = “‘jP I&VW - W>I, where the supremum is extended over all subintervals J of [0,l)“, F,(J) is N-’ times the number of points among to, II,. . . , tN-l falling into J, and V(J) denotes the s-dimensional volume of J. It is worth observing that the discrepancy of N true random points from [0,l>” is almost always of the order of magnitude N-“*(log logN)“* according to Kiefer’s probabilistic law of the iterated logarithm for discrepancies [ 121. The present paper deals with the discrepancy D;:,‘ 1Ib 3 .c = &(XfJ, Xl). . . ) X,-l ) of the lagged pairs X, =
(44,+2)E[O,
I>*,
of quadratic congruential Section 3 the main results results is given in Section illustrate their distribution
O
cm,
pseudorandom numbers. Section 2 contains several auxiliary results. In on the discrepancy D$,‘!,,, are established. A detailed discussion of these 4. Some figures -of the generated lagged pairs are included in order to in the unit square.
2. Auxiliary results First, some further notation is necessary. For integers k > 1 and q 22, let &(q) be the set of all nonzero lattice points (hr , . . . , hk) E Zk with -q/2 < hj
r(h,q) =
q W@llq) 1
forh E Cl(q), for h = 0,
and
for h=(hl,... , hk) E Ck(q). For real t, the abbreviation e(t) = e2nit is used, and u . v stands for the standard inner product of II, VERk. Subsequently, two known general results for estimating discrepancies are stated which follow from [ 14, Lemma 2.2; 19, Theorem 3. lo] and [ 16, Lemma 1; 19, Corollary 3.171, respectively.
J. Eichenauer-Herrmannl Journal of Computational and Applied Mathematics 79 (1997) 75-85
71
Lemma 1. Let N 2 1 and q > 2 be integers. Let t, = yn/q E [0,1)k with y,, E Zt for 0
Lemma 2. The discrepancy of N arbitrary points to, t,, . . . , t&l E [0, l)k satisjies rt DN(tO,tl,...,tN-I)>
2N((n:+ 1)’-
l)I$=,max(l,lhjl)
for any nonzero lattice point h= (h,, . . . , hk) E Zk, where l denotes the number of nonzero coordinates of h. The following
result can essentially
be deduced from Niederreiter
[ 15, Lemma 41.
Lemma 3. Let CI3 3 and c E 1 (mod 2) be integers. Then
kl
(mod2)
and
c’-
hEC,(Z’) h=c(mod 8)
r(h,
2” 1
<&
log2” + ;.
For integers u, v and a > 0 an exponential sum is defined by S(u, v; c; 2”) = c
e((uy + v(a3y4 + 2a2by3 + (2a2c + ab2 + ab)y2 + (2abc + b*)y))/2”),
.VEZ?T
where a, b,c are integers with a G 2 (mod4), b = 3 (mod4), and c E 1 (mod2). Some relevant properties of these sums are collected in the following two lemmas. These results can be deduced from [S, Lemmas 3-51. Lemma 4. Let u, v and IX> 1 be integers. (a) Ifu=vrO(mod2), then S(u,v;c;2”) = 2S(u/2,21/2;c;2”-‘). (b) If u + v = 1 (mod 2), then S(u, v; c; 2”) = 0. Lemma 5. Let u, v and c(> 3 be integers with u E v E 1 (mod 2). (a) Ifu + v=4(mod8), then
J. Eichenauer-Herrmann I Journal of Computational and Applied Mathematics 79 (1997)
78
754.5
(b) If u + ~$4 (mod S), then S(u, v; c; 2”) = 0.
3. Discrepancy bounds Theorem 6. The average value of the discrepancy Di($,c ential method over CE Z, with CE 1 (mod2)
Cal
of lagged pairs in the quadratic congru-
satisfies
(mod 2)
for any parameters a G 2 (mod 4) and b E 3 (mod 4).
Proof. First, Lemma 1 is applied with k = 2, N = q = m, and t,, = x, for 0
Since x, = ( yn, yn+2)/m and yn+l G ayi + by, + c (mod m) for ~12 0, it follows that m-l m-l c e(h. 1,) = c e((hly,
+ b
+ by, + cl2 + Nayi
+ by, + c> + c))lm>
n=O
n=O
{yo, yI, . . . , ym_l } = Z, implies that
and therefore m-l
I
I
x4.x,)
= IS(h,hz;c;m)l
n=O
for any It = (h, , h2) E C2(m), where S denotes the exponential obtains
sum of the second section. Hence, one
c D;:;lb + L --!-+(h,,hz;c;m)l > I Ic d 1 m m b=(h,,h>)EC2(m) r(h, m,
&lS(h,h;c;m)l m
’
v=O h=(h,,h>EC>(m) gcd(h,,hz,m)=2’
=f+
h=(h&(m) h~O(mod2”“2) h,+h2=0(modm)
rim)
J. Eichenauer-Herrmann I Journal of Computational and Applied Mathematics 79 (1997)
=--
2
m
+-$+$2P
75-85
19
1
c
IQll,g2;c;2”-“)I,
n=(q, 3. 42)EC*(2(‘J-v) r(g,2w-v)
v=o
where in the last two steps, Lemmas 4 and 5(b) have been applied. Therefore, the average value of the discrepancy D$z]qC over c E Z, with c E 1 (mod 2) satisfies
csl
(mod2)
c
I%hg2;
c; 2-l
lml>g2;
c; PJ-“)12,
cEZ,,,-, c=l (mod2)
X
where finally Schwa&s obtain 2
-
m
c CEL, Cal (mod2)
Dz:,‘,‘,
c <
&
c
1
cEZ,,.,-, c=l(mod2)
inequality has been applied. Now, Lemma 5(a) can be used in order to
’ + m
-$
+ q1~q2~1 (mod2)
I 2fi
ml/2
w--3 2-3~12 c
v=o
c u,ECI(2”-‘) y,=l (mod2)
g2&4-y1
(mod 8)
J. Eichenauer-Herrmannl
80
Journal of Computational and Applied Mathematics 79 (1997)
75-85
Hence, it follows from Lemma 3 that 2 -m
c CEG ccl
)(
(mod2)
m 2
<-+
m
which is the desired result.
0
Theorem 7. Let the parameters a E 2 (mod4) and b E 3 (mod 4) be jixed. Let 0 < a 6 1. Then there exist more than (1 - a)m/2 values of c E Z, with c -_ 1 (mod 2) such that the discrepancy
Di;z,lh,Cof lagged pairs in the quadratic congruential method satisfies 2 DKib,c
<
i1 ( 4Jz+2 7
Proof. Subsequently,
m -,,2 ( ;logm+; 1
) +f
) .
the abbreviation 2
M = 4Jz
+
2m-,/2
7
1 ;logm+:
+f )
(
is used. Suppose that there exist at most (1 - cr)m/2 values of c E Z, with c = 1 (mod2) and D~;~,lh,, < M/cc, i.e., there exist at least am/2 values of c E 7, with c = 1 (mod 2) and Dt;$C >M/a. Hence, one obtains
c
D$$,, B Mm/2
CEZ,,, csl
(mod2)
which contradicts
Theorem
6.
0
Theorem 8. Let co = 3v + p + 2 for suitable integers v> 1 and ,u E {0,1,2}. b=3 (mod4), and CE 1 (mod 2) be parameters with
4acE(b - 1)2 - 28 + 22”+4(mod2”-‘+I).
Let a s 2 (mod4),
J. Eichenauer-Herrmann I Journal of Computational and Applied Mathematics 79 (1997)
7545
81
of lagged pairs in the quadratic congruential method satisjies
Then the discrepancy D$$,r p-l);3 D”:31 m,u,h.L
> ’
-I.‘3 qn:
+
2)m
.
Proof. First, Lemma 2 is applied with k = 2, N = m, t, = x,, for Odn < m, and h = (27,l). yields 1
This
54(n:2)mlS(27J;c;m)l,
D”:31 > m3u~h~c ’ 54(rt + 2)m
where the last step follows as in the proof of Theorem 6 and S denotes the exponential sum of Section 2. Since 4v + 3 2 3v + 4 2 o and .z3EZ (mod 2), straightforward calculations show that c; m)
S(27,l; = 1 =
c
e((a3y4 + 2a2by3 + (2a2c + ab2 + ab)y2 + (2abc + b2 + 27)y)/2”) c
e((a3(x +
2”2)4
+
2a2b(x +
2”2)3
+ (2&
+ ab2 +
ab)(x + 2”z)2
+(2abc + b2 + 27)(x + 2”2))/2’“) = c e((a3x4 + 2a2bx3 + (2a2c + ab2 + ab)x2 + (2abc + b2 + 27)x)/2”‘) aEZ3, x c e(((4a3x3 + 6a2bx2 + (4a2c + 2ab2 + 2ab)x + 2abc + b2 + 27)~ zEZZ,v +2”(6a3x2 + 6a2bx + 2a2c + ab2 + ab)z2 + 22v+‘a2bz3)/2u’-“) = c e((a3x4 + 2a2bx3 + (2a2c + ab2 + ab)x2 + (2abc + b2 + 27)x)/2’“) XEZ28 x c e(((4a3x3 + 6a2bx2 + 3a(b2 - 9)x + (b3 - 27b + 54)/2)z ZEZ,,,-, +2”(6a3x2 + 6a2bx + 3a(b2 - 9)/2)~~)/2”‘-~) = 1 e((a3x4 + 2a2bx3 + (2a2c + ab2 + ab)x2 + (2abc + b2 + 27)x)/2”‘) .XE& x G(2”+‘3a(ax + (b - 3)/2)(ax + (b + 3)/2), 2(a.x + (b - 3)/2)2(2ax + b + 6); 2”“‘), where G denotes an exponential
sum defined by
G(u, u; 2”) = c e((Uz2 + 212)/2”) :tZIY for integers U, u and c(> 0. For fixed x E Z2,, let 5 E { 1,2,. . . , v + 2) be defined by gcd(ax + (b - 3)/2,2”+2) = 2’.
82
J. Eichenauer-Herrmann I Journal of Computational and Applied Mathematics 79 (1997)
75-85
Since ux + (b + 3)/2 = 1 (mod 2) and 2ax + b + 6 E 1 (mod 2), it follows that gcd(2”+‘3a(au
+ (b - 3)/2)(ax
+ (b + 3)/2), 2”‘-‘) = 2mi”(‘+S+2,“‘-r)
and gcd(2(ax + (b - 3)/2)2(2ax
+ b + 6), 2”‘-‘) = 2min(2~+‘~m-“‘.
If 5 dv, then 25 + 1 < v + 5 + 2 6 co - v and therefore G(2”+‘3a(ax + (b - 3)/2)(ax
[ 10, Lemma
61 implies that
+ (b + 3)/2), 2(ux + (b - 3)/2)‘(2ax
+ b + 6); 2,-“)
= 0.
+ b + 6); 2,-“)
= 2”-“;
If 5 = v + 1, then 25 + 1 = v + C$+ 2 > co - v - 1 which implies that G(2’+‘3a(ax
+ (b - 3)/2)(ax
+ (b + 3)/2), 2(ax + (b - 3)/2)2(2ax
if 4 = v + 2, then 25 + 1 > v + 4 + 2 > o - v which yields the same result. Since there exists exactly one x E Z2, with ax + (b - 3)/2 = 0 (mod 2’+‘), i.e., r 2 v + 1, it follows that JS(27,l; c; m)l = 2”-” and therefore p-’ jy:31
>
m~a~h,c ’
l
2-v
54(7c + 2)
which is the desired result.
113
zz
-113
27(rr + 2)m 0
4. Discussion Theorem 6 shows that in the quadratic congruential method for any parameters a 3 2 (mod 4) and b =: 3 (mod4) the discrepancy Di$,C of lagged pairs, on the average over the parameters c E 1 (mod2), is of an order of magnitude at most m -‘/2(log m)2. It should be observed that the upper bound in Theorem 6 is independent of the specific choice of the parameters a, b and that its asymptotic behaviour fits well the discrepancy of m true random points from [0, 1)2, which is almost always of the order of magnitude m-‘i2(log log m)“’ according to Kiefer’s probabilistic law of the iterated logarithm for discrepancies [ 121. Theorem 7 provides even more information, since it implies that for any parameters a, b only an arbitrarily small percentage of the parameters c may lead to a discrepancy of lagged pairs with an order of magnitude greater than m-‘i2(log m)2.On the other hand, Theorem 8 shows that there exist parameters a, b, c in the quadratic congruential method such that the discrepancy D~;$, is of an order of magnitude at least rn-li3, which is too large in view of the law of the iterated logarithm. It is still an unsolved problem to characterize all parameters a, b, c, for which the discrepancy of lagged pairs is of an order of magnitude considerably greater than m-“2. Figs. l-3 illustrate the distribution of lagged pairs of quadratic congruential pseudorandom numbers in the unit square. In these examples, always the modulus m = 219 = 524288 is considered and all generated points in the subsquare [0,0.1 )2 are plotted. Hence, the expected number of points under the uniform distribution is about 5243. It turned out that in the three examples the number of points within a period falling into the subregion [0,0.1 )2 is 5201, 5 128, and 524 1, respectively. Since co = 19 and v = 5, the crucial condition on the parameters a, b, c in Theorem 8 takes the form 4uc z (b - 1)2 + 16 356 (mod 32 768). This condition is satisfied in the first two examples. In the first
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x-i.
a = 2, b = 3, c = 2045.
example, the specific values of the parameters a, b, c have been chosen arbitrarily among all parameters satisfying the crucial condition in Theorem 8. In the second example, particularly small values of a, b, c have been selected. Figs. 1 and 2 show that in both cases nearly the same pronounced line structure appears. In the third example, the values of the parameters a, b, c differ only slightly from
J. Eichenauer-Herrmann I Journal of Computational and Applied Mathematics 79 (1997)
84
. ‘“.
. . .._ ..t:
. . :. -.
: .. . .y,._..,!.?.;p.. :.: ‘.. . ..&
Fig. 3. Distribution
of lagged pairs (x,,x,+~)
:$..;,:.;;*, :: .g...)..’ *‘:‘..:. :.,c.
“),
_.‘,‘> .,,
. . .
. :..
‘A. .‘.
,.
. . . ..
::..*
_.‘i.
‘i,?
7545
..$; g.$ ,>“‘::” .~.:”
. . .:;.
.‘! ~...“‘.
.
in [O,O.l)* for parameters
a = 462 270, b = 174403,
c = 341501.
those in the first example, but now the crucial condition in Theorem 8 is not satisfied. Fig. 3 shows that in this situation there is no obvious structure within the generated points and their distribution appears truly random. References sequences, Manuscripta Math. 58 (1987) 129-140. J. Eichenauer-Herrmann, A remark on the discrepancy of quadratic congruential pseudorandom numbers, J. Comput. Appl. Math. 43 (1992) 383-387. J. Eichenauer-Herrmann, Inversive congruential pseudorandom numbers: a tutorial, Znt. Statist. Rev. 60 (1992) 167-176. J. Eichenauer-Herrmann, Inversive congruential pseudorandom numbers, Z. Angew. Math. Mech. 73 (1993) T644-T647. J. Eichenauer-Herrmann, On the discrepancy of quadratic congruential pseudorandom numbers with power of two modulus, J. Comput. Appl. Math. 53 (1994) 371-376. J. Eichenauer-Herrmann, Pseudorandom number generation by nonlinear methods, Int. Statist. Rev. 63 (1995) 247255. J. Eichenauer-Herrmann, Discrepancy bounds for nonoverlapping pairs of quadratic congruential pseudorandom numbers, Arch. Math. 65 (1995) 362-368. J. Eichenauer-Herrmann, Quadratic congruential pseudorandom numbers: distribution of triples, J. Comput. Appl. Math. 62 (1995) 239-253. J. Eichenauer-Herrmann, Quadratic congruential pseudorandom numbers: distribution of triples, II, J. Comput. Appl. Math., to appear. J. Eichenauer-Herrmann and H. Niederreiter, On the discrepancy of quadratic congruential pseudorandom numbers, J. Comput. Appl. Math. 34 (1991) 243-249. J. Eichenauer-Herrmann and H. Niederreiter, An improved upper bound for the discrepancy of quadratic congruential pseudorandom numbers, Acta Arith. 69 (1995) 193-198.
[II J. Eichenauer and J. Lehn, On the structure of quadratic congruential PI [31 [41 [51 161 [71 181 [91 [lOI [ill
J. Eichenauer-Herrmann I Journal
qf Computational
and Applied Mathematics 79 (I 997) 75-6’5
85
[ 121 J. Kiefer, On large deviations of the empiric d.f. of vector chance variables and a law of the iterated logarithm, Pac$c J. Math. 11 (1961) 649-660. [13] D.E. Knuth, The Art of Computer Programming, Vol. 2, Seminumerical Algorithms (Addison-Wesley, Reading, MA, 2nd ed., 1981). [ 141 H. Niederreiter, Pseudo-random numbers and optimal coefficients, Adn. Math. 26 (1977) 99-181. [15] H. Niederreiter, The serial test for congruential pseudorandom numbers generated by inversions. Math. Comp. 52 (1989) 135-144. [16] H. Niederreiter, Lower bounds for the discrepancy of inversive congruential pseudorandom numbers, Math. Comp. 55 (1990) 277-287. [17] H. Niederreiter, Recent trends in random number and random vector generation, Ann. Oper. Rrs. 31 (1991) 323-345. [ 181 H. Niederreiter, Nonlinear methods for pseudorandom number and vector generation, in: G. Pflug and U. Dieter, Eds., Simulation and Optimization, Lecture Notes in Econom. and Math. Systems 374 (Springer, Berlin, 1992) 145-153. [ 191 H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods (SIAM, Philadelphia, PA, 1992). [20] H. Niederreiter, Finite fields, pseudorandom numbers, and quasirandom points, in: G.L. Mullen and P.J.S. Shiue, Eds., Finite Fields. Coding Theory, and Advances in Communications und Computing (Dekker, New York, 1993) 375-394. [21] H. Niederreiter, Pseudorandom numbers and quasirandom points, Z. Anyew. Math. Mech. 73 (1993) T648-T652. [22] H. Niederreiter, New developments in uniform pseudorandom number and vector generation, in: H. Niederreiter and P.J.-S. Shiue, Eds., Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, Lecture Notes in Statistics 106 (Springer, New York, 1995) 87-120.
EISEVIER
Journal of Computational
and Applied Mathematics
79 (1997) 75-85
Quadratic congruential pseudorandom numbers: Distribution of lagged pairs Jiirgen Eichenauer-Herrmann Fuchbereich Muthematik, Technische Hochschule Darmstadt, SchloJgartenstraJe Received
7, D-64289 Darmstadt, Germany
10 August 1995
Abstract The present paper deals with the quadratic congruential method with power of two modulus for generating uniform pseudorandom numbers. Statistical independence properties of the generated sequences (~,),~a are studied based on the distribution of lagged pairs (x,,x,+z). Upper and lower bounds for the discrepancy of the corresponding point sets in the unit square are established. Keywords: Uniform pseudorandom dence; Lagged pairs; Discrepancy;
numbers; Quadratic congruential Exponential sums
method; Power of two modulus;
Statistical
indepen-
AMS classjfication: 65C10, 1 lK45
1. Introduction Nonlinear congruential methods of generating uniform pseudorandom numbers in the interval [0, 1) have been studied intensively during the last years. Reviews of the rapid development of this progressively increasing field of research can be found in the survey articles [3,4,6,17,18,20-221 and in the excellent monograph of Niederreiter [ 191. The earliest nonlinear congruential approach is the quadratic congruential method proposed by Knuth [ 13, p. 251, which recently received considerable attention [ 1,2,5,7-l 11. The present paper concentrates on the particularly important case of a power of two modulus m = 2”’ with some integer u > 5. Let Z, = (0,1, . . . , n - 1} for integers n>l. For parameters a, b,c E Z, a quadratic congruential sequence (Y,,),,&~ of elements of Z, is defined recursively by yn+l-ay~+by,+c(modm),
n20.
of quadratic congruential pseudorandom numbers in the interval [0,1) is obtained by x, = y,/m for n 20. The sequences (x,,),~o and (yn)naO are purely periodic with the
A sequence (x, In30
0377-0427/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved PII SO377-0427(96)00158-6
76
J. Eichenauer-Herrmann I Journal of Computational and Applied Mathematics 79 (1997)
75-85
maximum possible period length m if and only if a E 0 (mod 2), b G a + 1 (mod 4), and c E 1 (mod 2) (cf. [13, p. 341). The results in [lo] on the distribution of pairs of successive quadratic congruential pseudorandom numbers indicate that a particularly interesting case is obtained if the parameters satisfy the more restrictive conditions a E 2 (mod 4), b E 3 (mod 4) and c = 1 (mod 2). From now on, it is always assumed that these conditions are satisfied. Statistical independence properties of the generated sequences are very important for their usability in stochastic simulations. They can be analysed based on the discrepancy of s-tuples of pseudorandom numbers. For N arbitrary points to, tl, . . . , tN_, E [0, 1 >”the discrepancy is defined by DN(f0, tl, **., G-l 1 = “‘jP I&VW - W>I, where the supremum is extended over all subintervals J of [0,l)“, F,(J) is N-’ times the number of points among to, II,. . . , tN-l falling into J, and V(J) denotes the s-dimensional volume of J. It is worth observing that the discrepancy of N true random points from [0,l>” is almost always of the order of magnitude N-“*(log logN)“* according to Kiefer’s probabilistic law of the iterated logarithm for discrepancies [ 121. The present paper deals with the discrepancy D;:,‘ 1Ib 3 .c = &(XfJ, Xl). . . ) X,-l ) of the lagged pairs X, =
(44,+2)E[O,
I>*,
of quadratic congruential Section 3 the main results results is given in Section illustrate their distribution
O
cm,
pseudorandom numbers. Section 2 contains several auxiliary results. In on the discrepancy D$,‘!,,, are established. A detailed discussion of these 4. Some figures -of the generated lagged pairs are included in order to in the unit square.
2. Auxiliary results First, some further notation is necessary. For integers k > 1 and q 22, let &(q) be the set of all nonzero lattice points (hr , . . . , hk) E Zk with -q/2 < hj
r(h,q) =
q W@llq) 1
forh E Cl(q), for h = 0,
and
for h=(hl,... , hk) E Ck(q). For real t, the abbreviation e(t) = e2nit is used, and u . v stands for the standard inner product of II, VERk. Subsequently, two known general results for estimating discrepancies are stated which follow from [ 14, Lemma 2.2; 19, Theorem 3. lo] and [ 16, Lemma 1; 19, Corollary 3.171, respectively.
J. Eichenauer-Herrmannl Journal of Computational and Applied Mathematics 79 (1997) 75-85
71
Lemma 1. Let N 2 1 and q > 2 be integers. Let t, = yn/q E [0,1)k with y,, E Zt for 0
Lemma 2. The discrepancy of N arbitrary points to, t,, . . . , t&l E [0, l)k satisjies rt DN(tO,tl,...,tN-I)>
2N((n:+ 1)’-
l)I$=,max(l,lhjl)
for any nonzero lattice point h= (h,, . . . , hk) E Zk, where l denotes the number of nonzero coordinates of h. The following
result can essentially
be deduced from Niederreiter
[ 15, Lemma 41.
Lemma 3. Let CI3 3 and c E 1 (mod 2) be integers. Then
kl
(mod2)
and
c’-
hEC,(Z’) h=c(mod 8)
r(h,
2” 1
<&
log2” + ;.
For integers u, v and a > 0 an exponential sum is defined by S(u, v; c; 2”) = c
e((uy + v(a3y4 + 2a2by3 + (2a2c + ab2 + ab)y2 + (2abc + b*)y))/2”),
.VEZ?T
where a, b,c are integers with a G 2 (mod4), b = 3 (mod4), and c E 1 (mod2). Some relevant properties of these sums are collected in the following two lemmas. These results can be deduced from [S, Lemmas 3-51. Lemma 4. Let u, v and IX> 1 be integers. (a) Ifu=vrO(mod2), then S(u,v;c;2”) = 2S(u/2,21/2;c;2”-‘). (b) If u + v = 1 (mod 2), then S(u, v; c; 2”) = 0. Lemma 5. Let u, v and c(> 3 be integers with u E v E 1 (mod 2). (a) Ifu + v=4(mod8), then
J. Eichenauer-Herrmann I Journal of Computational and Applied Mathematics 79 (1997)
78
754.5
(b) If u + ~$4 (mod S), then S(u, v; c; 2”) = 0.
3. Discrepancy bounds Theorem 6. The average value of the discrepancy Di($,c ential method over CE Z, with CE 1 (mod2)
Cal
of lagged pairs in the quadratic congru-
satisfies
(mod 2)
for any parameters a G 2 (mod 4) and b E 3 (mod 4).
Proof. First, Lemma 1 is applied with k = 2, N = q = m, and t,, = x, for 0
Since x, = ( yn, yn+2)/m and yn+l G ayi + by, + c (mod m) for ~12 0, it follows that m-l m-l c e(h. 1,) = c e((hly,
+ b
+ by, + cl2 + Nayi
+ by, + c> + c))lm>
n=O
n=O
{yo, yI, . . . , ym_l } = Z, implies that
and therefore m-l
I
I
x4.x,)
= IS(h,hz;c;m)l
n=O
for any It = (h, , h2) E C2(m), where S denotes the exponential obtains
sum of the second section. Hence, one
c D;:;lb + L --!-+(h,,hz;c;m)l > I Ic d 1 m m b=(h,,h>)EC2(m) r(h, m,
&lS(h,h;c;m)l m
’
v=O h=(h,,h>EC>(m) gcd(h,,hz,m)=2’
=f+
h=(h&(m) h~O(mod2”“2) h,+h2=0(modm)
rim)
J. Eichenauer-Herrmann I Journal of Computational and Applied Mathematics 79 (1997)
=--
2
m
+-$+$2P
75-85
19
1
c
IQll,g2;c;2”-“)I,
n=(q, 3. 42)EC*(2(‘J-v) r(g,2w-v)
v=o
where in the last two steps, Lemmas 4 and 5(b) have been applied. Therefore, the average value of the discrepancy D$z]qC over c E Z, with c E 1 (mod 2) satisfies
csl
(mod2)
c
I%hg2;
c; 2-l
lml>g2;
c; PJ-“)12,
cEZ,,,-, c=l (mod2)
X
where finally Schwa&s obtain 2
-
m
c CEL, Cal (mod2)
Dz:,‘,‘,
c <
&
c
1
cEZ,,.,-, c=l(mod2)
inequality has been applied. Now, Lemma 5(a) can be used in order to
’ + m
-$
+ q1~q2~1 (mod2)
I 2fi
ml/2
w--3 2-3~12 c
v=o
c u,ECI(2”-‘) y,=l (mod2)
g2&4-y1
(mod 8)
J. Eichenauer-Herrmannl
80
Journal of Computational and Applied Mathematics 79 (1997)
75-85
Hence, it follows from Lemma 3 that 2 -m
c CEG ccl
)(
(mod2)
m 2
<-+
m
which is the desired result.
0
Theorem 7. Let the parameters a E 2 (mod4) and b E 3 (mod 4) be jixed. Let 0 < a 6 1. Then there exist more than (1 - a)m/2 values of c E Z, with c -_ 1 (mod 2) such that the discrepancy
Di;z,lh,Cof lagged pairs in the quadratic congruential method satisfies 2 DKib,c
<
i1 ( 4Jz+2 7
Proof. Subsequently,
m -,,2 ( ;logm+; 1
) +f
) .
the abbreviation 2
M = 4Jz
+
2m-,/2
7
1 ;logm+:
+f )
(
is used. Suppose that there exist at most (1 - cr)m/2 values of c E Z, with c = 1 (mod2) and D~;~,lh,, < M/cc, i.e., there exist at least am/2 values of c E 7, with c = 1 (mod 2) and Dt;$C >M/a. Hence, one obtains
c
D$$,, B Mm/2
CEZ,,, csl
(mod2)
which contradicts
Theorem
6.
0
Theorem 8. Let co = 3v + p + 2 for suitable integers v> 1 and ,u E {0,1,2}. b=3 (mod4), and CE 1 (mod 2) be parameters with
4acE(b - 1)2 - 28 + 22”+4(mod2”-‘+I).
Let a s 2 (mod4),
J. Eichenauer-Herrmann I Journal of Computational and Applied Mathematics 79 (1997)
7545
81
of lagged pairs in the quadratic congruential method satisjies
Then the discrepancy D$$,r p-l);3 D”:31 m,u,h.L
> ’
-I.‘3 qn:
+
2)m
.
Proof. First, Lemma 2 is applied with k = 2, N = m, t, = x,, for Odn < m, and h = (27,l). yields 1
This
54(n:2)mlS(27J;c;m)l,
D”:31 > m3u~h~c ’ 54(rt + 2)m
where the last step follows as in the proof of Theorem 6 and S denotes the exponential sum of Section 2. Since 4v + 3 2 3v + 4 2 o and .z3EZ (mod 2), straightforward calculations show that c; m)
S(27,l; = 1 =
c
e((a3y4 + 2a2by3 + (2a2c + ab2 + ab)y2 + (2abc + b2 + 27)y)/2”) c
e((a3(x +
2”2)4
+
2a2b(x +
2”2)3
+ (2&
+ ab2 +
ab)(x + 2”z)2
+(2abc + b2 + 27)(x + 2”2))/2’“) = c e((a3x4 + 2a2bx3 + (2a2c + ab2 + ab)x2 + (2abc + b2 + 27)x)/2”‘) aEZ3, x c e(((4a3x3 + 6a2bx2 + (4a2c + 2ab2 + 2ab)x + 2abc + b2 + 27)~ zEZZ,v +2”(6a3x2 + 6a2bx + 2a2c + ab2 + ab)z2 + 22v+‘a2bz3)/2u’-“) = c e((a3x4 + 2a2bx3 + (2a2c + ab2 + ab)x2 + (2abc + b2 + 27)x)/2’“) XEZ28 x c e(((4a3x3 + 6a2bx2 + 3a(b2 - 9)x + (b3 - 27b + 54)/2)z ZEZ,,,-, +2”(6a3x2 + 6a2bx + 3a(b2 - 9)/2)~~)/2”‘-~) = 1 e((a3x4 + 2a2bx3 + (2a2c + ab2 + ab)x2 + (2abc + b2 + 27)x)/2”‘) .XE& x G(2”+‘3a(ax + (b - 3)/2)(ax + (b + 3)/2), 2(a.x + (b - 3)/2)2(2ax + b + 6); 2”“‘), where G denotes an exponential
sum defined by
G(u, u; 2”) = c e((Uz2 + 212)/2”) :tZIY for integers U, u and c(> 0. For fixed x E Z2,, let 5 E { 1,2,. . . , v + 2) be defined by gcd(ax + (b - 3)/2,2”+2) = 2’.
82
J. Eichenauer-Herrmann I Journal of Computational and Applied Mathematics 79 (1997)
75-85
Since ux + (b + 3)/2 = 1 (mod 2) and 2ax + b + 6 E 1 (mod 2), it follows that gcd(2”+‘3a(au
+ (b - 3)/2)(ax
+ (b + 3)/2), 2”‘-‘) = 2mi”(‘+S+2,“‘-r)
and gcd(2(ax + (b - 3)/2)2(2ax
+ b + 6), 2”‘-‘) = 2min(2~+‘~m-“‘.
If 5 dv, then 25 + 1 < v + 5 + 2 6 co - v and therefore G(2”+‘3a(ax + (b - 3)/2)(ax
[ 10, Lemma
61 implies that
+ (b + 3)/2), 2(ux + (b - 3)/2)‘(2ax
+ b + 6); 2,-“)
= 0.
+ b + 6); 2,-“)
= 2”-“;
If 5 = v + 1, then 25 + 1 = v + C$+ 2 > co - v - 1 which implies that G(2’+‘3a(ax
+ (b - 3)/2)(ax
+ (b + 3)/2), 2(ax + (b - 3)/2)2(2ax
if 4 = v + 2, then 25 + 1 > v + 4 + 2 > o - v which yields the same result. Since there exists exactly one x E Z2, with ax + (b - 3)/2 = 0 (mod 2’+‘), i.e., r 2 v + 1, it follows that JS(27,l; c; m)l = 2”-” and therefore p-’ jy:31
>
m~a~h,c ’
l
2-v
54(7c + 2)
which is the desired result.
113
zz
-113
27(rr + 2)m 0
4. Discussion Theorem 6 shows that in the quadratic congruential method for any parameters a 3 2 (mod 4) and b =: 3 (mod4) the discrepancy Di$,C of lagged pairs, on the average over the parameters c E 1 (mod2), is of an order of magnitude at most m -‘/2(log m)2. It should be observed that the upper bound in Theorem 6 is independent of the specific choice of the parameters a, b and that its asymptotic behaviour fits well the discrepancy of m true random points from [0, 1)2, which is almost always of the order of magnitude m-‘i2(log log m)“’ according to Kiefer’s probabilistic law of the iterated logarithm for discrepancies [ 121. Theorem 7 provides even more information, since it implies that for any parameters a, b only an arbitrarily small percentage of the parameters c may lead to a discrepancy of lagged pairs with an order of magnitude greater than m-‘i2(log m)2.On the other hand, Theorem 8 shows that there exist parameters a, b, c in the quadratic congruential method such that the discrepancy D~;$, is of an order of magnitude at least rn-li3, which is too large in view of the law of the iterated logarithm. It is still an unsolved problem to characterize all parameters a, b, c, for which the discrepancy of lagged pairs is of an order of magnitude considerably greater than m-“2. Figs. l-3 illustrate the distribution of lagged pairs of quadratic congruential pseudorandom numbers in the unit square. In these examples, always the modulus m = 219 = 524288 is considered and all generated points in the subsquare [0,0.1 )2 are plotted. Hence, the expected number of points under the uniform distribution is about 5243. It turned out that in the three examples the number of points within a period falling into the subregion [0,0.1 )2 is 5201, 5 128, and 524 1, respectively. Since co = 19 and v = 5, the crucial condition on the parameters a, b, c in Theorem 8 takes the form 4uc z (b - 1)2 + 16 356 (mod 32 768). This condition is satisfied in the first two examples. In the first
J. Eichenauer-Herrmann I Journal of Computational and Applied Mathematics 79 (1997) 0.’r
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in [O,O.1)2 for parameters
a = 462 270, h = 174403,
,
. . . ::. . .‘. ,-,..
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:‘. 1”. ..:.
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:
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of lagged pairs (xn,xn+2) in [0, 0.1)2 for parameters
..
x-i.
a = 2, b = 3, c = 2045.
example, the specific values of the parameters a, b, c have been chosen arbitrarily among all parameters satisfying the crucial condition in Theorem 8. In the second example, particularly small values of a, b, c have been selected. Figs. 1 and 2 show that in both cases nearly the same pronounced line structure appears. In the third example, the values of the parameters a, b, c differ only slightly from
J. Eichenauer-Herrmann I Journal of Computational and Applied Mathematics 79 (1997)
84
. ‘“.
. . .._ ..t:
. . :. -.
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Fig. 3. Distribution
of lagged pairs (x,,x,+~)
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in [O,O.l)* for parameters
a = 462 270, b = 174403,
c = 341501.
those in the first example, but now the crucial condition in Theorem 8 is not satisfied. Fig. 3 shows that in this situation there is no obvious structure within the generated points and their distribution appears truly random. References sequences, Manuscripta Math. 58 (1987) 129-140. J. Eichenauer-Herrmann, A remark on the discrepancy of quadratic congruential pseudorandom numbers, J. Comput. Appl. Math. 43 (1992) 383-387. J. Eichenauer-Herrmann, Inversive congruential pseudorandom numbers: a tutorial, Znt. Statist. Rev. 60 (1992) 167-176. J. Eichenauer-Herrmann, Inversive congruential pseudorandom numbers, Z. Angew. Math. Mech. 73 (1993) T644-T647. J. Eichenauer-Herrmann, On the discrepancy of quadratic congruential pseudorandom numbers with power of two modulus, J. Comput. Appl. Math. 53 (1994) 371-376. J. Eichenauer-Herrmann, Pseudorandom number generation by nonlinear methods, Int. Statist. Rev. 63 (1995) 247255. J. Eichenauer-Herrmann, Discrepancy bounds for nonoverlapping pairs of quadratic congruential pseudorandom numbers, Arch. Math. 65 (1995) 362-368. J. Eichenauer-Herrmann, Quadratic congruential pseudorandom numbers: distribution of triples, J. Comput. Appl. Math. 62 (1995) 239-253. J. Eichenauer-Herrmann, Quadratic congruential pseudorandom numbers: distribution of triples, II, J. Comput. Appl. Math., to appear. J. Eichenauer-Herrmann and H. Niederreiter, On the discrepancy of quadratic congruential pseudorandom numbers, J. Comput. Appl. Math. 34 (1991) 243-249. J. Eichenauer-Herrmann and H. Niederreiter, An improved upper bound for the discrepancy of quadratic congruential pseudorandom numbers, Acta Arith. 69 (1995) 193-198.
[II J. Eichenauer and J. Lehn, On the structure of quadratic congruential PI [31 [41 [51 161 [71 181 [91 [lOI [ill
J. Eichenauer-Herrmann I Journal
qf Computational
and Applied Mathematics 79 (I 997) 75-6’5
85
[ 121 J. Kiefer, On large deviations of the empiric d.f. of vector chance variables and a law of the iterated logarithm, Pac$c J. Math. 11 (1961) 649-660. [13] D.E. Knuth, The Art of Computer Programming, Vol. 2, Seminumerical Algorithms (Addison-Wesley, Reading, MA, 2nd ed., 1981). [ 141 H. Niederreiter, Pseudo-random numbers and optimal coefficients, Adn. Math. 26 (1977) 99-181. [15] H. Niederreiter, The serial test for congruential pseudorandom numbers generated by inversions. Math. Comp. 52 (1989) 135-144. [16] H. Niederreiter, Lower bounds for the discrepancy of inversive congruential pseudorandom numbers, Math. Comp. 55 (1990) 277-287. [17] H. Niederreiter, Recent trends in random number and random vector generation, Ann. Oper. Rrs. 31 (1991) 323-345. [ 181 H. Niederreiter, Nonlinear methods for pseudorandom number and vector generation, in: G. Pflug and U. Dieter, Eds., Simulation and Optimization, Lecture Notes in Econom. and Math. Systems 374 (Springer, Berlin, 1992) 145-153. [ 191 H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods (SIAM, Philadelphia, PA, 1992). [20] H. Niederreiter, Finite fields, pseudorandom numbers, and quasirandom points, in: G.L. Mullen and P.J.S. Shiue, Eds., Finite Fields. Coding Theory, and Advances in Communications und Computing (Dekker, New York, 1993) 375-394. [21] H. Niederreiter, Pseudorandom numbers and quasirandom points, Z. Anyew. Math. Mech. 73 (1993) T648-T652. [22] H. Niederreiter, New developments in uniform pseudorandom number and vector generation, in: H. Niederreiter and P.J.-S. Shiue, Eds., Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, Lecture Notes in Statistics 106 (Springer, New York, 1995) 87-120.