Node:Table_LCG, Next:Table_ICG, Up:Tables
y_n = a * y_{n-1} + b (mod p) n > 0
Hint: A rule of thumb suggests not to use more than sqrt(p) random
numbers from an LCG.
Notice that moduli larger than 2^32 require a computer with
sizeof(long)>32
.
Generators recommended by Park and Miller (1988),
"Random number generators: good ones are hard to find", Comm. ACM 31, pp. 1192-1201
(Minimal standard).
modul p | multiplicator a
| |
----------- | --------------------------
| |
2^31 - 1 = | 2147483647 | 16807 (b = 0)
|
Generators recommended by Fishman (1990),
"Multiplicative congruential random number generators with modulus
2^beta:
An exhaustive analysis for
beta=32
and a partial analysis for
beta=48",
Math. Comp. 54, pp. 331-344.
modul p | multiplicator a
| |
----------- | --------------------------
| |
2^31 - 1 = | 2147483647 | 950706376 (b = 0)
|
Generators recommended by L'Ecuyer (1999),
"Tables of linear congruential generators of different sizes and
good lattice structure", Math.Comp. 68, pp. 249-260.
(constant b = 0.)
Generators with short periods can be used for generating quasi-random numbers
(Quasi-Monte Carlo methods). In this case the whole period should be used.
(These figures are listed without warranty. Please see also the original paper.)
modul p | multiplicator a
| |
----------- | --------------------------
| |
2^8 - 5 = | 251 | 33
|
55
| ||
| ||
2^9 - 3 = | 509 | 25
|
110
| ||
273
| ||
349
| ||
| ||
2^10 - 3 = | 1021 | 65
|
331
| ||
| ||
2^11 - 9 = | 2039 | 995
|
328
| ||
393
| ||
| ||
2^12 - 3 = | 4093 | 209
|
235
| ||
219
| ||
3551
| ||
| ||
2^13 - 1 = | 8191 | 884
|
1716
| ||
2685
| ||
| ||
2^14 - 3 = | 16381 | 572
|
3007
| ||
665
| ||
12957
| ||
| ||
2^15 - 19 = | 32749 | 219
|
1944
| ||
9515
| ||
22661
| ||
| ||
2^16 - 15 = | 65521 | 17364
|
33285
| ||
2469
| ||
| ||
2^17 - 1 = | 131071 | 43165
|
29223
| ||
29803
| ||
| ||
2^18 - 5 = | 262139 | 92717
|
21876
| ||
| ||
2^19 - 1 = | 524287 | 283741
|
37698
| ||
155411
| ||
| ||
2^20 - 3 = | 1048573 | 380985
|
604211
| ||
100768
| ||
947805
| ||
22202
| ||
1026371
| ||
| ||
2^21 - 9 = | 2097143 | 360889
|
1043187
| ||
1939807
| ||
| ||
2^22 - 3 = | 4194301 | 914334
|
2788150
| ||
1731287
| ||
2463014
| ||
| ||
2^23 - 15 = | 8388593 | 653276
|
3219358
| ||
1706325
| ||
6682268
| ||
422527
| ||
7966066
| ||
| ||
2^24 - 3 = | 16777213 | 6423135
|
7050296
| ||
4408741
| ||
12368472
| ||
931724
| ||
15845489
| ||
| ||
2^25 - 39 = | 33554393 | 25907312
|
12836191
| ||
28133808
| ||
25612572
| ||
31693768
| ||
| ||
2^26 - 5 = | 67108859 | 26590841
|
19552116
| ||
66117721
| ||
| ||
2^27 - 39 = | 134217689 | 45576512
|
63826429
| ||
3162696
| ||
| ||
2^28 - 57 = | 268435399 | 246049789
|
140853223
| ||
29908911
| ||
104122896
| ||
| ||
2^29 - 3 = | 536870909 | 520332806
|
530877178
| ||
| ||
2^30 - 35 = | 1073741789 | 771645345
|
295397169
| ||
921746065
| ||
| ||
2^31 - 1 = | 2147483647 | 1583458089
|
784588716
| ||
| ||
2^32 - 5 = | 4294967291 | 1588635695
|
1223106847
| ||
279470273
| ||
| ||
2^33 - 9 = | 8589934583 | 7425194315
|
2278442619
| ||
7312638624
| ||
| ||
2^34 - 41 = | 17179869143 | 5295517759
|
473186378
| ||
| ||
2^35 - 31 = | 34359738337 | 3124199165
|
22277574834
| ||
8094871968
| ||
| ||
2^36 - 5 = | 68719476731 | 49865143810
|
45453986995
| ||
| ||
2^37 - 25 = | 137438953447 | 76886758244
|
2996735870
| ||
85876534675
| ||
| ||
2^38 - 45 = | 274877906899 | 17838542566
|
101262352583
| ||
24271817484
| ||
| ||
2^39 - 7 = | 549755813881 | 61992693052
|
486583348513
| ||
541240737696
| ||
| ||
2^40 - 87 = | 1099511627689 | 1038914804222
|
88718554611
| ||
937333352873
| ||
| ||
2^41 - 21 = | 2199023255531 | 140245111714
|
416480024109
| ||
1319743354064
| ||
| ||
2^42 - 11 = | 4398046511093 | 2214813540776
|
2928603677866
| ||
92644101553
| ||
| ||
2^43 - 57 = | 8796093022151 | 4928052325348
|
4204926164974
| ||
3663455557440
| ||
| ||
2^44 - 17 = | 17592186044399 | 6307617245999
|
11394954323348
| ||
949305806524
| ||
| ||
2^45 - 55 = | 35184372088777 | 25933916233908
|
18586042069168
| ||
20827157855185
| ||
| ||
2^46 - 21 = | 70368744177643 | 63975993200055
|
15721062042478
| ||
31895852118078
| ||
| ||
2^47 - 115 = | 140737488355213 | 72624924005429
|
47912952719020
| ||
106090059835221
| ||
| ||
2^48 - 59 = | 281474976710597 | 49235258628958
|
51699608632694
| ||
59279420901007
| ||
| ||
2^49 - 81 = | 562949953421231 | 265609885904224
|
480567615612976
| ||
305898857643681
| ||
| ||
2^50 - 27 = | 1125899906842597 | 1087141320185010
|
157252724901243
| ||
791038363307311
| ||
| ||
2^51 - 129 = | 2251799813685119 | 349044191547257
|
277678575478219
| ||
486848186921772
| ||
| ||
2^52 - 47 = | 4503599627370449 | 4359287924442956
|
3622689089018661
| ||
711667642880185
| ||
| ||
2^53 - 111 = | 9007199254740881 | 2082839274626558
|
4179081713689027
| ||
5667072534355537
| ||
| ||
2^54 - 33 = | 18014398509481951 | 9131148267933071
|
3819217137918427
| ||
11676603717543485
| ||
| ||
2^55 - 55 = | 36028797018963913 | 33266544676670489
|
19708881949174686
| ||
32075972421209701
| ||
| ||
2^56 - 5 = | 72057594037927931 | 4595551687825993
|
26093644409268278
| ||
4595551687828611
| ||
| ||
2^57 - 13 = | 144115188075855859 | 75953708294752990
|
95424006161758065
| ||
133686472073660397
| ||
| ||
2^58 - 27 = | 288230376151711717 | 101565695086122187
|
163847936876980536
| ||
206638310974457555
| ||
| ||
2^59 - 55 = | 576460752303423433 | 346764851511064641
|
124795884580648576
| ||
573223409952553925
| ||
| ||
2^60 - 93 = | 1152921504606846883 | 561860773102413563
|
439138238526007932
| ||
734022639675925522
| ||
| ||
2^61 - 1 = | 2305843009213693951 | 1351750484049952003
|
1070922063159934167
| ||
1267205010812451270
| ||
| ||
2^62 - 57 = | 4611686018427387847 | 2774243619903564593
|
431334713195186118
| ||
2192641879660214934
| ||
| ||
2^63 - 25 = | 9223372036854775783 | 4645906587823291368
|
2551091334535185398
| ||
4373305567859904186
| ||
| ||
2^64 - 59 = | 18446744073709551557 | 13891176665706064842
|
2227057010910366687
| ||
18263440312458789471
|