693:
of the generator itself, they can evidently still lead to significant errors.". This only refers to the standard LFG where each new number in the sequence depends on two previous numbers. A three-tap LFG has been shown to eliminate some statistical problems such as failing the
540:
observed very poor behavior with R(24, 55) and smaller generators, and advised against using generators of this type altogether. ... The basic problem of two-tap generators R(a, b) is that they have a built-in three-point correlation between
340:. If addition or subtraction is used, the maximum period is (2 − 1) × 2. If multiplication is used, the maximum period is (2 − 1) × 2, or 1/4 of period of the additive case. If bitwise xor is used, the maximum period is 2 − 1.
222:
536:, referring to LFGs that use the XOR operator, states that "It is now widely known that such generators, in particular with the two-tap rules such as R(103, 250), have serious deficiencies.
117:
691:
838:
632:
599:
338:
253:
566:
498:
Note that the smaller number have short periods (only a few "random" numbers are generated before the first "random" number is repeated and the sequence restarts).
907:
494:(24, 55), (38, 89), (37, 100), (30, 127), (83, 258), (107, 378), (273, 607), (1029, 2281), (576, 3217), (4187, 9689), (7083, 19937), (9739, 23209)
128:
363:
300:
842:
744:
486:
309:
267:). The theory of this type of generator is rather complex, and it may not be sufficient simply to choose random values for
751:
37:
29:
505:
values chosen to initialise the generator be odd. If multiplication is used, instead, it is required that all the first
793:
877:
808:
912:
122:
Hence, the new term is the sum of the last two terms in the sequence. This can be generalised to the sequence:
33:
57:
637:
533:
720:
48:
41:
734:
implements this generator in its DBMS_RANDOM package (available in Oracle 8 and newer versions).
260:
634:, simply given by the generator itself ... While these correlations are spread over the size
604:
571:
761:
537:
323:
305:
256:
238:
544:
731:
766:
710:
uses a lagged
Fibonacci generator with {j = 24, k = 55} for its random number generator.
901:
776:
771:
714:
695:
756:
521:
320:
The maximum period of lagged
Fibonacci generators depends on the binary operation
862:
781:
227:
In which case, the new term is some combination of any two previous terms.
889:
816:
289:
If the operation used is addition, then the generator is described as an
892:, Robert M. Ziff, Computers in Physics, 12(4), Jul/Aug 1998, pp. 385–392
724:
747:' lists other PRNGs including some with better statistical qualitites:
707:
217:{\displaystyle S_{n}\equiv S_{n-j}\star S_{n-k}{\pmod {m}},0<j<k}
308:
algorithm is a variation on a GFSR. The GFSR is also related to the
275:. These generators also tend to be very sensitive to initialisation.
501:
If addition is used, it is required that at least one of the first
374:
satisfying this constraint have been published in the literature.
259:. This may be either addition, subtraction, multiplication, or the
863:
Parameterizing
Parallel Multiplicative Lagged-Fibonacci Generators
509:
values be odd, and further that at least one of them is ±3 mod 8.
343:
For the generator to achieve this maximum period, the polynomial:
839:"SPRNG: Scalable Parallel Pseudo-Random Number Generator Library"
264:
890:"Four-tap shift-register-sequence random-number generators"
717:
includes an implementation of a lagged
Fibonacci generator.
723:, a lagged Fibonacci generator engine, is included in the
297:
or MLFG, and if the XOR operation is used, it is called a
878:"Uniform random number generators for supercomputers"
640:
607:
574:
547:
326:
241:
131:
60:
36:is aimed at being an improvement on the 'standard'
685:
626:
593:
560:
332:
247:
216:
111:
512:It has been suggested that good ratios between
47:The Fibonacci sequence may be described by the
378:Popular pairs of primitive polynomial degrees
40:. These are based on a generalisation of the
8:
293:or ALFG, if multiplication is used, it is a
794:Toward a universal random number generator
873:
871:
639:
612:
606:
579:
573:
552:
546:
325:
316:Properties of lagged Fibonacci generators
295:Multiplicative Lagged Fibonacci Generator
282:words of state (they 'remember' the last
240:
180:
168:
149:
136:
130:
97:
78:
65:
59:
532:In a paper on four-tap shift registers,
376:
800:
112:{\displaystyle S_{n}=S_{n-1}+S_{n-2}}
7:
686:{\displaystyle p=max(a,b,c,\ldots )}
476:Another list of possible values for
366:over the integers mod 2. Values of
301:generalised feedback shift register
291:Additive Lagged Fibonacci Generator
188:
14:
745:List of random number generators
487:The Art of Computer Programming
278:Generators of this type employ
181:
908:Pseudorandom number generators
698:and Generalized Triple tests.
680:
656:
310:linear-feedback shift register
192:
182:
1:
752:Linear congruential generator
484:is on page 29 of volume 2 of
38:linear congruential generator
30:pseudorandom number generator
865:, M. Mascagni, A. Srinivasan
255:operator denotes a general
929:
18:Lagged Fibonacci generator
231:is usually a power of 2 (
235:= 2), often 2 or 2. The
627:{\displaystyle x_{n-b}}
594:{\displaystyle x_{n-a}}
263:exclusive-or operator (
34:random number generator
796:, G.Marsaglia, A.Zaman
687:
628:
595:
562:
520:are approximately the
334:
333:{\displaystyle \star }
249:
248:{\displaystyle \star }
218:
113:
880:, Richard Brent, 1992
688:
629:
596:
563:
561:{\displaystyle x_{n}}
335:
250:
219:
114:
28:) is an example of a
638:
605:
572:
545:
324:
239:
129:
58:
721:Subtract with carry
379:
49:recurrence relation
683:
624:
591:
558:
528:Problems with LFGs
377:
330:
245:
214:
109:
42:Fibonacci sequence
913:Fibonacci numbers
696:Birthday Spacings
474:
473:
920:
893:
887:
881:
875:
866:
860:
854:
853:
851:
850:
841:. Archived from
835:
829:
828:
826:
824:
815:. Archived from
805:
762:Mersenne Twister
743:Knowledge page '
692:
690:
689:
684:
633:
631:
630:
625:
623:
622:
600:
598:
597:
592:
590:
589:
567:
565:
564:
559:
557:
556:
380:
339:
337:
336:
331:
306:Mersenne Twister
257:binary operation
254:
252:
251:
246:
223:
221:
220:
215:
195:
179:
178:
160:
159:
141:
140:
118:
116:
115:
110:
108:
107:
89:
88:
70:
69:
32:. This class of
928:
927:
923:
922:
921:
919:
918:
917:
898:
897:
896:
888:
884:
876:
869:
861:
857:
848:
846:
837:
836:
832:
822:
820:
819:on 9 March 2004
813:www.ccs.uky.edu
807:
806:
802:
790:
757:ACORN generator
741:
732:Oracle Database
704:
636:
635:
608:
603:
602:
575:
570:
569:
548:
543:
542:
530:
519:
515:
322:
321:
318:
274:
270:
237:
236:
164:
145:
132:
127:
126:
93:
74:
61:
56:
55:
12:
11:
5:
926:
924:
916:
915:
910:
900:
899:
895:
894:
882:
867:
855:
830:
799:
798:
797:
789:
786:
785:
784:
779:
774:
769:
764:
759:
754:
740:
737:
736:
735:
728:
718:
711:
703:
700:
682:
679:
676:
673:
670:
667:
664:
661:
658:
655:
652:
649:
646:
643:
621:
618:
615:
611:
588:
585:
582:
578:
555:
551:
534:Robert M. Ziff
529:
526:
517:
513:
496:
495:
472:
471:
468:
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459:
456:
453:
450:
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419:
416:
413:
410:
407:
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401:
398:
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389:
386:
360:
359:
329:
317:
314:
272:
268:
244:
225:
224:
213:
210:
207:
204:
201:
198:
194:
191:
187:
184:
177:
174:
171:
167:
163:
158:
155:
152:
148:
144:
139:
135:
120:
119:
106:
103:
100:
96:
92:
87:
84:
81:
77:
73:
68:
64:
13:
10:
9:
6:
4:
3:
2:
925:
914:
911:
909:
906:
905:
903:
891:
886:
883:
879:
874:
872:
868:
864:
859:
856:
845:on 2010-06-14
844:
840:
834:
831:
818:
814:
810:
804:
801:
795:
792:
791:
787:
783:
780:
778:
775:
773:
772:FISH (cipher)
770:
768:
767:Xoroshiro128+
765:
763:
760:
758:
755:
753:
750:
749:
748:
746:
738:
733:
729:
726:
722:
719:
716:
715:Boost library
712:
709:
706:
705:
701:
699:
697:
677:
674:
671:
668:
665:
662:
659:
653:
650:
647:
644:
641:
619:
616:
613:
609:
586:
583:
580:
576:
553:
549:
539:
535:
527:
525:
523:
510:
508:
504:
499:
493:
492:
491:
489:
488:
483:
479:
469:
466:
463:
460:
457:
454:
451:
448:
445:
442:
439:
436:
433:
431:
428:
427:
423:
420:
417:
414:
411:
408:
405:
402:
399:
396:
393:
390:
387:
385:
382:
381:
375:
373:
369:
365:
357:
353:
349:
346:
345:
344:
341:
327:
315:
313:
311:
307:
304:or GFSR. The
303:
302:
296:
292:
287:
285:
281:
276:
266:
262:
258:
242:
234:
230:
211:
208:
205:
202:
199:
196:
189:
185:
175:
172:
169:
165:
161:
156:
153:
150:
146:
142:
137:
133:
125:
124:
123:
104:
101:
98:
94:
90:
85:
82:
79:
75:
71:
66:
62:
54:
53:
52:
50:
45:
43:
39:
35:
31:
27:
24:or sometimes
23:
19:
885:
858:
847:. Retrieved
843:the original
833:
821:. Retrieved
817:the original
812:
809:"RN Chapter"
803:
742:
531:
522:golden ratio
511:
506:
502:
500:
497:
485:
481:
477:
475:
429:
383:
371:
367:
361:
355:
351:
347:
342:
319:
298:
294:
290:
288:
283:
279:
277:
232:
228:
226:
121:
46:
25:
21:
17:
15:
312:, or LFSR.
902:Categories
849:2005-04-11
823:13 January
788:References
782:VIC cipher
678:…
617:−
584:−
538:Marsaglia
364:primitive
328:⋆
286:values).
243:⋆
173:−
162:⋆
154:−
143:≡
102:−
83:−
739:See also
727:library.
362:must be
299:Two-tap
708:Freeciv
261:bitwise
601:, and
725:C++11
702:Usage
470:1279
825:2022
777:Pike
730:The
713:The
516:and
480:and
424:418
370:and
271:and
209:<
203:<
26:LFib
467:607
464:607
461:521
458:521
455:127
446:159
421:273
418:334
415:168
412:353
400:128
358:+ 1
265:XOR
186:mod
22:LFG
904::
870:^
811:.
568:,
524:.
490::
452:63
449:31
443:71
440:55
437:17
434:10
409:97
406:31
397:65
394:24
354:+
350:=
51::
44:.
16:A
852:.
827:.
681:)
675:,
672:c
669:,
666:b
663:,
660:a
657:(
654:x
651:a
648:m
645:=
642:p
620:b
614:n
610:x
587:a
581:n
577:x
554:n
550:x
518:k
514:j
507:k
503:k
482:k
478:j
430:k
403:6
391:5
388:7
384:j
372:k
368:j
356:x
352:x
348:y
284:k
280:k
273:k
269:j
233:m
229:m
212:k
206:j
200:0
197:,
193:)
190:m
183:(
176:k
170:n
166:S
157:j
151:n
147:S
138:n
134:S
105:2
99:n
95:S
91:+
86:1
80:n
76:S
72:=
67:n
63:S
20:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.