403:
116:
definition has only multiplication, addition and subtraction operations. Obviously the determinant is integer if all matrix entries are integer. However actual computation of the determinant using the definition or
151:
Fraction-free algorithm — uses division to keep the intermediate entries smaller, but due to the
Sylvester's Identity the transformation is still integer-preserving (the division has zero remainder).
144:
Bareiss brings up a question of performing an integer-preserving elimination while keeping the magnitudes of the intermediate coefficients reasonably small. Two algorithms are suggested:
141:
can be avoided if all the numbers are kept as integer fractions instead of floating point. But then the size of each element grows in size exponentially with the number of rows.
743:
163:
The program structure of this algorithm is a simple triple-loop, as in the standard
Gaussian elimination. However in this case the matrix is modified so that each
864:
509:
During execution of the
Bareiss algorithm, every integer that is computed is the determinant of a submatrix of the input matrix. This allows, using the
889:
98:
271:
884:
736:
645:
915:
843:
729:
874:
118:
801:
920:
833:
135:) complexity, but introduces division, which results in round-off errors when implemented using floating point numbers.
688:
787:
653:
925:
838:
752:
541:
94:
603:
Middeke, J.; Jeffrey, D.J.; Koutschan, C. (2020), "Common
Factors in Fraction-Free Matrix Decompositions",
173:
56:
910:
797:
513:, to bound the size of these integers. Otherwise, the Bareiss algorithm may be viewed as a variant of
148:
Division-free algorithm — performs matrix reduction to triangular form without any division operation.
67:
entries, avoiding the introduction of any round-off errors beyond those already present in the input.
792:
514:
128:
48:
80:
771:
583:
510:
155:
For completeness
Bareiss also suggests fraction-producing multiplication-free elimination methods.
32:
670:
612:
414:
807:
662:
622:
848:
138:
83:
63:). The method can also be used to compute the determinant of matrices with (approximated)
766:
529:
904:
812:
528:
matrix of maximum (absolute) value 2 for each entry, the
Bareiss algorithm runs in
76:
44:
17:
113:
64:
40:
627:
828:
646:"Sylvester's Identity and multistep integer-preserving Gaussian elimination"
398:{\displaystyle M_{i,j}={\frac {M_{i,j}M_{k,k}-M_{i,k}M_{k,j}}{M_{k-1,k-1}}}}
60:
36:
721:
82:
The general
Bareiss algorithm is distinct from the Bareiss algorithm for
540: 2) bound on the absolute value of intermediate values needed. Its
453:
If the assumption about principal minors turns out to be false, e.g. if
674:
52:
879:
869:
102:
666:
89:
In some
Spanish-speaking countries, this algorithm is also known as
617:
725:
517:
and needs roughly the same number of arithmetic operations.
59:
that are performed are guaranteed to be exact (there is no
181:. Algorithm correctness is easily shown by induction on
699:(Contains a clearer picture of the operations sequence)
274:
857:
821:
780:
759:
397:
690:MULTISTEP INTEGER-PRESERVING GAUSSIAN ELIMINATION
501:-th row and change the sign of the final answer.
105:, who popularized the method among his students.
737:
8:
712:Fundamental Problems of Algorithmic Algebra
55:entries using only integer arithmetic; any
744:
730:
722:
75:The algorithm was originally announced by
639:
637:
626:
616:
559:)) when using elementary arithmetic or O(
443:contains the determinant of the original
369:
352:
336:
317:
301:
294:
279:
273:
875:Basic Linear Algebra Subprograms (BLAS)
595:
201:assuming its leading principal minors
172:entry contains the leading principal
7:
99:Universidad Autónoma de Nuevo León
25:
428:entry contains the leading minor
121:is impractical, as it requires O(
536:elementary operations with an O(
605:Mathematics in Computer Science
413:Output: The matrix is modified
1:
493:) then we can exchange the
942:
788:System of linear equations
687:Bareiss, Erwin H. (1966),
654:Mathematics of Computation
644:Bareiss, Erwin H. (1968),
628:10.1007/s11786-020-00495-9
79:in 1966 published in 1967.
839:Cache-oblivious algorithm
714:, Oxford University Press
497:−1-th row with the
95:René Mario Montante Pardo
916:Numerical linear algebra
890:General purpose software
753:Numerical linear algebra
542:computational complexity
520:It follows that, for an
710:Yap, Chee Keng (2000),
399:
224:is a special variable)
885:Specialized libraries
798:Matrix multiplication
793:Matrix decompositions
400:
97:, a professor of the
515:Gaussian elimination
272:
129:Gaussian elimination
27:In mathematics, the
921:Exchange algorithms
772:Numerical stability
584:fast multiplication
511:Hadamard inequality
395:
898:
897:
393:
207:are all non-zero.
84:Toeplitz matrices
39:to calculate the
29:Bareiss algorithm
18:Bareiss Algorithm
16:(Redirected from
933:
926:Computer algebra
808:Matrix splitting
746:
739:
732:
723:
716:
715:
707:
701:
696:
695:
684:
678:
677:
661:(103): 565–578,
650:
641:
632:
631:
630:
620:
600:
500:
496:
492:
488:
484:
478:
474:
470:
464:
460:
456:
446:
442:
432:
426:
422:
404:
402:
401:
396:
394:
392:
391:
364:
363:
362:
347:
346:
328:
327:
312:
311:
295:
290:
289:
264:
260:
256:
249:
245:
241:
234:
230:
205:
198:
194:
184:
179:
170:
166:
139:Round-off errors
91:Bareiss-Montante
21:
941:
940:
936:
935:
934:
932:
931:
930:
901:
900:
899:
894:
853:
849:Multiprocessing
817:
813:Sparse problems
776:
755:
750:
720:
719:
709:
708:
704:
693:
686:
685:
681:
667:10.2307/2004533
648:
643:
642:
635:
602:
601:
597:
592:
574:) log(log(
507:
498:
494:
490:
486:
482:
480:
476:
472:
468:
466:
462:
458:
454:
451:
450:
444:
441:
437:
435:
433:
430:
427:
424:
420:
418:
365:
348:
332:
313:
297:
296:
275:
270:
269:
262:
258:
254:
247:
243:
239:
232:
228:
223:
216:
206:
203:
200:
196:
192:
182:
180:
177:
171:
168:
164:
161:
119:Leibniz formula
111:
73:
23:
22:
15:
12:
11:
5:
939:
937:
929:
928:
923:
918:
913:
903:
902:
896:
895:
893:
892:
887:
882:
877:
872:
867:
861:
859:
855:
854:
852:
851:
846:
841:
836:
831:
825:
823:
819:
818:
816:
815:
810:
805:
795:
790:
784:
782:
778:
777:
775:
774:
769:
767:Floating point
763:
761:
757:
756:
751:
749:
748:
741:
734:
726:
718:
717:
702:
679:
633:
611:(4): 589–608,
594:
593:
591:
588:
578:) +
570:) +
555:) +
506:
503:
471:
457:
449:
448:
439:
429:
423:
411:
410:
409:
408:
407:
406:
405:
390:
387:
384:
381:
378:
375:
372:
368:
361:
358:
355:
351:
345:
342:
339:
335:
331:
326:
323:
320:
316:
310:
307:
304:
300:
293:
288:
285:
282:
278:
225:
221:
214:
208:
202:
199:-square matrix
188:
187:
176:
167:
160:
157:
153:
152:
149:
125:) operations.
110:
107:
72:
69:
31:, named after
24:
14:
13:
10:
9:
6:
4:
3:
2:
938:
927:
924:
922:
919:
917:
914:
912:
909:
908:
906:
891:
888:
886:
883:
881:
878:
876:
873:
871:
868:
866:
863:
862:
860:
856:
850:
847:
845:
842:
840:
837:
835:
832:
830:
827:
826:
824:
820:
814:
811:
809:
806:
803:
799:
796:
794:
791:
789:
786:
785:
783:
779:
773:
770:
768:
765:
764:
762:
758:
754:
747:
742:
740:
735:
733:
728:
727:
724:
713:
706:
703:
700:
692:
691:
683:
680:
676:
672:
668:
664:
660:
656:
655:
647:
640:
638:
634:
629:
624:
619:
614:
610:
606:
599:
596:
589:
587:
585:
582:))) by using
581:
577:
573:
569:
565:
562:
558:
554:
550:
547:
543:
539:
535:
533:
527:
523:
518:
516:
512:
504:
502:
467:= 0 and some
416:
412:
388:
385:
382:
379:
376:
373:
370:
366:
359:
356:
353:
349:
343:
340:
337:
333:
329:
324:
321:
318:
314:
308:
305:
302:
298:
291:
286:
283:
280:
276:
267:
266:
252:
251:
237:
236:
226:
220:
213:
209:
190:
189:
186:
175:
159:The algorithm
158:
156:
150:
147:
146:
145:
142:
140:
136:
134:
130:
126:
124:
120:
115:
108:
106:
104:
100:
96:
93:, because of
92:
87:
85:
81:
78:
70:
68:
66:
62:
58:
54:
50:
46:
42:
38:
34:
33:Erwin Bareiss
30:
19:
911:Determinants
760:Key concepts
711:
705:
698:
689:
682:
658:
652:
608:
604:
598:
579:
575:
571:
567:
563:
560:
556:
552:
548:
545:
537:
531:
525:
521:
519:
508:
452:
218:
211:
162:
154:
143:
137:
132:
127:
122:
112:
90:
88:
77:Jack Edmonds
74:
45:echelon form
28:
26:
566: (log(
551: (log(
217:= 1 (Note:
114:Determinant
41:determinant
905:Categories
802:algorithms
618:2005.12380
590:References
544:is thus O(
235:−1:
231:from 1 to
829:CPU cache
461:−1,
386:−
374:−
330:−
61:remainder
57:divisions
37:algorithm
858:Software
822:Hardware
781:Problems
505:Analysis
479:−1
465:−1
415:in-place
109:Overview
35:, is an
675:2004533
191:Input:
71:History
53:integer
43:or the
880:LAPACK
870:MATLAB
673:
436:entry
261:+1 to
246:+1 to
131:has O(
103:Mexico
49:matrix
865:ATLAS
694:(PDF)
671:JSTOR
649:(PDF)
613:arXiv
489:,...,
481:≠ 0 (
419:each
257:from
242:from
195:— an
174:minor
51:with
47:of a
844:SIMD
268:Set
253:For
238:For
227:For
210:Let
65:real
834:TLB
663:doi
623:doi
440:n,n
431:k,k
425:k,k
222:0,0
215:0,0
204:k,k
178:k,k
169:k,k
907::
697:.
669:,
659:22
657:,
651:,
636:^
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609:15
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586:.
530:O(
524:×
485:=
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800:(
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