1097:
740:
405:
auxiliary memory to compute, similar to other rank-revealing decompositions. Crucially however, if a row/column is added or removed or the matrix is perturbed by a rank-one matrix, its decomposition can be updated in
1307:
1410:
991:
1232:
963:
1697:
1358:
1198:
1152:
895:
839:
790:
638:
478:
364:
299:
270:
106:
403:
1126:
325:
925:
869:
1412:, is a special case of the UTV decomposition. Computing the SVD is slightly more expensive than the UTV decomposition, but has a stronger rank-revealing property.
569:
436:
153:
764:
1172:
983:
813:
658:
612:
589:
542:
502:
244:
224:
201:
177:
126:
63:
1818:
1843:
666:
1838:
1690:
32:, but typically somewhat cheaper to compute and in particular much cheaper and easier to update when the original matrix is slightly altered.
1240:
1869:
1797:
1683:
1613:
1494:
1469:
1828:
1755:
1588:
Adams, G.; Griffin, M.F.; Stewart, G.W. (1991). "Direction-of-arrival estimation using the rank-revealing URV decomposition".
1864:
1655:
1363:
1787:
1318:
29:
1741:
1792:
1706:
366:
1547:
1092:{\displaystyle {\begin{bmatrix}R_{11}^{*}\\R_{12}^{*}\end{bmatrix}}=V'{\begin{bmatrix}T^{*}\\0\end{bmatrix}}}
1431:
897:
1590:[Proceedings] ICASSP 91: 1991 International Conference on Acoustics, Speech, and Signal Processing
1751:
1592:. Proc. Of IEEE Internat. Conf. On Acoustics, Speech, and Signal Processing. pp. 1385-1388 vol.2.
1746:
43:
25:
1725:
1426:
930:
1619:
1570:
1421:
792:
1324:
1177:
1131:
874:
818:
769:
617:
444:
330:
278:
249:
72:
1609:
1490:
1465:
1203:
372:
180:
304:
1761:
1601:
1593:
1562:
1528:
903:
847:
521:
547:
409:
131:
1802:
1314:
749:
484:. Depending on whether a left-triangular or right-triangular matrix is used in place of
1106:
1720:
1157:
968:
841:
798:
643:
597:
574:
527:
487:
229:
209:
186:
162:
156:
111:
66:
48:
40:
17:
1858:
1766:
36:
1574:
1623:
204:
1597:
735:{\displaystyle A\Pi =U{\begin{bmatrix}R_{11}&R_{12}\\0&0\end{bmatrix}}}
1566:
1532:
1782:
1637:
1546:
Fierro, Ricardo D.; Hansen, Per
Christian; Hansen, Peter Søren Kirk (1999).
524:: one QR decomposition is applied to the matrix from the left, which yields
1675:
1519:
Decomposition Solver for
Hierarchically Semiseparable Representations".
1302:{\displaystyle A=U{\begin{bmatrix}T&0\\0&0\end{bmatrix}}V^{*}}
965:
matrix. One then performs another QR decomposition on the adjoint of
1833:
1823:
1605:
1548:"UTV Tools: Matlab templates for rank-revealing UTV decompositions"
520:
The UTV decomposition is usually computed by means of a pair of
1679:
660:. One first performs a QR decomposition with column pivoting:
1515:
Chandrasekaran, S.; Gu, M.; Pals, T. (January 2006). "A Fast
272:
block is nonzero, making the decomposition rank-revealing.
1258:
1061:
1000:
687:
1460:
Golub, Gene; van Loon, Charles F. (15 October 1996).
1366:
1327:
1243:
1206:
1180:
1160:
1134:
1109:
994:
971:
933:
906:
877:
850:
821:
801:
772:
752:
669:
646:
620:
600:
577:
550:
530:
490:
447:
412:
375:
333:
307:
281:
252:
232:
212:
189:
165:
134:
114:
75:
51:
1234:
yields the complete orthogonal (UTV) decomposition:
35:
Specifically, the complete orthogonal decomposition
1811:
1775:
1734:
1713:
1405:{\displaystyle S_{11}\geq S_{22}\geq \ldots \geq 0}
1464:(Third ed.). Johns Hopkins University Press.
1404:
1352:
1301:
1226:
1192:
1166:
1146:
1120:
1091:
977:
957:
919:
889:
863:
833:
807:
784:
758:
734:
652:
632:
606:
583:
563:
536:
496:
472:
430:
397:
358:
319:
293:
264:
238:
218:
195:
171:
147:
120:
100:
57:
327:, the complete orthogonal decomposition requires
1521:SIAM Journal on Matrix Analysis and Applications
1510:
1508:
1506:
1455:
1453:
1451:
1449:
1447:
544:, another applied from the right, which yields
1691:
8:
1638:"LAPACK – Complete Orthogonal Factorization"
1487:Numerical methods for least squares problems
1698:
1684:
1676:
246:can be chosen such that only its top-left
1384:
1371:
1365:
1344:
1326:
1293:
1253:
1242:
1205:
1179:
1159:
1133:
1108:
1068:
1056:
1031:
1026:
1012:
1007:
995:
993:
970:
932:
911:
905:
876:
855:
849:
820:
800:
771:
751:
706:
694:
682:
668:
645:
619:
599:
576:
555:
549:
529:
489:
464:
446:
411:
386:
374:
347:
332:
306:
280:
251:
231:
211:
188:
164:
139:
133:
113:
92:
74:
50:
1656:"Eigen::CompleteOrthogonalDecomposition"
1200:lower (left) triangular matrix. Setting
1829:Basic Linear Algebra Subprograms (BLAS)
1443:
571:, which "sandwiches" triangular matrix
480:, the decomposition is also known as
7:
1317:is by construction triangular, the
1213:
753:
673:
14:
1660:Eigen 3.3 reference documentation
22:complete orthogonal decomposition
1422:Rank-revealing QR decomposition
952:
940:
425:
416:
392:
379:
353:
337:
1:
1485:Björck, Åke (December 1996).
958:{\displaystyle r\times (n-r)}
1319:singular value decomposition
504:, it is also referred to as
30:singular value decomposition
1886:
1742:System of linear equations
1598:10.1109/icassp.1991.150681
1793:Cache-oblivious algorithm
1533:10.1137/S0895479803436652
1353:{\displaystyle A=USV^{*}}
1193:{\displaystyle r\times r}
1147:{\displaystyle n\times n}
890:{\displaystyle r\times r}
834:{\displaystyle m\times m}
785:{\displaystyle n\times n}
633:{\displaystyle m\times n}
473:{\displaystyle A=UTV^{*}}
367:floating point operations
359:{\displaystyle O(mn^{2})}
294:{\displaystyle m\times n}
265:{\displaystyle r\times r}
101:{\displaystyle A=UTV^{*}}
1870:Numerical linear algebra
1844:General purpose software
1707:Numerical linear algebra
1227:{\displaystyle V=\Pi V'}
398:{\displaystyle O(m^{2})}
226:, the triangular matrix
1567:10.1023/A:1019112103049
1432:Online machine learning
898:upper triangular matrix
320:{\displaystyle m\geq n}
28:. It is similar to the
1406:
1354:
1303:
1228:
1194:
1168:
1148:
1122:
1093:
979:
959:
921:
920:{\displaystyle R_{12}}
891:
865:
864:{\displaystyle R_{11}}
835:
809:
786:
760:
736:
654:
634:
608:
585:
565:
538:
498:
474:
432:
399:
360:
321:
295:
266:
240:
220:
197:
173:
149:
122:
102:
59:
1865:Matrix decompositions
1839:Specialized libraries
1752:Matrix multiplication
1747:Matrix decompositions
1407:
1355:
1304:
1229:
1195:
1169:
1149:
1123:
1094:
980:
960:
922:
892:
866:
836:
810:
787:
761:
737:
655:
635:
609:
586:
566:
564:{\displaystyle V^{*}}
539:
499:
475:
441:Because of its form,
433:
431:{\displaystyle O(mn)}
400:
361:
322:
296:
275:For a matrix of size
267:
241:
221:
198:
174:
150:
148:{\displaystyle V^{*}}
123:
103:
60:
1555:Numerical Algorithms
1364:
1325:
1241:
1204:
1178:
1158:
1132:
1107:
992:
969:
931:
904:
875:
848:
819:
799:
770:
759:{\displaystyle \Pi }
750:
667:
644:
618:
598:
575:
548:
528:
488:
445:
410:
373:
331:
305:
279:
250:
230:
210:
187:
163:
132:
112:
73:
49:
26:matrix decomposition
1726:Numerical stability
1462:Matrix Computations
1427:Schur decomposition
1154:unitary matrix and
1036:
1017:
69:of three matrices,
1402:
1350:
1299:
1283:
1224:
1190:
1164:
1144:
1121:{\displaystyle V'}
1118:
1089:
1083:
1039:
1022:
1003:
975:
955:
917:
887:
861:
831:
805:
793:permutation matrix
782:
756:
732:
726:
650:
630:
604:
581:
561:
534:
494:
470:
428:
395:
356:
317:
291:
262:
236:
216:
193:
169:
145:
118:
98:
55:
1852:
1851:
1167:{\displaystyle T}
978:{\displaystyle R}
808:{\displaystyle U}
653:{\displaystyle r}
607:{\displaystyle A}
584:{\displaystyle T}
537:{\displaystyle U}
522:QR decompositions
510:URV decomposition
506:ULV decomposition
497:{\displaystyle T}
482:UTV decomposition
239:{\displaystyle T}
219:{\displaystyle r}
196:{\displaystyle A}
181:triangular matrix
172:{\displaystyle T}
121:{\displaystyle U}
58:{\displaystyle A}
1877:
1762:Matrix splitting
1700:
1693:
1686:
1677:
1670:
1669:
1667:
1666:
1652:
1646:
1645:
1634:
1628:
1627:
1585:
1579:
1578:
1561:(2/3): 165–194.
1552:
1543:
1537:
1536:
1512:
1501:
1500:
1482:
1476:
1475:
1457:
1411:
1409:
1408:
1403:
1389:
1388:
1376:
1375:
1359:
1357:
1356:
1351:
1349:
1348:
1308:
1306:
1305:
1300:
1298:
1297:
1288:
1287:
1233:
1231:
1230:
1225:
1223:
1199:
1197:
1196:
1191:
1173:
1171:
1170:
1165:
1153:
1151:
1150:
1145:
1127:
1125:
1124:
1119:
1117:
1098:
1096:
1095:
1090:
1088:
1087:
1073:
1072:
1055:
1044:
1043:
1035:
1030:
1016:
1011:
984:
982:
981:
976:
964:
962:
961:
956:
926:
924:
923:
918:
916:
915:
896:
894:
893:
888:
870:
868:
867:
862:
860:
859:
840:
838:
837:
832:
814:
812:
811:
806:
791:
789:
788:
783:
765:
763:
762:
757:
741:
739:
738:
733:
731:
730:
711:
710:
699:
698:
659:
657:
656:
651:
639:
637:
636:
631:
613:
611:
610:
605:
590:
588:
587:
582:
570:
568:
567:
562:
560:
559:
543:
541:
540:
535:
512:, respectively.
503:
501:
500:
495:
479:
477:
476:
471:
469:
468:
437:
435:
434:
429:
404:
402:
401:
396:
391:
390:
365:
363:
362:
357:
352:
351:
326:
324:
323:
318:
300:
298:
297:
292:
271:
269:
268:
263:
245:
243:
242:
237:
225:
223:
222:
217:
202:
200:
199:
194:
183:. For a matrix
178:
176:
175:
170:
157:unitary matrices
154:
152:
151:
146:
144:
143:
127:
125:
124:
119:
107:
105:
104:
99:
97:
96:
64:
62:
61:
56:
1885:
1884:
1880:
1879:
1878:
1876:
1875:
1874:
1855:
1854:
1853:
1848:
1807:
1803:Multiprocessing
1771:
1767:Sparse problems
1730:
1709:
1704:
1674:
1673:
1664:
1662:
1654:
1653:
1649:
1636:
1635:
1631:
1616:
1587:
1586:
1582:
1550:
1545:
1544:
1540:
1514:
1513:
1504:
1497:
1484:
1483:
1479:
1472:
1459:
1458:
1445:
1440:
1418:
1380:
1367:
1362:
1361:
1340:
1323:
1322:
1315:diagonal matrix
1289:
1282:
1281:
1276:
1270:
1269:
1264:
1254:
1239:
1238:
1216:
1202:
1201:
1176:
1175:
1156:
1155:
1130:
1129:
1110:
1105:
1104:
1082:
1081:
1075:
1074:
1064:
1057:
1048:
1038:
1037:
1019:
1018:
996:
990:
989:
967:
966:
929:
928:
907:
902:
901:
873:
872:
851:
846:
845:
817:
816:
797:
796:
768:
767:
748:
747:
725:
724:
719:
713:
712:
702:
700:
690:
683:
665:
664:
642:
641:
640:matrix of rank
616:
615:
596:
595:
591:in the middle.
573:
572:
551:
546:
545:
526:
525:
518:
486:
485:
460:
443:
442:
408:
407:
382:
371:
370:
343:
329:
328:
303:
302:
277:
276:
248:
247:
228:
227:
208:
207:
185:
184:
161:
160:
135:
130:
129:
110:
109:
88:
71:
70:
47:
46:
12:
11:
5:
1883:
1881:
1873:
1872:
1867:
1857:
1856:
1850:
1849:
1847:
1846:
1841:
1836:
1831:
1826:
1821:
1815:
1813:
1809:
1808:
1806:
1805:
1800:
1795:
1790:
1785:
1779:
1777:
1773:
1772:
1770:
1769:
1764:
1759:
1749:
1744:
1738:
1736:
1732:
1731:
1729:
1728:
1723:
1721:Floating point
1717:
1715:
1711:
1710:
1705:
1703:
1702:
1695:
1688:
1680:
1672:
1671:
1647:
1629:
1614:
1580:
1538:
1527:(3): 603–622.
1502:
1495:
1477:
1470:
1442:
1441:
1439:
1436:
1435:
1434:
1429:
1424:
1417:
1414:
1401:
1398:
1395:
1392:
1387:
1383:
1379:
1374:
1370:
1347:
1343:
1339:
1336:
1333:
1330:
1311:
1310:
1296:
1292:
1286:
1280:
1277:
1275:
1272:
1271:
1268:
1265:
1263:
1260:
1259:
1257:
1252:
1249:
1246:
1222:
1219:
1215:
1212:
1209:
1189:
1186:
1183:
1163:
1143:
1140:
1137:
1116:
1113:
1101:
1100:
1086:
1080:
1077:
1076:
1071:
1067:
1063:
1062:
1060:
1054:
1051:
1047:
1042:
1034:
1029:
1025:
1021:
1020:
1015:
1010:
1006:
1002:
1001:
999:
974:
954:
951:
948:
945:
942:
939:
936:
914:
910:
886:
883:
880:
858:
854:
842:unitary matrix
830:
827:
824:
804:
781:
778:
775:
755:
744:
743:
729:
723:
720:
718:
715:
714:
709:
705:
701:
697:
693:
689:
688:
686:
681:
678:
675:
672:
649:
629:
626:
623:
603:
580:
558:
554:
533:
517:
514:
493:
467:
463:
459:
456:
453:
450:
427:
424:
421:
418:
415:
394:
389:
385:
381:
378:
355:
350:
346:
342:
339:
336:
316:
313:
310:
290:
287:
284:
261:
258:
255:
235:
215:
192:
168:
142:
138:
117:
95:
91:
87:
84:
81:
78:
54:
18:linear algebra
13:
10:
9:
6:
4:
3:
2:
1882:
1871:
1868:
1866:
1863:
1862:
1860:
1845:
1842:
1840:
1837:
1835:
1832:
1830:
1827:
1825:
1822:
1820:
1817:
1816:
1814:
1810:
1804:
1801:
1799:
1796:
1794:
1791:
1789:
1786:
1784:
1781:
1780:
1778:
1774:
1768:
1765:
1763:
1760:
1757:
1753:
1750:
1748:
1745:
1743:
1740:
1739:
1737:
1733:
1727:
1724:
1722:
1719:
1718:
1716:
1712:
1708:
1701:
1696:
1694:
1689:
1687:
1682:
1681:
1678:
1661:
1657:
1651:
1648:
1643:
1639:
1633:
1630:
1625:
1621:
1617:
1615:0-7803-0003-3
1611:
1607:
1603:
1599:
1595:
1591:
1584:
1581:
1576:
1572:
1568:
1564:
1560:
1556:
1549:
1542:
1539:
1534:
1530:
1526:
1522:
1518:
1511:
1509:
1507:
1503:
1498:
1496:0-89871-360-9
1492:
1488:
1481:
1478:
1473:
1471:0-8018-5414-8
1467:
1463:
1456:
1454:
1452:
1450:
1448:
1444:
1437:
1433:
1430:
1428:
1425:
1423:
1420:
1419:
1415:
1413:
1399:
1396:
1393:
1390:
1385:
1381:
1377:
1372:
1368:
1345:
1341:
1337:
1334:
1331:
1328:
1320:
1316:
1294:
1290:
1284:
1278:
1273:
1266:
1261:
1255:
1250:
1247:
1244:
1237:
1236:
1235:
1220:
1217:
1210:
1207:
1187:
1184:
1181:
1161:
1141:
1138:
1135:
1114:
1111:
1084:
1078:
1069:
1065:
1058:
1052:
1049:
1045:
1040:
1032:
1027:
1023:
1013:
1008:
1004:
997:
988:
987:
986:
972:
949:
946:
943:
937:
934:
912:
908:
899:
884:
881:
878:
856:
852:
843:
828:
825:
822:
802:
794:
779:
776:
773:
727:
721:
716:
707:
703:
695:
691:
684:
679:
676:
670:
663:
662:
661:
647:
627:
624:
621:
601:
592:
578:
556:
552:
531:
523:
515:
513:
511:
507:
491:
483:
465:
461:
457:
454:
451:
448:
439:
422:
419:
413:
387:
383:
376:
368:
348:
344:
340:
334:
314:
311:
308:
288:
285:
282:
273:
259:
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