1742:
1105:
649:
303:
412:
748:
816:
909:
1261:
1639:
1306:
1267:
473:
901:
143:
1510:
1159:
105:
1634:
846:
465:
78:
1179:
1127:
435:
131:
2230:
2128:
1503:
1414:
36:) finds the solution to the secant equation that is closest to the current estimate and satisfies the curvature condition. It was the first
318:
2209:
1671:
1723:
1584:
1296:
1691:
1479:
1460:
1441:
1350:
660:
1802:
1496:
1100:{\displaystyle H_{k+1}=H_{k}-{\frac {H_{k}y_{k}y_{k}^{T}H_{k}}{y_{k}^{T}H_{k}y_{k}}}+{\frac {s_{k}s_{k}^{T}}{y_{k}^{T}s_{k}}}.}
754:
2079:
1741:
29:
2187:
1807:
1187:
2123:
2091:
2172:
1797:
440:
The DFP formula finds a solution that is symmetric, positive-definite and closest to the current approximate value of
2118:
2074:
1676:
1967:
1696:
1857:
644:{\displaystyle B_{k+1}=(I-\gamma _{k}y_{k}s_{k}^{T})B_{k}(I-\gamma _{k}s_{k}y_{k}^{T})+\gamma _{k}y_{k}y_{k}^{T},}
1519:
2042:
849:
108:
1904:
2086:
1985:
1701:
1579:
1321:
2177:
2162:
2052:
1930:
1556:
1523:
2066:
2032:
1935:
1877:
1758:
1564:
1544:
1271:
298:{\displaystyle f(x_{k}+s_{k})=f(x_{k})+\nabla f(x_{k})^{T}s_{k}+{\frac {1}{2}}s_{k}^{T}{B}s_{k}+\dots ,}
858:
2113:
1940:
1852:
33:
1488:
2182:
2047:
2000:
1990:
1842:
1830:
1643:
1626:
1531:
1311:
1301:
44:
to a multidimensional problem. This update maintains the symmetry and positive definiteness of the
37:
1917:
1886:
1872:
1862:
1653:
1316:
25:
1569:
1291:
1433:
1426:
1132:
1925:
1603:
1475:
1456:
1437:
1410:
1346:
87:
2005:
1995:
1899:
1776:
1681:
1663:
1616:
1527:
1387:
1379:
1371:
824:
443:
2021:
54:
2009:
1894:
1781:
1715:
1686:
1403:
1164:
1112:
420:
116:
111:
45:
2224:
2167:
2151:
309:
134:
41:
2105:
1611:
1392:
2192:
1574:
1594:
81:
1914:
1266:
The DFP formula is quite effective, but it was soon superseded by the
1383:
407:{\displaystyle \nabla f(x_{k}+s_{k})=\nabla f(x_{k})+Bs_{k}+\dots }
2149:
1965:
1828:
1756:
1542:
1492:
855:
The corresponding update to the inverse
Hessian approximation
1740:
743:{\displaystyle y_{k}=\nabla f(x_{k}+s_{k})-\nabla f(x_{k}),}
811:{\displaystyle \gamma _{k}={\frac {1}{y_{k}^{T}s_{k}}},}
1190:
1167:
1135:
1115:
912:
861:
827:
757:
663:
476:
446:
423:
321:
146:
119:
90:
57:
1256:{\displaystyle s_{k}^{T}y_{k}=s_{k}^{T}Bs_{k}>0.}
1129:
is assumed to be positive-definite, and the vectors
2104:
2065:
2031:
2020:
1978:
1913:
1885:
1871:
1841:
1790:
1769:
1714:
1662:
1625:
1602:
1593:
1555:
1474:. London: John Wiley & Sons. pp. 110–120.
1425:
1402:
1255:
1173:
1153:
1121:
1099:
895:
840:
810:
742:
643:
459:
429:
406:
297:
125:
99:
72:
1409:(2nd ed.). New York: John Wiley & Sons.
1428:Methods for Unconstrained Optimization Problems
1307:Broyden–Fletcher–Goldfarb–Shanno (BFGS) method
1504:
8:
1376:AEC Research and Development Report ANL-5990
1451:Nocedal, Jorge; Wright, Stephen J. (1999).
1343:Nonlinear Programming: Analysis and Methods
2146:
2062:
2028:
1975:
1962:
1882:
1838:
1825:
1766:
1753:
1599:
1552:
1539:
1511:
1497:
1489:
1391:
1372:"Variable Metric Method for Minimization"
1241:
1228:
1223:
1210:
1200:
1195:
1189:
1166:
1145:
1140:
1134:
1114:
1085:
1075:
1070:
1058:
1053:
1043:
1036:
1024:
1014:
1004:
999:
987:
977:
972:
962:
952:
945:
936:
917:
911:
884:
879:
866:
860:
832:
826:
796:
786:
781:
771:
762:
756:
728:
703:
690:
668:
662:
632:
627:
617:
607:
591:
586:
576:
566:
547:
534:
529:
519:
509:
481:
475:
451:
445:
422:
392:
373:
348:
335:
320:
312:of the gradient itself (secant equation)
280:
271:
265:
260:
246:
237:
227:
217:
192:
170:
157:
145:
118:
89:
56:
1745:Optimization computes maxima and minima.
1268:Broyden–Fletcher–Goldfarb–Shanno formula
1333:
1941:Principal pivoting algorithm of Lemke
1181:must satisfy the curvature condition
7:
1424:Kowalik, J.; Osborne, M. R. (1968).
2231:Optimization algorithms and methods
1345:. Prentice-Hall. pp. 352–353.
1585:Successive parabolic interpolation
715:
677:
360:
322:
204:
91:
14:
1905:Projective algorithm of Karmarkar
1405:Practical methods of optimization
1900:Ellipsoid algorithm of Khachiyan
1803:Sequential quadratic programming
1640:Broyden–Fletcher–Goldfarb–Shanno
896:{\displaystyle H_{k}=B_{k}^{-1}}
1432:. New York: Elsevier. pp.
1297:Newton's method in optimization
18:Davidon–Fletcher–Powell formula
1858:Reduced gradient (Frank–Wolfe)
734:
721:
709:
683:
597:
553:
540:
496:
379:
366:
354:
328:
224:
210:
198:
185:
176:
150:
67:
61:
1:
2188:Spiral optimization algorithm
1808:Successive linear programming
1926:Simplex algorithm of Dantzig
1798:Augmented Lagrangian methods
1274:(interchanging the roles of
2247:
1317:Symmetric rank-one formula
1312:Limited-memory BFGS method
2205:
2158:
2145:
2129:Push–relabel maximum flow
1974:
1961:
1931:Revised simplex algorithm
1837:
1824:
1765:
1752:
1738:
1551:
1538:
1341:Avriel, Mordecai (1976).
1154:{\displaystyle s_{k}^{T}}
1654:Symmetric rank-one (SR1)
1635:Berndt–Hall–Hall–Hausman
1401:Fletcher, Roger (1987).
850:positive-definite matrix
100:{\displaystyle \nabla f}
2178:Parallel metaheuristics
1986:Approximation algorithm
1697:Powell's dog leg method
1649:Davidon–Fletcher–Powell
1545:Unconstrained nonlinear
1472:Methods of Optimization
1393:2027/mdp.39015078508226
1370:Davidon, W. C. (1959).
2163:Evolutionary algorithm
1746:
1453:Numerical Optimization
1257:
1175:
1155:
1123:
1101:
897:
842:
812:
744:
645:
461:
431:
408:
299:
127:
101:
74:
1936:Criss-cross algorithm
1759:Constrained nonlinear
1744:
1565:Golden-section search
1470:Walsh, G. R. (1975).
1258:
1176:
1156:
1124:
1102:
898:
843:
841:{\displaystyle B_{k}}
813:
745:
646:
462:
460:{\displaystyle B_{k}}
432:
409:
300:
128:
102:
75:
1853:Cutting-plane method
1188:
1165:
1133:
1113:
910:
859:
825:
755:
661:
474:
444:
421:
319:
144:
117:
88:
73:{\displaystyle f(x)}
55:
34:Michael J. D. Powell
2183:Simulated annealing
2001:Integer programming
1991:Dynamic programming
1831:Convex optimization
1692:Levenberg–Marquardt
1455:. Springer-Verlag.
1302:Quasi-Newton method
1233:
1205:
1150:
1080:
1063:
1009:
982:
892:
848:is a symmetric and
791:
637:
596:
539:
270:
38:quasi-Newton method
1863:Subgradient method
1747:
1672:Conjugate gradient
1580:Nelder–Mead method
1322:Nelder–Mead method
1253:
1219:
1191:
1171:
1151:
1136:
1119:
1097:
1066:
1049:
995:
968:
893:
875:
838:
808:
777:
740:
641:
623:
582:
525:
457:
427:
417:is used to update
404:
295:
256:
123:
97:
70:
40:to generalize the
26:William C. Davidon
2218:
2217:
2201:
2200:
2141:
2140:
2137:
2136:
2100:
2099:
2061:
2060:
1957:
1956:
1953:
1952:
1949:
1948:
1820:
1819:
1816:
1815:
1736:
1735:
1732:
1731:
1710:
1709:
1416:978-0-471-91547-8
1174:{\displaystyle y}
1122:{\displaystyle B}
1092:
1031:
803:
430:{\displaystyle B}
254:
126:{\displaystyle B}
109:positive-definite
51:Given a function
2238:
2147:
2063:
2029:
2006:Branch and bound
1996:Greedy algorithm
1976:
1963:
1883:
1839:
1826:
1767:
1754:
1702:Truncated Newton
1617:Wolfe conditions
1600:
1553:
1540:
1513:
1506:
1499:
1490:
1485:
1466:
1447:
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1408:
1397:
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1338:
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1245:
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1215:
1214:
1204:
1199:
1180:
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1177:
1172:
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1157:
1152:
1149:
1144:
1128:
1126:
1125:
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1103:
1098:
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1089:
1079:
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1057:
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1029:
1028:
1019:
1018:
1008:
1003:
993:
992:
991:
981:
976:
967:
966:
957:
956:
946:
941:
940:
928:
927:
902:
900:
899:
894:
891:
883:
871:
870:
847:
845:
844:
839:
837:
836:
817:
815:
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809:
804:
802:
801:
800:
790:
785:
772:
767:
766:
749:
747:
746:
741:
733:
732:
708:
707:
695:
694:
673:
672:
650:
648:
647:
642:
636:
631:
622:
621:
612:
611:
595:
590:
581:
580:
571:
570:
552:
551:
538:
533:
524:
523:
514:
513:
492:
491:
466:
464:
463:
458:
456:
455:
436:
434:
433:
428:
413:
411:
410:
405:
397:
396:
378:
377:
353:
352:
340:
339:
304:
302:
301:
296:
285:
284:
275:
269:
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255:
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241:
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222:
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197:
196:
175:
174:
162:
161:
132:
130:
129:
124:
106:
104:
103:
98:
79:
77:
76:
71:
2246:
2245:
2241:
2240:
2239:
2237:
2236:
2235:
2221:
2220:
2219:
2214:
2197:
2154:
2133:
2096:
2057:
2034:
2023:
2016:
1970:
1945:
1909:
1876:
1867:
1844:
1833:
1812:
1786:
1782:Penalty methods
1777:Barrier methods
1761:
1748:
1728:
1724:Newton's method
1706:
1658:
1621:
1589:
1570:Powell's method
1547:
1534:
1517:
1482:
1469:
1463:
1450:
1444:
1423:
1417:
1400:
1384:10.2172/4252678
1369:
1366:
1364:Further reading
1361:
1360:
1353:
1340:
1339:
1335:
1330:
1292:Newton's method
1288:
1270:, which is its
1237:
1206:
1186:
1185:
1163:
1162:
1131:
1130:
1111:
1110:
1081:
1065:
1039:
1038:
1020:
1010:
994:
983:
958:
948:
947:
932:
913:
908:
907:
862:
857:
856:
828:
823:
822:
792:
776:
758:
753:
752:
724:
699:
686:
664:
659:
658:
613:
603:
572:
562:
543:
515:
505:
477:
472:
471:
447:
442:
441:
419:
418:
388:
369:
344:
331:
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276:
233:
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188:
166:
153:
142:
141:
115:
114:
86:
85:
53:
52:
12:
11:
5:
2244:
2242:
2234:
2233:
2223:
2222:
2216:
2215:
2213:
2212:
2206:
2203:
2202:
2199:
2198:
2196:
2195:
2190:
2185:
2180:
2175:
2170:
2165:
2159:
2156:
2155:
2152:Metaheuristics
2150:
2143:
2142:
2139:
2138:
2135:
2134:
2132:
2131:
2126:
2124:Ford–Fulkerson
2121:
2116:
2110:
2108:
2102:
2101:
2098:
2097:
2095:
2094:
2092:Floyd–Warshall
2089:
2084:
2083:
2082:
2071:
2069:
2059:
2058:
2056:
2055:
2050:
2045:
2039:
2037:
2026:
2018:
2017:
2015:
2014:
2013:
2012:
1998:
1993:
1988:
1982:
1980:
1972:
1971:
1966:
1959:
1958:
1955:
1954:
1951:
1950:
1947:
1946:
1944:
1943:
1938:
1933:
1928:
1922:
1920:
1911:
1910:
1908:
1907:
1902:
1897:
1895:Affine scaling
1891:
1889:
1887:Interior point
1880:
1869:
1868:
1866:
1865:
1860:
1855:
1849:
1847:
1835:
1834:
1829:
1822:
1821:
1818:
1817:
1814:
1813:
1811:
1810:
1805:
1800:
1794:
1792:
1791:Differentiable
1788:
1787:
1785:
1784:
1779:
1773:
1771:
1763:
1762:
1757:
1750:
1749:
1739:
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1567:
1561:
1559:
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1548:
1543:
1536:
1535:
1518:
1516:
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1508:
1501:
1493:
1487:
1486:
1480:
1467:
1461:
1448:
1442:
1421:
1415:
1398:
1365:
1362:
1359:
1358:
1351:
1332:
1331:
1329:
1326:
1325:
1324:
1319:
1314:
1309:
1304:
1299:
1294:
1287:
1284:
1264:
1263:
1252:
1249:
1244:
1240:
1236:
1231:
1226:
1222:
1218:
1213:
1209:
1203:
1198:
1194:
1170:
1148:
1143:
1139:
1118:
1108:
1107:
1096:
1088:
1084:
1078:
1073:
1069:
1061:
1056:
1052:
1046:
1042:
1035:
1027:
1023:
1017:
1013:
1007:
1002:
998:
990:
986:
980:
975:
971:
965:
961:
955:
951:
944:
939:
935:
931:
926:
923:
920:
916:
890:
887:
882:
878:
874:
869:
865:
835:
831:
819:
818:
807:
799:
795:
789:
784:
780:
775:
770:
765:
761:
750:
739:
736:
731:
727:
723:
720:
717:
714:
711:
706:
702:
698:
693:
689:
685:
682:
679:
676:
671:
667:
652:
651:
640:
635:
630:
626:
620:
616:
610:
606:
602:
599:
594:
589:
585:
579:
575:
569:
565:
561:
558:
555:
550:
546:
542:
537:
532:
528:
522:
518:
512:
508:
504:
501:
498:
495:
490:
487:
484:
480:
454:
450:
426:
415:
414:
403:
400:
395:
391:
387:
384:
381:
376:
372:
368:
365:
362:
359:
356:
351:
347:
343:
338:
334:
330:
327:
324:
306:
305:
294:
291:
288:
283:
279:
274:
268:
263:
259:
253:
250:
245:
240:
236:
230:
226:
220:
216:
212:
209:
206:
203:
200:
195:
191:
187:
184:
181:
178:
173:
169:
165:
160:
156:
152:
149:
122:
112:Hessian matrix
96:
93:
69:
66:
63:
60:
46:Hessian matrix
30:Roger Fletcher
24:; named after
13:
10:
9:
6:
4:
3:
2:
2243:
2232:
2229:
2228:
2226:
2211:
2208:
2207:
2204:
2194:
2191:
2189:
2186:
2184:
2181:
2179:
2176:
2174:
2171:
2169:
2168:Hill climbing
2166:
2164:
2161:
2160:
2157:
2153:
2148:
2144:
2130:
2127:
2125:
2122:
2120:
2117:
2115:
2112:
2111:
2109:
2107:
2106:Network flows
2103:
2093:
2090:
2088:
2085:
2081:
2078:
2077:
2076:
2073:
2072:
2070:
2068:
2067:Shortest path
2064:
2054:
2051:
2049:
2046:
2044:
2041:
2040:
2038:
2036:
2035:spanning tree
2030:
2027:
2025:
2019:
2011:
2007:
2004:
2003:
2002:
1999:
1997:
1994:
1992:
1989:
1987:
1984:
1983:
1981:
1977:
1973:
1969:
1968:Combinatorial
1964:
1960:
1942:
1939:
1937:
1934:
1932:
1929:
1927:
1924:
1923:
1921:
1919:
1916:
1912:
1906:
1903:
1901:
1898:
1896:
1893:
1892:
1890:
1888:
1884:
1881:
1879:
1874:
1870:
1864:
1861:
1859:
1856:
1854:
1851:
1850:
1848:
1846:
1840:
1836:
1832:
1827:
1823:
1809:
1806:
1804:
1801:
1799:
1796:
1795:
1793:
1789:
1783:
1780:
1778:
1775:
1774:
1772:
1768:
1764:
1760:
1755:
1751:
1743:
1725:
1722:
1721:
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1717:
1713:
1703:
1700:
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1688:
1685:
1683:
1680:
1678:
1675:
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1670:
1669:
1667:
1665:
1664:Other methods
1661:
1655:
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1628:
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1533:
1529:
1525:
1521:
1514:
1509:
1507:
1502:
1500:
1495:
1494:
1491:
1483:
1481:0-471-91922-5
1477:
1473:
1468:
1464:
1462:0-387-98793-2
1458:
1454:
1449:
1445:
1443:0-444-00041-0
1439:
1435:
1430:
1429:
1422:
1418:
1412:
1407:
1406:
1399:
1394:
1389:
1385:
1381:
1377:
1373:
1368:
1367:
1363:
1354:
1352:0-13-623603-0
1348:
1344:
1337:
1334:
1327:
1323:
1320:
1318:
1315:
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1310:
1308:
1305:
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1300:
1298:
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1290:
1289:
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1281:
1277:
1273:
1269:
1250:
1247:
1242:
1238:
1234:
1229:
1224:
1220:
1216:
1211:
1207:
1201:
1196:
1192:
1184:
1183:
1182:
1168:
1146:
1141:
1137:
1116:
1094:
1086:
1082:
1076:
1071:
1067:
1059:
1054:
1050:
1044:
1040:
1033:
1025:
1021:
1015:
1011:
1005:
1000:
996:
988:
984:
978:
973:
969:
963:
959:
953:
949:
942:
937:
933:
929:
924:
921:
918:
914:
906:
905:
904:
888:
885:
880:
876:
872:
867:
863:
853:
851:
833:
829:
805:
797:
793:
787:
782:
778:
773:
768:
763:
759:
751:
737:
729:
725:
718:
712:
704:
700:
696:
691:
687:
680:
674:
669:
665:
657:
656:
655:
638:
633:
628:
624:
618:
614:
608:
604:
600:
592:
587:
583:
577:
573:
567:
563:
559:
556:
548:
544:
535:
530:
526:
520:
516:
510:
506:
502:
499:
493:
488:
485:
482:
478:
470:
469:
468:
452:
448:
438:
424:
401:
398:
393:
389:
385:
382:
374:
370:
363:
357:
349:
345:
341:
336:
332:
325:
315:
314:
313:
311:
310:Taylor series
292:
289:
286:
281:
277:
272:
266:
261:
257:
251:
248:
243:
238:
234:
228:
218:
214:
207:
201:
193:
189:
182:
179:
171:
167:
163:
158:
154:
147:
140:
139:
138:
136:
135:Taylor series
120:
113:
110:
94:
83:
64:
58:
49:
47:
43:
42:secant method
39:
35:
31:
27:
23:
19:
2173:Local search
2119:Edmonds–Karp
2075:Bellman–Ford
1845:minimization
1677:Gauss–Newton
1648:
1627:Quasi–Newton
1612:Trust region
1520:Optimization
1471:
1452:
1427:
1404:
1375:
1342:
1336:
1279:
1275:
1265:
1109:
903:is given by
854:
820:
653:
439:
416:
307:
50:
21:
17:
15:
2193:Tabu search
1604:Convergence
1575:Line search
2024:algorithms
1532:heuristics
1524:Algorithms
1328:References
1979:Paradigms
1878:quadratic
1595:Gradients
1557:Functions
943:−
886:−
760:γ
716:∇
713:−
678:∇
605:γ
564:γ
560:−
507:γ
503:−
402:…
361:∇
323:∇
290:…
205:∇
92:∇
2225:Category
2210:Software
2087:Dijkstra
1918:exchange
1716:Hessians
1682:Gradient
1286:See also
308:and the
82:gradient
2053:Kruskal
2043:Borůvka
2033:Minimum
1770:General
1528:methods
107:), and
1915:Basis-
1873:Linear
1843:Convex
1687:Mirror
1644:L-BFGS
1530:, and
1478:
1459:
1440:
1413:
1349:
654:where
133:, the
80:, its
32:, and
2114:Dinic
2022:Graph
1434:45–48
2080:SPFA
2048:Prim
1642:and
1476:ISBN
1457:ISBN
1438:ISBN
1411:ISBN
1347:ISBN
1278:and
1272:dual
1248:>
1161:and
821:and
437:.
20:(or
16:The
2010:cut
1875:and
1388:hdl
1380:doi
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137:is
22:DFP
2227::
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1436:.
1386:.
1378:.
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852:.
467::
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28:,
2008:/
1512:e
1505:t
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1484:.
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1446:.
1419:.
1396:.
1390::
1382::
1355:.
1280:s
1276:y
1243:k
1239:s
1235:B
1230:T
1225:k
1221:s
1217:=
1212:k
1208:y
1202:T
1197:k
1193:s
1169:y
1147:T
1142:k
1138:s
1117:B
1095:.
1087:k
1083:s
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1072:k
1068:y
1060:T
1055:k
1051:s
1045:k
1041:s
1034:+
1026:k
1022:y
1016:k
1012:H
1006:T
1001:k
997:y
989:k
985:H
979:T
974:k
970:y
964:k
960:y
954:k
950:H
938:k
934:H
930:=
925:1
922:+
919:k
915:H
889:1
881:k
877:B
873:=
868:k
864:H
834:k
830:B
806:,
798:k
794:s
788:T
783:k
779:y
774:1
769:=
764:k
738:,
735:)
730:k
726:x
722:(
719:f
710:)
705:k
701:s
697:+
692:k
688:x
684:(
681:f
675:=
670:k
666:y
639:,
634:T
629:k
625:y
619:k
615:y
609:k
601:+
598:)
593:T
588:k
584:y
578:k
574:s
568:k
557:I
554:(
549:k
545:B
541:)
536:T
531:k
527:s
521:k
517:y
511:k
500:I
497:(
494:=
489:1
486:+
483:k
479:B
453:k
449:B
425:B
399:+
394:k
390:s
386:B
383:+
380:)
375:k
371:x
367:(
364:f
358:=
355:)
350:k
346:s
342:+
337:k
333:x
329:(
326:f
293:,
287:+
282:k
278:s
273:B
267:T
262:k
258:s
252:2
249:1
244:+
239:k
235:s
229:T
225:)
219:k
215:x
211:(
208:f
202:+
199:)
194:k
190:x
186:(
183:f
180:=
177:)
172:k
168:s
164:+
159:k
155:x
151:(
148:f
121:B
95:f
84:(
68:)
65:x
62:(
59:f
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