1753:
1116:
660:
314:
423:
759:
827:
920:
1272:
1650:
1317:
1278:
484:
912:
154:
1521:
1170:
116:
1645:
857:
476:
89:
1190:
1138:
446:
142:
2241:
2139:
1514:
1425:
47:) finds the solution to the secant equation that is closest to the current estimate and satisfies the curvature condition. It was the first
329:
2220:
1682:
1734:
1595:
1307:
1702:
1490:
1471:
1452:
1361:
671:
1813:
1507:
1111:{\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}}}.}
765:
2090:
1752:
40:
2198:
1818:
1198:
2134:
2102:
2183:
1808:
451:
The DFP formula finds a solution that is symmetric, positive-definite and closest to the current approximate value of
2129:
2085:
1687:
1978:
1707:
1868:
655:{\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},}
1530:
2053:
860:
119:
1915:
2097:
1996:
1712:
1590:
1332:
2188:
2173:
2063:
1941:
1567:
1534:
2077:
2043:
1946:
1888:
1769:
1575:
1555:
1282:
309:{\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 ,}
869:
2124:
1951:
1863:
44:
1499:
2193:
2058:
2011:
2001:
1853:
1841:
1654:
1637:
1542:
1322:
1312:
55:
to a multidimensional problem. This update maintains the symmetry and positive definiteness of the
48:
1928:
1897:
1883:
1873:
1664:
1327:
36:
1580:
1302:
1444:
1437:
1143:
17:
1936:
1614:
1486:
1467:
1448:
1421:
1357:
98:
2016:
2006:
1910:
1787:
1692:
1674:
1627:
1538:
1398:
1390:
1382:
835:
454:
2032:
65:
2020:
1905:
1792:
1726:
1697:
1414:
1175:
1123:
431:
127:
122:
56:
2235:
2178:
2162:
320:
145:
52:
2116:
1622:
1403:
2203:
1585:
1605:
92:
1925:
1277:
The DFP formula is quite effective, but it was soon superseded by the
1394:
418:{\displaystyle \nabla f(x_{k}+s_{k})=\nabla f(x_{k})+Bs_{k}+\dots }
2160:
1976:
1839:
1767:
1553:
1503:
866:
The corresponding update to the inverse
Hessian approximation
1751:
754:{\displaystyle y_{k}=\nabla f(x_{k}+s_{k})-\nabla f(x_{k}),}
822:{\displaystyle \gamma _{k}={\frac {1}{y_{k}^{T}s_{k}}},}
1201:
1178:
1146:
1126:
923:
872:
838:
768:
674:
487:
457:
434:
332:
157:
130:
101:
68:
1267:{\displaystyle s_{k}^{T}y_{k}=s_{k}^{T}Bs_{k}>0.}
1140:
is assumed to be positive-definite, and the vectors
2115:
2076:
2042:
2031:
1989:
1924:
1896:
1882:
1852:
1801:
1780:
1725:
1673:
1636:
1613:
1604:
1566:
1485:. London: John Wiley & Sons. pp. 110–120.
1436:
1413:
1266:
1184:
1164:
1132:
1110:
906:
851:
821:
753:
654:
470:
440:
417:
308:
136:
110:
83:
1420:(2nd ed.). New York: John Wiley & Sons.
1439:Methods for Unconstrained Optimization Problems
1318:Broyden–Fletcher–Goldfarb–Shanno (BFGS) method
1515:
8:
1387:AEC Research and Development Report ANL-5990
1462:Nocedal, Jorge; Wright, Stephen J. (1999).
1354:Nonlinear Programming: Analysis and Methods
2157:
2073:
2039:
1986:
1973:
1893:
1849:
1836:
1777:
1764:
1610:
1563:
1550:
1522:
1508:
1500:
1402:
1383:"Variable Metric Method for Minimization"
1252:
1239:
1234:
1221:
1211:
1206:
1200:
1177:
1156:
1151:
1145:
1125:
1096:
1086:
1081:
1069:
1064:
1054:
1047:
1035:
1025:
1015:
1010:
998:
988:
983:
973:
963:
956:
947:
928:
922:
895:
890:
877:
871:
843:
837:
807:
797:
792:
782:
773:
767:
739:
714:
701:
679:
673:
643:
638:
628:
618:
602:
597:
587:
577:
558:
545:
540:
530:
520:
492:
486:
462:
456:
433:
403:
384:
359:
346:
331:
323:of the gradient itself (secant equation)
291:
282:
276:
271:
257:
248:
238:
228:
203:
181:
168:
156:
129:
100:
67:
1756:Optimization computes maxima and minima.
1279:Broyden–Fletcher–Goldfarb–Shanno formula
1344:
1952:Principal pivoting algorithm of Lemke
1192:must satisfy the curvature condition
7:
1435:Kowalik, J.; Osborne, M. R. (1968).
2242:Optimization algorithms and methods
1356:. Prentice-Hall. pp. 352–353.
1596:Successive parabolic interpolation
726:
688:
371:
333:
215:
102:
25:
1916:Projective algorithm of Karmarkar
1416:Practical methods of optimization
1911:Ellipsoid algorithm of Khachiyan
1814:Sequential quadratic programming
1651:Broyden–Fletcher–Goldfarb–Shanno
907:{\displaystyle H_{k}=B_{k}^{-1}}
1443:. New York: Elsevier. pp.
1308:Newton's method in optimization
29:Davidon–Fletcher–Powell formula
18:Davidon-Fletcher-Powell formula
1869:Reduced gradient (Frank–Wolfe)
745:
732:
720:
694:
608:
564:
551:
507:
390:
377:
365:
339:
235:
221:
209:
196:
187:
161:
78:
72:
1:
2199:Spiral optimization algorithm
1819:Successive linear programming
1937:Simplex algorithm of Dantzig
1809:Augmented Lagrangian methods
1285:(interchanging the roles of
2258:
1328:Symmetric rank-one formula
1323:Limited-memory BFGS method
2216:
2169:
2156:
2140:Push–relabel maximum flow
1985:
1972:
1942:Revised simplex algorithm
1848:
1835:
1776:
1763:
1749:
1562:
1549:
1352:Avriel, Mordecai (1976).
1165:{\displaystyle s_{k}^{T}}
1665:Symmetric rank-one (SR1)
1646:Berndt–Hall–Hall–Hausman
1412:Fletcher, Roger (1987).
861:positive-definite matrix
111:{\displaystyle \nabla f}
2189:Parallel metaheuristics
1997:Approximation algorithm
1708:Powell's dog leg method
1660:Davidon–Fletcher–Powell
1556:Unconstrained nonlinear
1483:Methods of Optimization
1404:2027/mdp.39015078508226
1381:Davidon, W. C. (1959).
2174:Evolutionary algorithm
1757:
1464:Numerical Optimization
1268:
1186:
1166:
1134:
1112:
908:
853:
823:
755:
656:
472:
442:
419:
310:
138:
112:
85:
1947:Criss-cross algorithm
1770:Constrained nonlinear
1755:
1576:Golden-section search
1481:Walsh, G. R. (1975).
1269:
1187:
1167:
1135:
1113:
909:
854:
852:{\displaystyle B_{k}}
824:
756:
657:
473:
471:{\displaystyle B_{k}}
443:
420:
311:
139:
113:
86:
1864:Cutting-plane method
1199:
1176:
1144:
1124:
921:
870:
836:
766:
672:
485:
455:
432:
330:
155:
128:
99:
84:{\displaystyle f(x)}
66:
45:Michael J. D. Powell
2194:Simulated annealing
2012:Integer programming
2002:Dynamic programming
1842:Convex optimization
1703:Levenberg–Marquardt
1466:. Springer-Verlag.
1313:Quasi-Newton method
1244:
1216:
1161:
1091:
1074:
1020:
993:
903:
859:is a symmetric and
802:
648:
607:
550:
281:
49:quasi-Newton method
1874:Subgradient method
1758:
1683:Conjugate gradient
1591:Nelder–Mead method
1333:Nelder–Mead method
1264:
1230:
1202:
1182:
1162:
1147:
1130:
1108:
1077:
1060:
1006:
979:
904:
886:
849:
819:
788:
751:
652:
634:
593:
536:
468:
438:
428:is used to update
415:
306:
267:
134:
108:
81:
51:to generalize the
37:William C. Davidon
2229:
2228:
2212:
2211:
2152:
2151:
2148:
2147:
2111:
2110:
2072:
2071:
1968:
1967:
1964:
1963:
1960:
1959:
1831:
1830:
1827:
1826:
1747:
1746:
1743:
1742:
1721:
1720:
1427:978-0-471-91547-8
1185:{\displaystyle y}
1133:{\displaystyle B}
1103:
1042:
814:
441:{\displaystyle B}
265:
137:{\displaystyle B}
120:positive-definite
62:Given a function
16:(Redirected from
2249:
2158:
2074:
2040:
2017:Branch and bound
2007:Greedy algorithm
1987:
1974:
1894:
1850:
1837:
1778:
1765:
1713:Truncated Newton
1628:Wolfe conditions
1611:
1564:
1551:
1524:
1517:
1510:
1501:
1496:
1477:
1458:
1442:
1431:
1419:
1408:
1406:
1368:
1367:
1349:
1273:
1271:
1270:
1265:
1257:
1256:
1243:
1238:
1226:
1225:
1215:
1210:
1191:
1189:
1188:
1183:
1171:
1169:
1168:
1163:
1160:
1155:
1139:
1137:
1136:
1131:
1117:
1115:
1114:
1109:
1104:
1102:
1101:
1100:
1090:
1085:
1075:
1073:
1068:
1059:
1058:
1048:
1043:
1041:
1040:
1039:
1030:
1029:
1019:
1014:
1004:
1003:
1002:
992:
987:
978:
977:
968:
967:
957:
952:
951:
939:
938:
913:
911:
910:
905:
902:
894:
882:
881:
858:
856:
855:
850:
848:
847:
828:
826:
825:
820:
815:
813:
812:
811:
801:
796:
783:
778:
777:
760:
758:
757:
752:
744:
743:
719:
718:
706:
705:
684:
683:
661:
659:
658:
653:
647:
642:
633:
632:
623:
622:
606:
601:
592:
591:
582:
581:
563:
562:
549:
544:
535:
534:
525:
524:
503:
502:
477:
475:
474:
469:
467:
466:
447:
445:
444:
439:
424:
422:
421:
416:
408:
407:
389:
388:
364:
363:
351:
350:
315:
313:
312:
307:
296:
295:
286:
280:
275:
266:
258:
253:
252:
243:
242:
233:
232:
208:
207:
186:
185:
173:
172:
143:
141:
140:
135:
117:
115:
114:
109:
90:
88:
87:
82:
21:
2257:
2256:
2252:
2251:
2250:
2248:
2247:
2246:
2232:
2231:
2230:
2225:
2208:
2165:
2144:
2107:
2068:
2045:
2034:
2027:
1981:
1956:
1920:
1887:
1878:
1855:
1844:
1823:
1797:
1793:Penalty methods
1788:Barrier methods
1772:
1759:
1739:
1735:Newton's method
1717:
1669:
1632:
1600:
1581:Powell's method
1558:
1545:
1528:
1493:
1480:
1474:
1461:
1455:
1434:
1428:
1411:
1395:10.2172/4252678
1380:
1377:
1375:Further reading
1372:
1371:
1364:
1351:
1350:
1346:
1341:
1303:Newton's method
1299:
1281:, which is its
1248:
1217:
1197:
1196:
1174:
1173:
1142:
1141:
1122:
1121:
1092:
1076:
1050:
1049:
1031:
1021:
1005:
994:
969:
959:
958:
943:
924:
919:
918:
873:
868:
867:
839:
834:
833:
803:
787:
769:
764:
763:
735:
710:
697:
675:
670:
669:
624:
614:
583:
573:
554:
526:
516:
488:
483:
482:
458:
453:
452:
430:
429:
399:
380:
355:
342:
328:
327:
287:
244:
234:
224:
199:
177:
164:
153:
152:
126:
125:
97:
96:
64:
63:
23:
22:
15:
12:
11:
5:
2255:
2253:
2245:
2244:
2234:
2233:
2227:
2226:
2224:
2223:
2217:
2214:
2213:
2210:
2209:
2207:
2206:
2201:
2196:
2191:
2186:
2181:
2176:
2170:
2167:
2166:
2163:Metaheuristics
2161:
2154:
2153:
2150:
2149:
2146:
2145:
2143:
2142:
2137:
2135:Ford–Fulkerson
2132:
2127:
2121:
2119:
2113:
2112:
2109:
2108:
2106:
2105:
2103:Floyd–Warshall
2100:
2095:
2094:
2093:
2082:
2080:
2070:
2069:
2067:
2066:
2061:
2056:
2050:
2048:
2037:
2029:
2028:
2026:
2025:
2024:
2023:
2009:
2004:
1999:
1993:
1991:
1983:
1982:
1977:
1970:
1969:
1966:
1965:
1962:
1961:
1958:
1957:
1955:
1954:
1949:
1944:
1939:
1933:
1931:
1922:
1921:
1919:
1918:
1913:
1908:
1906:Affine scaling
1902:
1900:
1898:Interior point
1891:
1880:
1879:
1877:
1876:
1871:
1866:
1860:
1858:
1846:
1845:
1840:
1833:
1832:
1829:
1828:
1825:
1824:
1822:
1821:
1816:
1811:
1805:
1803:
1802:Differentiable
1799:
1798:
1796:
1795:
1790:
1784:
1782:
1774:
1773:
1768:
1761:
1760:
1750:
1748:
1745:
1744:
1741:
1740:
1738:
1737:
1731:
1729:
1723:
1722:
1719:
1718:
1716:
1715:
1710:
1705:
1700:
1695:
1690:
1685:
1679:
1677:
1671:
1670:
1668:
1667:
1662:
1657:
1648:
1642:
1640:
1634:
1633:
1631:
1630:
1625:
1619:
1617:
1608:
1602:
1601:
1599:
1598:
1593:
1588:
1583:
1578:
1572:
1570:
1560:
1559:
1554:
1547:
1546:
1529:
1527:
1526:
1519:
1512:
1504:
1498:
1497:
1491:
1478:
1472:
1459:
1453:
1432:
1426:
1409:
1376:
1373:
1370:
1369:
1362:
1343:
1342:
1340:
1337:
1336:
1335:
1330:
1325:
1320:
1315:
1310:
1305:
1298:
1295:
1275:
1274:
1263:
1260:
1255:
1251:
1247:
1242:
1237:
1233:
1229:
1224:
1220:
1214:
1209:
1205:
1181:
1159:
1154:
1150:
1129:
1119:
1118:
1107:
1099:
1095:
1089:
1084:
1080:
1072:
1067:
1063:
1057:
1053:
1046:
1038:
1034:
1028:
1024:
1018:
1013:
1009:
1001:
997:
991:
986:
982:
976:
972:
966:
962:
955:
950:
946:
942:
937:
934:
931:
927:
901:
898:
893:
889:
885:
880:
876:
846:
842:
830:
829:
818:
810:
806:
800:
795:
791:
786:
781:
776:
772:
761:
750:
747:
742:
738:
734:
731:
728:
725:
722:
717:
713:
709:
704:
700:
696:
693:
690:
687:
682:
678:
663:
662:
651:
646:
641:
637:
631:
627:
621:
617:
613:
610:
605:
600:
596:
590:
586:
580:
576:
572:
569:
566:
561:
557:
553:
548:
543:
539:
533:
529:
523:
519:
515:
512:
509:
506:
501:
498:
495:
491:
465:
461:
437:
426:
425:
414:
411:
406:
402:
398:
395:
392:
387:
383:
379:
376:
373:
370:
367:
362:
358:
354:
349:
345:
341:
338:
335:
317:
316:
305:
302:
299:
294:
290:
285:
279:
274:
270:
264:
261:
256:
251:
247:
241:
237:
231:
227:
223:
220:
217:
214:
211:
206:
202:
198:
195:
192:
189:
184:
180:
176:
171:
167:
163:
160:
133:
123:Hessian matrix
107:
104:
80:
77:
74:
71:
57:Hessian matrix
41:Roger Fletcher
35:; named after
24:
14:
13:
10:
9:
6:
4:
3:
2:
2254:
2243:
2240:
2239:
2237:
2222:
2219:
2218:
2215:
2205:
2202:
2200:
2197:
2195:
2192:
2190:
2187:
2185:
2182:
2180:
2179:Hill climbing
2177:
2175:
2172:
2171:
2168:
2164:
2159:
2155:
2141:
2138:
2136:
2133:
2131:
2128:
2126:
2123:
2122:
2120:
2118:
2117:Network flows
2114:
2104:
2101:
2099:
2096:
2092:
2089:
2088:
2087:
2084:
2083:
2081:
2079:
2078:Shortest path
2075:
2065:
2062:
2060:
2057:
2055:
2052:
2051:
2049:
2047:
2046:spanning tree
2041:
2038:
2036:
2030:
2022:
2018:
2015:
2014:
2013:
2010:
2008:
2005:
2003:
2000:
1998:
1995:
1994:
1992:
1988:
1984:
1980:
1979:Combinatorial
1975:
1971:
1953:
1950:
1948:
1945:
1943:
1940:
1938:
1935:
1934:
1932:
1930:
1927:
1923:
1917:
1914:
1912:
1909:
1907:
1904:
1903:
1901:
1899:
1895:
1892:
1890:
1885:
1881:
1875:
1872:
1870:
1867:
1865:
1862:
1861:
1859:
1857:
1851:
1847:
1843:
1838:
1834:
1820:
1817:
1815:
1812:
1810:
1807:
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1800:
1794:
1791:
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1779:
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1754:
1736:
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1699:
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1676:
1675:Other methods
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1536:
1532:
1525:
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1518:
1513:
1511:
1506:
1505:
1502:
1494:
1492:0-471-91922-5
1488:
1484:
1479:
1475:
1473:0-387-98793-2
1469:
1465:
1460:
1456:
1454:0-444-00041-0
1450:
1446:
1441:
1440:
1433:
1429:
1423:
1418:
1417:
1410:
1405:
1400:
1396:
1392:
1388:
1384:
1379:
1378:
1374:
1365:
1363:0-13-623603-0
1359:
1355:
1348:
1345:
1338:
1334:
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1329:
1326:
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1300:
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1249:
1245:
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1235:
1231:
1227:
1222:
1218:
1212:
1207:
1203:
1195:
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1193:
1179:
1157:
1152:
1148:
1127:
1105:
1097:
1093:
1087:
1082:
1078:
1070:
1065:
1061:
1055:
1051:
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1036:
1032:
1026:
1022:
1016:
1011:
1007:
999:
995:
989:
984:
980:
974:
970:
964:
960:
953:
948:
944:
940:
935:
932:
929:
925:
917:
916:
915:
899:
896:
891:
887:
883:
878:
874:
864:
862:
844:
840:
816:
808:
804:
798:
793:
789:
784:
779:
774:
770:
762:
748:
740:
736:
729:
723:
715:
711:
707:
702:
698:
691:
685:
680:
676:
668:
667:
666:
649:
644:
639:
635:
629:
625:
619:
615:
611:
603:
598:
594:
588:
584:
578:
574:
570:
567:
559:
555:
546:
541:
537:
531:
527:
521:
517:
513:
510:
504:
499:
496:
493:
489:
481:
480:
479:
463:
459:
449:
435:
412:
409:
404:
400:
396:
393:
385:
381:
374:
368:
360:
356:
352:
347:
343:
336:
326:
325:
324:
322:
321:Taylor series
303:
300:
297:
292:
288:
283:
277:
272:
268:
262:
259:
254:
249:
245:
239:
229:
225:
218:
212:
204:
200:
193:
190:
182:
178:
174:
169:
165:
158:
151:
150:
149:
147:
146:Taylor series
131:
124:
121:
105:
94:
75:
69:
60:
58:
54:
53:secant method
50:
46:
42:
38:
34:
30:
19:
2184:Local search
2130:Edmonds–Karp
2086:Bellman–Ford
1856:minimization
1688:Gauss–Newton
1659:
1638:Quasi–Newton
1623:Trust region
1531:Optimization
1482:
1463:
1438:
1415:
1386:
1353:
1347:
1290:
1286:
1276:
1120:
914:is given by
865:
831:
664:
450:
427:
318:
61:
32:
28:
26:
2204:Tabu search
1615:Convergence
1586:Line search
2035:algorithms
1543:heuristics
1535:Algorithms
1339:References
1990:Paradigms
1889:quadratic
1606:Gradients
1568:Functions
954:−
897:−
771:γ
727:∇
724:−
689:∇
616:γ
575:γ
571:−
518:γ
514:−
413:…
372:∇
334:∇
301:…
216:∇
103:∇
2236:Category
2221:Software
2098:Dijkstra
1929:exchange
1727:Hessians
1693:Gradient
1297:See also
319:and the
93:gradient
2064:Kruskal
2054:Borůvka
2044:Minimum
1781:General
1539:methods
118:), and
1926:Basis-
1884:Linear
1854:Convex
1698:Mirror
1655:L-BFGS
1541:, and
1489:
1470:
1451:
1424:
1360:
665:where
144:, the
91:, its
43:, and
2125:Dinic
2033:Graph
1445:45–48
2091:SPFA
2059:Prim
1653:and
1487:ISBN
1468:ISBN
1449:ISBN
1422:ISBN
1358:ISBN
1289:and
1283:dual
1259:>
1172:and
832:and
448:.
31:(or
27:The
2021:cut
1886:and
1399:hdl
1391:doi
1293:).
148:is
33:DFP
2238::
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1533::
1447:.
1397:.
1389:.
1385:.
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863:.
478::
59:.
39:,
2019:/
1523:e
1516:t
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1495:.
1476:.
1457:.
1430:.
1407:.
1401::
1393::
1366:.
1291:s
1287:y
1254:k
1250:s
1246:B
1241:T
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1232:s
1228:=
1223:k
1219:y
1213:T
1208:k
1204:s
1180:y
1158:T
1153:k
1149:s
1128:B
1106:.
1098:k
1094:s
1088:T
1083:k
1079:y
1071:T
1066:k
1062:s
1056:k
1052:s
1045:+
1037:k
1033:y
1027:k
1023:H
1017:T
1012:k
1008:y
1000:k
996:H
990:T
985:k
981:y
975:k
971:y
965:k
961:H
949:k
945:H
941:=
936:1
933:+
930:k
926:H
900:1
892:k
888:B
884:=
879:k
875:H
845:k
841:B
817:,
809:k
805:s
799:T
794:k
790:y
785:1
780:=
775:k
749:,
746:)
741:k
737:x
733:(
730:f
721:)
716:k
712:s
708:+
703:k
699:x
695:(
692:f
686:=
681:k
677:y
650:,
645:T
640:k
636:y
630:k
626:y
620:k
612:+
609:)
604:T
599:k
595:y
589:k
585:s
579:k
568:I
565:(
560:k
556:B
552:)
547:T
542:k
538:s
532:k
528:y
522:k
511:I
508:(
505:=
500:1
497:+
494:k
490:B
464:k
460:B
436:B
410:+
405:k
401:s
397:B
394:+
391:)
386:k
382:x
378:(
375:f
369:=
366:)
361:k
357:s
353:+
348:k
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340:(
337:f
304:,
298:+
293:k
289:s
284:B
278:T
273:k
269:s
263:2
260:1
255:+
250:k
246:s
240:T
236:)
230:k
226:x
222:(
219:f
213:+
210:)
205:k
201:x
197:(
194:f
191:=
188:)
183:k
179:s
175:+
170:k
166:x
162:(
159:f
132:B
106:f
95:(
79:)
76:x
73:(
70:f
20:)
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