77:(or semi-definite) quadratic form is defined in much the same way, except that "always positive" and "always negative" are replaced by "never negative" and "never positive", respectively. In other words, it may take on zero values for some non-zero vectors of
312:
983:
1852:
528:
138:
1545:
1196:
1091:
1268:
1991:
1338:
1462:
1926:
390:
1653:
1730:
432:
1126:
1021:
133:
1372:
824:
705:
592:
1602:
1575:
2157:
17:
853:
2019:, in which a vector of estimated parameters is sought which minimizes the sum of squared deviations from a perfect fit within the dataset.
1759:
has any non-zero off-diagonal elements, the non-matrix form will also contain some terms involving products of two different variables.
2012:
for a minimum are met at this point. If the quadratic form is negative-definite, the second-order conditions for a maximum are met.
1550:
In two dimensions this means that the distance between two points is the square root of the sum of the squared distances along the
2139:
1799:
440:
1480:
1131:
1026:
2165:
2105:
1201:
1937:
1762:
Positive or negative-definiteness or semi-definiteness, or indefiniteness, of this quadratic form is equivalent to
1273:
1395:
1380:
centered on the origin. If the general quadratic form above is equated to 0, the resulting equation is that of an
2009:
2134:, vol. 211 (Corrected fourth printing, revised third ed.), New York: Springer-Verlag, p. 578,
2131:
53:
1879:
320:
1622:
1672:
402:
2033:
1667:
1755:
this is equivalent to a non-matrix form containing solely terms involving squared variables; but if
1096:
991:
307:{\displaystyle {\begin{aligned}Q(x)&=B(x,x)\\B(x,y)&=B(y,x)={\tfrac {1}{2}}~.\end{aligned}}}
2192:
2075:
2038:
2028:
1763:
89:
735:
is indefinite and sometimes evaluates to a positive number and sometimes to a negative number. If
2197:
108:
1343:
758:
639:
536:
2043:
1866:
1613:
1580:
1553:
755:
the quadratic form is negative-definite and always evaluates to a negative number whenever
2175:
2115:
8:
2016:
2001:
833:
In general a quadratic form in two variables will also involve a cross-product term in
2161:
2135:
2101:
1870:
2171:
2111:
1744:
1740:
711:
is positive semidefinite and always evaluates to either 0 or a positive number. If
1778:
1752:
830:
is negative semidefinite and always evaluates to either 0 or a negative number.
1793:
problems. Suppose the matrix quadratic form is augmented with linear terms, as
1467:
39:
2100:. Cambridge Tracts in Mathematics. Vol. 106. Cambridge University Press.
2186:
2079:
1377:
96:
1790:
88:
quadratic form takes on both positive and negative values and is called an
46:
2149:
978:{\displaystyle Q(x)=c_{1}{x_{1}}^{2}+c_{2}{x_{2}}^{2}+2c_{3}x_{1}x_{2}~.}
43:
31:
2123:
1770:
1385:
119:, etc. according to its associated quadratic form. A quadratic form
111:
over the same space. A symmetric bilinear form is also described as
95:
More generally, these definitions apply to any vector space over an
1474:-dimensional space, the most commonly used measure of distance, is
1389:
1340:
with the sign of the semidefiniteness coinciding with the sign of
56:(always positive or always negative) for every non-zero vector of
1381:
826:
And if one of the constants is negative and the other is 0, then
707:
If one of the constants is positive and the other is 0, then
1384:
if the quadratic form is positive or negative-definite, a
1376:
This bivariate quadratic form appears in the context of
2015:
An important example of such an optimization arises in
1847:{\displaystyle x^{\mathsf {T}}A\,x+b^{\mathsf {T}}x\;,}
62:. According to that sign, the quadratic form is called
1951:
523:{\displaystyle Q(x)=c_{1}{x_{1}}^{2}+c_{2}{x_{2}}^{2}}
229:
1940:
1882:
1802:
1675:
1625:
1583:
1556:
1483:
1398:
1346:
1276:
1204:
1134:
1099:
1029:
994:
856:
761:
642:
539:
443:
405:
323:
136:
102:
1789:
Definite quadratic forms lend themselves readily to
1985:
1920:
1869:for a maximum or minimum are found by setting the
1846:
1724:
1647:
1596:
1569:
1539:
1456:
1366:
1332:
1262:
1190:
1120:
1085:
1015:
977:
818:
699:
586:
522:
426:
384:
306:
1540:{\displaystyle {x_{1}}^{2}+\cdots +{x_{n}}^{2}~.}
2184:
2158:Ergebnisse der Mathematik und ihrer Grenzgebiete
2148:
2062:
1191:{\displaystyle \,c_{1}c_{2}-{c_{3}}^{2}>0\;,}
1086:{\displaystyle \,c_{1}c_{2}-{c_{3}}^{2}>0\;,}
1263:{\displaystyle \;c_{1}c_{2}-{c_{3}}^{2}<0~.}
1986:{\displaystyle x=-{\tfrac {1}{2}}\,A^{-1}b\;,}
1333:{\displaystyle \;c_{1}c_{2}-{c_{3}}^{2}=0\;,}
1612:A quadratic form can be written in terms of
1457:{\displaystyle \;c_{1}c_{2}-{c_{3}}^{2}=0~.}
988:This quadratic form is positive-definite if
1270:It is positive or negative semidefinite if
123:and its associated symmetric bilinear form
2082:, which necessarily have characteristic 0.
1979:
1914:
1840:
1769:, which can be checked by considering all
1721:
1676:
1399:
1347:
1326:
1277:
1205:
1184:
1117:
1100:
1079:
1012:
995:
762:
643:
324:
27:Type of homogeneous polynomial of degree 2
1962:
1889:
1818:
1641:
1135:
1030:
414:
317:The latter formula arises from expanding
107:Quadratic forms correspond one-to-one to
2078:other than 2, but here we consider only
1732:in which at least one element is not 0;
636:evaluates to a positive number whenever
127:are related by the following equations:
2095:
1777:or by checking the signs of all of its
14:
2185:
1831:
1809:
1715:
1632:
1921:{\displaystyle 2A\,x+b={\vec {0}}\;,}
385:{\displaystyle \;Q(x+y)=B(x+y,x+y)~.}
2122:
2004:. If the quadratic form, and hence
1648:{\displaystyle x^{\mathsf {T}}A\,x}
24:
2074:This is true only over a field of
1725:{\displaystyle \;^{\mathsf {T}}\;}
434:, and consider the quadratic form
427:{\displaystyle V=\mathbb {R} ^{2}}
103:Associated symmetric bilinear form
25:
2209:
18:Positive-definite quadratic form
1784:
2068:
2056:
1908:
1710:
1677:
1607:
1121:{\displaystyle \;c_{1}<0\;}
1016:{\displaystyle \;c_{1}>0\;}
866:
860:
807:
795:
789:
763:
688:
676:
670:
644:
575:
549:
453:
447:
373:
349:
340:
328:
291:
288:
282:
273:
267:
258:
246:
240:
222:
210:
197:
185:
175:
163:
150:
144:
13:
1:
2132:Graduate Texts in Mathematics
2098:Arithmetic of quadratic forms
2089:
1743:; and superscript denotes a
2063:Milnor & Husemoller 1973
2008:, is positive-definite, the
1865:×1 vector of constants. The
7:
2096:Kitaoka, Yoshiyuki (1993).
2022:
1388:if it is indefinite, and a
394:
10:
2214:
2160:. Vol. 73. Springer.
2034:Positive-definite function
2152:; Husemoller, D. (1973).
1367:{\displaystyle \;c_{1}~.}
819:{\displaystyle \;\neq ~.}
700:{\displaystyle \;\neq ~.}
632:is positive-definite, so
2154:Symmetric Bilinear Forms
2049:
2039:Positive-definite matrix
2029:Isotropic quadratic form
587:{\displaystyle ~x=\in V}
109:symmetric bilinear forms
90:isotropic quadratic form
2010:second-order conditions
36:definite quadratic form
1987:
1922:
1867:first-order conditions
1848:
1726:
1649:
1598:
1571:
1541:
1458:
1368:
1334:
1264:
1192:
1122:
1087:
1017:
979:
820:
701:
588:
524:
428:
386:
308:
2044:Polarization identity
1988:
1923:
1849:
1764:the same property of
1727:
1650:
1599:
1597:{\displaystyle x_{2}}
1572:
1570:{\displaystyle x_{1}}
1542:
1459:
1369:
1335:
1265:
1193:
1123:
1093:negative-definite if
1088:
1018:
980:
821:
702:
589:
525:
429:
387:
309:
1938:
1880:
1873:to the zero vector:
1800:
1673:
1623:
1581:
1554:
1481:
1396:
1344:
1274:
1202:
1132:
1097:
1027:
992:
854:
759:
731:or vice versa, then
640:
537:
441:
403:
321:
134:
2017:multiple regression
628:the quadratic form
399:As an example, let
1983:
1960:
1918:
1844:
1722:
1645:
1594:
1567:
1537:
1466:The square of the
1454:
1364:
1330:
1260:
1198:and indefinite if
1188:
1118:
1083:
1013:
975:
816:
697:
608:are constants. If
584:
520:
424:
382:
304:
302:
238:
52:that has the same
2141:978-0-387-95385-4
1959:
1911:
1871:matrix derivative
1533:
1450:
1360:
1256:
971:
812:
693:
542:
378:
296:
237:
68:negative-definite
64:positive-definite
16:(Redirected from
2205:
2179:
2144:
2119:
2083:
2072:
2066:
2060:
2007:
1999:
1992:
1990:
1989:
1984:
1975:
1974:
1961:
1952:
1927:
1925:
1924:
1919:
1913:
1912:
1904:
1864:
1860:
1853:
1851:
1850:
1845:
1836:
1835:
1834:
1814:
1813:
1812:
1779:principal minors
1776:
1767:
1758:
1750:
1745:matrix transpose
1741:symmetric matrix
1739:
1735:
1731:
1729:
1728:
1723:
1720:
1719:
1718:
1708:
1707:
1689:
1688:
1668:Cartesian vector
1665:
1661:
1654:
1652:
1651:
1646:
1637:
1636:
1635:
1603:
1601:
1600:
1595:
1593:
1592:
1576:
1574:
1573:
1568:
1566:
1565:
1546:
1544:
1543:
1538:
1531:
1530:
1529:
1524:
1523:
1522:
1502:
1501:
1496:
1495:
1494:
1473:
1463:
1461:
1460:
1455:
1448:
1441:
1440:
1435:
1434:
1433:
1419:
1418:
1409:
1408:
1373:
1371:
1370:
1365:
1358:
1357:
1356:
1339:
1337:
1336:
1331:
1319:
1318:
1313:
1312:
1311:
1297:
1296:
1287:
1286:
1269:
1267:
1266:
1261:
1254:
1247:
1246:
1241:
1240:
1239:
1225:
1224:
1215:
1214:
1197:
1195:
1194:
1189:
1177:
1176:
1171:
1170:
1169:
1155:
1154:
1145:
1144:
1127:
1125:
1124:
1119:
1110:
1109:
1092:
1090:
1089:
1084:
1072:
1071:
1066:
1065:
1064:
1050:
1049:
1040:
1039:
1022:
1020:
1019:
1014:
1005:
1004:
984:
982:
981:
976:
969:
968:
967:
958:
957:
948:
947:
932:
931:
926:
925:
924:
913:
912:
900:
899:
894:
893:
892:
881:
880:
843:
836:
829:
825:
823:
822:
817:
810:
788:
787:
775:
774:
754:
749:
744:
739:
734:
730:
725:
720:
715:
710:
706:
704:
703:
698:
691:
669:
668:
656:
655:
631:
627:
622:
617:
612:
604:
597:
593:
591:
590:
585:
574:
573:
561:
560:
540:
529:
527:
526:
521:
519:
518:
513:
512:
511:
500:
499:
487:
486:
481:
480:
479:
468:
467:
433:
431:
430:
425:
423:
422:
417:
391:
389:
388:
383:
376:
313:
311:
310:
305:
303:
294:
239:
230:
126:
122:
80:
61:
51:
21:
2213:
2212:
2208:
2207:
2206:
2204:
2203:
2202:
2193:Quadratic forms
2183:
2182:
2168:
2142:
2108:
2092:
2087:
2086:
2073:
2069:
2061:
2057:
2052:
2025:
2005:
1997:
1967:
1963:
1950:
1939:
1936:
1935:
1903:
1902:
1881:
1878:
1877:
1862:
1858:
1830:
1829:
1825:
1808:
1807:
1803:
1801:
1798:
1797:
1787:
1774:
1765:
1756:
1748:
1737:
1733:
1714:
1713:
1709:
1703:
1699:
1684:
1680:
1674:
1671:
1670:
1663:
1659:
1631:
1630:
1626:
1624:
1621:
1620:
1610:
1588:
1584:
1582:
1579:
1578:
1561:
1557:
1555:
1552:
1551:
1525:
1518:
1514:
1513:
1512:
1497:
1490:
1486:
1485:
1484:
1482:
1479:
1478:
1471:
1436:
1429:
1425:
1424:
1423:
1414:
1410:
1404:
1400:
1397:
1394:
1393:
1352:
1348:
1345:
1342:
1341:
1314:
1307:
1303:
1302:
1301:
1292:
1288:
1282:
1278:
1275:
1272:
1271:
1242:
1235:
1231:
1230:
1229:
1220:
1216:
1210:
1206:
1203:
1200:
1199:
1172:
1165:
1161:
1160:
1159:
1150:
1146:
1140:
1136:
1133:
1130:
1129:
1105:
1101:
1098:
1095:
1094:
1067:
1060:
1056:
1055:
1054:
1045:
1041:
1035:
1031:
1028:
1025:
1024:
1000:
996:
993:
990:
989:
963:
959:
953:
949:
943:
939:
927:
920:
916:
915:
914:
908:
904:
895:
888:
884:
883:
882:
876:
872:
855:
852:
851:
846:
841:
839:
834:
827:
783:
779:
770:
766:
760:
757:
756:
752:
747:
746:
742:
737:
736:
732:
728:
723:
722:
718:
713:
712:
708:
664:
660:
651:
647:
641:
638:
637:
629:
625:
620:
619:
615:
610:
609:
607:
602:
600:
595:
569:
565:
556:
552:
538:
535:
534:
514:
507:
503:
502:
501:
495:
491:
482:
475:
471:
470:
469:
463:
459:
442:
439:
438:
418:
413:
412:
404:
401:
400:
397:
322:
319:
318:
301:
300:
228:
200:
179:
178:
153:
137:
135:
132:
131:
124:
120:
105:
78:
57:
49:
28:
23:
22:
15:
12:
11:
5:
2211:
2201:
2200:
2198:Linear algebra
2195:
2181:
2180:
2166:
2146:
2140:
2120:
2106:
2091:
2088:
2085:
2084:
2080:ordered fields
2076:characteristic
2067:
2054:
2053:
2051:
2048:
2047:
2046:
2041:
2036:
2031:
2024:
2021:
1994:
1993:
1982:
1978:
1973:
1970:
1966:
1958:
1955:
1949:
1946:
1943:
1929:
1928:
1917:
1910:
1907:
1901:
1898:
1895:
1892:
1888:
1885:
1855:
1854:
1843:
1839:
1833:
1828:
1824:
1821:
1817:
1811:
1806:
1786:
1783:
1717:
1712:
1706:
1702:
1698:
1695:
1692:
1687:
1683:
1679:
1656:
1655:
1644:
1640:
1634:
1629:
1609:
1606:
1591:
1587:
1564:
1560:
1548:
1547:
1536:
1528:
1521:
1517:
1511:
1508:
1505:
1500:
1493:
1489:
1468:Euclidean norm
1453:
1447:
1444:
1439:
1432:
1428:
1422:
1417:
1413:
1407:
1403:
1378:conic sections
1363:
1355:
1351:
1329:
1325:
1322:
1317:
1310:
1306:
1300:
1295:
1291:
1285:
1281:
1259:
1253:
1250:
1245:
1238:
1234:
1228:
1223:
1219:
1213:
1209:
1187:
1183:
1180:
1175:
1168:
1164:
1158:
1153:
1149:
1143:
1139:
1116:
1113:
1108:
1104:
1082:
1078:
1075:
1070:
1063:
1059:
1053:
1048:
1044:
1038:
1034:
1011:
1008:
1003:
999:
986:
985:
974:
966:
962:
956:
952:
946:
942:
938:
935:
930:
923:
919:
911:
907:
903:
898:
891:
887:
879:
875:
871:
868:
865:
862:
859:
844:
837:
815:
809:
806:
803:
800:
797:
794:
791:
786:
782:
778:
773:
769:
765:
750:
740:
726:
716:
696:
690:
687:
684:
681:
678:
675:
672:
667:
663:
659:
654:
650:
646:
623:
613:
605:
598:
583:
580:
577:
572:
568:
564:
559:
555:
551:
548:
545:
531:
530:
517:
510:
506:
498:
494:
490:
485:
478:
474:
466:
462:
458:
455:
452:
449:
446:
421:
416:
411:
408:
396:
393:
381:
375:
372:
369:
366:
363:
360:
357:
354:
351:
348:
345:
342:
339:
336:
333:
330:
327:
315:
314:
299:
293:
290:
287:
284:
281:
278:
275:
272:
269:
266:
263:
260:
257:
254:
251:
248:
245:
242:
236:
233:
227:
224:
221:
218:
215:
212:
209:
206:
203:
201:
199:
196:
193:
190:
187:
184:
181:
180:
177:
174:
171:
168:
165:
162:
159:
156:
154:
152:
149:
146:
143:
140:
139:
104:
101:
40:quadratic form
26:
9:
6:
4:
3:
2:
2210:
2199:
2196:
2194:
2191:
2190:
2188:
2177:
2173:
2169:
2167:3-540-06009-X
2163:
2159:
2155:
2151:
2147:
2143:
2137:
2133:
2129:
2125:
2121:
2117:
2113:
2109:
2107:0-521-40475-4
2103:
2099:
2094:
2093:
2081:
2077:
2071:
2065:, p. 61.
2064:
2059:
2055:
2045:
2042:
2040:
2037:
2035:
2032:
2030:
2027:
2026:
2020:
2018:
2013:
2011:
2003:
1980:
1976:
1971:
1968:
1964:
1956:
1953:
1947:
1944:
1941:
1934:
1933:
1932:
1915:
1905:
1899:
1896:
1893:
1890:
1886:
1883:
1876:
1875:
1874:
1872:
1868:
1841:
1837:
1826:
1822:
1819:
1815:
1804:
1796:
1795:
1794:
1792:
1782:
1780:
1772:
1768:
1760:
1754:
1746:
1742:
1704:
1700:
1696:
1693:
1690:
1685:
1681:
1669:
1642:
1638:
1627:
1619:
1618:
1617:
1615:
1605:
1589:
1585:
1577:axis and the
1562:
1558:
1534:
1526:
1519:
1515:
1509:
1506:
1503:
1498:
1491:
1487:
1477:
1476:
1475:
1469:
1464:
1451:
1445:
1442:
1437:
1430:
1426:
1420:
1415:
1411:
1405:
1401:
1391:
1387:
1383:
1379:
1374:
1361:
1353:
1349:
1327:
1323:
1320:
1315:
1308:
1304:
1298:
1293:
1289:
1283:
1279:
1257:
1251:
1248:
1243:
1236:
1232:
1226:
1221:
1217:
1211:
1207:
1185:
1181:
1178:
1173:
1166:
1162:
1156:
1151:
1147:
1141:
1137:
1114:
1111:
1106:
1102:
1080:
1076:
1073:
1068:
1061:
1057:
1051:
1046:
1042:
1036:
1032:
1009:
1006:
1001:
997:
972:
964:
960:
954:
950:
944:
940:
936:
933:
928:
921:
917:
909:
905:
901:
896:
889:
885:
877:
873:
869:
863:
857:
850:
849:
848:
831:
813:
804:
801:
798:
792:
784:
780:
776:
771:
767:
694:
685:
682:
679:
673:
665:
661:
657:
652:
648:
635:
581:
578:
570:
566:
562:
557:
553:
546:
543:
515:
508:
504:
496:
492:
488:
483:
476:
472:
464:
460:
456:
450:
444:
437:
436:
435:
419:
409:
406:
392:
379:
370:
367:
364:
361:
358:
355:
352:
346:
343:
337:
334:
331:
325:
297:
285:
279:
276:
270:
264:
261:
255:
252:
249:
243:
234:
231:
225:
219:
216:
213:
207:
204:
202:
194:
191:
188:
182:
172:
169:
166:
160:
157:
155:
147:
141:
130:
129:
128:
118:
114:
110:
100:
98:
97:ordered field
93:
91:
87:
82:
76:
71:
69:
65:
60:
55:
48:
45:
41:
37:
33:
19:
2153:
2127:
2097:
2070:
2058:
2014:
1995:
1930:
1856:
1791:optimization
1788:
1785:Optimization
1761:
1657:
1611:
1549:
1465:
1375:
987:
832:
633:
532:
398:
316:
117:semidefinite
116:
112:
106:
94:
85:
83:
75:semidefinite
74:
72:
67:
63:
58:
47:vector space
35:
29:
2124:Lang, Serge
2002:nonsingular
1771:eigenvalues
1608:Matrix form
32:mathematics
2187:Categories
2176:0292.10016
2150:Milnor, J.
2116:0785.11021
2090:References
86:indefinite
42:over some
1996:assuming
1969:−
1948:−
1909:→
1694:⋯
1507:⋯
1421:−
1386:hyperbola
1299:−
1227:−
1157:−
1052:−
793:≠
674:≠
579:∈
277:−
262:−
2126:(2004),
2023:See also
1753:diagonal
1614:matrices
1390:parabola
753:< 0 ,
729:< 0 ,
626:> 0 ,
395:Examples
113:definite
2128:Algebra
1931:giving
1662:is any
1382:ellipse
743:< 0
719:> 0
616:> 0
2174:
2164:
2138:
2114:
2104:
1861:is an
1857:where
1738:n × n
1736:is an
1658:where
1604:axis.
1532:
1449:
1359:
1255:
970:
811:
692:
541:
533:where
377:
295:
2050:Notes
1747:. If
38:is a
2162:ISBN
2136:ISBN
2102:ISBN
1249:<
1179:>
1128:and
1112:<
1074:>
1023:and
1007:>
745:and
721:and
618:and
601:and
594:and
54:sign
44:real
34:, a
2172:Zbl
2112:Zbl
2000:is
1773:of
1751:is
1666:×1
1616:as
1470:in
1392:if
84:An
66:or
30:In
2189::
2170:.
2156:.
2130:,
2110:.
1781:.
847::
115:,
99:.
92:.
81:.
73:A
70:.
2178:.
2145:.
2118:.
2006:A
1998:A
1981:,
1977:b
1972:1
1965:A
1957:2
1954:1
1945:=
1942:x
1916:,
1906:0
1900:=
1897:b
1894:+
1891:x
1887:A
1884:2
1863:n
1859:b
1842:,
1838:x
1832:T
1827:b
1823:+
1820:x
1816:A
1810:T
1805:x
1775:A
1766:A
1757:A
1749:A
1734:A
1716:T
1711:]
1705:n
1701:x
1697:,
1691:,
1686:1
1682:x
1678:[
1664:n
1660:x
1643:x
1639:A
1633:T
1628:x
1590:2
1586:x
1563:1
1559:x
1535:.
1527:2
1520:n
1516:x
1510:+
1504:+
1499:2
1492:1
1488:x
1472:n
1452:.
1446:0
1443:=
1438:2
1431:3
1427:c
1416:2
1412:c
1406:1
1402:c
1362:.
1354:1
1350:c
1328:,
1324:0
1321:=
1316:2
1309:3
1305:c
1294:2
1290:c
1284:1
1280:c
1258:.
1252:0
1244:2
1237:3
1233:c
1222:2
1218:c
1212:1
1208:c
1186:,
1182:0
1174:2
1167:3
1163:c
1152:2
1148:c
1142:1
1138:c
1115:0
1107:1
1103:c
1081:,
1077:0
1069:2
1062:3
1058:c
1047:2
1043:c
1037:1
1033:c
1010:0
1002:1
998:c
973:.
965:2
961:x
955:1
951:x
945:3
941:c
937:2
934:+
929:2
922:2
918:x
910:2
906:c
902:+
897:2
890:1
886:x
878:1
874:c
870:=
867:)
864:x
861:(
858:Q
845:2
842:x
840:·
838:1
835:x
828:Q
814:.
808:]
805:0
802:,
799:0
796:[
790:]
785:2
781:x
777:,
772:1
768:x
764:[
751:2
748:c
741:1
738:c
733:Q
727:2
724:c
717:1
714:c
709:Q
695:.
689:]
686:0
683:,
680:0
677:[
671:]
666:2
662:x
658:,
653:1
649:x
645:[
634:Q
630:Q
624:2
621:c
614:1
611:c
606:2
603:c
599:1
596:c
582:V
576:]
571:2
567:x
563:,
558:1
554:x
550:[
547:=
544:x
516:2
509:2
505:x
497:2
493:c
489:+
484:2
477:1
473:x
465:1
461:c
457:=
454:)
451:x
448:(
445:Q
420:2
415:R
410:=
407:V
380:.
374:)
371:y
368:+
365:x
362:,
359:y
356:+
353:x
350:(
347:B
344:=
341:)
338:y
335:+
332:x
329:(
326:Q
298:.
292:]
289:)
286:y
283:(
280:Q
274:)
271:x
268:(
265:Q
259:)
256:y
253:+
250:x
247:(
244:Q
241:[
235:2
232:1
226:=
223:)
220:x
217:,
214:y
211:(
208:B
205:=
198:)
195:y
192:,
189:x
186:(
183:B
176:)
173:x
170:,
167:x
164:(
161:B
158:=
151:)
148:x
145:(
142:Q
125:B
121:Q
79:V
59:V
50:V
20:)
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