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R\\,x^{(k)}",[574,1133,1135,1158,1231,1250],{"className":1134,"ariaHidden":615},[614],[574,1136,1138,1142,1146,1149,1152,1155],{"className":1137},[619],[574,1139],{"className":1140,"style":1141},[623],"height:1em;vertical-align:-0.25em;",[574,1143,1075],{"className":1144},[1145],"mopen",[574,1147,969],{"className":1148,"style":1015},[628,629],[574,1150],{"className":1151,"style":1019},[636],[574,1153,972],{"className":1154},[927],[574,1156],{"className":1157,"style":1019},[636],[574,1159,1161,1165,1168,1172,1175,1222,1225,1228],{"className":1160},[619],[574,1162],{"className":1163,"style":1164},[623],"height:1.188em;vertical-align:-0.25em;",[574,1166,975],{"className":1167},[628,629],[574,1169,1084],{"className":1170},[1171],"mclose",[574,1173],{"className":1174,"style":878},[636],[574,1176,1178,1181],{"className":1177},[628],[574,1179,598],{"className":1180},[628,629],[574,1182,1184],{"className":1183},[728],[574,1185,1187],{"className":1186},[732],[574,1188,1190],{"className":1189},[737],[574,1191,1194],{"className":1192,"style":1193},[741],"height:0.938em;",[574,1195,1197,1200],{"style":1196},"top:-3.113em;margin-right:0.05em;",[574,1198],{"className":1199,"style":750},[749],[574,1201,1203],{"className":1202},[754,755,756,757],[574,1204,1206,1209,1213,1216,1219],{"className":1205},[628,757],[574,1207,1075],{"className":1208},[1145,757],[574,1210,1100],{"className":1211,"style":1212},[628,629,757],"margin-right:0.0315em;",[574,1214,972],{"className":1215},[927,757],[574,1217,795],{"className":1218},[628,757],[574,1220,1084],{"className":1221},[1171,757],[574,1223],{"className":1224,"style":637},[636],[574,1226,602],{"className":1227},[641],[574,1229],{"className":1230,"style":637},[636],[574,1232,1234,1238,1241,1244,1247],{"className":1233},[619],[574,1235],{"className":1236,"style":1237},[623],"height:0.7778em;vertical-align:-0.0833em;",[574,1239,605],{"className":1240},[628,629],[574,1242],{"className":1243,"style":1019},[636],[574,1245,814],{"className":1246},[927],[574,1248],{"className":1249,"style":1019},[636],[574,1251,1253,1256,1259,1262],{"className":1252},[619],[574,1254],{"className":1255,"style":1193},[623],[574,1257,980],{"className":1258,"style":1053},[628,629],[574,1260],{"className":1261,"style":878},[636],[574,1263,1265,1268],{"className":1264},[628],[574,1266,598],{"className":1267},[628,629],[574,1269,1271],{"className":1270},[728],[574,1272,1274],{"className":1273},[732],[574,1275,1277],{"className":1276},[737],[574,1278,1280],{"className":1279,"style":1193},[741],[574,1281,1282,1285],{"style":1196},[574,1283],{"className":1284,"style":750},[749],[574,1286,1288],{"className":1287},[754,755,756,757],[574,1289,1291,1294,1297],{"className":1290},[628,757],[574,1292,1075],{"className":1293},[1145,757],[574,1295,1100],{"className":1296,"style":1212},[628,629,757],[574,1298,1084],{"className":1299},[1171,757],[563,1301,1302],{},"Man löst also in jedem Schritt ein Dreieckssystem, was durch Vorwärtseinsetzen von oben nach unten geschieht.",[563,1304,1305],{},"Konvergenz:",[1307,1308,1309,1350,1497],"ul",{},[1310,1311,1312,1313,1341,1342,1345,1346,1349],"li",{},"Hinreichend ist, dass ",[574,1314,1316,1329],{"className":1315},[577],[574,1317,1319],{"className":1318},[581],[583,1320,1321],{"xmlns":585},[587,1322,1323,1327],{},[590,1324,1325],{},[593,1326,595],{},[607,1328,595],{"encoding":609},[574,1330,1332],{"className":1331,"ariaHidden":615},[614],[574,1333,1335,1338],{"className":1334},[619],[574,1336],{"className":1337,"style":624},[623],[574,1339,595],{"className":1340},[628,629]," ",[567,1343,1344],{},"strikt diagonaldominant"," ist oder ",[567,1347,1348],{},"symmetrisch positiv definit",".",[1310,1351,1352,1353,1467,1468,1349],{},"Notwendig und hinreichend ist ein Spektralradius der Iterationsmatrix ",[574,1354,1356,1390],{"className":1355},[577],[574,1357,1359],{"className":1358},[581],[583,1360,1361],{"xmlns":585},[587,1362,1363,1387],{},[590,1364,1365,1367,1369,1371,1373,1375,1385],{},[600,1366,814],{},[600,1368,1075],{"stretchy":1074},[593,1370,969],{},[600,1372,972],{},[593,1374,975],{},[1090,1376,1377,1379],{},[600,1378,1084],{"stretchy":1074},[590,1380,1381,1383],{},[600,1382,814],{},[793,1384,795],{},[593,1386,980],{},[607,1388,1389],{"encoding":609},"-(D + L)^{-1} R",[574,1391,1393,1417],{"className":1392,"ariaHidden":615},[614],[574,1394,1396,1399,1402,1405,1408,1411,1414],{"className":1395},[619],[574,1397],{"className":1398,"style":1141},[623],[574,1400,814],{"className":1401},[628],[574,1403,1075],{"className":1404},[1145],[574,1406,969],{"className":1407,"style":1015},[628,629],[574,1409],{"className":1410,"style":1019},[636],[574,1412,972],{"className":1413},[927],[574,1415],{"className":1416,"style":1019},[636],[574,1418,1420,1424,1427,1464],{"className":1419},[619],[574,1421],{"className":1422,"style":1423},[623],"height:1.0641em;vertical-align:-0.25em;",[574,1425,975],{"className":1426},[628,629],[574,1428,1430,1433],{"className":1429},[1171],[574,1431,1084],{"className":1432},[1171],[574,1434,1436],{"className":1435},[728],[574,1437,1439],{"className":1438},[732],[574,1440,1442],{"className":1441},[737],[574,1443,1446],{"className":1444,"style":1445},[741],"height:0.8141em;",[574,1447,1449,1452],{"style":1448},"top:-3.063em;margin-right:0.05em;",[574,1450],{"className":1451,"style":750},[749],[574,1453,1455],{"className":1454},[754,755,756,757],[574,1456,1458,1461],{"className":1457},[628,757],[574,1459,814],{"className":1460},[628,757],[574,1462,795],{"className":1463},[628,757],[574,1465,980],{"className":1466,"style":1053},[628,629]," kleiner als ",[574,1469,1471,1484],{"className":1470},[577],[574,1472,1474],{"className":1473},[581],[583,1475,1476],{"xmlns":585},[587,1477,1478,1482],{},[590,1479,1480],{},[793,1481,795],{},[607,1483,795],{"encoding":609},[574,1485,1487],{"className":1486,"ariaHidden":615},[614],[574,1488,1490,1494],{"className":1489},[619],[574,1491],{"className":1492,"style":1493},[623],"height:0.6444em;",[574,1495,795],{"className":1496},[628],[1310,1498,1499,1500,1503,1504,1649,1650,1819],{},"Konvergieren beide Verfahren, ist Gauß-Seidel oft schneller als Jacobi. Nur für ",[567,1501,1502],{},"konsistent geordnete"," Matrizen (z.B. tridiagonale Matrizen mit ",[574,1505,1507,1537],{"className":1506},[577],[574,1508,1510],{"className":1509},[581],[583,1511,1512],{"xmlns":585},[587,1513,1514,1534],{},[590,1515,1516,1527,1531],{},[700,1517,1518,1521],{},[593,1519,1520],{},"a",[590,1522,1523,1525],{},[593,1524,670],{},[593,1526,670],{},[600,1528,1530],{"mathvariant":1529},"normal","≠",[793,1532,1533],{},"0",[607,1535,1536],{"encoding":609},"a_{ii} \\neq 0",[574,1538,1540,1640],{"className":1539,"ariaHidden":615},[614],[574,1541,1543,1547,1591,1594,1637],{"className":1542},[619],[574,1544],{"className":1545,"style":1546},[623],"height:0.8889em;vertical-align:-0.1944em;",[574,1548,1550,1553],{"className":1549},[628],[574,1551,1520],{"className":1552},[628,629],[574,1554,1556],{"className":1555},[728],[574,1557,1559,1583],{"className":1558},[732,733],[574,1560,1562,1580],{"className":1561},[737],[574,1563,1565],{"className":1564,"style":742},[741],[574,1566,1567,1570],{"style":745},[574,1568],{"className":1569,"style":750},[749],[574,1571,1573],{"className":1572},[754,755,756,757],[574,1574,1576],{"className":1575},[628,757],[574,1577,1579],{"className":1578},[628,629,757],"ii",[574,1581,765],{"className":1582},[764],[574,1584,1586],{"className":1585},[737],[574,1587,1589],{"className":1588,"style":772},[741],[574,1590],{},[574,1592],{"className":1593,"style":637},[636],[574,1595,1597,1630,1634],{"className":1596},[641],[574,1598,1600],{"className":1599},[641],[574,1601,1604],{"className":1602},[628,1603],"vbox",[574,1605,1608],{"className":1606},[1607],"thinbox",[574,1609,1612,1615,1626],{"className":1610},[1611],"rlap",[574,1613],{"className":1614,"style":1546},[623],[574,1616,1619],{"className":1617},[1618],"inner",[574,1620,1622],{"className":1621},[628],[574,1623,1625],{"className":1624},[641],"",[574,1627],{"className":1628},[1629],"fix",[574,1631],{"className":1632},[636,1633],"nobreak",[574,1635,602],{"className":1636},[641],[574,1638],{"className":1639,"style":637},[636],[574,1641,1643,1646],{"className":1642},[619],[574,1644],{"className":1645,"style":1493},[623],[574,1647,1533],{"className":1648},[628],") gilt ",[574,1651,1653,1686],{"className":1652},[577],[574,1654,1656],{"className":1655},[581],[583,1657,1658],{"xmlns":585},[587,1659,1660,1683],{},[590,1661,1662,1670,1672],{},[700,1663,1664,1667],{},[593,1665,1666],{},"ρ",[1086,1668,1669],{},"GS",[600,1671,602],{},[1673,1674,1675,1677,1680],"msubsup",{},[593,1676,1666],{},[1086,1678,1679],{},"J",[793,1681,1682],{},"2",[607,1684,1685],{"encoding":609},"\\rho_{\\text{GS}} = \\rho_{\\text{J}}^2",[574,1687,1689,1753],{"className":1688,"ariaHidden":615},[614],[574,1690,1692,1696,1744,1747,1750],{"className":1691},[619],[574,1693],{"className":1694,"style":1695},[623],"height:0.625em;vertical-align:-0.1944em;",[574,1697,1699,1702],{"className":1698},[628],[574,1700,1666],{"className":1701},[628,629],[574,1703,1705],{"className":1704},[728],[574,1706,1708,1736],{"className":1707},[732,733],[574,1709,1711,1733],{"className":1710},[737],[574,1712,1715],{"className":1713,"style":1714},[741],"height:0.3283em;",[574,1716,1717,1720],{"style":745},[574,1718],{"className":1719,"style":750},[749],[574,1721,1723],{"className":1722},[754,755,756,757],[574,1724,1726],{"className":1725},[628,757],[574,1727,1730],{"className":1728},[628,1729,757],"text",[574,1731,1669],{"className":1732},[628,757],[574,1734,765],{"className":1735},[764],[574,1737,1739],{"className":1738},[737],[574,1740,1742],{"className":1741,"style":772},[741],[574,1743],{},[574,1745],{"className":1746,"style":637},[636],[574,1748,602],{"className":1749},[641],[574,1751],{"className":1752,"style":637},[636],[574,1754,1756,1760],{"className":1755},[619],[574,1757],{"className":1758,"style":1759},[623],"height:1.0894em;vertical-align:-0.2753em;",[574,1761,1763,1766],{"className":1762},[628],[574,1764,1666],{"className":1765},[628,629],[574,1767,1769],{"className":1768},[728],[574,1770,1772,1810],{"className":1771},[732,733],[574,1773,1775,1807],{"className":1774},[737],[574,1776,1778,1796],{"className":1777,"style":1445},[741],[574,1779,1781,1784],{"style":1780},"top:-2.4247em;margin-left:0em;margin-right:0.05em;",[574,1782],{"className":1783,"style":750},[749],[574,1785,1787],{"className":1786},[754,755,756,757],[574,1788,1790],{"className":1789},[628,757],[574,1791,1793],{"className":1792},[628,1729,757],[574,1794,1679],{"className":1795},[628,757],[574,1797,1798,1801],{"style":1448},[574,1799],{"className":1800,"style":750},[749],[574,1802,1804],{"className":1803},[754,755,756,757],[574,1805,1682],{"className":1806},[628,757],[574,1808,765],{"className":1809},[764],[574,1811,1813],{"className":1812},[737],[574,1814,1817],{"className":1815,"style":1816},[741],"height:0.2753em;",[574,1818],{},". Dann braucht Gauß-Seidel für dieselbe Genauigkeit etwa halb so viele Schritte. Allgemein gibt es keinen solchen Zusammenhang.",[563,1821,1822],{},"Ein Nachteil: Die Komponenten hängen voneinander ab, das Verfahren lässt sich schlechter parallelisieren. Außerdem hängt das Ergebnis eines Schritts von der Reihenfolge der Gleichungen ab.",[558,1824,1826],{"id":1825},"beispiele","Beispiele",[563,1828,1829,1830,1993,1994,2157,2158,2210],{},"Dasselbe System wie beim Jacobi-Verfahren, ",[574,1831,1833,1867],{"className":1832},[577],[574,1834,1836],{"className":1835},[581],[583,1837,1838],{"xmlns":585},[587,1839,1840,1864],{},[590,1841,1842,1845,1851,1853,1859,1861],{},[793,1843,1844],{},"4",[700,1846,1847,1849],{},[593,1848,598],{},[793,1850,795],{},[600,1852,972],{},[700,1854,1855,1857],{},[593,1856,598],{},[793,1858,1682],{},[600,1860,602],{},[793,1862,1863],{},"6",[607,1865,1866],{"encoding":609},"4x_1 + x_2 = 6",[574,1868,1870,1929,1984],{"className":1869,"ariaHidden":615},[614],[574,1871,1873,1877,1880,1920,1923,1926],{"className":1872},[619],[574,1874],{"className":1875,"style":1876},[623],"height:0.7944em;vertical-align:-0.15em;",[574,1878,1844],{"className":1879},[628],[574,1881,1883,1886],{"className":1882},[628],[574,1884,598],{"className":1885},[628,629],[574,1887,1889],{"className":1888},[728],[574,1890,1892,1912],{"className":1891},[732,733],[574,1893,1895,1909],{"className":1894},[737],[574,1896,1898],{"className":1897,"style":848},[741],[574,1899,1900,1903],{"style":745},[574,1901],{"className":1902,"style":750},[749],[574,1904,1906],{"className":1905},[754,755,756,757],[574,1907,795],{"className":1908},[628,757],[574,1910,765],{"className":1911},[764],[574,1913,1915],{"className":1914},[737],[574,1916,1918],{"className":1917,"style":772},[741],[574,1919],{},[574,1921],{"className":1922,"style":1019},[636],[574,1924,972],{"className":1925},[927],[574,1927],{"className":1928,"style":1019},[636],[574,1930,1932,1935,1975,1978,1981],{"className":1931},[619],[574,1933],{"className":1934,"style":718},[623],[574,1936,1938,1941],{"className":1937},[628],[574,1939,598],{"className":1940},[628,629],[574,1942,1944],{"className":1943},[728],[574,1945,1947,1967],{"className":1946},[732,733],[574,1948,1950,1964],{"className":1949},[737],[574,1951,1953],{"className":1952,"style":848},[741],[574,1954,1955,1958],{"style":745},[574,1956],{"className":1957,"style":750},[749],[574,1959,1961],{"className":1960},[754,755,756,757],[574,1962,1682],{"className":1963},[628,757],[574,1965,765],{"className":1966},[764],[574,1968,1970],{"className":1969},[737],[574,1971,1973],{"className":1972,"style":772},[741],[574,1974],{},[574,1976],{"className":1977,"style":637},[636],[574,1979,602],{"className":1980},[641],[574,1982],{"className":1983,"style":637},[636],[574,1985,1987,1990],{"className":1986},[619],[574,1988],{"className":1989,"style":1493},[623],[574,1991,1863],{"className":1992},[628],", ",[574,1995,1997,2031],{"className":1996},[577],[574,1998,2000],{"className":1999},[581],[583,2001,2002],{"xmlns":585},[587,2003,2004,2028],{},[590,2005,2006,2012,2014,2017,2023,2025],{},[700,2007,2008,2010],{},[593,2009,598],{},[793,2011,795],{},[600,2013,972],{},[793,2015,2016],{},"3",[700,2018,2019,2021],{},[593,2020,598],{},[793,2022,1682],{},[600,2024,602],{},[793,2026,2027],{},"7",[607,2029,2030],{"encoding":609},"x_1 + 3x_2 = 7",[574,2032,2034,2090,2148],{"className":2033,"ariaHidden":615},[614],[574,2035,2037,2041,2081,2084,2087],{"className":2036},[619],[574,2038],{"className":2039,"style":2040},[623],"height:0.7333em;vertical-align:-0.15em;",[574,2042,2044,2047],{"className":2043},[628],[574,2045,598],{"className":2046},[628,629],[574,2048,2050],{"className":2049},[728],[574,2051,2053,2073],{"className":2052},[732,733],[574,2054,2056,2070],{"className":2055},[737],[574,2057,2059],{"className":2058,"style":848},[741],[574,2060,2061,2064],{"style":745},[574,2062],{"className":2063,"style":750},[749],[574,2065,2067],{"className":2066},[754,755,756,757],[574,2068,795],{"className":2069},[628,757],[574,2071,765],{"className":2072},[764],[574,2074,2076],{"className":2075},[737],[574,2077,2079],{"className":2078,"style":772},[741],[574,2080],{},[574,2082],{"className":2083,"style":1019},[636],[574,2085,972],{"className":2086},[927],[574,2088],{"className":2089,"style":1019},[636],[574,2091,2093,2096,2099,2139,2142,2145],{"className":2092},[619],[574,2094],{"className":2095,"style":1876},[623],[574,2097,2016],{"className":2098},[628],[574,2100,2102,2105],{"className":2101},[628],[574,2103,598],{"className":2104},[628,629],[574,2106,2108],{"className":2107},[728],[574,2109,2111,2131],{"className":2110},[732,733],[574,2112,2114,2128],{"className":2113},[737],[574,2115,2117],{"className":2116,"style":848},[741],[574,2118,2119,2122],{"style":745},[574,2120],{"className":2121,"style":750},[749],[574,2123,2125],{"className":2124},[754,755,756,757],[574,2126,1682],{"className":2127},[628,757],[574,2129,765],{"className":2130},[764],[574,2132,2134],{"className":2133},[737],[574,2135,2137],{"className":2136,"style":772},[741],[574,2138],{},[574,2140],{"className":2141,"style":637},[636],[574,2143,602],{"className":2144},[641],[574,2146],{"className":2147,"style":637},[636],[574,2149,2151,2154],{"className":2150},[619],[574,2152],{"className":2153,"style":1493},[623],[574,2155,2027],{"className":2156},[628],", mit Lösung ",[574,2159,2161,2183],{"className":2160},[577],[574,2162,2164],{"className":2163},[581],[583,2165,2166],{"xmlns":585},[587,2167,2168,2180],{},[590,2169,2170,2172,2174,2176,2178],{},[600,2171,1075],{"stretchy":1074},[793,2173,795],{},[600,2175,798],{"separator":615},[793,2177,1682],{},[600,2179,1084],{"stretchy":1074},[607,2181,2182],{"encoding":609},"(1, 2)",[574,2184,2186],{"className":2185,"ariaHidden":615},[614],[574,2187,2189,2192,2195,2198,2201,2204,2207],{"className":2188},[619],[574,2190],{"className":2191,"style":1141},[623],[574,2193,1075],{"className":2194},[1145],[574,2196,795],{"className":2197},[628],[574,2199,798],{"className":2200},[874],[574,2202],{"className":2203,"style":878},[636],[574,2205,1682],{"className":2206},[628],[574,2208,1084],{"className":2209},[1171],":",[574,2212,2214],{"className":2213},[1058],[574,2215,2217,2327],{"className":2216},[577],[574,2218,2220],{"className":2219},[581],[583,2221,2222],{"xmlns":585,"display":1067},[587,2223,2224,2324],{},[590,2225,2226,2244,2246,2271,2273,2276,2294,2296],{},[1673,2227,2228,2230,2232],{},[593,2229,598],{},[793,2231,795],{},[590,2233,2234,2236,2238,2240,2242],{},[600,2235,1075],{"stretchy":1074},[593,2237,1100],{},[600,2239,972],{},[793,2241,795],{},[600,2243,1084],{"stretchy":1074},[600,2245,602],{},[2247,2248,2249,2269],"mfrac",{},[590,2250,2251,2253,2255],{},[793,2252,1863],{},[600,2254,814],{},[1673,2256,2257,2259,2261],{},[593,2258,598],{},[793,2260,1682],{},[590,2262,2263,2265,2267],{},[600,2264,1075],{"stretchy":1074},[593,2266,1100],{},[600,2268,1084],{"stretchy":1074},[793,2270,1844],{},[600,2272,798],{"separator":615},[636,2274],{"width":2275},"2em",[1673,2277,2278,2280,2282],{},[593,2279,598],{},[793,2281,1682],{},[590,2283,2284,2286,2288,2290,2292],{},[600,2285,1075],{"stretchy":1074},[593,2287,1100],{},[600,2289,972],{},[793,2291,795],{},[600,2293,1084],{"stretchy":1074},[600,2295,602],{},[2247,2297,2298,2322],{},[590,2299,2300,2302,2304],{},[793,2301,2027],{},[600,2303,814],{},[1673,2305,2306,2308,2310],{},[593,2307,598],{},[793,2309,795],{},[590,2311,2312,2314,2316,2318,2320],{},[600,2313,1075],{"stretchy":1074},[593,2315,1100],{},[600,2317,972],{},[793,2319,795],{},[600,2321,1084],{"stretchy":1074},[793,2323,2016],{},[607,2325,2326],{"encoding":609},"x_1^{(k+1)} = \\frac{6 - x_2^{(k)}}{4}, \\qquad x_2^{(k+1)} = \\frac{7 - x_1^{(k+1)}}{3}",[574,2328,2330,2416,2648],{"className":2329,"ariaHidden":615},[614],[574,2331,2333,2337,2407,2410,2413],{"className":2332},[619],[574,2334],{"className":2335,"style":2336},[623],"height:1.3111em;vertical-align:-0.2663em;",[574,2338,2340,2343],{"className":2339},[628],[574,2341,598],{"className":2342},[628,629],[574,2344,2346],{"className":2345},[728],[574,2347,2349,2398],{"className":2348},[732,733],[574,2350,2352,2395],{"className":2351},[737],[574,2353,2356,2368],{"className":2354,"style":2355},[741],"height:1.0448em;",[574,2357,2359,2362],{"style":2358},"top:-2.4337em;margin-left:0em;margin-right:0.05em;",[574,2360],{"className":2361,"style":750},[749],[574,2363,2365],{"className":2364},[754,755,756,757],[574,2366,795],{"className":2367},[628,757],[574,2369,2371,2374],{"style":2370},"top:-3.2198em;margin-right:0.05em;",[574,2372],{"className":2373,"style":750},[749],[574,2375,2377],{"className":2376},[754,755,756,757],[574,2378,2380,2383,2386,2389,2392],{"className":2379},[628,757],[574,2381,1075],{"className":2382},[1145,757],[574,2384,1100],{"className":2385,"style":1212},[628,629,757],[574,2387,972],{"className":2388},[927,757],[574,2390,795],{"className":2391},[628,757],[574,2393,1084],{"className":2394},[1171,757],[574,2396,765],{"className":2397},[764],[574,2399,2401],{"className":2400},[737],[574,2402,2405],{"className":2403,"style":2404},[741],"height:0.2663em;",[574,2406],{},[574,2408],{"className":2409,"style":637},[636],[574,2411,602],{"className":2412},[641],[574,2414],{"className":2415,"style":637},[636],[574,2417,2419,2423,2563,2566,2570,2573,2639,2642,2645],{"className":2418},[619],[574,2420],{"className":2421,"style":2422},[623],"height:2.4078em;vertical-align:-0.686em;",[574,2424,2426,2430,2560],{"className":2425},[628],[574,2427],{"className":2428},[1145,2429],"nulldelimiter",[574,2431,2433],{"className":2432},[2247],[574,2434,2436,2551],{"className":2435},[732,733],[574,2437,2439,2548],{"className":2438},[737],[574,2440,2443,2456,2467],{"className":2441,"style":2442},[741],"height:1.7218em;",[574,2444,2446,2450],{"style":2445},"top:-2.3588em;",[574,2447],{"className":2448,"style":2449},[749],"height:3.0448em;",[574,2451,2453],{"className":2452},[628],[574,2454,1844],{"className":2455},[628],[574,2457,2459,2462],{"style":2458},"top:-3.2748em;",[574,2460],{"className":2461,"style":2449},[749],[574,2463],{"className":2464,"style":2466},[2465],"frac-line","border-bottom-width:0.04em;",[574,2468,2470,2473],{"style":2469},"top:-3.7218em;",[574,2471],{"className":2472,"style":2449},[749],[574,2474,2476,2479,2482,2485,2488],{"className":2475},[628],[574,2477,1863],{"className":2478},[628],[574,2480],{"className":2481,"style":1019},[636],[574,2483,814],{"className":2484},[927],[574,2486],{"className":2487,"style":1019},[636],[574,2489,2491,2494],{"className":2490},[628],[574,2492,598],{"className":2493},[628,629],[574,2495,2497],{"className":2496},[728],[574,2498,2500,2540],{"className":2499},[732,733],[574,2501,2503,2537],{"className":2502},[737],[574,2504,2506,2517],{"className":2505,"style":2355},[741],[574,2507,2508,2511],{"style":2358},[574,2509],{"className":2510,"style":750},[749],[574,2512,2514],{"className":2513},[754,755,756,757],[574,2515,1682],{"className":2516},[628,757],[574,2518,2519,2522],{"style":2370},[574,2520],{"className":2521,"style":750},[749],[574,2523,2525],{"className":2524},[754,755,756,757],[574,2526,2528,2531,2534],{"className":2527},[628,757],[574,2529,1075],{"className":2530},[1145,757],[574,2532,1100],{"className":2533,"style":1212},[628,629,757],[574,2535,1084],{"className":2536},[1171,757],[574,2538,765],{"className":2539},[764],[574,2541,2543],{"className":2542},[737],[574,2544,2546],{"className":2545,"style":2404},[741],[574,2547],{},[574,2549,765],{"className":2550},[764],[574,2552,2554],{"className":2553},[737],[574,2555,2558],{"className":2556,"style":2557},[741],"height:0.686em;",[574,2559],{},[574,2561],{"className":2562},[1171,2429],[574,2564,798],{"className":2565},[874],[574,2567],{"className":2568,"style":2569},[636],"margin-right:2em;",[574,2571],{"className":2572,"style":878},[636],[574,2574,2576,2579],{"className":2575},[628],[574,2577,598],{"className":2578},[628,629],[574,2580,2582],{"className":2581},[728],[574,2583,2585,2631],{"className":2584},[732,733],[574,2586,2588,2628],{"className":2587},[737],[574,2589,2591,2602],{"className":2590,"style":2355},[741],[574,2592,2593,2596],{"style":2358},[574,2594],{"className":2595,"style":750},[749],[574,2597,2599],{"className":2598},[754,755,756,757],[574,2600,1682],{"className":2601},[628,757],[574,2603,2604,2607],{"style":2370},[574,2605],{"className":2606,"style":750},[749],[574,2608,2610],{"className":2609},[754,755,756,757],[574,2611,2613,2616,2619,2622,2625],{"className":2612},[628,757],[574,2614,1075],{"className":2615},[1145,757],[574,2617,1100],{"className":2618,"style":1212},[628,629,757],[574,2620,972],{"className":2621},[927,757],[574,2623,795],{"className":2624},[628,757],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},"1,5",[3160,3166,3167],{},"1,8333",[2921,3169,3170,3172,3175],{},[3160,3171,1682],{},[3160,3173,3174],{},"1,0417",[3160,3176,3177],{},"1,9861",[2921,3179,3180,3182,3185],{},[3160,3181,2016],{},[3160,3183,3184],{},"1,0035",[3160,3186,3187],{},"1,9988",[563,3189,3190,3191,3261,3262,3303,3304,3344,3345,3485,3486,3589,3590,3747],{},"Nach drei Schritten ist der Fehler in ",[574,3192,3194,3212],{"className":3193},[577],[574,3195,3197],{"className":3196},[581],[583,3198,3199],{"xmlns":585},[587,3200,3201,3209],{},[590,3202,3203],{},[700,3204,3205,3207],{},[593,3206,598],{},[793,3208,795],{},[607,3210,3211],{"encoding":609},"x_1",[574,3213,3215],{"className":3214,"ariaHidden":615},[614],[574,3216,3218,3221],{"className":3217},[619],[574,3219],{"className":3220,"style":718},[623],[574,3222,3224,3227],{"className":3223},[628],[574,3225,598],{"className":3226},[628,629],[574,3228,3230],{"className":3229},[728],[574,3231,3233,3253],{"className":3232},[732,733],[574,3234,3236,3250],{"className":3235},[737],[574,3237,3239],{"className":3238,"style":848},[741],[574,3240,3241,3244],{"style":745},[574,3242],{"className":3243,"style":750},[749],[574,3245,3247],{"className":3246},[754,755,756,757],[574,3248,795],{"className":3249},[628,757],[574,3251,765],{"className":3252},[764],[574,3254,3256],{"className":3255},[737],[574,3257,3259],{"className":3258,"style":772},[741],[574,3260],{}," etwa ",[574,3263,3265,3280],{"className":3264},[577],[574,3266,3268],{"className":3267},[581],[583,3269,3270],{"xmlns":585},[587,3271,3272,3277],{},[590,3273,3274],{},[793,3275,3276],{},"0,0035",[607,3278,3279],{"encoding":609},"0{,}0035",[574,3281,3283],{"className":3282,"ariaHidden":615},[614],[574,3284,3286,3290,3293,3299],{"className":3285},[619],[574,3287],{"className":3288,"style":3289},[623],"height:0.8389em;vertical-align:-0.1944em;",[574,3291,1533],{"className":3292},[628],[574,3294,3296],{"className":3295},[628],[574,3297,798],{"className":3298},[874],[574,3300,3302],{"className":3301},[628],"0035",", beim Jacobi-Verfahren noch ",[574,3305,3307,3322],{"className":3306},[577],[574,3308,3310],{"className":3309},[581],[583,3311,3312],{"xmlns":585},[587,3313,3314,3319],{},[590,3315,3316],{},[793,3317,3318],{},"0,0417",[607,3320,3321],{"encoding":609},"0{,}0417",[574,3323,3325],{"className":3324,"ariaHidden":615},[614],[574,3326,3328,3331,3334,3340],{"className":3327},[619],[574,3329],{"className":3330,"style":3289},[623],[574,3332,1533],{"className":3333},[628],[574,3335,3337],{"className":3336},[628],[574,3338,798],{"className":3339},[874],[574,3341,3343],{"className":3342},[628],"0417",". Der Spektralradius ist hier ",[574,3346,3348,3373],{"className":3347},[577],[574,3349,3351],{"className":3350},[581],[583,3352,3353],{"xmlns":585},[587,3354,3355,3370],{},[590,3356,3357,3364,3367],{},[2247,3358,3359,3361],{},[793,3360,795],{},[793,3362,3363],{},"12",[600,3365,3366],{},"≈",[793,3368,3369],{},"0,0833",[607,3371,3372],{"encoding":609},"\\frac{1}{12} \\approx 0{,}0833",[574,3374,3376,3466],{"className":3375,"ariaHidden":615},[614],[574,3377,3379,3383,3457,3460,3463],{"className":3378},[619],[574,3380],{"className":3381,"style":3382},[623],"height:1.1901em;vertical-align:-0.345em;",[574,3384,3386,3389,3454],{"className":3385},[628],[574,3387],{"className":3388},[1145,2429],[574,3390,3392],{"className":3391},[2247],[574,3393,3395,3445],{"className":3394},[732,733],[574,3396,3398,3442],{"className":3397},[737],[574,3399,3402,3418,3427],{"className":3400,"style":3401},[741],"height:0.8451em;",[574,3403,3405,3409],{"style":3404},"top:-2.655em;",[574,3406],{"className":3407,"style":3408},[749],"height:3em;",[574,3410,3412],{"className":3411},[754,755,756,757],[574,3413,3415],{"className":3414},[628,757],[574,3416,3363],{"className":3417},[628,757],[574,3419,3421,3424],{"style":3420},"top:-3.23em;",[574,3422],{"className":3423,"style":3408},[749],[574,3425],{"className":3426,"style":2466},[2465],[574,3428,3430,3433],{"style":3429},"top:-3.394em;",[574,3431],{"className":3432,"style":3408},[749],[574,3434,3436],{"className":3435},[754,755,756,757],[574,3437,3439],{"className":3438},[628,757],[574,3440,795],{"className":3441},[628,757],[574,3443,765],{"className":3444},[764],[574,3446,3448],{"className":3447},[737],[574,3449,3452],{"className":3450,"style":3451},[741],"height:0.345em;",[574,3453],{},[574,3455],{"className":3456},[1171,2429],[574,3458],{"className":3459,"style":637},[636],[574,3461,3366],{"className":3462},[641],[574,3464],{"className":3465,"style":637},[636],[574,3467,3469,3472,3475,3481],{"className":3468},[619],[574,3470],{"className":3471,"style":3289},[623],[574,3473,1533],{"className":3474},[628],[574,3476,3478],{"className":3477},[628],[574,3479,798],{"className":3480},[874],[574,3482,3484],{"className":3483},[628],"0833",", das Quadrat des Jacobi-Wertes ",[574,3487,3489,3513],{"className":3488},[577],[574,3490,3492],{"className":3491},[581],[583,3493,3494],{"xmlns":585},[587,3495,3496,3510],{},[590,3497,3498],{},[3499,3500,3501],"msqrt",{},[590,3502,3503,3505,3508],{},[793,3504,795],{},[593,3506,3507],{"mathvariant":1529},"\u002F",[793,3509,3363],{},[607,3511,3512],{"encoding":609},"\\sqrt{1\u002F12}",[574,3514,3516],{"className":3515,"ariaHidden":615},[614],[574,3517,3519,3523],{"className":3518},[619],[574,3520],{"className":3521,"style":3522},[623],"height:1.24em;vertical-align:-0.305em;",[574,3524,3527],{"className":3525},[628,3526],"sqrt",[574,3528,3530,3580],{"className":3529},[732,733],[574,3531,3533,3577],{"className":3532},[737],[574,3534,3537,3554],{"className":3535,"style":3536},[741],"height:0.935em;",[574,3538,3542,3546],{"className":3539,"style":3541},[3540],"svg-align","top:-3.2em;",[574,3543],{"className":3544,"style":3545},[749],"height:3.2em;",[574,3547,3550],{"className":3548,"style":3549},[628],"padding-left:1em;",[574,3551,3553],{"className":3552},[628],"1\u002F12",[574,3555,3557,3560],{"style":3556},"top:-2.895em;",[574,3558],{"className":3559,"style":3545},[749],[574,3561,3565],{"className":3562,"style":3564},[3563],"hide-tail","min-width:1.02em;height:1.28em;",[3566,3567,3573],"svg",{"xmlns":3568,"width":3569,"height":3570,"viewBox":3571,"preserveAspectRatio":3572},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","1.28em","0 0 400000 1296","xMinYMin slice",[3574,3575],"path",{"d":3576},"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z",[574,3578,765],{"className":3579},[764],[574,3581,3583],{"className":3582},[737],[574,3584,3587],{"className":3585,"style":3586},[741],"height:0.305em;",[574,3588],{},". Die Matrix ist tridiagonal, die Beziehung ",[574,3591,3593,3620],{"className":3592},[577],[574,3594,3596],{"className":3595},[581],[583,3597,3598],{"xmlns":585},[587,3599,3600,3618],{},[590,3601,3602,3608,3610],{},[700,3603,3604,3606],{},[593,3605,1666],{},[1086,3607,1669],{},[600,3609,602],{},[1673,3611,3612,3614,3616],{},[593,3613,1666],{},[1086,3615,1679],{},[793,3617,1682],{},[607,3619,1685],{"encoding":609},[574,3621,3623,3684],{"className":3622,"ariaHidden":615},[614],[574,3624,3626,3629,3675,3678,3681],{"className":3625},[619],[574,3627],{"className":3628,"style":1695},[623],[574,3630,3632,3635],{"className":3631},[628],[574,3633,1666],{"className":3634},[628,629],[574,3636,3638],{"className":3637},[728],[574,3639,3641,3667],{"className":3640},[732,733],[574,3642,3644,3664],{"className":3643},[737],[574,3645,3647],{"className":3646,"style":1714},[741],[574,3648,3649,3652],{"style":745},[574,3650],{"className":3651,"style":750},[749],[574,3653,3655],{"className":3654},[754,755,756,757],[574,3656,3658],{"className":3657},[628,757],[574,3659,3661],{"className":3660},[628,1729,757],[574,3662,1669],{"className":3663},[628,757],[574,3665,765],{"className":3666},[764],[574,3668,3670],{"className":3669},[737],[574,3671,3673],{"className":3672,"style":772},[741],[574,3674],{},[574,3676],{"className":3677,"style":637},[636],[574,3679,602],{"className":3680},[641],[574,3682],{"className":3683,"style":637},[636],[574,3685,3687,3690],{"className":3686},[619],[574,3688],{"className":3689,"style":1759},[623],[574,3691,3693,3696],{"className":3692},[628],[574,3694,1666],{"className":3695},[628,629],[574,3697,3699],{"className":3698},[728],[574,3700,3702,3739],{"className":3701},[732,733],[574,3703,3705,3736],{"className":3704},[737],[574,3706,3708,3725],{"className":3707,"style":1445},[741],[574,3709,3710,3713],{"style":1780},[574,3711],{"className":3712,"style":750},[749],[574,3714,3716],{"className":3715},[754,755,756,757],[574,3717,3719],{"className":3718},[628,757],[574,3720,3722],{"className":3721},[628,1729,757],[574,3723,1679],{"className":3724},[628,757],[574,3726,3727,3730],{"style":1448},[574,3728],{"className":3729,"style":750},[749],[574,3731,3733],{"className":3732},[754,755,756,757],[574,3734,1682],{"className":3735},[628,757],[574,3737,765],{"className":3738},[764],[574,3740,3742],{"className":3741},[737],[574,3743,3745],{"className":3744,"style":1816},[741],[574,3746],{}," gilt also.",[558,3749,3751],{"id":3750},"typische-fehler","Typische Fehler",[1307,3753,3754,3757,3760,3763],{},[1310,3755,3756],{},"Doch die alten Werte verwenden und damit versehentlich das Jacobi-Verfahren rechnen.",[1310,3758,3759],{},"Die Reihenfolge der Gleichungen ändern, ohne zu prüfen, ob die Diagonaleinträge dann noch passen. Ein Zeilentausch kann Diagonaldominanz herstellen oder zerstören.",[1310,3761,3762],{},"Annehmen, Gauß-Seidel konvergiere immer, wenn Jacobi konvergiert. Das gilt für viele, aber nicht für alle Matrizen.",[1310,3764,3765,3766,3865],{},"Bei ",[574,3767,3769,3795],{"className":3768},[577],[574,3770,3772],{"className":3771},[581],[583,3773,3774],{"xmlns":585},[587,3775,3776,3792],{},[590,3777,3778,3788,3790],{},[700,3779,3780,3782],{},[593,3781,1520],{},[590,3783,3784,3786],{},[593,3785,670],{},[593,3787,670],{},[600,3789,602],{},[793,3791,1533],{},[607,3793,3794],{"encoding":609},"a_{ii} = 0",[574,3796,3798,3856],{"className":3797,"ariaHidden":615},[614],[574,3799,3801,3804,3847,3850,3853],{"className":3800},[619],[574,3802],{"className":3803,"style":718},[623],[574,3805,3807,3810],{"className":3806},[628],[574,3808,1520],{"className":3809},[628,629],[574,3811,3813],{"className":3812},[728],[574,3814,3816,3839],{"className":3815},[732,733],[574,3817,3819,3836],{"className":3818},[737],[574,3820,3822],{"className":3821,"style":742},[741],[574,3823,3824,3827],{"style":745},[574,3825],{"className":3826,"style":750},[749],[574,3828,3830],{"className":3829},[754,755,756,757],[574,3831,3833],{"className":3832},[628,757],[574,3834,1579],{"className":3835},[628,629,757],[574,3837,765],{"className":3838},[764],[574,3840,3842],{"className":3841},[737],[574,3843,3845],{"className":3844,"style":772},[741],[574,3846],{},[574,3848],{"className":3849,"style":637},[636],[574,3851,602],{"className":3852},[641],[574,3854],{"className":3855,"style":637},[636],[574,3857,3859,3862],{"className":3858},[619],[574,3860],{"className":3861,"style":1493},[623],[574,3863,1533],{"className":3864},[628]," rechnen. Dann ist die Vorschrift nicht definiert.",{"title":3867,"searchDepth":3868,"depth":3868,"links":3869},"",2,[3870,3871,3872],{"id":560,"depth":3868,"text":561},{"id":1825,"depth":3868,"text":1826},{"id":3750,"depth":3868,"text":3751},[3874,3876,3878,3881,3884,3887,3890,3893,3896,3899,3902,3905,3908,3911,3913,3916],{"lhs":3875,"rhs":795},"(6 - 2)\u002F4",{"lhs":3877,"rhs":1682},"(7 - 1)\u002F3",{"lhs":3879,"rhs":3880},"(6 - 0)\u002F4","1.5",{"lhs":3882,"rhs":3883},"(7 - 1.5)\u002F3","11\u002F6",{"lhs":3885,"rhs":3886},"round(11\u002F6, 4)","1.8333",{"lhs":3888,"rhs":3889},"(6 - 11\u002F6)\u002F4","25\u002F24",{"lhs":3891,"rhs":3892},"round(25\u002F24, 4)","1.0417",{"lhs":3894,"rhs":3895},"(7 - 25\u002F24)\u002F3","143\u002F72",{"lhs":3897,"rhs":3898},"round(143\u002F72, 4)","1.9861",{"lhs":3900,"rhs":3901},"(6 - 143\u002F72)\u002F4","289\u002F288",{"lhs":3903,"rhs":3904},"round(289\u002F288, 4)","1.0035",{"lhs":3906,"rhs":3907},"(7 - 289\u002F288)\u002F3","1727\u002F864",{"lhs":3909,"rhs":3910},"round(1727\u002F864, 4)","1.9988",{"lhs":3912,"rhs":3553},"-multiply(inv([[4, 0], [1, 3]]), [[0, 1], [0, 0]])[2, 2]",{"lhs":3914,"rhs":3915},"-multiply(inv([[4, 0], [1, 3]]), [[0, 1], [0, 0]])[1, 2]","-1\u002F4",{"lhs":3917,"rhs":3918},"round(1\u002F12, 4)","0.0833","A x = b,\\ a_{ii} \\neq 0,\\ i = 1, \\ldots, n","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>A\u003C\u002Fmi>\u003Cmi>x\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>b\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">A x = b,\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>a\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo mathvariant=\"normal\">≠\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">a_{ii} \\neq 0,\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">ii\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">\u003Cspan class=\"mrel\">\u003Cspan class=\"mord katex-vbox\">\u003Cspan class=\"katex-thinbox\">\u003Cspan class=\"rlap\">\u003Cspan class=\"katex-strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-inner\">\u003Cspan class=\"mord\">\u003Cspan class=\"mrel\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-fix\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace nobreak\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmo>…\u003C\u002Fmo>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">i = 1, \\ldots, n\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6595em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">i\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"minner\">…\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",null,"md","gauss-seidel-method","x_i^{(k+1)} = \\frac{1}{a_{ii}} \\Bigl( b_i - \\sum_{j \u003C i} a_{ij}\\,x_j^{(k+1)} - \\sum_{j > i} a_{ij}\\,x_j^{(k)} \\Bigr)","\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmsubsup>\u003Cmi>x\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>k\u003C\u002Fmi>\u003Cmo>+\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmsubsup>\u003Cmo>=\u003C\u002Fmo>\u003Cmfrac>\u003Cmn>1\u003C\u002Fmn>\u003Cmsub>\u003Cmi>a\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmfrac>\u003Cmo fence=\"true\" stretchy=\"true\" 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stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmsubsup>\u003Cmo>−\u003C\u002Fmo>\u003Cmunder>\u003Cmo>∑\u003C\u002Fmo>\u003Cmrow>\u003Cmi>j\u003C\u002Fmi>\u003Cmo>>\u003C\u002Fmo>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmunder>\u003Cmsub>\u003Cmi>a\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>j\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmtext> \u003C\u002Fmtext>\u003Cmsubsup>\u003Cmi>x\u003C\u002Fmi>\u003Cmi>j\u003C\u002Fmi>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>k\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmsubsup>\u003Cmo fence=\"true\" stretchy=\"true\" minsize=\"1.8em\" maxsize=\"1.8em\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">x_i^{(k+1)} = \\frac{1}{a_{ii}} \\Bigl( b_i - \\sum_{j < i} a_{ij}\\,x_j^{(k+1)} - \\sum_{j > i} a_{ij}\\,x_j^{(k)} 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