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powGenerator",[943,1132,1012],{"class":953},[943,1134,1005],{"class":986},[943,1136,1137],{"class":1018}," n",[943,1139,1038],{"class":953},[943,1141,1041],{"class":953},[943,1143,1145,1147,1150,1153,1155,1157,1159,1162,1165,1168,1171,1174,1176,1178,1180,1182,1185],{"class":945,"line":1144},11,[943,1146,1047],{"class":949},[943,1148,1149],{"class":953}," [=]",[943,1151,1152],{"class":953}," (",[943,1154,1015],{"class":986},[943,1156,1019],{"class":1018},[943,1158,1038],{"class":953},[943,1160,1161],{"class":953}," {",[943,1163,1164],{"class":949}," return",[943,1166,1167],{"class":996}," std",[943,1169,1170],{"class":953},"::",[943,1172,1173],{"class":1008},"pow",[943,1175,1012],{"class":953},[943,1177,1074],{"class":1050},[943,1179,1022],{"class":953},[943,1181,1137],{"class":1050},[943,1183,1184],{"class":953},");",[943,1186,1187],{"class":953}," };\n",[943,1189,1191],{"class":945,"line":1190},12,[943,1192,1109],{"class":953},[943,1194,1196],{"class":945,"line":1195},13,[943,1197,980],{"emptyLinePlaceholder":979},[943,1199,1201,1203,1206,1209],{"class":945,"line":1200},14,[943,1202,1015],{"class":986},[943,1204,1205],{"class":1008}," main",[943,1207,1208],{"class":953},"()",[943,1210,1041],{"class":953},[943,1212,1214,1217,1219],{"class":945,"line":1213},15,[943,1215,1216],{"class":986},"    int",[943,1218,1137],{"class":1050},[943,1220,1221],{"class":953},";\n",[943,1223,1225,1228,1230,1233,1236,1238],{"class":945,"line":1224},16,[943,1226,1227],{"class":996},"    std",[943,1229,1170],{"class":953},[943,1231,1232],{"class":1050},"cin ",[943,1234,1235],{"class":953},">>",[943,1237,1137],{"class":1050},[943,1239,1221],{"class":953},[943,1241,1243,1245,1247,1250,1253,1255,1257,1260,1262,1265,1267,1269,1271,1274,1277,1280,1282,1284,1287],{"class":945,"line":1242},17,[943,1244,1227],{"class":996},[943,1246,1170],{"class":953},[943,1248,1249],{"class":1050},"cout ",[943,1251,1252],{"class":953},"\u003C\u003C",[943,1254,1009],{"class":1008},[943,1256,1012],{"class":953},[943,1258,1259],{"class":1063},"1",[943,1261,1022],{"class":953},[943,1263,1264],{"class":1063}," 5",[943,1266,1022],{"class":953},[943,1268,1130],{"class":1008},[943,1270,1012],{"class":953},[943,1272,1273],{"class":1050},"n",[943,1275,1276],{"class":953},"))",[943,1278,1279],{"class":953}," \u003C\u003C",[943,1281,1167],{"class":996},[943,1283,1170],{"class":953},[943,1285,1286],{"class":1050},"endl",[943,1288,1221],{"class":953},[943,1290,1292],{"class":945,"line":1291},18,[943,1293,1109],{"class":953},[918,1295,1296,1297,1300,1301,1303,1304,1307,1308,1311],{},"我来仔细解读以下这个代码。它的核心是使用了 ",[925,1298,1299],{},"powGenerator"," 这个函数。它会根据传入的参数 ",[925,1302,1273],{}," 返回对应的 Lambda 表达式 ",[925,1305,1306],{},"[=] (int a) { return std::pow(a, n); }","，而这正是上一种解法中的 Lambda 表达式。所以，只要通过 ",[925,1309,1310],{},"powGenerator(n)"," 的调用，就能获取到我们想要的 Lambda 表达式。",[918,1313,1314,1315,928,1317,1319],{},"语法上注意一点：由于 Lambda 表达式的类型是匿名的，所以我们不得不用 ",[925,1316,1127],{},[925,1318,1299],{}," 的返回值类型。如果你的编译器不支持这个语法，那这里只能用模板类型了。",[918,1321,1322,1323,1325,1326,1328],{},"尽管这种写法代码稍微多了一点，但如果 ",[925,1324,1306],{}," 这个东西在整个代码里多次出现的话，那 ",[925,1327,1299],{}," 就会减少这些重复，方便代码调试和重构。",[918,1330,1331],{},"像这种，返回函数的函数有一个专业的术语——“高阶函数”（Higher order function）。它其实在数学里很常见。比如函数的复合运算 $\\circ$ 就是一个高阶函数。函数的复合运算是一个二元运算，它接受两个函数 $f$ 和 $g$，$f\\circ g$ 生成一个新的函数，作为复合运算的结果。$f\\circ g$ 的定义是，$(f\\circ g)(x)=f(g(x))$。",[918,1333,1334,1335,1338],{},"可以在 C++ 中实现简单的复合运算。下面的函数 ",[925,1336,1337],{},"comb"," 就是这样的：",[934,1340,1342],{"className":936,"code":1341,"language":938,"meta":939,"style":939},"#include \u003Ciostream>\n\ntemplate\u003Ctypename F, typename G>\nauto comb(F f, G g) {\n    return [=](\u002Fint x) { return f(g(x)); };\n}\n\nint f(int x) { return x + 1; }\nint g(int x) { return x * 2; }\nint main() {\n    std::cout \u003C\u003C comb(f, g)(3) \u003C\u003C std::endl;\n    std::cout \u003C\u003C comb(g, f)(3) \u003C\u003C std::endl;\n}\n",[925,1343,1344,1354,1358,1378,1404,1442,1446,1450,1481,1511,1521,1560,1596],{"__ignoreMap":939},[943,1345,1346,1348,1350,1352],{"class":945,"line":946},[943,1347,950],{"class":949},[943,1349,954],{"class":953},[943,1351,958],{"class":957},[943,1353,961],{"class":953},[943,1355,1356],{"class":945,"line":964},[943,1357,980],{"emptyLinePlaceholder":979},[943,1359,1360,1362,1364,1366,1368,1370,1373,1376],{"class":945,"line":976},[943,1361,987],{"class":986},[943,1363,990],{"class":953},[943,1365,993],{"class":986},[943,1367,997],{"class":996},[943,1369,1022],{"class":953},[943,1371,1372],{"class":986}," typename",[943,1374,1375],{"class":996}," G",[943,1377,961],{"class":953},[943,1379,1380,1382,1385,1387,1390,1393,1395,1397,1400,1402],{"class":945,"line":983},[943,1381,1127],{"class":986},[943,1383,1384],{"class":1008}," comb",[943,1386,1012],{"class":953},[943,1388,1389],{"class":996},"F",[943,1391,1392],{"class":1018}," f",[943,1394,1022],{"class":953},[943,1396,1375],{"class":996},[943,1398,1399],{"class":1018}," g",[943,1401,1038],{"class":953},[943,1403,1041],{"class":953},[943,1405,1406,1408,1411,1414,1416,1419,1421,1423,1425,1427,1429,1432,1434,1437,1440],{"class":945,"line":1002},[943,1407,1047],{"class":949},[943,1409,1410],{"class":953}," [=](",[943,1412,1413],{"class":1050},"\u002F",[943,1415,1015],{"class":986},[943,1417,1418],{"class":1018}," x",[943,1420,1038],{"class":953},[943,1422,1161],{"class":953},[943,1424,1164],{"class":949},[943,1426,1392],{"class":1008},[943,1428,1012],{"class":953},[943,1430,1431],{"class":1008},"g",[943,1433,1012],{"class":953},[943,1435,1436],{"class":1050},"x",[943,1438,1439],{"class":953},"));",[943,1441,1187],{"class":953},[943,1443,1444],{"class":945,"line":1044},[943,1445,1109],{"class":953},[943,1447,1448],{"class":945,"line":1106},[943,1449,980],{"emptyLinePlaceholder":979},[943,1451,1452,1454,1456,1458,1460,1462,1464,1466,1468,1471,1473,1475,1478],{"class":945,"line":1112},[943,1453,1015],{"class":986},[943,1455,1392],{"class":1008},[943,1457,1012],{"class":953},[943,1459,1015],{"class":986},[943,1461,1418],{"class":1018},[943,1463,1038],{"class":953},[943,1465,1161],{"class":953},[943,1467,1164],{"class":949},[943,1469,1470],{"class":1050}," x ",[943,1472,1089],{"class":953},[943,1474,1092],{"class":1063},[943,1476,1477],{"class":953},";",[943,1479,1480],{"class":953}," }\n",[943,1482,1483,1485,1487,1489,1491,1493,1495,1497,1499,1501,1504,1507,1509],{"class":945,"line":1117},[943,1484,1015],{"class":986},[943,1486,1399],{"class":1008},[943,1488,1012],{"class":953},[943,1490,1015],{"class":986},[943,1492,1418],{"class":1018},[943,1494,1038],{"class":953},[943,1496,1161],{"class":953},[943,1498,1164],{"class":949},[943,1500,1470],{"class":1050},[943,1502,1503],{"class":953},"*",[943,1505,1506],{"class":1063}," 2",[943,1508,1477],{"class":953},[943,1510,1480],{"class":953},[943,1512,1513,1515,1517,1519],{"class":945,"line":1124},[943,1514,1015],{"class":986},[943,1516,1205],{"class":1008},[943,1518,1208],{"class":953},[943,1520,1041],{"class":953},[943,1522,1523,1525,1527,1529,1531,1533,1535,1538,1540,1542,1545,1548,1550,1552,1554,1556,1558],{"class":945,"line":1144},[943,1524,1227],{"class":996},[943,1526,1170],{"class":953},[943,1528,1249],{"class":1050},[943,1530,1252],{"class":953},[943,1532,1384],{"class":1008},[943,1534,1012],{"class":953},[943,1536,1537],{"class":1050},"f",[943,1539,1022],{"class":953},[943,1541,1399],{"class":1050},[943,1543,1544],{"class":953},")(",[943,1546,1547],{"class":1063},"3",[943,1549,1038],{"class":953},[943,1551,1279],{"class":953},[943,1553,1167],{"class":996},[943,1555,1170],{"class":953},[943,1557,1286],{"class":1050},[943,1559,1221],{"class":953},[943,1561,1562,1564,1566,1568,1570,1572,1574,1576,1578,1580,1582,1584,1586,1588,1590,1592,1594],{"class":945,"line":1190},[943,1563,1227],{"class":996},[943,1565,1170],{"class":953},[943,1567,1249],{"class":1050},[943,1569,1252],{"class":953},[943,1571,1384],{"class":1008},[943,1573,1012],{"class":953},[943,1575,1431],{"class":1050},[943,1577,1022],{"class":953},[943,1579,1392],{"class":1050},[943,1581,1544],{"class":953},[943,1583,1547],{"class":1063},[943,1585,1038],{"class":953},[943,1587,1279],{"class":953},[943,1589,1167],{"class":996},[943,1591,1170],{"class":953},[943,1593,1286],{"class":1050},[943,1595,1221],{"class":953},[943,1597,1598],{"class":945,"line":1195},[943,1599,1109],{"class":953},[918,1601,1602],{},"这份代码中，给出了函数 $f$ 和 $g$ 的定义，然后计算了 $(f\\circ g)(3)$ 和 $(g\\circ f)(3)$ 的值。运行程序，发现它们不相等，证实了 $\\circ$ 不满足交换律。",[918,1604,1605],{},"但 $\\circ$ 是满足结合律的；下面的代码尝试去验证这个事实：",[934,1607,1609],{"className":936,"code":1608,"language":938,"meta":939,"style":939},"#include \u003Ciostream>\n\ntemplate\u003Ctypename F, typename G>\nauto operator*(F f, G g) {\n    return [=](\u002Fint x) { return f(g(x)); };\n}\n\nauto f = [](\u002Fint x) { return x + 1; };\nauto g = [](\u002Fint x) { return x * 2; };\nauto h = [](\u002Fint x) { return x * x; };\nint main() {\n    std::cout \u003C\u003C ((f * g) * h)(3) \u003C\u003C std::endl;\n    std::cout \u003C\u003C (f * (g * h))(3) \u003C\u003C std::endl;\n}\n",[925,1610,1611,1621,1625,1643,1668,1700,1704,1708,1743,1776,1809,1819,1863,1905],{"__ignoreMap":939},[943,1612,1613,1615,1617,1619],{"class":945,"line":946},[943,1614,950],{"class":949},[943,1616,954],{"class":953},[943,1618,958],{"class":957},[943,1620,961],{"class":953},[943,1622,1623],{"class":945,"line":964},[943,1624,980],{"emptyLinePlaceholder":979},[943,1626,1627,1629,1631,1633,1635,1637,1639,1641],{"class":945,"line":976},[943,1628,987],{"class":986},[943,1630,990],{"class":953},[943,1632,993],{"class":986},[943,1634,997],{"class":996},[943,1636,1022],{"class":953},[943,1638,1372],{"class":986},[943,1640,1375],{"class":996},[943,1642,961],{"class":953},[943,1644,1645,1647,1650,1652,1654,1656,1658,1660,1662,1664,1666],{"class":945,"line":983},[943,1646,1127],{"class":986},[943,1648,1649],{"class":1063}," operator",[943,1651,1503],{"class":996},[943,1653,1012],{"class":953},[943,1655,1389],{"class":996},[943,1657,1392],{"class":1018},[943,1659,1022],{"class":953},[943,1661,1375],{"class":996},[943,1663,1399],{"class":1018},[943,1665,1038],{"class":953},[943,1667,1041],{"class":953},[943,1669,1670,1672,1674,1676,1678,1680,1682,1684,1686,1688,1690,1692,1694,1696,1698],{"class":945,"line":1002},[943,1671,1047],{"class":949},[943,1673,1410],{"class":953},[943,1675,1413],{"class":1050},[943,1677,1015],{"class":986},[943,1679,1418],{"class":1018},[943,1681,1038],{"class":953},[943,1683,1161],{"class":953},[943,1685,1164],{"class":949},[943,1687,1392],{"class":1008},[943,1689,1012],{"class":953},[943,1691,1431],{"class":1008},[943,1693,1012],{"class":953},[943,1695,1436],{"class":1050},[943,1697,1439],{"class":953},[943,1699,1187],{"class":953},[943,1701,1702],{"class":945,"line":1044},[943,1703,1109],{"class":953},[943,1705,1706],{"class":945,"line":1106},[943,1707,980],{"emptyLinePlaceholder":979},[943,1709,1710,1712,1715,1718,1721,1723,1725,1727,1729,1731,1733,1735,1737,1739,1741],{"class":945,"line":1112},[943,1711,1127],{"class":986},[943,1713,1714],{"class":1050}," f ",[943,1716,1717],{"class":953},"=",[943,1719,1720],{"class":953}," [](",[943,1722,1413],{"class":1050},[943,1724,1015],{"class":986},[943,1726,1418],{"class":1018},[943,1728,1038],{"class":953},[943,1730,1161],{"class":953},[943,1732,1164],{"class":949},[943,1734,1470],{"class":1050},[943,1736,1089],{"class":953},[943,1738,1092],{"class":1063},[943,1740,1477],{"class":953},[943,1742,1187],{"class":953},[943,1744,1745,1747,1750,1752,1754,1756,1758,1760,1762,1764,1766,1768,1770,1772,1774],{"class":945,"line":1117},[943,1746,1127],{"class":986},[943,1748,1749],{"class":1050}," g ",[943,1751,1717],{"class":953},[943,1753,1720],{"class":953},[943,1755,1413],{"class":1050},[943,1757,1015],{"class":986},[943,1759,1418],{"class":1018},[943,1761,1038],{"class":953},[943,1763,1161],{"class":953},[943,1765,1164],{"class":949},[943,1767,1470],{"class":1050},[943,1769,1503],{"class":953},[943,1771,1506],{"class":1063},[943,1773,1477],{"class":953},[943,1775,1187],{"class":953},[943,1777,1778,1780,1783,1785,1787,1789,1791,1793,1795,1797,1799,1801,1803,1805,1807],{"class":945,"line":1124},[943,1779,1127],{"class":986},[943,1781,1782],{"class":1050}," h 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",[943,1836,1503],{"class":953},[943,1838,1399],{"class":1050},[943,1840,1038],{"class":953},[943,1842,1843],{"class":953}," *",[943,1845,1846],{"class":1050}," h",[943,1848,1544],{"class":953},[943,1850,1547],{"class":1063},[943,1852,1038],{"class":953},[943,1854,1279],{"class":953},[943,1856,1167],{"class":996},[943,1858,1170],{"class":953},[943,1860,1286],{"class":1050},[943,1862,1221],{"class":953},[943,1864,1865,1867,1869,1871,1873,1875,1877,1879,1881,1884,1886,1888,1891,1893,1895,1897,1899,1901,1903],{"class":945,"line":1195},[943,1866,1227],{"class":996},[943,1868,1170],{"class":953},[943,1870,1249],{"class":1050},[943,1872,1252],{"class":953},[943,1874,1152],{"class":953},[943,1876,1834],{"class":1050},[943,1878,1503],{"class":953},[943,1880,1152],{"class":953},[943,1882,1883],{"class":1050},"g ",[943,1885,1503],{"class":953},[943,1887,1846],{"class":1050},[943,1889,1890],{"class":953},"))(",[943,1892,1547],{"class":1063},[943,1894,1038],{"class":953},[943,1896,1279],{"class":953},[943,1898,1167],{"class":996},[943,1900,1170],{"class":953},[943,1902,1286],{"class":1050},[943,1904,1221],{"class":953},[943,1906,1907],{"class":945,"line":1200},[943,1908,1109],{"class":953},[918,1910,1911,1912,1915,1916,1919,1920,1922,1923,1926,1927,1930],{},"我在这里玩了个花样：将 ",[925,1913,1914],{},"comp"," 改名为 ",[925,1917,1918],{},"operator*","，也就是重载二元的 ",[925,1921,1503],{}," 运算符。这样，原先 ",[925,1924,1925],{},"comp(f, g)"," 就可以写成 ",[925,1928,1929],{},"f * g"," 这种更易读的形式。",[1932,1933,1934],"blockquote",{},[918,1935,1936,1937,1939,1940,1939,1942,1945,1946,1950],{},"但为了让重载生效，我需要将 ",[925,1938,1537],{}," ",[925,1941,1431],{},[925,1943,1944],{},"h"," 改成 Lambda 表达式而非函数。因为运算符重载只在类类型之间生效（回忆",[1074,1947,1949],{"href":1948},"\u002Fch06\u002Fnonmember_operator#%E9%9D%9E%E6%88%90%E5%91%98%E7%9A%84%E8%BF%90%E7%AE%97%E7%AC%A6%E9%87%8D%E8%BD%BD%E5%BD%A2%E5%BC%8F","非成员的运算符","这一节，运算符重载总要求至少一个参数是类类型），故函数指针之间的运算不会考虑自定义的重载。而 Lambda 表达式作为匿名类类型，会考虑这些重载。",[918,1952,1953],{},"还有一个常见的高阶函数是微分算子 $\\dfrac{\\mathrm d}{\\mathrm dx}$。$\\dfrac{\\mathrm d}{\\mathrm dx}$ 可作用在一个函数 $f$ 上，定义 $\\displaystyle\\frac{\\mathrm d}{\\mathrm dx}f(x)=f'(x)=\\lim_{\\Delta x\\to0}\\frac{f(x+\\Delta x) - f(x)}{\\Delta x}$。大家都知道的，称 $\\dfrac{\\mathrm d}{\\mathrm dx}f$ 这个运算结果为 $f$ 的导函数。",[918,1955,1956,1957,1960,1961,1964],{},"在代码中很难模拟极限过程，但如果取这个 $\\Delta x$ 为比较小的值，那或许能得到近似的 $\\dfrac{\\mathrm d}{\\mathrm dx}f$ 的结果。下面的函数 ",[925,1958,1959],{},"d"," 实现了近似的微分算子 $\\dfrac{\\mathrm d}{\\mathrm dx}$。（其中，取 $\\Delta x=10^{-6}$，即 ",[925,1962,1963],{},"DX","。）",[934,1966,1968],{"className":936,"code":1967,"language":938,"meta":939,"style":939},"#include \u003Ciostream>\n\nconstexpr double DX{1e-6};\n\ntemplate\u003Ctypename F>\nauto d(F f) {\n    return [=](\u002Fdouble x) {\n        return (f(x + DX) - f(x)) \u002F DX;\n    };\n}\n\nauto f = [](\u002Fdouble x) { return x * x; };\nint main() {\n    std::cout \u003C\u003C f(5) \u003C\u003C std::endl;\n    std::cout \u003C\u003C d(f)(5) \u003C\u003C std::endl;\n}\n",[925,1969,1970,1980,1984,2010,2014,2026,2043,2059,2097,2102,2106,2110,2142,2152,2181,2213],{"__ignoreMap":939},[943,1971,1972,1974,1976,1978],{"class":945,"line":946},[943,1973,950],{"class":949},[943,1975,954],{"class":953},[943,1977,958],{"class":957},[943,1979,961],{"class":953},[943,1981,1982],{"class":945,"line":964},[943,1983,980],{"emptyLinePlaceholder":979},[943,1985,1986,1989,1992,1995,1998,2001,2004,2007],{"class":945,"line":976},[943,1987,1988],{"class":986},"constexpr",[943,1990,1991],{"class":986}," double",[943,1993,1994],{"class":1050}," 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$f(x)=x^2$ 时，数学上即可知道 $\\dfrac{\\mathrm d}{\\mathrm dx}f(x)=2x$。上面的代码验证了这个结果。可以见得，函数式编程和数学的关系非常紧密。因此数学家往往偏爱函数式编程；数学中的大量理论（比如抽象代数）也在函数式编程中得以应用。",[918,2221,2222],{},"我们现在已经熟悉了函数式编程的一些基本语法。接下来，我将介绍 C++ 标准库的另一重要组件——STL 算法。在使用 STL 算法的过程中，程序员总是或多或少地会用到一些函数式思想，因此我将它放到这里来介绍。",[1932,2224,2225],{},[918,2226,2227,2228,2232,2233,2236,2237,2239,2240,2242,2243,2245,2246,2248,2249,2251,2252,2254],{},"本节中的所有 Lambda 表达式都使用复制捕获。这是因为，当引用捕获的 Lambda 出现在返回值时，相当于返回了",[1074,2229,2231],{"href":2230},"\u002Fch05\u002Freference#%E6%82%AC%E5%9E%82%E5%BC%95%E7%94%A8","悬垂引用","；这是显然错误的。比如，若 ",[925,2234,2235],{},"[&] (int a) { return std::pow(a, n); }"," 引用捕获了 ",[925,2238,1299],{}," 的形参 ",[925,2241,1273],{},"，那等到这个 Lambda 表达式被调用之时（即在 ",[925,2244,931],{}," 函数执行时），",[925,2247,1273],{}," 所在的 ",[925,2250,1299],{}," 函数已经释放，",[925,2253,1273],{}," 已经不存在了。",[2256,2257,2258],"style",{},"html pre.shiki code .s7zQu, html code.shiki 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