/usr/include/specialfunctions.h is in libalglib-dev 3.8.2-1.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 1377 1378 1379 1380 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 1423 1424 1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456 1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488 1489 1490 1491 1492 1493 1494 1495 1496 1497 1498 1499 1500 1501 1502 1503 1504 1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 1521 1522 1523 1524 1525 1526 1527 1528 1529 1530 1531 1532 1533 1534 1535 1536 1537 1538 1539 1540 1541 1542 1543 1544 1545 1546 1547 1548 1549 1550 1551 1552 1553 1554 1555 1556 1557 1558 1559 1560 1561 1562 1563 1564 1565 1566 1567 1568 1569 1570 1571 1572 1573 1574 1575 1576 1577 1578 1579 1580 1581 1582 1583 1584 1585 1586 1587 1588 1589 1590 1591 1592 1593 1594 1595 1596 1597 1598 1599 1600 1601 1602 1603 1604 1605 1606 1607 1608 1609 1610 1611 1612 1613 1614 1615 1616 1617 1618 1619 1620 1621 1622 1623 1624 1625 1626 1627 1628 1629 1630 1631 1632 1633 1634 1635 1636 1637 1638 1639 1640 1641 1642 1643 1644 1645 1646 1647 1648 1649 1650 1651 1652 1653 1654 1655 1656 1657 1658 1659 1660 1661 1662 1663 1664 1665 1666 1667 1668 1669 1670 1671 1672 1673 1674 1675 1676 1677 1678 1679 1680 1681 1682 1683 1684 1685 1686 1687 1688 1689 1690 1691 1692 1693 1694 1695 1696 1697 1698 1699 1700 1701 1702 1703 1704 1705 1706 1707 1708 1709 1710 1711 1712 1713 1714 1715 1716 1717 1718 1719 1720 1721 1722 1723 1724 1725 1726 1727 1728 1729 1730 1731 1732 1733 1734 1735 1736 1737 1738 1739 1740 1741 1742 1743 1744 1745 1746 1747 1748 1749 1750 1751 1752 1753 1754 1755 1756 1757 1758 1759 1760 1761 1762 1763 1764 1765 1766 1767 1768 1769 1770 1771 1772 1773 1774 1775 1776 1777 1778 1779 1780 1781 1782 1783 1784 1785 1786 1787 1788 1789 1790 1791 1792 1793 1794 1795 1796 1797 1798 1799 1800 1801 1802 1803 1804 1805 1806 1807 1808 1809 1810 1811 1812 1813 1814 1815 1816 1817 1818 1819 1820 1821 1822 1823 1824 1825 1826 1827 1828 1829 1830 1831 1832 1833 1834 1835 1836 1837 1838 1839 1840 1841 1842 1843 1844 1845 1846 1847 1848 1849 1850 1851 1852 1853 1854 1855 1856 1857 1858 1859 1860 1861 1862 1863 1864 1865 1866 1867 1868 1869 1870 1871 1872 1873 1874 1875 1876 1877 1878 1879 1880 1881 1882 1883 1884 1885 1886 1887 1888 1889 1890 1891 1892 1893 1894 1895 1896 1897 1898 1899 1900 1901 1902 1903 1904 1905 1906 1907 1908 1909 1910 1911 1912 1913 1914 1915 1916 1917 1918 1919 1920 1921 1922 1923 1924 1925 1926 1927 1928 1929 1930 1931 1932 1933 1934 1935 1936 1937 1938 1939 1940 1941 1942 1943 1944 1945 1946 1947 1948 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 | /*************************************************************************
Copyright (c) Sergey Bochkanov (ALGLIB project).
>>> SOURCE LICENSE >>>
This program is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation (www.fsf.org); either version 2 of the
License, or (at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
A copy of the GNU General Public License is available at
http://www.fsf.org/licensing/licenses
>>> END OF LICENSE >>>
*************************************************************************/
#ifndef _specialfunctions_pkg_h
#define _specialfunctions_pkg_h
#include "ap.h"
#include "alglibinternal.h"
/////////////////////////////////////////////////////////////////////////
//
// THIS SECTION CONTAINS COMPUTATIONAL CORE DECLARATIONS (DATATYPES)
//
/////////////////////////////////////////////////////////////////////////
namespace alglib_impl
{
}
/////////////////////////////////////////////////////////////////////////
//
// THIS SECTION CONTAINS C++ INTERFACE
//
/////////////////////////////////////////////////////////////////////////
namespace alglib
{
/*************************************************************************
Gamma function
Input parameters:
X - argument
Domain:
0 < X < 171.6
-170 < X < 0, X is not an integer.
Relative error:
arithmetic domain # trials peak rms
IEEE -170,-33 20000 2.3e-15 3.3e-16
IEEE -33, 33 20000 9.4e-16 2.2e-16
IEEE 33, 171.6 20000 2.3e-15 3.2e-16
Cephes Math Library Release 2.8: June, 2000
Original copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
Translated to AlgoPascal by Bochkanov Sergey (2005, 2006, 2007).
*************************************************************************/
double gammafunction(const double x);
/*************************************************************************
Natural logarithm of gamma function
Input parameters:
X - argument
Result:
logarithm of the absolute value of the Gamma(X).
Output parameters:
SgnGam - sign(Gamma(X))
Domain:
0 < X < 2.55e305
-2.55e305 < X < 0, X is not an integer.
ACCURACY:
arithmetic domain # trials peak rms
IEEE 0, 3 28000 5.4e-16 1.1e-16
IEEE 2.718, 2.556e305 40000 3.5e-16 8.3e-17
The error criterion was relative when the function magnitude
was greater than one but absolute when it was less than one.
The following test used the relative error criterion, though
at certain points the relative error could be much higher than
indicated.
IEEE -200, -4 10000 4.8e-16 1.3e-16
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
Translated to AlgoPascal by Bochkanov Sergey (2005, 2006, 2007).
*************************************************************************/
double lngamma(const double x, double &sgngam);
/*************************************************************************
Error function
The integral is
x
-
2 | | 2
erf(x) = -------- | exp( - t ) dt.
sqrt(pi) | |
-
0
For 0 <= |x| < 1, erf(x) = x * P4(x**2)/Q5(x**2); otherwise
erf(x) = 1 - erfc(x).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0,1 30000 3.7e-16 1.0e-16
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double errorfunction(const double x);
/*************************************************************************
Complementary error function
1 - erf(x) =
inf.
-
2 | | 2
erfc(x) = -------- | exp( - t ) dt
sqrt(pi) | |
-
x
For small x, erfc(x) = 1 - erf(x); otherwise rational
approximations are computed.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0,26.6417 30000 5.7e-14 1.5e-14
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double errorfunctionc(const double x);
/*************************************************************************
Normal distribution function
Returns the area under the Gaussian probability density
function, integrated from minus infinity to x:
x
-
1 | | 2
ndtr(x) = --------- | exp( - t /2 ) dt
sqrt(2pi) | |
-
-inf.
= ( 1 + erf(z) ) / 2
= erfc(z) / 2
where z = x/sqrt(2). Computation is via the functions
erf and erfc.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE -13,0 30000 3.4e-14 6.7e-15
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double normaldistribution(const double x);
/*************************************************************************
Inverse of the error function
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double inverf(const double e);
/*************************************************************************
Inverse of Normal distribution function
Returns the argument, x, for which the area under the
Gaussian probability density function (integrated from
minus infinity to x) is equal to y.
For small arguments 0 < y < exp(-2), the program computes
z = sqrt( -2.0 * log(y) ); then the approximation is
x = z - log(z)/z - (1/z) P(1/z) / Q(1/z).
There are two rational functions P/Q, one for 0 < y < exp(-32)
and the other for y up to exp(-2). For larger arguments,
w = y - 0.5, and x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0.125, 1 20000 7.2e-16 1.3e-16
IEEE 3e-308, 0.135 50000 4.6e-16 9.8e-17
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double invnormaldistribution(const double y0);
/*************************************************************************
Incomplete gamma integral
The function is defined by
x
-
1 | | -t a-1
igam(a,x) = ----- | e t dt.
- | |
| (a) -
0
In this implementation both arguments must be positive.
The integral is evaluated by either a power series or
continued fraction expansion, depending on the relative
values of a and x.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0,30 200000 3.6e-14 2.9e-15
IEEE 0,100 300000 9.9e-14 1.5e-14
Cephes Math Library Release 2.8: June, 2000
Copyright 1985, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double incompletegamma(const double a, const double x);
/*************************************************************************
Complemented incomplete gamma integral
The function is defined by
igamc(a,x) = 1 - igam(a,x)
inf.
-
1 | | -t a-1
= ----- | e t dt.
- | |
| (a) -
x
In this implementation both arguments must be positive.
The integral is evaluated by either a power series or
continued fraction expansion, depending on the relative
values of a and x.
ACCURACY:
Tested at random a, x.
a x Relative error:
arithmetic domain domain # trials peak rms
IEEE 0.5,100 0,100 200000 1.9e-14 1.7e-15
IEEE 0.01,0.5 0,100 200000 1.4e-13 1.6e-15
Cephes Math Library Release 2.8: June, 2000
Copyright 1985, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double incompletegammac(const double a, const double x);
/*************************************************************************
Inverse of complemented imcomplete gamma integral
Given p, the function finds x such that
igamc( a, x ) = p.
Starting with the approximate value
3
x = a t
where
t = 1 - d - ndtri(p) sqrt(d)
and
d = 1/9a,
the routine performs up to 10 Newton iterations to find the
root of igamc(a,x) - p = 0.
ACCURACY:
Tested at random a, p in the intervals indicated.
a p Relative error:
arithmetic domain domain # trials peak rms
IEEE 0.5,100 0,0.5 100000 1.0e-14 1.7e-15
IEEE 0.01,0.5 0,0.5 100000 9.0e-14 3.4e-15
IEEE 0.5,10000 0,0.5 20000 2.3e-13 3.8e-14
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double invincompletegammac(const double a, const double y0);
/*************************************************************************
Airy function
Solution of the differential equation
y"(x) = xy.
The function returns the two independent solutions Ai, Bi
and their first derivatives Ai'(x), Bi'(x).
Evaluation is by power series summation for small x,
by rational minimax approximations for large x.
ACCURACY:
Error criterion is absolute when function <= 1, relative
when function > 1, except * denotes relative error criterion.
For large negative x, the absolute error increases as x^1.5.
For large positive x, the relative error increases as x^1.5.
Arithmetic domain function # trials peak rms
IEEE -10, 0 Ai 10000 1.6e-15 2.7e-16
IEEE 0, 10 Ai 10000 2.3e-14* 1.8e-15*
IEEE -10, 0 Ai' 10000 4.6e-15 7.6e-16
IEEE 0, 10 Ai' 10000 1.8e-14* 1.5e-15*
IEEE -10, 10 Bi 30000 4.2e-15 5.3e-16
IEEE -10, 10 Bi' 30000 4.9e-15 7.3e-16
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
void airy(const double x, double &ai, double &aip, double &bi, double &bip);
/*************************************************************************
Bessel function of order zero
Returns Bessel function of order zero of the argument.
The domain is divided into the intervals [0, 5] and
(5, infinity). In the first interval the following rational
approximation is used:
2 2
(w - r ) (w - r ) P (w) / Q (w)
1 2 3 8
2
where w = x and the two r's are zeros of the function.
In the second interval, the Hankel asymptotic expansion
is employed with two rational functions of degree 6/6
and 7/7.
ACCURACY:
Absolute error:
arithmetic domain # trials peak rms
IEEE 0, 30 60000 4.2e-16 1.1e-16
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double besselj0(const double x);
/*************************************************************************
Bessel function of order one
Returns Bessel function of order one of the argument.
The domain is divided into the intervals [0, 8] and
(8, infinity). In the first interval a 24 term Chebyshev
expansion is used. In the second, the asymptotic
trigonometric representation is employed using two
rational functions of degree 5/5.
ACCURACY:
Absolute error:
arithmetic domain # trials peak rms
IEEE 0, 30 30000 2.6e-16 1.1e-16
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double besselj1(const double x);
/*************************************************************************
Bessel function of integer order
Returns Bessel function of order n, where n is a
(possibly negative) integer.
The ratio of jn(x) to j0(x) is computed by backward
recurrence. First the ratio jn/jn-1 is found by a
continued fraction expansion. Then the recurrence
relating successive orders is applied until j0 or j1 is
reached.
If n = 0 or 1 the routine for j0 or j1 is called
directly.
ACCURACY:
Absolute error:
arithmetic range # trials peak rms
IEEE 0, 30 5000 4.4e-16 7.9e-17
Not suitable for large n or x. Use jv() (fractional order) instead.
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double besseljn(const ae_int_t n, const double x);
/*************************************************************************
Bessel function of the second kind, order zero
Returns Bessel function of the second kind, of order
zero, of the argument.
The domain is divided into the intervals [0, 5] and
(5, infinity). In the first interval a rational approximation
R(x) is employed to compute
y0(x) = R(x) + 2 * log(x) * j0(x) / PI.
Thus a call to j0() is required.
In the second interval, the Hankel asymptotic expansion
is employed with two rational functions of degree 6/6
and 7/7.
ACCURACY:
Absolute error, when y0(x) < 1; else relative error:
arithmetic domain # trials peak rms
IEEE 0, 30 30000 1.3e-15 1.6e-16
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double bessely0(const double x);
/*************************************************************************
Bessel function of second kind of order one
Returns Bessel function of the second kind of order one
of the argument.
The domain is divided into the intervals [0, 8] and
(8, infinity). In the first interval a 25 term Chebyshev
expansion is used, and a call to j1() is required.
In the second, the asymptotic trigonometric representation
is employed using two rational functions of degree 5/5.
ACCURACY:
Absolute error:
arithmetic domain # trials peak rms
IEEE 0, 30 30000 1.0e-15 1.3e-16
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double bessely1(const double x);
/*************************************************************************
Bessel function of second kind of integer order
Returns Bessel function of order n, where n is a
(possibly negative) integer.
The function is evaluated by forward recurrence on
n, starting with values computed by the routines
y0() and y1().
If n = 0 or 1 the routine for y0 or y1 is called
directly.
ACCURACY:
Absolute error, except relative
when y > 1:
arithmetic domain # trials peak rms
IEEE 0, 30 30000 3.4e-15 4.3e-16
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double besselyn(const ae_int_t n, const double x);
/*************************************************************************
Modified Bessel function of order zero
Returns modified Bessel function of order zero of the
argument.
The function is defined as i0(x) = j0( ix ).
The range is partitioned into the two intervals [0,8] and
(8, infinity). Chebyshev polynomial expansions are employed
in each interval.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0,30 30000 5.8e-16 1.4e-16
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double besseli0(const double x);
/*************************************************************************
Modified Bessel function of order one
Returns modified Bessel function of order one of the
argument.
The function is defined as i1(x) = -i j1( ix ).
The range is partitioned into the two intervals [0,8] and
(8, infinity). Chebyshev polynomial expansions are employed
in each interval.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0, 30 30000 1.9e-15 2.1e-16
Cephes Math Library Release 2.8: June, 2000
Copyright 1985, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double besseli1(const double x);
/*************************************************************************
Modified Bessel function, second kind, order zero
Returns modified Bessel function of the second kind
of order zero of the argument.
The range is partitioned into the two intervals [0,8] and
(8, infinity). Chebyshev polynomial expansions are employed
in each interval.
ACCURACY:
Tested at 2000 random points between 0 and 8. Peak absolute
error (relative when K0 > 1) was 1.46e-14; rms, 4.26e-15.
Relative error:
arithmetic domain # trials peak rms
IEEE 0, 30 30000 1.2e-15 1.6e-16
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double besselk0(const double x);
/*************************************************************************
Modified Bessel function, second kind, order one
Computes the modified Bessel function of the second kind
of order one of the argument.
The range is partitioned into the two intervals [0,2] and
(2, infinity). Chebyshev polynomial expansions are employed
in each interval.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0, 30 30000 1.2e-15 1.6e-16
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double besselk1(const double x);
/*************************************************************************
Modified Bessel function, second kind, integer order
Returns modified Bessel function of the second kind
of order n of the argument.
The range is partitioned into the two intervals [0,9.55] and
(9.55, infinity). An ascending power series is used in the
low range, and an asymptotic expansion in the high range.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0,30 90000 1.8e-8 3.0e-10
Error is high only near the crossover point x = 9.55
between the two expansions used.
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1988, 2000 by Stephen L. Moshier
*************************************************************************/
double besselkn(const ae_int_t nn, const double x);
/*************************************************************************
Beta function
- -
| (a) | (b)
beta( a, b ) = -----------.
-
| (a+b)
For large arguments the logarithm of the function is
evaluated using lgam(), then exponentiated.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0,30 30000 8.1e-14 1.1e-14
Cephes Math Library Release 2.0: April, 1987
Copyright 1984, 1987 by Stephen L. Moshier
*************************************************************************/
double beta(const double a, const double b);
/*************************************************************************
Incomplete beta integral
Returns incomplete beta integral of the arguments, evaluated
from zero to x. The function is defined as
x
- -
| (a+b) | | a-1 b-1
----------- | t (1-t) dt.
- - | |
| (a) | (b) -
0
The domain of definition is 0 <= x <= 1. In this
implementation a and b are restricted to positive values.
The integral from x to 1 may be obtained by the symmetry
relation
1 - incbet( a, b, x ) = incbet( b, a, 1-x ).
The integral is evaluated by a continued fraction expansion
or, when b*x is small, by a power series.
ACCURACY:
Tested at uniformly distributed random points (a,b,x) with a and b
in "domain" and x between 0 and 1.
Relative error
arithmetic domain # trials peak rms
IEEE 0,5 10000 6.9e-15 4.5e-16
IEEE 0,85 250000 2.2e-13 1.7e-14
IEEE 0,1000 30000 5.3e-12 6.3e-13
IEEE 0,10000 250000 9.3e-11 7.1e-12
IEEE 0,100000 10000 8.7e-10 4.8e-11
Outputs smaller than the IEEE gradual underflow threshold
were excluded from these statistics.
Cephes Math Library, Release 2.8: June, 2000
Copyright 1984, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double incompletebeta(const double a, const double b, const double x);
/*************************************************************************
Inverse of imcomplete beta integral
Given y, the function finds x such that
incbet( a, b, x ) = y .
The routine performs interval halving or Newton iterations to find the
root of incbet(a,b,x) - y = 0.
ACCURACY:
Relative error:
x a,b
arithmetic domain domain # trials peak rms
IEEE 0,1 .5,10000 50000 5.8e-12 1.3e-13
IEEE 0,1 .25,100 100000 1.8e-13 3.9e-15
IEEE 0,1 0,5 50000 1.1e-12 5.5e-15
With a and b constrained to half-integer or integer values:
IEEE 0,1 .5,10000 50000 5.8e-12 1.1e-13
IEEE 0,1 .5,100 100000 1.7e-14 7.9e-16
With a = .5, b constrained to half-integer or integer values:
IEEE 0,1 .5,10000 10000 8.3e-11 1.0e-11
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1996, 2000 by Stephen L. Moshier
*************************************************************************/
double invincompletebeta(const double a, const double b, const double y);
/*************************************************************************
Binomial distribution
Returns the sum of the terms 0 through k of the Binomial
probability density:
k
-- ( n ) j n-j
> ( ) p (1-p)
-- ( j )
j=0
The terms are not summed directly; instead the incomplete
beta integral is employed, according to the formula
y = bdtr( k, n, p ) = incbet( n-k, k+1, 1-p ).
The arguments must be positive, with p ranging from 0 to 1.
ACCURACY:
Tested at random points (a,b,p), with p between 0 and 1.
a,b Relative error:
arithmetic domain # trials peak rms
For p between 0.001 and 1:
IEEE 0,100 100000 4.3e-15 2.6e-16
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double binomialdistribution(const ae_int_t k, const ae_int_t n, const double p);
/*************************************************************************
Complemented binomial distribution
Returns the sum of the terms k+1 through n of the Binomial
probability density:
n
-- ( n ) j n-j
> ( ) p (1-p)
-- ( j )
j=k+1
The terms are not summed directly; instead the incomplete
beta integral is employed, according to the formula
y = bdtrc( k, n, p ) = incbet( k+1, n-k, p ).
The arguments must be positive, with p ranging from 0 to 1.
ACCURACY:
Tested at random points (a,b,p).
a,b Relative error:
arithmetic domain # trials peak rms
For p between 0.001 and 1:
IEEE 0,100 100000 6.7e-15 8.2e-16
For p between 0 and .001:
IEEE 0,100 100000 1.5e-13 2.7e-15
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double binomialcdistribution(const ae_int_t k, const ae_int_t n, const double p);
/*************************************************************************
Inverse binomial distribution
Finds the event probability p such that the sum of the
terms 0 through k of the Binomial probability density
is equal to the given cumulative probability y.
This is accomplished using the inverse beta integral
function and the relation
1 - p = incbi( n-k, k+1, y ).
ACCURACY:
Tested at random points (a,b,p).
a,b Relative error:
arithmetic domain # trials peak rms
For p between 0.001 and 1:
IEEE 0,100 100000 2.3e-14 6.4e-16
IEEE 0,10000 100000 6.6e-12 1.2e-13
For p between 10^-6 and 0.001:
IEEE 0,100 100000 2.0e-12 1.3e-14
IEEE 0,10000 100000 1.5e-12 3.2e-14
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double invbinomialdistribution(const ae_int_t k, const ae_int_t n, const double y);
/*************************************************************************
Calculation of the value of the Chebyshev polynomials of the
first and second kinds.
Parameters:
r - polynomial kind, either 1 or 2.
n - degree, n>=0
x - argument, -1 <= x <= 1
Result:
the value of the Chebyshev polynomial at x
*************************************************************************/
double chebyshevcalculate(const ae_int_t r, const ae_int_t n, const double x);
/*************************************************************************
Summation of Chebyshev polynomials using Clenshaw�s recurrence formula.
This routine calculates
c[0]*T0(x) + c[1]*T1(x) + ... + c[N]*TN(x)
or
c[0]*U0(x) + c[1]*U1(x) + ... + c[N]*UN(x)
depending on the R.
Parameters:
r - polynomial kind, either 1 or 2.
n - degree, n>=0
x - argument
Result:
the value of the Chebyshev polynomial at x
*************************************************************************/
double chebyshevsum(const real_1d_array &c, const ae_int_t r, const ae_int_t n, const double x);
/*************************************************************************
Representation of Tn as C[0] + C[1]*X + ... + C[N]*X^N
Input parameters:
N - polynomial degree, n>=0
Output parameters:
C - coefficients
*************************************************************************/
void chebyshevcoefficients(const ae_int_t n, real_1d_array &c);
/*************************************************************************
Conversion of a series of Chebyshev polynomials to a power series.
Represents A[0]*T0(x) + A[1]*T1(x) + ... + A[N]*Tn(x) as
B[0] + B[1]*X + ... + B[N]*X^N.
Input parameters:
A - Chebyshev series coefficients
N - degree, N>=0
Output parameters
B - power series coefficients
*************************************************************************/
void fromchebyshev(const real_1d_array &a, const ae_int_t n, real_1d_array &b);
/*************************************************************************
Chi-square distribution
Returns the area under the left hand tail (from 0 to x)
of the Chi square probability density function with
v degrees of freedom.
x
-
1 | | v/2-1 -t/2
P( x | v ) = ----------- | t e dt
v/2 - | |
2 | (v/2) -
0
where x is the Chi-square variable.
The incomplete gamma integral is used, according to the
formula
y = chdtr( v, x ) = igam( v/2.0, x/2.0 ).
The arguments must both be positive.
ACCURACY:
See incomplete gamma function
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double chisquaredistribution(const double v, const double x);
/*************************************************************************
Complemented Chi-square distribution
Returns the area under the right hand tail (from x to
infinity) of the Chi square probability density function
with v degrees of freedom:
inf.
-
1 | | v/2-1 -t/2
P( x | v ) = ----------- | t e dt
v/2 - | |
2 | (v/2) -
x
where x is the Chi-square variable.
The incomplete gamma integral is used, according to the
formula
y = chdtr( v, x ) = igamc( v/2.0, x/2.0 ).
The arguments must both be positive.
ACCURACY:
See incomplete gamma function
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double chisquarecdistribution(const double v, const double x);
/*************************************************************************
Inverse of complemented Chi-square distribution
Finds the Chi-square argument x such that the integral
from x to infinity of the Chi-square density is equal
to the given cumulative probability y.
This is accomplished using the inverse gamma integral
function and the relation
x/2 = igami( df/2, y );
ACCURACY:
See inverse incomplete gamma function
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double invchisquaredistribution(const double v, const double y);
/*************************************************************************
Dawson's Integral
Approximates the integral
x
-
2 | | 2
dawsn(x) = exp( -x ) | exp( t ) dt
| |
-
0
Three different rational approximations are employed, for
the intervals 0 to 3.25; 3.25 to 6.25; and 6.25 up.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0,10 10000 6.9e-16 1.0e-16
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double dawsonintegral(const double x);
/*************************************************************************
Complete elliptic integral of the first kind
Approximates the integral
pi/2
-
| |
| dt
K(m) = | ------------------
| 2
| | sqrt( 1 - m sin t )
-
0
using the approximation
P(x) - log x Q(x).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0,1 30000 2.5e-16 6.8e-17
Cephes Math Library, Release 2.8: June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double ellipticintegralk(const double m);
/*************************************************************************
Complete elliptic integral of the first kind
Approximates the integral
pi/2
-
| |
| dt
K(m) = | ------------------
| 2
| | sqrt( 1 - m sin t )
-
0
where m = 1 - m1, using the approximation
P(x) - log x Q(x).
The argument m1 is used rather than m so that the logarithmic
singularity at m = 1 will be shifted to the origin; this
preserves maximum accuracy.
K(0) = pi/2.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0,1 30000 2.5e-16 6.8e-17
Cephes Math Library, Release 2.8: June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double ellipticintegralkhighprecision(const double m1);
/*************************************************************************
Incomplete elliptic integral of the first kind F(phi|m)
Approximates the integral
phi
-
| |
| dt
F(phi_\m) = | ------------------
| 2
| | sqrt( 1 - m sin t )
-
0
of amplitude phi and modulus m, using the arithmetic -
geometric mean algorithm.
ACCURACY:
Tested at random points with m in [0, 1] and phi as indicated.
Relative error:
arithmetic domain # trials peak rms
IEEE -10,10 200000 7.4e-16 1.0e-16
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double incompleteellipticintegralk(const double phi, const double m);
/*************************************************************************
Complete elliptic integral of the second kind
Approximates the integral
pi/2
-
| | 2
E(m) = | sqrt( 1 - m sin t ) dt
| |
-
0
using the approximation
P(x) - x log x Q(x).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0, 1 10000 2.1e-16 7.3e-17
Cephes Math Library, Release 2.8: June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double ellipticintegrale(const double m);
/*************************************************************************
Incomplete elliptic integral of the second kind
Approximates the integral
phi
-
| |
| 2
E(phi_\m) = | sqrt( 1 - m sin t ) dt
|
| |
-
0
of amplitude phi and modulus m, using the arithmetic -
geometric mean algorithm.
ACCURACY:
Tested at random arguments with phi in [-10, 10] and m in
[0, 1].
Relative error:
arithmetic domain # trials peak rms
IEEE -10,10 150000 3.3e-15 1.4e-16
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1993, 2000 by Stephen L. Moshier
*************************************************************************/
double incompleteellipticintegrale(const double phi, const double m);
/*************************************************************************
Exponential integral Ei(x)
x
- t
| | e
Ei(x) = -|- --- dt .
| | t
-
-inf
Not defined for x <= 0.
See also expn.c.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0,100 50000 8.6e-16 1.3e-16
Cephes Math Library Release 2.8: May, 1999
Copyright 1999 by Stephen L. Moshier
*************************************************************************/
double exponentialintegralei(const double x);
/*************************************************************************
Exponential integral En(x)
Evaluates the exponential integral
inf.
-
| | -xt
| e
E (x) = | ---- dt.
n | n
| | t
-
1
Both n and x must be nonnegative.
The routine employs either a power series, a continued
fraction, or an asymptotic formula depending on the
relative values of n and x.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0, 30 10000 1.7e-15 3.6e-16
Cephes Math Library Release 2.8: June, 2000
Copyright 1985, 2000 by Stephen L. Moshier
*************************************************************************/
double exponentialintegralen(const double x, const ae_int_t n);
/*************************************************************************
F distribution
Returns the area from zero to x under the F density
function (also known as Snedcor's density or the
variance ratio density). This is the density
of x = (u1/df1)/(u2/df2), where u1 and u2 are random
variables having Chi square distributions with df1
and df2 degrees of freedom, respectively.
The incomplete beta integral is used, according to the
formula
P(x) = incbet( df1/2, df2/2, (df1*x/(df2 + df1*x) ).
The arguments a and b are greater than zero, and x is
nonnegative.
ACCURACY:
Tested at random points (a,b,x).
x a,b Relative error:
arithmetic domain domain # trials peak rms
IEEE 0,1 0,100 100000 9.8e-15 1.7e-15
IEEE 1,5 0,100 100000 6.5e-15 3.5e-16
IEEE 0,1 1,10000 100000 2.2e-11 3.3e-12
IEEE 1,5 1,10000 100000 1.1e-11 1.7e-13
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double fdistribution(const ae_int_t a, const ae_int_t b, const double x);
/*************************************************************************
Complemented F distribution
Returns the area from x to infinity under the F density
function (also known as Snedcor's density or the
variance ratio density).
inf.
-
1 | | a-1 b-1
1-P(x) = ------ | t (1-t) dt
B(a,b) | |
-
x
The incomplete beta integral is used, according to the
formula
P(x) = incbet( df2/2, df1/2, (df2/(df2 + df1*x) ).
ACCURACY:
Tested at random points (a,b,x) in the indicated intervals.
x a,b Relative error:
arithmetic domain domain # trials peak rms
IEEE 0,1 1,100 100000 3.7e-14 5.9e-16
IEEE 1,5 1,100 100000 8.0e-15 1.6e-15
IEEE 0,1 1,10000 100000 1.8e-11 3.5e-13
IEEE 1,5 1,10000 100000 2.0e-11 3.0e-12
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double fcdistribution(const ae_int_t a, const ae_int_t b, const double x);
/*************************************************************************
Inverse of complemented F distribution
Finds the F density argument x such that the integral
from x to infinity of the F density is equal to the
given probability p.
This is accomplished using the inverse beta integral
function and the relations
z = incbi( df2/2, df1/2, p )
x = df2 (1-z) / (df1 z).
Note: the following relations hold for the inverse of
the uncomplemented F distribution:
z = incbi( df1/2, df2/2, p )
x = df2 z / (df1 (1-z)).
ACCURACY:
Tested at random points (a,b,p).
a,b Relative error:
arithmetic domain # trials peak rms
For p between .001 and 1:
IEEE 1,100 100000 8.3e-15 4.7e-16
IEEE 1,10000 100000 2.1e-11 1.4e-13
For p between 10^-6 and 10^-3:
IEEE 1,100 50000 1.3e-12 8.4e-15
IEEE 1,10000 50000 3.0e-12 4.8e-14
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double invfdistribution(const ae_int_t a, const ae_int_t b, const double y);
/*************************************************************************
Fresnel integral
Evaluates the Fresnel integrals
x
-
| |
C(x) = | cos(pi/2 t**2) dt,
| |
-
0
x
-
| |
S(x) = | sin(pi/2 t**2) dt.
| |
-
0
The integrals are evaluated by a power series for x < 1.
For x >= 1 auxiliary functions f(x) and g(x) are employed
such that
C(x) = 0.5 + f(x) sin( pi/2 x**2 ) - g(x) cos( pi/2 x**2 )
S(x) = 0.5 - f(x) cos( pi/2 x**2 ) - g(x) sin( pi/2 x**2 )
ACCURACY:
Relative error.
Arithmetic function domain # trials peak rms
IEEE S(x) 0, 10 10000 2.0e-15 3.2e-16
IEEE C(x) 0, 10 10000 1.8e-15 3.3e-16
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
void fresnelintegral(const double x, double &c, double &s);
/*************************************************************************
Calculation of the value of the Hermite polynomial.
Parameters:
n - degree, n>=0
x - argument
Result:
the value of the Hermite polynomial Hn at x
*************************************************************************/
double hermitecalculate(const ae_int_t n, const double x);
/*************************************************************************
Summation of Hermite polynomials using Clenshaw�s recurrence formula.
This routine calculates
c[0]*H0(x) + c[1]*H1(x) + ... + c[N]*HN(x)
Parameters:
n - degree, n>=0
x - argument
Result:
the value of the Hermite polynomial at x
*************************************************************************/
double hermitesum(const real_1d_array &c, const ae_int_t n, const double x);
/*************************************************************************
Representation of Hn as C[0] + C[1]*X + ... + C[N]*X^N
Input parameters:
N - polynomial degree, n>=0
Output parameters:
C - coefficients
*************************************************************************/
void hermitecoefficients(const ae_int_t n, real_1d_array &c);
/*************************************************************************
Jacobian Elliptic Functions
Evaluates the Jacobian elliptic functions sn(u|m), cn(u|m),
and dn(u|m) of parameter m between 0 and 1, and real
argument u.
These functions are periodic, with quarter-period on the
real axis equal to the complete elliptic integral
ellpk(1.0-m).
Relation to incomplete elliptic integral:
If u = ellik(phi,m), then sn(u|m) = sin(phi),
and cn(u|m) = cos(phi). Phi is called the amplitude of u.
Computation is by means of the arithmetic-geometric mean
algorithm, except when m is within 1e-9 of 0 or 1. In the
latter case with m close to 1, the approximation applies
only for phi < pi/2.
ACCURACY:
Tested at random points with u between 0 and 10, m between
0 and 1.
Absolute error (* = relative error):
arithmetic function # trials peak rms
IEEE phi 10000 9.2e-16* 1.4e-16*
IEEE sn 50000 4.1e-15 4.6e-16
IEEE cn 40000 3.6e-15 4.4e-16
IEEE dn 10000 1.3e-12 1.8e-14
Peak error observed in consistency check using addition
theorem for sn(u+v) was 4e-16 (absolute). Also tested by
the above relation to the incomplete elliptic integral.
Accuracy deteriorates when u is large.
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
void jacobianellipticfunctions(const double u, const double m, double &sn, double &cn, double &dn, double &ph);
/*************************************************************************
Calculation of the value of the Laguerre polynomial.
Parameters:
n - degree, n>=0
x - argument
Result:
the value of the Laguerre polynomial Ln at x
*************************************************************************/
double laguerrecalculate(const ae_int_t n, const double x);
/*************************************************************************
Summation of Laguerre polynomials using Clenshaw�s recurrence formula.
This routine calculates c[0]*L0(x) + c[1]*L1(x) + ... + c[N]*LN(x)
Parameters:
n - degree, n>=0
x - argument
Result:
the value of the Laguerre polynomial at x
*************************************************************************/
double laguerresum(const real_1d_array &c, const ae_int_t n, const double x);
/*************************************************************************
Representation of Ln as C[0] + C[1]*X + ... + C[N]*X^N
Input parameters:
N - polynomial degree, n>=0
Output parameters:
C - coefficients
*************************************************************************/
void laguerrecoefficients(const ae_int_t n, real_1d_array &c);
/*************************************************************************
Calculation of the value of the Legendre polynomial Pn.
Parameters:
n - degree, n>=0
x - argument
Result:
the value of the Legendre polynomial Pn at x
*************************************************************************/
double legendrecalculate(const ae_int_t n, const double x);
/*************************************************************************
Summation of Legendre polynomials using Clenshaw�s recurrence formula.
This routine calculates
c[0]*P0(x) + c[1]*P1(x) + ... + c[N]*PN(x)
Parameters:
n - degree, n>=0
x - argument
Result:
the value of the Legendre polynomial at x
*************************************************************************/
double legendresum(const real_1d_array &c, const ae_int_t n, const double x);
/*************************************************************************
Representation of Pn as C[0] + C[1]*X + ... + C[N]*X^N
Input parameters:
N - polynomial degree, n>=0
Output parameters:
C - coefficients
*************************************************************************/
void legendrecoefficients(const ae_int_t n, real_1d_array &c);
/*************************************************************************
Poisson distribution
Returns the sum of the first k+1 terms of the Poisson
distribution:
k j
-- -m m
> e --
-- j!
j=0
The terms are not summed directly; instead the incomplete
gamma integral is employed, according to the relation
y = pdtr( k, m ) = igamc( k+1, m ).
The arguments must both be positive.
ACCURACY:
See incomplete gamma function
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double poissondistribution(const ae_int_t k, const double m);
/*************************************************************************
Complemented Poisson distribution
Returns the sum of the terms k+1 to infinity of the Poisson
distribution:
inf. j
-- -m m
> e --
-- j!
j=k+1
The terms are not summed directly; instead the incomplete
gamma integral is employed, according to the formula
y = pdtrc( k, m ) = igam( k+1, m ).
The arguments must both be positive.
ACCURACY:
See incomplete gamma function
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double poissoncdistribution(const ae_int_t k, const double m);
/*************************************************************************
Inverse Poisson distribution
Finds the Poisson variable x such that the integral
from 0 to x of the Poisson density is equal to the
given probability y.
This is accomplished using the inverse gamma integral
function and the relation
m = igami( k+1, y ).
ACCURACY:
See inverse incomplete gamma function
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double invpoissondistribution(const ae_int_t k, const double y);
/*************************************************************************
Psi (digamma) function
d -
psi(x) = -- ln | (x)
dx
is the logarithmic derivative of the gamma function.
For integer x,
n-1
-
psi(n) = -EUL + > 1/k.
-
k=1
This formula is used for 0 < n <= 10. If x is negative, it
is transformed to a positive argument by the reflection
formula psi(1-x) = psi(x) + pi cot(pi x).
For general positive x, the argument is made greater than 10
using the recurrence psi(x+1) = psi(x) + 1/x.
Then the following asymptotic expansion is applied:
inf. B
- 2k
psi(x) = log(x) - 1/2x - > -------
- 2k
k=1 2k x
where the B2k are Bernoulli numbers.
ACCURACY:
Relative error (except absolute when |psi| < 1):
arithmetic domain # trials peak rms
IEEE 0,30 30000 1.3e-15 1.4e-16
IEEE -30,0 40000 1.5e-15 2.2e-16
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double psi(const double x);
/*************************************************************************
Student's t distribution
Computes the integral from minus infinity to t of the Student
t distribution with integer k > 0 degrees of freedom:
t
-
| |
- | 2 -(k+1)/2
| ( (k+1)/2 ) | ( x )
---------------------- | ( 1 + --- ) dx
- | ( k )
sqrt( k pi ) | ( k/2 ) |
| |
-
-inf.
Relation to incomplete beta integral:
1 - stdtr(k,t) = 0.5 * incbet( k/2, 1/2, z )
where
z = k/(k + t**2).
For t < -2, this is the method of computation. For higher t,
a direct method is derived from integration by parts.
Since the function is symmetric about t=0, the area under the
right tail of the density is found by calling the function
with -t instead of t.
ACCURACY:
Tested at random 1 <= k <= 25. The "domain" refers to t.
Relative error:
arithmetic domain # trials peak rms
IEEE -100,-2 50000 5.9e-15 1.4e-15
IEEE -2,100 500000 2.7e-15 4.9e-17
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double studenttdistribution(const ae_int_t k, const double t);
/*************************************************************************
Functional inverse of Student's t distribution
Given probability p, finds the argument t such that stdtr(k,t)
is equal to p.
ACCURACY:
Tested at random 1 <= k <= 100. The "domain" refers to p:
Relative error:
arithmetic domain # trials peak rms
IEEE .001,.999 25000 5.7e-15 8.0e-16
IEEE 10^-6,.001 25000 2.0e-12 2.9e-14
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double invstudenttdistribution(const ae_int_t k, const double p);
/*************************************************************************
Sine and cosine integrals
Evaluates the integrals
x
-
| cos t - 1
Ci(x) = eul + ln x + | --------- dt,
| t
-
0
x
-
| sin t
Si(x) = | ----- dt
| t
-
0
where eul = 0.57721566490153286061 is Euler's constant.
The integrals are approximated by rational functions.
For x > 8 auxiliary functions f(x) and g(x) are employed
such that
Ci(x) = f(x) sin(x) - g(x) cos(x)
Si(x) = pi/2 - f(x) cos(x) - g(x) sin(x)
ACCURACY:
Test interval = [0,50].
Absolute error, except relative when > 1:
arithmetic function # trials peak rms
IEEE Si 30000 4.4e-16 7.3e-17
IEEE Ci 30000 6.9e-16 5.1e-17
Cephes Math Library Release 2.1: January, 1989
Copyright 1984, 1987, 1989 by Stephen L. Moshier
*************************************************************************/
void sinecosineintegrals(const double x, double &si, double &ci);
/*************************************************************************
Hyperbolic sine and cosine integrals
Approximates the integrals
x
-
| | cosh t - 1
Chi(x) = eul + ln x + | ----------- dt,
| | t
-
0
x
-
| | sinh t
Shi(x) = | ------ dt
| | t
-
0
where eul = 0.57721566490153286061 is Euler's constant.
The integrals are evaluated by power series for x < 8
and by Chebyshev expansions for x between 8 and 88.
For large x, both functions approach exp(x)/2x.
Arguments greater than 88 in magnitude return MAXNUM.
ACCURACY:
Test interval 0 to 88.
Relative error:
arithmetic function # trials peak rms
IEEE Shi 30000 6.9e-16 1.6e-16
Absolute error, except relative when |Chi| > 1:
IEEE Chi 30000 8.4e-16 1.4e-16
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
void hyperbolicsinecosineintegrals(const double x, double &shi, double &chi);
}
/////////////////////////////////////////////////////////////////////////
//
// THIS SECTION CONTAINS COMPUTATIONAL CORE DECLARATIONS (FUNCTIONS)
//
/////////////////////////////////////////////////////////////////////////
namespace alglib_impl
{
double gammafunction(double x, ae_state *_state);
double lngamma(double x, double* sgngam, ae_state *_state);
double errorfunction(double x, ae_state *_state);
double errorfunctionc(double x, ae_state *_state);
double normaldistribution(double x, ae_state *_state);
double inverf(double e, ae_state *_state);
double invnormaldistribution(double y0, ae_state *_state);
double incompletegamma(double a, double x, ae_state *_state);
double incompletegammac(double a, double x, ae_state *_state);
double invincompletegammac(double a, double y0, ae_state *_state);
void airy(double x,
double* ai,
double* aip,
double* bi,
double* bip,
ae_state *_state);
double besselj0(double x, ae_state *_state);
double besselj1(double x, ae_state *_state);
double besseljn(ae_int_t n, double x, ae_state *_state);
double bessely0(double x, ae_state *_state);
double bessely1(double x, ae_state *_state);
double besselyn(ae_int_t n, double x, ae_state *_state);
double besseli0(double x, ae_state *_state);
double besseli1(double x, ae_state *_state);
double besselk0(double x, ae_state *_state);
double besselk1(double x, ae_state *_state);
double besselkn(ae_int_t nn, double x, ae_state *_state);
double beta(double a, double b, ae_state *_state);
double incompletebeta(double a, double b, double x, ae_state *_state);
double invincompletebeta(double a, double b, double y, ae_state *_state);
double binomialdistribution(ae_int_t k,
ae_int_t n,
double p,
ae_state *_state);
double binomialcdistribution(ae_int_t k,
ae_int_t n,
double p,
ae_state *_state);
double invbinomialdistribution(ae_int_t k,
ae_int_t n,
double y,
ae_state *_state);
double chebyshevcalculate(ae_int_t r,
ae_int_t n,
double x,
ae_state *_state);
double chebyshevsum(/* Real */ ae_vector* c,
ae_int_t r,
ae_int_t n,
double x,
ae_state *_state);
void chebyshevcoefficients(ae_int_t n,
/* Real */ ae_vector* c,
ae_state *_state);
void fromchebyshev(/* Real */ ae_vector* a,
ae_int_t n,
/* Real */ ae_vector* b,
ae_state *_state);
double chisquaredistribution(double v, double x, ae_state *_state);
double chisquarecdistribution(double v, double x, ae_state *_state);
double invchisquaredistribution(double v, double y, ae_state *_state);
double dawsonintegral(double x, ae_state *_state);
double ellipticintegralk(double m, ae_state *_state);
double ellipticintegralkhighprecision(double m1, ae_state *_state);
double incompleteellipticintegralk(double phi, double m, ae_state *_state);
double ellipticintegrale(double m, ae_state *_state);
double incompleteellipticintegrale(double phi, double m, ae_state *_state);
double exponentialintegralei(double x, ae_state *_state);
double exponentialintegralen(double x, ae_int_t n, ae_state *_state);
double fdistribution(ae_int_t a, ae_int_t b, double x, ae_state *_state);
double fcdistribution(ae_int_t a, ae_int_t b, double x, ae_state *_state);
double invfdistribution(ae_int_t a,
ae_int_t b,
double y,
ae_state *_state);
void fresnelintegral(double x, double* c, double* s, ae_state *_state);
double hermitecalculate(ae_int_t n, double x, ae_state *_state);
double hermitesum(/* Real */ ae_vector* c,
ae_int_t n,
double x,
ae_state *_state);
void hermitecoefficients(ae_int_t n,
/* Real */ ae_vector* c,
ae_state *_state);
void jacobianellipticfunctions(double u,
double m,
double* sn,
double* cn,
double* dn,
double* ph,
ae_state *_state);
double laguerrecalculate(ae_int_t n, double x, ae_state *_state);
double laguerresum(/* Real */ ae_vector* c,
ae_int_t n,
double x,
ae_state *_state);
void laguerrecoefficients(ae_int_t n,
/* Real */ ae_vector* c,
ae_state *_state);
double legendrecalculate(ae_int_t n, double x, ae_state *_state);
double legendresum(/* Real */ ae_vector* c,
ae_int_t n,
double x,
ae_state *_state);
void legendrecoefficients(ae_int_t n,
/* Real */ ae_vector* c,
ae_state *_state);
double poissondistribution(ae_int_t k, double m, ae_state *_state);
double poissoncdistribution(ae_int_t k, double m, ae_state *_state);
double invpoissondistribution(ae_int_t k, double y, ae_state *_state);
double psi(double x, ae_state *_state);
double studenttdistribution(ae_int_t k, double t, ae_state *_state);
double invstudenttdistribution(ae_int_t k, double p, ae_state *_state);
void sinecosineintegrals(double x,
double* si,
double* ci,
ae_state *_state);
void hyperbolicsinecosineintegrals(double x,
double* shi,
double* chi,
ae_state *_state);
}
#endif
|