1098:
652:
524:
1300:
861:
867:
777:
535:
114:
1701:
1330:
321:
1395:
329:
1902:
1353:
1104:
208:
175:
146:
231:
251:
1428:
Filon quadrature is widely used in physics and engineering for robust computation of
Fourier-type integrals. Applications include evaluation of oscillatory
2057:
1645:
DomĂnguez, V.; Graham, I. G.; Smyshlyaev, V. P. (2011). "Stability and error estimates for Filon–Clenshaw–Curtis rules for highly oscillatory integrals".
1699:
Mosig, J. R.; Gardiol, F. E. (1983). "Analytical and numerical techniques in the Green's function treatment of microstrip antennas and scatterers".
1818:
Fedotov, A.; Ilderton, A.; Karbstein, F.; King, B.; Seipt, D.; Taya, H.; Torgrimsson, G. (2023). "Advances in QED with intense background fields".
783:
1980:
1789:
Grimley, David I.; Wright, Adrian C.; Sinclair, Roger N. (1990). "Neutron scattering from vitreous silica IV. Time-of-flight diffraction".
1335:
Explicit Filon integration formulas for sine and complex exponential functions can be derived similarly. The formulas above fail for small
181:
or a complex exponential that causes the rapid oscillation of the integrand, particularly for high frequencies. In Filon quadrature, the
1895:
1760:
Dennis, S. C. R.; Chang, Gau-Zu (1970). "Numerical solutions for steady flow past a circular cylinder at
Reynolds numbers up to 100".
1791:
1744:
1683:
1647:
1542:
1530:
2021:
1995:
1417:
1093:{\displaystyle C_{2n}={\frac {1}{2}}f(a)\cos(ka)+f(a+2h)\cos(k(a+2h))+f(a+4h)\cos(k(a+4h))+\ldots +{\frac {1}{2}}f(b)\cos(kb)}
1888:
1433:
1990:
1985:
1857:
Thouless, M. D.; Evans, A. G.; Ashby, M. F.; Hutchinson, J. W. (1987). "The edge cracking and spalling of brittle plates".
1970:
1942:
1975:
1587:
Iserles, A.; Nørsett, S. P. (2004). "On quadrature methods for highly oscillatory integrals and their implementation".
40:
1762:
2011:
1919:
647:{\displaystyle \alpha =\left(\theta ^{2}+\theta \sin(\theta )\cos(\theta )-2\sin ^{2}(\theta )\right)/\theta ^{3}}
1589:
1560:
1356:
36:
658:
57:
1911:
1618:
32:
2036:
1469:
270:
2016:
1962:
1952:
1441:
1429:
1409:
2031:
1829:
1449:
1405:
266:
20:
1306:
519:{\displaystyle \int _{a}^{b}f(x)\cos(kx)dx\approx h(\alpha \left+\beta C_{2n}+\gamma C_{2n-1})}
276:
1937:
1932:
1740:
1736:
1679:
1538:
1453:
262:
1366:
1927:
1866:
1839:
1800:
1771:
1710:
1656:
1627:
1598:
1569:
1501:
1413:
1295:{\displaystyle C_{2n-1}=f(a+h)\cos(k(a+h))+f(a+3h)\cos(k(a+3h))+\ldots +f(b-h)\cos(k(b-h))}
1820:
1526:
1445:
178:
1947:
1338:
1404:
Modifications, extensions and generalizations of Filon quadrature have been reported in
184:
151:
122:
2051:
1870:
1804:
1728:
1360:
254:
1492:
Filon, L. N. G. (1930). "III.—On a
Quadrature Formula for Trigonometric Integrals".
213:
1412:
literature; these are known as Filon-type integration methods. These include Filon-
236:
1843:
1558:
Chase, Stephen M.; Fosdick, Lloyd D. (1969). "An algorithm for Filon quadrature".
1398:
1775:
1631:
1602:
1505:
1457:
1437:
1714:
1660:
1573:
856:{\displaystyle \gamma =4(\sin(\theta )-\theta \cos(\theta ))/\theta ^{3}}
258:
1880:
1363:
approximations must be in such cases to mitigate numerical errors, with
51:
The method is applied to oscillatory definite integrals in the form:
1834:
1884:
1616:
Xiang, Shuhuang (2007). "Efficient Filon-type methods for".
1397:
being recommended as a possible switchover point for 44-bit
1440:
problems in layered media and numerical solution to steady
1521:
1519:
1517:
1515:
1369:
1341:
279:
239:
216:
187:
154:
125:
1309:
1107:
870:
786:
661:
538:
332:
60:
2004:
1961:
1918:
39:integrals. It is named after English mathematician
1389:
1347:
1324:
1294:
1092:
855:
771:
646:
518:
315:
245:
225:
202:
169:
140:
108:
1678:. University of Toronto Press. pp. 287–289.
261:. Since each subinterval is now converted into a
1537:(2 ed.). Academic Press. pp. 151–160.
1896:
1494:Proceedings of the Royal Society of Edinburgh
8:
1674:ÄŚervenĂ˝, Vlastislav; Ravindra, Ravi (1971).
1487:
1485:
148:is a relatively slowly-varying function and
16:Integration method for oscillatory integrals
1448:, as well as various different problems in
269:, these can be evaluated in closed-form by
1903:
1889:
1881:
43:, who first described the method in 1934.
1833:
1379:
1368:
1340:
1308:
1112:
1106:
1050:
887:
875:
869:
847:
838:
785:
772:{\displaystyle \beta =2\left/\theta ^{3}}
763:
754:
692:
660:
638:
629:
606:
554:
537:
498:
479:
342:
337:
331:
278:
238:
215:
186:
153:
124:
70:
65:
59:
1733:Waves and Fields in Inhomogeneous Media
1481:
323:, the integration formula is given as:
109:{\displaystyle \int _{a}^{b}f(x)g(x)dx}
7:
2058:Numerical integration (quadrature)
14:
1792:Journal of Non-Crystalline Solids
1648:IMA Journal of Numerical Analysis
1981:Gauss–Kronrod quadrature formula
1535:Methods of Numerical Integration
1289:
1286:
1274:
1268:
1259:
1247:
1232:
1229:
1214:
1208:
1199:
1184:
1175:
1172:
1160:
1154:
1145:
1133:
1087:
1078:
1069:
1063:
1038:
1035:
1020:
1014:
1005:
990:
981:
978:
963:
957:
948:
933:
924:
915:
906:
900:
835:
832:
826:
811:
805:
796:
746:
740:
731:
725:
710:
707:
701:
679:
621:
615:
593:
587:
578:
572:
513:
461:
452:
443:
437:
428:
419:
410:
404:
390:
375:
366:
357:
351:
310:
301:
289:
283:
197:
191:
164:
158:
135:
129:
97:
91:
85:
79:
1:
1844:10.1016/j.physrep.2023.01.003
1871:10.1016/0001-6160(87)90015-0
1805:10.1016/0022-3093(90)90240-M
1676:Theory of Seismic Head Waves
41:Louis Napoleon George Filon
2074:
2022:Clenshaw–Curtis quadrature
1996:Chebyshev–Gauss quadrature
1763:Journal of Fluid Mechanics
1325:{\displaystyle \theta =kh}
316:{\textstyle g(x)=\cos(kx)}
1991:Gauss–Legendre quadrature
1986:Gauss–Laguerre quadrature
1943:Adaptive Simpson's method
1776:10.1017/S0022112070001428
1632:10.1007/s00211-006-0051-0
1603:10.1007/s10543-004-5243-3
1590:BIT Numerical Mathematics
1561:Communications of the ACM
1506:10.1017/S0370164600026262
1357:catastrophic cancellation
1971:Gauss–Hermite quadrature
1715:10.1049/ip-h-1.1983.0029
1390:{\textstyle \theta =1/6}
1976:Gauss–Jacobi quadrature
233:subintervals of length
1391:
1349:
1326:
1296:
1094:
857:
773:
648:
520:
317:
247:
227:
204:
171:
142:
110:
2012:Barnes–Hut simulation
1920:Newton–Cotes formulas
1912:Numerical integration
1737:Van Nostrand Reinhold
1661:10.1093/imanum/drq036
1619:Numerische Mathematik
1574:10.1145/363196.363209
1392:
1350:
1327:
1297:
1095:
858:
774:
649:
521:
318:
267:quadratic polynomials
248:
228:
205:
172:
143:
111:
33:numerical integration
2037:Tanh-sinh quadrature
1470:Tanh-sinh quadrature
1430:Sommerfeld integrals
1367:
1348:{\textstyle \theta }
1339:
1307:
1105:
868:
784:
659:
536:
330:
277:
271:integration by parts
237:
214:
185:
152:
123:
58:
2017:Bayesian quadrature
1963:Gaussian quadrature
1442:incompressible flow
1410:applied mathematics
347:
75:
31:is a technique for
2032:Lebedev quadrature
1938:Simpson's 3/8 rule
1531:Rabinowitz, Philip
1450:neutron scattering
1406:numerical analysis
1387:
1345:
1322:
1292:
1090:
853:
769:
644:
516:
333:
313:
273:. For the case of
243:
223:
200:
167:
138:
106:
61:
21:numerical analysis
2045:
2044:
1859:Acta Metallurgica
1702:IEE Proceedings H
1454:quantum mechanics
1058:
895:
253:, which are then
203:{\textstyle f(x)}
170:{\textstyle g(x)}
141:{\textstyle f(x)}
2065:
2027:Filon quadrature
1953:Romberg's method
1928:Trapezoidal rule
1905:
1898:
1891:
1882:
1875:
1874:
1865:(6): 1333–1341.
1854:
1848:
1847:
1837:
1815:
1809:
1808:
1786:
1780:
1779:
1757:
1751:
1750:
1725:
1719:
1718:
1696:
1690:
1689:
1671:
1665:
1664:
1655:(4): 1253–1280.
1642:
1636:
1635:
1613:
1607:
1606:
1584:
1578:
1577:
1555:
1549:
1548:
1527:Davis, Philip J.
1523:
1510:
1509:
1489:
1396:
1394:
1393:
1388:
1383:
1354:
1352:
1351:
1346:
1331:
1329:
1328:
1323:
1301:
1299:
1298:
1293:
1126:
1125:
1099:
1097:
1096:
1091:
1059:
1051:
896:
888:
883:
882:
862:
860:
859:
854:
852:
851:
842:
778:
776:
775:
770:
768:
767:
758:
753:
749:
697:
696:
653:
651:
650:
645:
643:
642:
633:
628:
624:
611:
610:
559:
558:
525:
523:
522:
517:
512:
511:
487:
486:
468:
464:
346:
341:
322:
320:
319:
314:
263:Fourier integral
252:
250:
249:
244:
232:
230:
229:
224:
210:is divided into
209:
207:
206:
201:
176:
174:
173:
168:
147:
145:
144:
139:
115:
113:
112:
107:
74:
69:
25:Filon quadrature
2073:
2072:
2068:
2067:
2066:
2064:
2063:
2062:
2048:
2047:
2046:
2041:
2000:
1957:
1914:
1909:
1879:
1878:
1856:
1855:
1851:
1821:Physics Reports
1817:
1816:
1812:
1788:
1787:
1783:
1759:
1758:
1754:
1747:
1739:. p. 118.
1727:
1726:
1722:
1698:
1697:
1693:
1686:
1673:
1672:
1668:
1644:
1643:
1639:
1615:
1614:
1610:
1586:
1585:
1581:
1557:
1556:
1552:
1545:
1525:
1524:
1513:
1491:
1490:
1483:
1478:
1466:
1446:fluid mechanics
1434:electromagnetic
1426:
1418:Clenshaw–Curtis
1365:
1364:
1337:
1336:
1305:
1304:
1108:
1103:
1102:
871:
866:
865:
843:
782:
781:
759:
688:
675:
671:
657:
656:
634:
602:
550:
549:
545:
534:
533:
494:
475:
400:
396:
328:
327:
275:
274:
235:
234:
226:{\textstyle 2N}
212:
211:
183:
182:
150:
149:
121:
120:
56:
55:
49:
17:
12:
11:
5:
2071:
2069:
2061:
2060:
2050:
2049:
2043:
2042:
2040:
2039:
2034:
2029:
2024:
2019:
2014:
2008:
2006:
2002:
2001:
1999:
1998:
1993:
1988:
1983:
1978:
1973:
1967:
1965:
1959:
1958:
1956:
1955:
1950:
1945:
1940:
1935:
1933:Simpson's rule
1930:
1924:
1922:
1916:
1915:
1910:
1908:
1907:
1900:
1893:
1885:
1877:
1876:
1849:
1810:
1781:
1770:(3): 471–489.
1752:
1745:
1729:Chew, Weng Cho
1720:
1709:(2): 175–182.
1691:
1684:
1666:
1637:
1608:
1579:
1568:(8): 453–457.
1550:
1543:
1511:
1480:
1479:
1477:
1474:
1473:
1472:
1465:
1462:
1425:
1422:
1386:
1382:
1378:
1375:
1372:
1355:values due to
1344:
1333:
1332:
1321:
1318:
1315:
1312:
1302:
1291:
1288:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1129:
1124:
1121:
1118:
1115:
1111:
1100:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1057:
1054:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
998:
995:
992:
989:
986:
983:
980:
977:
974:
971:
968:
965:
962:
959:
956:
953:
950:
947:
944:
941:
938:
935:
932:
929:
926:
923:
920:
917:
914:
911:
908:
905:
902:
899:
894:
891:
886:
881:
878:
874:
863:
850:
846:
841:
837:
834:
831:
828:
825:
822:
819:
816:
813:
810:
807:
804:
801:
798:
795:
792:
789:
779:
766:
762:
757:
752:
748:
745:
742:
739:
736:
733:
730:
727:
724:
721:
718:
715:
712:
709:
706:
703:
700:
695:
691:
687:
684:
681:
678:
674:
670:
667:
664:
654:
641:
637:
632:
627:
623:
620:
617:
614:
609:
605:
601:
598:
595:
592:
589:
586:
583:
580:
577:
574:
571:
568:
565:
562:
557:
553:
548:
544:
541:
527:
526:
515:
510:
507:
504:
501:
497:
493:
490:
485:
482:
478:
474:
471:
467:
463:
460:
457:
454:
451:
448:
445:
442:
439:
436:
433:
430:
427:
424:
421:
418:
415:
412:
409:
406:
403:
399:
395:
392:
389:
386:
383:
380:
377:
374:
371:
368:
365:
362:
359:
356:
353:
350:
345:
340:
336:
312:
309:
306:
303:
300:
297:
294:
291:
288:
285:
282:
246:{\textstyle h}
242:
222:
219:
199:
196:
193:
190:
179:sine or cosine
166:
163:
160:
157:
137:
134:
131:
128:
117:
116:
105:
102:
99:
96:
93:
90:
87:
84:
81:
78:
73:
68:
64:
48:
45:
29:Filon's method
15:
13:
10:
9:
6:
4:
3:
2:
2070:
2059:
2056:
2055:
2053:
2038:
2035:
2033:
2030:
2028:
2025:
2023:
2020:
2018:
2015:
2013:
2010:
2009:
2007:
2003:
1997:
1994:
1992:
1989:
1987:
1984:
1982:
1979:
1977:
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1972:
1969:
1968:
1966:
1964:
1960:
1954:
1951:
1949:
1946:
1944:
1941:
1939:
1936:
1934:
1931:
1929:
1926:
1925:
1923:
1921:
1917:
1913:
1906:
1901:
1899:
1894:
1892:
1887:
1886:
1883:
1872:
1868:
1864:
1860:
1853:
1850:
1845:
1841:
1836:
1831:
1827:
1823:
1822:
1814:
1811:
1806:
1802:
1798:
1794:
1793:
1785:
1782:
1777:
1773:
1769:
1765:
1764:
1756:
1753:
1748:
1746:9780780347496
1742:
1738:
1734:
1730:
1724:
1721:
1716:
1712:
1708:
1704:
1703:
1695:
1692:
1687:
1685:9780802000491
1681:
1677:
1670:
1667:
1662:
1658:
1654:
1650:
1649:
1641:
1638:
1633:
1629:
1625:
1621:
1620:
1612:
1609:
1604:
1600:
1596:
1592:
1591:
1583:
1580:
1575:
1571:
1567:
1563:
1562:
1554:
1551:
1546:
1544:9781483264288
1540:
1536:
1532:
1528:
1522:
1520:
1518:
1516:
1512:
1507:
1503:
1499:
1495:
1488:
1486:
1482:
1475:
1471:
1468:
1467:
1463:
1461:
1459:
1455:
1451:
1447:
1443:
1439:
1435:
1431:
1423:
1421:
1419:
1415:
1411:
1407:
1402:
1400:
1384:
1380:
1376:
1373:
1370:
1362:
1361:Taylor series
1358:
1342:
1319:
1316:
1313:
1310:
1303:
1283:
1280:
1277:
1271:
1265:
1262:
1256:
1253:
1250:
1244:
1241:
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1235:
1226:
1223:
1220:
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1211:
1205:
1202:
1196:
1193:
1190:
1187:
1181:
1178:
1169:
1166:
1163:
1157:
1151:
1148:
1142:
1139:
1136:
1130:
1127:
1122:
1119:
1116:
1113:
1109:
1101:
1084:
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1075:
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1066:
1060:
1055:
1052:
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1044:
1041:
1032:
1029:
1026:
1023:
1017:
1011:
1008:
1002:
999:
996:
993:
987:
984:
975:
972:
969:
966:
960:
954:
951:
945:
942:
939:
936:
930:
927:
921:
918:
912:
909:
903:
897:
892:
889:
884:
879:
876:
872:
864:
848:
844:
839:
829:
823:
820:
817:
814:
808:
802:
799:
793:
790:
787:
780:
764:
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743:
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584:
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326:
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307:
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62:
54:
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38:
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30:
26:
22:
2026:
1948:Boole's rule
1862:
1858:
1852:
1825:
1819:
1813:
1799:(1): 49–64.
1796:
1790:
1784:
1767:
1761:
1755:
1735:. New York:
1732:
1723:
1706:
1700:
1694:
1675:
1669:
1652:
1646:
1640:
1623:
1617:
1611:
1594:
1588:
1582:
1565:
1559:
1553:
1534:
1497:
1493:
1444:problems in
1427:
1424:Applications
1403:
1334:
528:
255:interpolated
118:
50:
28:
24:
18:
1626:: 633–658.
1597:: 755–772.
1414:trapezoidal
47:Description
37:oscillatory
1835:2203.00019
1476:References
1458:metallurgy
1416:and Filon–
177:is either
1828:: 1–138.
1500:: 38–47.
1420:methods.
1371:θ
1343:θ
1311:θ
1281:−
1266:
1254:−
1239:…
1206:
1152:
1120:−
1076:
1045:…
1012:
955:
913:
845:θ
830:θ
824:
818:θ
815:−
809:θ
803:
788:γ
761:θ
744:θ
738:
729:θ
723:
714:−
705:θ
699:
677:θ
663:β
636:θ
619:θ
613:
597:−
591:θ
585:
576:θ
570:
564:θ
552:θ
540:α
506:−
492:γ
473:β
450:
432:−
417:
394:α
385:≈
364:
335:∫
299:
259:parabolas
63:∫
2052:Category
1731:(1990).
1533:(1984).
1464:See also
1399:mantissa
1438:seismic
529:where
1743:
1682:
1541:
119:where
2005:Other
1830:arXiv
1826:1010
1741:ISBN
1680:ISBN
1539:ISBN
1456:and
1436:and
1432:for
1408:and
1867:doi
1840:doi
1801:doi
1797:119
1772:doi
1711:doi
1707:130
1657:doi
1628:doi
1624:105
1599:doi
1570:doi
1502:doi
1263:cos
1203:cos
1149:cos
1073:cos
1009:cos
952:cos
910:cos
821:cos
800:sin
735:cos
720:sin
690:cos
604:sin
582:cos
567:sin
447:sin
414:sin
361:cos
296:cos
265:of
257:by
35:of
27:or
19:In
2054::
1863:35
1861:.
1838:.
1824:.
1795:.
1768:42
1766:.
1705:.
1653:31
1651:.
1622:.
1595:44
1593:.
1566:12
1564:.
1529:;
1514:^
1498:49
1496:.
1484:^
1460:.
1452:,
1401:.
1359:;
23:,
1904:e
1897:t
1890:v
1873:.
1869::
1846:.
1842::
1832::
1807:.
1803::
1778:.
1774::
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1601::
1576:.
1572::
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1504::
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1317:k
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1278:b
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1269:(
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1248:(
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1200:)
1197:h
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1070:)
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1030:4
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1024:a
1021:(
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1015:(
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1003:h
1000:4
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958:(
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928:+
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919:k
916:(
907:)
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901:(
898:f
893:2
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877:2
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791:=
765:3
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741:(
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616:(
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588:(
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573:(
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509:1
503:n
500:2
496:C
489:+
484:n
481:2
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420:(
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408:b
405:(
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302:(
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241:h
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192:(
189:f
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162:x
159:(
156:g
136:)
133:x
130:(
127:f
104:x
101:d
98:)
95:x
92:(
89:g
86:)
83:x
80:(
77:f
72:b
67:a
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