On the numerical quadrature of highly-oscillating integrals I: Fourier transforms

IMA Journal of Numerical Analysis

Volumn 24, Issue 3, 2004, Pages 365-391

  Iserles, Arieh ^a  

^a UNIVERSITY OF CAMBRIDGE   (United Kingdom)

Author keywords

High oscillation; Lie group methods; Quadrature

Indexed keywords

FOURIER TRANSFORMS; LIE GROUPS;

FREQUENCY REGIMES; HIGH OSCILLATION; HIGHLY OSCILLATORY INTEGRALS; LIE GROUP METHOD; NUMERICAL QUADRATURE; QUADRATURE; QUADRATURE METHODS;

ORDINARY DIFFERENTIAL EQUATIONS;

EID: 4043159167     PISSN: 02724979     EISSN: None     Source Type: Journal    
DOI: 10.1093/imanum/24.3.365     Document Type: Article

References (22)

1. BAKHVALOV, N. S. & VASILČEVA, L. G. (1968) Evaluation of the integrals of oscillating functions by interpolation at nodes of Gaussian quadrature. USSR Comp. Math. Phys., 8, 241-249.
2. CLENDENIN, W. W. (1966) A method for numerical calculation of Fourier integrals. Numerische Mathematik, 8, 422-436.
3. DAVIS, P. J. & RABINOWITZ, P. (1980) Methods of Numerical Integration 2nd edn. Orlando, FL: Academic.
4. ENGELS, H. (1980) Numerical Quadrature and Cubature. London: Academic.
5. EVANS, G. A. & CHUNG, K. C. (2003) Theoretical aspects of generalised quadrature methods. J. Complexity, 19, 272-285.
6. FlLON, L. N. G. (1928) On a quadrature formula for trigonometric integrals. Proc. R. Soc. Edinburgh, 49, 38-47.
7. FLINN, E. A. (1960) A modification of Filon's method of numerical integration. J. ACM, 7, 181-184.
8. FOSDICK, L. D. (1968) A special case of the Filon quadrature formula. Math. Comput., 22, 77-81.
9. GAUTSCHI, W. (1968) Construction of Gauss-Christoffel quadrature formulas. Math. Comput., 22, 251-270.
10. GAUTSCHI, W. (1981) A survey of Gauss-Christoffel quadrature formulae. E. B. Christoffel. The Influence of His Work on Mathematics and the Physical Sciences. (P. Butzer & F. Fehér, eds). Basel: Birkhäuser, pp. 72-147.
11. GOLDBERG, R. R. & VARGA, R. S. (1956) Moebius inversion of Fourier transforms. Duke Math. J., 23, 553-559.
12. HAIRER, E., NØRSETT, S. P. & WANNER, G. (1993) Solving Ordinary Differential Equations I: Nonstiff Problems 2nd revised edn. Berlin: Springer.
13. ISERLES, A. (2002a) On the global error of discretization methods for highly-oscillatory ordinary differential equations. BIT, 42, 561-599.
14. ISERLES, A. (2002b) Think globally, act locally: solving highly-oscillatory ordinary differential equations. Appl. Numer. Anal., 43, 145-160.
15. ISERLES, A., MUNTHE-KAAS, H. Z., NØRSETT, S. P. & ZANNA, A. (2000) Lie-group methods. Acta Numer., 9, 215-365.
16. LEVIN, D. (1982) Procedure for computing one- and two-dimensional integrals of functions with rapid irregular oscillations. Math. Comput., 38, 531-538.
17. LEVIN, D. (1996) Fast integration of rapidly oscillatory functions. J. Comput. Appl. Math., 67, 95-101.
18. LEVIN, D. (1997) Analysis of a collocation method for integrating rapidly oscillatory functions. J. Comput. Appl. Math., 78, 131-138.
19. LEVIN, D., REICHEL, L. & RINGHOFFER, C. (1984) Analysis of an integration method for rapidly oscillating integrands. Technical Report MRC 2670 Madison: University of Wisconsin.
20. LUKE, Y. L. (1954) On the computation of oscillatory integrals. Proc. Cambridge Phil. Soc., 50, 269-277.
21. LYNESS, J. N. (1971) Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature. Math. Comput., 25, 87-104.
22. ZAMFIRESCU, I. (1963) An extension of Gauss's method for the calculation of improper integrals. Acad. R.P. Romîne Stud. Cerc. Mat., 14, 615-631 (in Romanian).