| Title:
|
On numerical evaluation of integrals involving Bessel functions (English) |
| Author:
|
Bezvoda, Václav |
| Author:
|
Farzan, Ruszlán |
| Author:
|
Segeth, Karel |
| Author:
|
Takó, Galina |
| Language:
|
English |
| Journal:
|
Aplikace matematiky |
| ISSN:
|
0373-6725 |
| Volume:
|
31 |
| Issue:
|
5 |
| Year:
|
1986 |
| Pages:
|
396-410 |
| Summary lang:
|
English |
| Summary lang:
|
Russian |
| Summary lang:
|
Czech |
| . |
| Category:
|
math |
| . |
| Summary:
|
The paper is concerned with the efficient evaluation of the integral $\int^\infty_0 f(x)J_n(rx)dx$, where $J_n$ is the Bessel function of index $n$ and $n$ is a nonnegative integer, for a given sequence of values of a real parameter $r$.
Two procedures are proposed and compared. One of them consists in a direct generalization of a procedure for the evaluation of of a similar integral with the weight function exp $(irx), which employs the fast Fourier transform. The other approach is based on the construction of a special Gaussian quadrature formula where $J_n$ appears as a weight.
The results of the comparison show that the application of the Gaussian formula is much more efficient. (English) |
| Keyword:
|
integrals with Bessel functions |
| Keyword:
|
fast Fourier transform |
| Keyword:
|
Gaussian integration formula |
| Keyword:
|
five-point Gauss rule |
| Keyword:
|
error analysis |
| Keyword:
|
numerical quadrature |
| MSC:
|
33C10 |
| MSC:
|
42A16 |
| MSC:
|
65D20 |
| MSC:
|
65D30 |
| MSC:
|
65D32 |
| MSC:
|
65T40 |
| idZBL:
|
Zbl 0614.65012 |
| idMR:
|
MR0863034 |
| DOI:
|
10.21136/AM.1986.104216 |
| . |
| Date available:
|
2008-05-20T18:30:47Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/104216 |
| . |
| Reference:
|
[1] M. Ambrožová: Using fast Fourier transform for evaluation of the integral of an oscillating function on the infinite integral.(Czech.) RNDr. thesis. Matematicko-fyzikální fakulta Univerzity Karlovy, Praha 1979. |
| Reference:
|
[2] N. S. Bahvalov: Numerical Methods.(Russian.) Nauka, Moskva 1973. MR 0362811 |
| Reference:
|
[3] I. S. Berezin I. P. Židkov: Methods of Computation.(Russian.) Vol. 1. Nauka, Moskva 1966. |
| Reference:
|
[4] V. Bezvoda K. Segeth: A contribution to the theory of electromagnetic induction of a line source.Studia Geod. et Geoph. 20 (1976), 366-377. 10.1007/BF01617648 |
| Reference:
|
[5] V. Bezvoda K. Segeth: Mathematical Modeling in Electromagnetic Prospecting Methods.Univerzita Karlova, Praha 1982. |
| Reference:
|
[6] P. I. Davis P. Rabinowitz: Methods of Numerical Integration.Academic Press, New York 1975. MR 0448814 |
| Reference:
|
[7] R. H. Farzan: Propagation of electromagnetic waves in stratified media with local inhomogeneity.Mathematical Models in Physics and Chemistry and Numerical Methods of Their Realization. Teubner Texte zur Mathematik 61. Teubner, Leipzig 1984, 237-247. |
| Reference:
|
[8] L. N. G. Filon: On a quadrature formula for trigonometric integrals.Proc. Roy. Soc. Edinburgh 49 (1928), 38-47. |
| Reference:
|
[9] I. S. Gradštein I. M. Ryžik: Tables of Integrals, Sums, Series, and Products.(Russian.) 5th ed. Nauka, Moskva 1971. |
| Reference:
|
[10] S.-Å. Gustafson G. Dahlqaist: On the computation of slowly convergent Fourier integrals.Methoden und Verfahren der mathematischen Physik. Band 6. Bibliographisches Institut, Mannheim 1972, 93-112. MR 0359377 |
| Reference:
|
[11] T. Kaneko B. Liu: Accumulation of round-off error in fast Fourier transforms.J. Assoc. Comput. Mach. 17 (1970), 637-654. MR 0275710, 10.1145/321607.321613 |
| Reference:
|
[12] V. I. Krylov: Approximate Computation of Integrals.(Russian.) Nauka, Moskva 1967. MR 0218015 |
| Reference:
|
[13] I. M. Longman: Note on a method for computing infinite integrals of oscillatory functions.Proc. Camb. Phil. Soc. 52 (1956), 764-768. Zbl 0072.33803, MR 0082193, 10.1017/S030500410003187X |
| Reference:
|
[14] R. Piessens E. de Doncker-Kapenga C. W. Überhuber D. K. Kahaner: QUADPACK. A Subroutine Package for Automatic Integration.Springer Series in Computational Mathematics 1. Springer-Verlag, Berlin 1983. MR 0712135 |
| Reference:
|
[15] A. Ralston: A First Course in Numerical Analysis.McGraw-Hill, New York 1965. Zbl 0139.31603, MR 0191070 |
| Reference:
|
[16] K. Segeth: Roundoff errors in the fast computation of discrete convolutions.Apl. Mat. 26 (1981), 241-262. Zbl 0474.65025, MR 0623505 |
| Reference:
|
[17] H. J. Stetter: Numerical approximation of Fourier transforms.Numer. Math. 8 (1966), 235-249. Zbl 0163.39503, MR 0198716, 10.1007/BF02162560 |
| . |
