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Advances in Mathematical Physics
Volume 2015 (2015), Article ID 103074, 7 pages
http://dx.doi.org/10.1155/2015/103074
Research Article

A Simpler GMRES Method for Oscillatory Integrals with Irregular Oscillations

1Department of Mathematics and Computational Science, Hunan University of Science and Engineering, Yongzhou 425100, China
2International College, Central South University of Forestry and Technology, Changsha 410083, China

Received 2 April 2015; Accepted 4 June 2015

Academic Editor: Ming Mei

Copyright © 2015 Qinghua Wu and Meiying Xiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. Iserles and S. P. Nørsett, “On quadrature methods for highly oscillatory integrals and their implementation,” BIT Numerical Mathematics, vol. 44, no. 4, pp. 755–772, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. A. Iserles and S. P. Nørsett, “Efficient quadrature of highly oscillatory integrals using derivatives,” Proceedings of the Royal Society of London: Series A, vol. 461, no. 2057, pp. 1383–1399, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. D. Levin, “Fast integration of rapidly oscillatory functions,” Journal of Computational and Applied Mathematics, vol. 67, no. 1, pp. 95–101, 1996. View at Publisher · View at Google Scholar
  4. S. Xiang, Y. J. Cho, H. Wang, and H. Brunner, “Clenshaw-curtis-filon-type methods for highly oscillatory Bessel transforms and applications,” IMA Journal of Numerical Analysis, vol. 31, no. 4, pp. 1281–1314, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  5. G. A. Evans and K. C. Chung, “Some theoretical aspects of generalised quadrature methods,” Journal of Complexity, vol. 19, no. 3, pp. 272–285, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  6. D. Huybrechs and S. Vandewalle, “On the evaluation of highly oscillatory integrals by analytic continuation,” SIAM Journal on Numerical Analysis, vol. 44, no. 3, pp. 1026–1048, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. S. Xiang, “Efficient Filon-type methods for abfxeiwgxdx,” Numerische Mathematik, vol. 105, pp. 633–658, 2007. View at Google Scholar
  8. S. Xiang and Q. Wu, “Numerical solutions to Volterra integral equations of the second kind with oscillatory trigonometric kernels,” Applied Mathematics and Computation, vol. 223, pp. 34–44, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. S. Olver, “Moment-free numerical integration of highly oscillatory functions,” IMA Journal of Numerical Analysis, vol. 26, no. 2, pp. 213–227, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. S. Olver, “Moment-free numerical approximation of highly oscillatory integrals with stationary points,” European Journal of Applied Mathematics, vol. 18, no. 4, pp. 435–447, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  11. S. Olver, “GMRES for the differentiation operator,” SIAM Journal on Numerical Analysis, vol. 47, no. 5, pp. 3359–3373, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. S. Olver, “Shifted GMRES for oscillatory integrals,” Numerische Mathematik, vol. 114, no. 4, pp. 607–628, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. Y. Saad and M. H. Schultz, “GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM Journal on Scientific and Statistical Computing, vol. 7, no. 3, pp. 856–869, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  14. H. F. Walker and L. Zhou, “A simpler GMRES,” Numerical Linear Algebra with Applications, vol. 1, no. 6, pp. 571–581, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  15. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, USA, 1964.