Skip to main content
SpringerLink
Log in

Fast algorithms for the integral equations of the inverse scattering problem

  • Short Communications
  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

Abstract

The Gelfand-Levitan and Marchenko equations of inverse scattering theory are integral equations with Toeplitz and Hankel kernels respectively. It is shown that these facts can be used to reduce the integral equations to differential equations which can be solved with an order of magnitude less computation than generally envisaged.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. K. Chadan and P. C. Sabatier, The Inverse Problem of Quantum Scattering Theory, New York: Springer-Verlag, 1977.

    Google Scholar 

  2. L. D. Faddeyev, "The Inverse Problem in the Quantum Theory of Scattering," J. Math. Phys., vol. 4, no. 1, pp. 72–104, January 1963.

    Google Scholar 

  3. M. G. Krein, "Continuous Analogs of Propositions on Polynomials Orthogonal on the Unit Circle," Dokl. Akad. Nauk SSSR, vol. 105, no. 4, pp. 637–640, 1955.

    Google Scholar 

  4. T. Kailath, L. Ljung and M. Morf, "Generalized Krein-Levinson Equations for the Efficient Computation of Fredholm Resolvents of Nondisplacement Kernels," in Surveys in Mathematical Analysis: Essays Dedicated to M. G. Krein, New York: Academic Press, 1978.

    Google Scholar 

  5. B. Levy, M. Morf and T. Kailath "Fast Algorithm for Integral Equations with Toeplitz and/or Hankel Kernels," to be submitted for publication.

Download references

Author information

Authors and Affiliations

  1. Dept. of Electrical Engineering, The University of Newcastle, 2308, New So. Wales, Australia

    B. D. O. Anderson

  2. Department of Electrical Engineering, Stanform University, 94305, Stanford, CA, USA

    T. Kailath

Authors
  1. B. D. O. Anderson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. Kailath
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This work was supported by the Army Research Office under Contract DAAG29-77-C-0042, by the Air Force Office of Scientific Research, Air Force Systems Command, under Contract AF44-620-74-C-0068 and the Australian Research Grants Committee.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Anderson, B.D.O., Kailath, T. Fast algorithms for the integral equations of the inverse scattering problem. Integr equ oper theory 1, 132–136 (1978). https://doi.org/10.1007/BF01682743

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01682743

Keywords

Search

Navigation