An overview of invariant imbedding algorithms and two-point boundary-value problems

  • E. D. Denman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 76)


It is somewhat difficult to draw conclusions on invariant imbedding algorithms, particularly with regard to numerical accuracy and efficiency of an algorithm. The only conclusion that this writer has reached in his work with II algorithms is that no particular algorithm is ideally suited for all two-point boundary-value problems. There has not been an in-depth study of all of the algorithms such as has been carried out by Enright, Hull and Lindberg [1975] on stiff system of equations. Recent developments may modify the conclusion reached from such a study if it were carried out at this time.


Optimal Control Problem Riccati Equation Recursive Equation State Transition Matrix Signal Flow Graph 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1].
    Chandrasekhar, S., Radiative Transfer, Dover, New York, 1960.Google Scholar
  2. [2].
    Preisendorfer, R. W., Radiative Transfer in Discrete Spaces, Pergamon, Oxford, 1965.Google Scholar
  3. [3].
    Redheffer, R. M., Difference Equations and Functional Equations in Transmission Line Theory, in Modern Mathematics for Engineers, (Beckenbach, Editor) Second Series, McGraw-Hill, New York, 1961.Google Scholar
  4. [4].
    Bellman, R., R. Kalaba, and M. Prestrud, Invariant Imbedding and Radiative Transfer in Slabs of Finite Thickness, American Elsevier, New York, 1963.Google Scholar
  5. [5].
    Wing, G. M., An Introduction to Transport Theory, John Wiley and Sons, New York, 1962.Google Scholar
  6. [6].
    Bellman, R. and G. M. Wing, An Introduction to Invariant Imbedding, John Wiley and Sons, New York, 1975.Google Scholar
  7. [7].
    Shimizu, R. and R. Aoki, Application of Invariant Imbedding to Reactor Physics, Academic, New York, 1972.Google Scholar
  8. [8].
    Adams, R. and E. D. Denman, Wave Propagation and Turbulent Media, American Elsevier, New York, 1966.Google Scholar
  9. [9].
    Denman, E. D., Coupled Modes in Plasmas, Elastic Media and Parametric Amplifiers, American Elsevier, New York, 1970.Google Scholar
  10. [10].
    Mingle, J. O., The Invariant Imbedding Theory of Nuclear Transport, American Elsevier, New York, 1973.Google Scholar
  11. [11].
    Case, K. M. and P. Zweifel, Linear Transport Problems, Addison-Wesley, Reading, Massachusetts, 1967.Google Scholar
  12. [12].
    Ribaric, M., Functional-Analytic Concepts and Structures of Neutron Transport Theory, Slovene Academy of Sciences and Arts, Ljublyana, Yugoslavia, 1973.Google Scholar
  13. [13].
    Scott, M., Invariant Imbedding and its Application to Ordinary Differential Equations, Addison-Wesley, Reading, Massachusetts, 1973.Google Scholar
  14. [14].
    Meyer, G. H., Initial Value Methods for Boundary Value Problems, Academic Press, New York, 1973.Google Scholar
  15. [15].
    Casti, J., and R. Kalaba, Imbedding Methods in Applied Mathematics, Addison-Wesley, Reading, Massachusetts, 1973.Google Scholar
  16. [16].
    Golberg, M. A., Some Functional Relationships for Two-Point Boundary-Value Problems, J. M. A. A., 45, 1974, 199–209.Google Scholar
  17. [17].
    Rybicki, G. and P. Usher, The Generalized Riccati Transformation as a Simple Alternative to Invariant Imbedding, Astrophy., 146, 1966, 871–879.Google Scholar
  18. [18].
    Allen, R. C. and G. M. Wing, An Invariant Imbedding Algorithm for the Solution of Inhomogeneous Two-Point Boundary-Value Problem, J. Comp. Phy., 14, 1974, 40–58.Google Scholar
  19. [19].
    Nelson, P., and M. Scott, The Relationship Between Two Variants of Invariant Imbedding, J. M. A. A., 37, 1972, 501–505.Google Scholar
  20. [20].
    Scott, M., A Bibliography on Invariant Imbedding and Related Topics, Sandia Laboratories, Rept. SC-B-71 0886, 1971.Google Scholar
  21. [21].
    Vandevender, W., On the Stability of an Invariant Imbedding Algorithm for the Solution of Two-Point Boundary-Value Problems, Sandia Laboratories, Rept. SAND 77-1107, 1977.Google Scholar
  22. [22].
    Yoo, K., Development of A Numerical Algorithm for Uncoupling of Constant Coefficient State Equations of Control Theory, Ph. D. Dissertation, University of Houston, 1974.Google Scholar
  23. [23].
    Musial, W. T., A Numerical Investigation of the P-Equations Relating The Matrix Riccati Equation to a Linear State Equations, M. S. Thesis, University of Houston, 1972.Google Scholar
  24. [24].
    Denman, E., and P. Nelson, Comment on "Comparison of Linear and Riccati Equations Used to Solve Optimal Control Problems," AIAA J., 12, 1974, 575–576.Google Scholar
  25. [25].
    Tapley, B. D. and W. E. Williamson, Comparison of Linear and Riccati Equations Used to Solve Optimal Control Problems, AIAA J., 10, 1972, 1154–1159.Google Scholar
  26. [26].
    Ljung, L., T. Kailath and B. Friedlander, Scattering Theory and Linear Least Squares Estimation, Part I: Continuous Time Problems, Proc. IEEE, 64, 131–139.Google Scholar
  27. [27].
    Lainiotis, D. G., Partitioned Estimation Algorithms, II: Linear Estimation, Info. Sci., 7, 1974, 317–340.Google Scholar
  28. [28].
    Denman, E. D. and A. N. Beavers, Jr., The Matrix Sign Function and Computations in Systems, Appl. Math and Comp., 2, 1976, 63–94.Google Scholar
  29. [29].
    Enright, W. H., T. E. Hull and B. Lindberg, Comparing Numerical Methods for Stiff Systems of O. D. E.'s, BIT, 15, 1975, 10–48.Google Scholar
  30. [30].
    Scott, M. R. and W. H. Vandevender, A Comparison of Several Invariant Imbedding Algorithms for the Solution of Two-Point Boundary-Value Problems, Appl. Math. and Comp., 1, 1975, 187–218.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1979

Authors and Affiliations

  • E. D. Denman
    • 1
  1. 1.University of HoustonUSA

Personalised recommendations