Numerical iterative methods and repetitive processes

  • Rudolf RabensteinEmail author
  • Peter Steffen


Implicit schemes are a popular approach to the discretization of linear partial differential equations by finite differences. They require to solve a set of linear equations in each time step. Since finite difference discretizations lead to a local coupling, these systems of equations are sparse and can be effectively solved by iterative methods. Numerical procedures of this type are known in control theory as repetitive processes. They have mostly been used to solve problems in control like processes where passes of finite length are repeated over and over. This paper shows how the implicit discretization of partial differential equations can be cast into the framework of repetitive processes. Thus it establishes a link between yet unrelated results in numerical mathematics and control theory.


Numerical mathematics Iterative methods Repetitive processes Partial differential equations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bristow D. A., Tharayil M., Alleyne A. (2006) A survey of iterative learning control. IEEE Control Systems Magazine 26(3): 96–114CrossRefGoogle Scholar
  2. Cichy, B., Galkowski, K., Rogers, E., & Owens, D. (2005). Control of a class of ’wave’ discrete linear repetitive processes. In Proceedings of the 2005 IEEE international symposium on intelligent control and 2005 mediterranean conference on control and automation. Limassol, Cyprus.Google Scholar
  3. Cichy, B., Galkowski, K., Rogers, E., & Kummert, A. (2009). Iterative learning control for the ’wave’ linear repetitive processes. In 6th Interational workshop on multidimensional (nd) systems (nDS09). Thessaloniki, June/July.Google Scholar
  4. Collatz L. (1968) Funktionalanalysis und numerische Mathematik. Springer, BerlinGoogle Scholar
  5. Galkowski, K., Paszke, W., Rogers, E., & Owens, D. (2001). The state-space model of wave linear repetitive processes. In Proceedings of the international symposium on mathematical models in automation and robotics (Vol. 1, pp. 119–123).Google Scholar
  6. Garcia A. L. (1994) Numerical methods for physics. Prentice-Hall, Englewood Cliffs, New JerseyGoogle Scholar
  7. Golub G. H., van Loan C. F. (1989) Matrix computations. Johns Hopkins University Press, Baltimore, LondonzbMATHGoogle Scholar
  8. Golub G. H., Ortega J. M. (1996) Scientific computing: Eine Einführung in das wissenschaftliche Rechnen und Parallele Numerik. B. G. Teubner, StuttgartzbMATHGoogle Scholar
  9. Grenander U., Szegö G. (1958) Toeplitz forms and their applications. University of California Press, Berkeley and Los AngeleszbMATHGoogle Scholar
  10. Hildebrand F. B. (1968) Finite-difference equations and simulations. Prentice-Hall, Englewood Cliffs, New JerseyzbMATHGoogle Scholar
  11. Mitchell A. R., Griffiths D. F. (1980) The finite difference method in partial differential equations. Wiley, LondonzbMATHGoogle Scholar
  12. Rabenstein, R., & Steffen, P. (2009). Implicit discretization of linear partial differential equations and repetitive processes. In 6th International workshop on multidimensional (nd) systems (nDS 09) (pp. 113–119). Thessaloniki, June/July.Google Scholar
  13. Rabenstein, R., Steffen, P., & Galkowski, K. (2007) Stability of two-step repetitive processes based on matrix formulation. In 5th International workshop on multidimensional (nd) systems (nDS 07). Aveiro, June.Google Scholar
  14. Rogers E., Galkowski K., Owens D. (2007) Control systems theory and applications for linear repetitive processes. Springer, BerlinzbMATHGoogle Scholar
  15. Smith D. (1969) Numerical solution of partial differential equations. Oxford University Press, LondonGoogle Scholar
  16. Varga R. S. (2000) Matrix iterative analysis. Springer, BerlinCrossRefzbMATHGoogle Scholar
  17. Zurmühl R. (1964) Matrizen. Springer, BerlinzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.University Erlangen-Nuremberg, Chair of Multimedia Communications and Signal ProcessingErlangenGermany

Personalised recommendations