Staircase Algorithm and Boundary Valued Convex Interpolation by Gregory’s Splines

  • Jochen W. Schmidt
Conference paper


In 1984, the staircase algorithm was introduced as an abstract concept for solving weakly coupled systems of inequalities. Originally this algorithm was described by means of projections of relations. Recently a composition based form was proposed allowing further applications. Now, using this new proposal we derive a concrete algorithm for the problem of boundary valued convex interpolation applying Gregory’s rational cubic splines. It turns out that these splines always guarantee success under natural compatibility conditions.


Rationality Parameter Spline Interpolation Strict Convexity Convex Position Convexity Relation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bastian-Walther, M., Schmidt, J. W. (1999) Range restricted interpolation using Gregory’s rational cubic splines. J. Comput. Appl. Math. 103, 221–237CrossRefGoogle Scholar
  2. 2.
    Costantini, P. (1986) On monotone and convex spline interpolation. Math. Corp. 46, 203–214CrossRefGoogle Scholar
  3. 3.
    Costantini, P. (1996) Abstract schemes for functional shape preserving interpolation. In: Advanced Course on Fairshape (Hoschek, J., Kaklis, P., eds.) Teubner, Stuttgart, pp. 185–199CrossRefGoogle Scholar
  4. 4.
    Costantini, P., Morandi, R. (1984) Monotone and convex cubic spline interpolation. Calcolo 21, 281–294CrossRefGoogle Scholar
  5. 5.
    Gregory, J. A. (1986) Shape preserving spline interpolation. Computer Aided Design 18, 53–57CrossRefGoogle Scholar
  6. 6.
    Mulansky, B., Schmidt, J. W. (1999) Convex interval interpolation using a three-term staircase algorithm. Numer. Math. 82, 313–337CrossRefGoogle Scholar
  7. 7.
    Mulansky, B., Schmidt, J. W. (2000) Composition based staircase algorithm and constrained interpolation with boundary conditions. Numer. Math., to appearGoogle Scholar
  8. 8.
    Schmidt, J. W. (1996) Staircase algorithm and construction of convex spline interpolants up to the continuityC 3. Comput. Math. Appl. 31, 67–79CrossRefGoogle Scholar
  9. 9.
    Schmidt, J. W., Heß, W. (1984) Schwach verkoppelte Ungleichungssysteme und konvexe Spline-Interpolation. Elem. Math. 39, 85–95Google Scholar
  10. 10.
    Späth, H. (1995) One Dimensional Spline Interpolation Algorithms. A.K. Peters, WellesleyGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Jochen W. Schmidt
    • 1
  1. 1.Institute of Numerical MathematicsTechnical University of DresdenDresdenGermany

Personalised recommendations