Skip to main content

Shape-Preservation Conditions for Cubic Spline Interpolation

Abstract

We consider the problem on shape-preserving interpolation by classical cubic splines. Namely, we consider conditions guaranteeing that, for a positive function (or a function whose kth derivative is positive), the cubic spline (respectively, its kth derivative) is positive. We present a survey of known results, completely describe the cases in which boundary conditions are formulated in terms of the first derivative, and obtain similar results for the second derivative. We discuss in detail mathematical methods for obtaining sufficient conditions for shape-preserving interpolation. We also develop such methods, which allows us to obtain general conditions for a spline and its derivative to be positive. We prove that, for a strictly positive function (or a function whose derivative is positive), it is possible to find an interpolant of the same sign as the initial function (respectively, its derivative) by thickening the mesh.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    J. H. Ahlberg, E. N. Nilson, and J. L. Walsh, The Theory of Splines and Their Applications (Academic Press, New York-London, 1967).

    MATH  Google Scholar 

  2. 2.

    V. V. Bogdanov, “Sufficient conditions for the comonotone interpolation of cubic C 2-splines,” Mat. Trudy 14, 3 (2011) [Siberian Adv. Math. 22, 153 (2012)].

    MathSciNet  MATH  Google Scholar 

  3. 3.

    V. V. Bogdanov, “Sufficient conditions for the nonnegativity of solutions to a system of equations with a nonstrictly Jacobian matrix,” Sib. Matem. Zh. 54, 544 (2013) [Siberian Math. J. 54, 425 (2013)].

    MathSciNet  Google Scholar 

  4. 4.

    V. V. Bogdanov and Yu. S. Volkov, “Selection of parameters of generalized cubic splines with convexity preserving interpolation,” Sib. Zh. Vychisl. Mat. 9, 5 (2006) [in Russian].

    MATH  Google Scholar 

  5. 5.

    V. V. Bogdanov and Yu. S. Volkov, “On shape-preservation conditions under interpolation by Subbotin parabolic splines,” Trudy Inst. Mat. Mekh. (Ekaterinburg) 22, no. 4, 102 (2016) [in Russian].

    MathSciNet  Google Scholar 

  6. 6.

    C. de Boor, A Practical Guide to Splines (Springer, New York, 2001).

    MATH  Google Scholar 

  7. 7.

    L. Collatz, Funktionalanalysis und Numerische Mathematik (Springer-Verlag, Berlin-Göttingen-Heidelberg, 1964) [Functional Analysis and Numerical Mathematics (Academic Press, New York-London, 1966)].

    MATH  Book  Google Scholar 

  8. 8.

    H. Dauner and C. H. Reinsch, “An analysis of two algorithms for shape-preserving cubic spline interpolation,” IMA J. Numer. Anal. 9, 299 (1989).

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    J.-C. Dupin and A. Fréville, “Shape preserving interpolating cubic splines with geometric mesh,” Appl. Numer. Math. 9, 447 (1992).

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    J. C. Fiorot and J. Tabka, “Shape-preserving C 2 cubic polynomial interpolating splines,” Math. Comput. 57, 291 (1991).

    MATH  Google Scholar 

  11. 11.

    F. N. Fritsch and R. E. Carlson, “Monotone piecewise cubic interpolation,” SIAM J. Numer. Anal. 17, 238 (1980).

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    V. M. Galkin and Yu. S. Volkov, “Elements of nozzle design optimization,” in: Computational Optimization: New Research Developments, 97 (Nova Science Publishers, New York, 2010).

    Google Scholar 

  13. 13.

    J. C. Holladay, “A smoothest curve approximation,” Math. Tables Aids Comput. 11, 233 (1957).

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    U. Hornung, “Monotone Spline-Interpolation,” in: Numerische Methoden der Approximationstheorie, 172 (Birkhäuser, Basel, 1978) [in German].

    Chapter  Google Scholar 

  15. 15.

    U. Hornung, “Interpolation by smooth functions under restrictions on the derivatives,” J. Approx. Theory 28, 227 (1980).

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    B. I. Kvasov, Methods of Shape-Preserving Spline Approximation (World Scientific, Singapore, 2000).

    MATH  Book  Google Scholar 

  17. 17.

    B. I. Kvasov, Methods of Isogeometric Approximation by Splines (Fizmatlit, Moscow, 2006) [in Russian].

    Google Scholar 

  18. 18.

    V. L. Miroshnichenko, “Convex and monotone spline interpolation,” in: Constructive Theory of Functions’ 84, 610 (Publ. House Bulgar. Acad. Sci., Sofia, 1984).

    Google Scholar 

  19. 19.

    V. L. Miroshnichenko, “Sufficient conditions for monotonicity and convexity of cubic splines of class C 2,” Vychisl. Sist. 137, 31 (1990) [Siberian Adv. Math. 2, 147 (1992)].

    MATH  Google Scholar 

  20. 20.

    V. L. Miroshnichenko, “Sufficient conditions of monotonicity and convexity of parabolic spline interpolants,” Vychisl. Sist. 142, 3 (1991) [Siberian Adv. Math. 3, 101 (1993)].

    MATH  Google Scholar 

  21. 21.

    V. L. Miroshnichenko, “Optimization of the form of a rational spline,” Vychisl. Sist. 159, 87 (1997) [in Russian].

    MathSciNet  MATH  Google Scholar 

  22. 22.

    E. Neuman, “Convex interpolating splines of arbitrary degree,” in: Numerische Methoden der Approximationstheorie, 211 (Birkhäuser Verlag, Basel-Boston-Stuttgart, 1980).

    Chapter  Google Scholar 

  23. 23.

    G. Opfer and H. J. Oberle, “The derivation of cubic splines with obstacles by methods of optimization and optimal control,” Numer. Math. 52, 17 (1988).

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    E. Passow, “Piecewise monotone spline interpolation,” J. Approx. Theory 12, 240 (1974).

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    E. Passow, “Monotone quadratic spline interpolation,” J. Approx. Theory 19, 143 (1977).

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    E. Passow and J. A. Roulier, “Monotone and convex spline interpolation,” SIAM J. Numer. Anal. 14, 904 (1977).

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    V. I. Pinchukov, “A monotone nonlocal cubic spline,” Zh. Vychisl. Mat. Mat. Fiz. 41, 200 (2001) [Comput. Math. Math. Phys. 41, 180 (2001)].

    MathSciNet  MATH  Google Scholar 

  28. 28.

    T. Popoviciu, “Sur le prolongement des fonctions convexes d’ordre supérieur,” Bull. Math. Soc. Roum. Sci. 36, 75 (1934) [in French].

    MATH  Google Scholar 

  29. 29.

    J. W. Schmidt, “On the convex cubic C 2-spline interpolation,” in: Numerical Methods of Approximation Theory, 213 (Birkhäuser, Basel, 1987).

    Google Scholar 

  30. 30.

    J. W. Schmidt and W. Heß, “Positivity of cubic polynomials on intervals and positive spline interpolation,” BIT 28, 340 (1988).

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    D. G. Schweikert, “An interpolation curve using a spline in tension,” J. Math. Phys. 45, 312 (1966).

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    O. Shisha, “Monotone approximation,” Pacific J. Math. 15, 667 (1965).

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    H. Späth, Spline-Algorithmen zur Konstruktion glatter Kurven und Flächen (R. Oldenbourg Verlag, München-Wien, 1973) [Spline Algorithms for Curves and Surfaces (Ultilitas Mathematica Publishing, Winnipeg, 1974)].

    MATH  Google Scholar 

  34. 34.

    Yu. S. Volkov, “Application of rational cubic splines for computing dynamical characteristics of an engine,” Vychisl. Sist. 154, 65 (1995) [in Russian].

    Google Scholar 

  35. 35.

    Yu. S. Volkov, “On the construction of interpolation polynomial splines,” Vychisl. Sist. 159, 3 (1997) [in Russian].

    MathSciNet  Google Scholar 

  36. 36.

    Yu. S. Volkov, “On a nonnegative solution of a system of equations with a symmetric circulant matrix,” Mat. Zametki 70, 170 (2001) [Math. Notes 70, 154 (2001)].

    MathSciNet  Article  Google Scholar 

  37. 37.

    Yu. S. Volkov, “On monotone interpolation by cubic splines,” Vychisl. Tekhnol. 6, no. 6, 14 (2001) [in Russian].

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Yu. S. Volkov, “A new method for constructing interpolation cubic splines,” Dokl. Ross. Akad. Nauk 382, 155 (2002) [Doklady Math. 65, 13 (2002)].

    MathSciNet  Google Scholar 

  39. 39.

    Yu. S. Volkov, “A new method for constructing cubic interpolating splines,” Zh. Vychisl. Mat. Mat. Fiz. 44, 231 (2004) [Comput. Math. Math. Phys. 44, 215 (2004)].

    MATH  Google Scholar 

  40. 40.

    Yu. S. Volkov, “Totally positive matrices in the methods of constructing interpolation splines of odd degree,” Mat. Trudy 7, no. 2, 3 (2004) [Siberian Adv. Math. 15, no. 4, 96 (2005)].

    MathSciNet  MATH  Google Scholar 

  41. 41.

    Yu. S. Volkov, “On complete interpolation spline finding via B-splines,” Sib. Elektron. Mat. Izv. 5, 334 (2008) [in Russian].

    MATH  Google Scholar 

  42. 42.

    Yu. S. Volkov, “Obtaining a banded system of equations in complete spline interpolation problem via B-spline basis,” Cent. Eur. J. Math. 10, 352 (2012).

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Yu. S. Volkov, “Convergence analysis of an interpolation process for the derivatives of a complete spline,” Urk. Mat. Visn. 9, 278 (2012) [J. Math. Sci. 187, 101 (2012)].

    Google Scholar 

  44. 44.

    Yu. S. Volkov, V. V. Bogdanov, V. L. Miroshnichenko, and V. T. Shevaldin, “Shape-preserving interpolation by cubic splines,” Mat. Zametki 88, 836 (2010) [Math. Notes 88, 798 (2010)].

    MathSciNet  MATH  Article  Google Scholar 

  45. 45.

    Yu. S. Volkov and V. M. Galkin, “On the choice of approximations in direct problems of nozzle design,” Zh. Vychisl. Mat. Mat. Fiz. 47, 923 (2007) [Comput. Math. Math. Phys. 47, 882 (2007)].

    MathSciNet  MATH  Google Scholar 

  46. 46.

    Yu. S. Volkov and V. M. Galkin, “Optimal nozzle design with monotonicity constraints,” in: Computer Design and Computational Defense Systems, 93 (Nova Science Publishers, New York, 2011).

    Google Scholar 

  47. 47.

    Yu. S. Volkov and V. L. Miroshnichenko, “Approximation of derivatives by jumps of interpolating splines,” Mat. Zametki 89, 127 (2011) Math. Notes 89, 138 (2011).

    MATH  Article  Google Scholar 

  48. 48.

    Yu. S. Volkov and V. T. Shevaldin, “Shape-preservation conditions for quadratic spline interpolation in the sense of Subbotin and Marsden,” Trudy Inst. Mat. Mekh. (Ekaterinburg) 18, no. 4, 145 (2012) [in Russian].

    Google Scholar 

  49. 49.

    Yu. S. Volkov and Yu. N. Subbotin, “Fifty years of Schoenberg’s problem on the convergence of spline interpolation,” Trudy Inst. Mat. Mekh. (Ekaterinburg) 20, no. 1, 52 (2014) [Proc. Steklov Inst. Math. 288, Suppl. 1, 222 (2015)].

    Google Scholar 

  50. 50.

    Yu. S. Zav’yalov, “On a nonnegative solution of a system of equations with a nonstrictly Jacobian matrix,” Sib. Matem. Zh. 37, 1303 (1996) [Siberian Math. J. 37, 1143 (1996)].

    MATH  Google Scholar 

  51. 51.

    Yu. S. Zav’yalov, B. I. Kvasov, and V. L. Miroshnichenko, Methods of Spline Functions (Nauka, Moscow, 1980) [in Russian].

    MATH  Google Scholar 

  52. 52.

    T. Zhanlav, “Some estimations for the approximation of second derivatives with the help of cubic interpolational splines,” Vychisl. Sist. 81, 12 (1979) [in Russian].

    MATH  Google Scholar 

Download references

Funding

The study was carried out within the framework of the state contract of the Sobolev Institute of Mathematics, SB RAS (project 0314-2016-0013).

Author information

Affiliations

Authors

Corresponding authors

Correspondence to V. V. Bogdanov or Yu. S. Volkov.

Additional information

Russian Text © The Author(s), 2019, published in Matematicheskie Trudy, 2019, Vol. 22, No. 1, pp. 19–67.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bogdanov, V.V., Volkov, Y.S. Shape-Preservation Conditions for Cubic Spline Interpolation. Sib. Adv. Math. 29, 231–262 (2019). https://doi.org/10.3103/S1055134419040011

Download citation

Keywords

  • cubic spline
  • shape-preserving interpolation
  • monotonicity
  • convexity