Skip to main content
SpringerLink
Log in

\(C^2\) Tension Splines Construction Based on a Class of Sixth-Order Ordinary Differential Equations

  • Original Paper
  • Published:
Bulletin of the Iranian Mathematical Society Aims and scope Submit manuscript

Abstract

In this work, we construct a class of Hermite-type interpolation basis functions based on the sixth-order ordinary differential equation \({S^{(6)}}(\mathrm{{t}}) - {\tau }^4{S^{(2)}}(t) = 0\). Using them, we propose a kind of \(C^2\) tension interpolation splines with a local tension parameter \(\tau _i\). For \(C^2\) interpolation, the given interpolant has \(O(h^2)\) convergence. Some applications of the \(C^2\) tension interpolation splines on the construction of interest rate term structure in Chinese financial market are given. Moreover, a kind of generalized non-uniform B-splines of the space spanned by \(\text {span}\left\{ {1,t, \ldots ,{t^{n - 4}},\sin (\tau t),\cos (\tau t),\sinh (\tau t),\cosh (\tau t)} \right\} \) is constructed.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Schweikert, D.G.: An interpolation curve using a spline in tension. J. Math. Phys. 45, 312–317 (1966)

    Article  MathSciNet  Google Scholar 

  2. Späth, H.: Exponential spline interpolation. Computing 4, 225–233 (1969)

    Article  MathSciNet  Google Scholar 

  3. Hess, W., Schmidt, J.W.: Convexity preserving interpolation with exponential splines. Computing 36, 335–342 (1986)

    Article  MathSciNet  Google Scholar 

  4. Renka, R.J.: Interpolation tension splines with automatic selection of tension factors. SIAM J. Sci. Stat. Comput. 8, 393–415 (1987)

    Article  MathSciNet  Google Scholar 

  5. Sapidis, N.S., Kaklis, P.D.: An algorithm for constructing convexity and monotonicity-preserving splines in tension. Comput. Aided Geom. Des. 5, 127–137 (1988)

    Article  MathSciNet  Google Scholar 

  6. Pruess, S.: Properties of splines in tension. J. Approx. Theory 5, 86–96 (1976)

    Article  MathSciNet  Google Scholar 

  7. Costantini, P., Kvasov, B.I., Manni, C.: On discrete hyperbolic tension splines. Adv. Comput. Math. 11, 331–354 (1999)

    Article  MathSciNet  Google Scholar 

  8. Mcculloch, J.H.: The tax adjusted yield curve. J. Financ. 30, 811–830 (1975)

    Article  Google Scholar 

  9. Vasicek, O.A., Fong, H.G.: Term structure modeling using exponential splines. J. Financ. 37, 339–348 (1982)

    Article  Google Scholar 

  10. Nelson, C.R., Siegel, A.F.: Parsimonious modeling of yield curves. J. Bus. 60, 473–489 (1987)

    Article  Google Scholar 

  11. Chambers, D.R., Carleton, W.T., Waldman, D.W.: A new approach to estimation of the term structure of interest rates. J. Financ. Quant. Anal. 19, 233–252 (1984)

    Article  Google Scholar 

  12. Svensson, L.E.O.: Estimating and interpreting forward interest rates: Sweden 1992–1994. NBER working paper no. 4871 (1994)

  13. Shea, G.S.: Interest rate term structure estimation with exponential splines. J. Financ. 40, 319–325 (1985)

    Article  Google Scholar 

  14. Steeley, J.M.: Estimating the gilt-edged term structure: basis splines and confidence intervals. J. Bus. Financ. Account. 18, 513–529 (1991)

    Article  Google Scholar 

  15. Lin, B.H., Paxson, D.A.: Valuing the “new-issue” quality option in bund futures. Rev. Futures Mark. 12, 349–388 (1993)

    Google Scholar 

  16. Barzanti, L., Corradi, C.: A note on interest rate term structure estimation using tension splines. Insur. Math. Econ. 22, 139–143 (1998)

    Article  Google Scholar 

  17. Andersen, L.: Discount curve construction with tension splines. Rev. Deriv. Res. 10, 227–267 (2007)

    Article  Google Scholar 

  18. Blomvall, J.: Measurement of interest rates using a convex optimization model. Eur. J. Oper. Res. 256, 308–316 (2017)

    Article  MathSciNet  Google Scholar 

  19. Bosner, T., Rogina, M.: Non-uniform exponential tension splines. Numer. Algorithm 46, 265–294 (2007)

    Article  MathSciNet  Google Scholar 

  20. Lü, Y.G., Wang, G.Z., Yang, Y.N.: Uniform hyperbolic polynomial B-spline curves. Comput. Aided Geom. Des. 19, 379–393 (2002)

    Article  MathSciNet  Google Scholar 

  21. Qian, J., Tang, Y.H.: On non-uniform algebraic-hyperbolic (NUAH) B-splines. Numer. Math. A J. Chin. Univ. 15, 320–335 (2006)

    MathSciNet  MATH  Google Scholar 

  22. Mainar, E., Peña, J.M.: Optimal bases for a class of mixed spaces and their associated spline spaces. Comput. Math. Appl. 59, 1509–1523 (2010)

    Article  MathSciNet  Google Scholar 

  23. Brilleaud, M., Mazure, M.L.: Mixed hyperbolic/trigonometric spaces for design. Comput. Math. Appl. 64, 2459–2477 (2012)

    Article  MathSciNet  Google Scholar 

  24. Manni, C., Pelosi, F., Sampoli, M.L.: Generalized B-splines as a tool in isogeometric analysis. Comput. Methods Appl. Mech. Eng. 200, 867–881 (2011)

    Article  MathSciNet  Google Scholar 

  25. Ersoy, O., Korkmaz, A., Dağ, I.: Exponential B-splines for numerical solutions to some Boussinesq systems for water waves. Mediterr. J. Math. 13, 4975–4994 (2016)

    Article  MathSciNet  Google Scholar 

  26. Xu, G., Sun, N.G., Xu, J.L., Hui, K.C., Wang, G.Z.: A unified approach to construct generalized B-splines for isogeometric applications. J. Syst. Sci. Complex. 30, 983–998 (2017)

    Article  MathSciNet  Google Scholar 

  27. Wong, P.J.Y., Agarwal, R.P.L.: Explicit error estimates for quintic and biquintic spline interpolation I and II. Comput. Math. Appl. 28, 51–69 (1994)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We wish to express our gratitude to the editor and referees for their valuable remarks for improvements. The research is supported by the National Natural Science Foundation of China (nos. 61802129, 11771453), and the Natural Science Foundation Guangdong Province, China (no. 2018A030310381).

Author information

Authors and Affiliations

  1. School of Mathematics, South China University of Technology, Guangzhou, 510641, Guangdong, People’s Republic of China

    Yuanpeng Zhu

  2. School of Business Administration, South China University of Technology, Guangzhou, 510641, Guangdong, People’s Republic of China

    Zhenbiao Chen

  3. Department of Mathematics, Southern University of Science and Technology, Shenzhen, 518055, Guangdong, People’s Republic of China

    Xuli Han

Authors
  1. Yuanpeng Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhenbiao Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xuli Han
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuanpeng Zhu.

Additional information

Communicated by Behnam Hashemi.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhu, Y., Chen, Z. & Han, X. \(C^2\) Tension Splines Construction Based on a Class of Sixth-Order Ordinary Differential Equations. Bull. Iran. Math. Soc. 48, 127–150 (2022). https://doi.org/10.1007/s41980-020-00505-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41980-020-00505-3

Keywords

Mathematics Subject Classification

Search

Navigation