Practical spline approximation

  • M. G. Cox
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 965)

Abstract

This two-part paper describes the use of polynomial spline functions for purposes of interpolation and approximation. The emphasis is on practical utility rather than detailed theory. Part I introduces polynomial splines, defines B-splines and treats the representation of splines in terms of B-splines. Part II deals with the statement and solution of spline interpolation and least squares spline approximation problems. It also discusses strategies for selecting particular solutions to spline approximation problems having nonunique solutions and techniques for automatic knot placement.

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References

  1. BARRODALE, I. and YOUNG, A. 1967 A note on numerical procedures for approximation by spline functions. Comput. J. 9, 318–320.MathSciNetCrossRefMATHGoogle Scholar
  2. BUTTERFIELD, K.R. 1976 The computation of all the derivatives of a B-spline basis. J. Inst. Math. Appl. 17, 15–25.MathSciNetCrossRefMATHGoogle Scholar
  3. COX, M.G. 1971 Curve fitting with piecewise polynomials. J. Inst. Math. Appl. 8, 36–52.MathSciNetCrossRefMATHGoogle Scholar
  4. COX, M.G. 1972 The numerical evaluation of B-splines. J. Inst. Math. Appl. 10, 134–149.MathSciNetCrossRefMATHGoogle Scholar
  5. COX, M.G. 1975a Numerical methods for the interpolation and approximation of data by spline functions. London, City University, PhD Thesis.Google Scholar
  6. COX, M.G. 1975b An algorithm for spline interpolation, J. Inst. Math. Appl. 15, 95–108.MathSciNetCrossRefMATHGoogle Scholar
  7. COX, M.G. 1976 A survey of numerical methods for data and function approximation. The State of the Art in Numerical Analysis, D.A.H. Jacobs, Ed., London, Academic Press, 627–668.Google Scholar
  8. COX, M.G. 1978a The representation of polynomials in terms of B-splines. Proc. Seventh Manitoba Conference on Numerical Mathematics and Computing, D. McCarthy and H.C. Williams, Eds., Winnepeg, University of Manitoba, 73–105.Google Scholar
  9. COX, M.G. 1978b The incorporation of boundary conditions in spline approximation problems. Lecture Notes in Mathematics 630: Numerical Analysis, G.A. Watson, Ed., Berlin, Springer-Verlag, 51–63.Google Scholar
  10. COX, M.G. 1978c The numerical evaluation of a spline from its B-spline representation. J. Inst. Math. Appl. 21, 135–143.MathSciNetCrossRefMATHGoogle Scholar
  11. COX, M.G. 1981 The least squares solution of overdetermined linear equations having band or augmented band structure. IMA J. Numer. Anal. 1, 3–22.MathSciNetCrossRefMATHGoogle Scholar
  12. CURRY, H.B. and SCHOENBERG, I.J. 1966 On Pólya frequency functions IV: the fundamental spline functions and their limits. J. Analyse Math. 17, 71–107.MathSciNetCrossRefMATHGoogle Scholar
  13. DE BOOR, C. 1972 On calculating with B-splines. J. Approx. Theory, 6, 50–62.MathSciNetCrossRefMATHGoogle Scholar
  14. DE BOOR, C. 1974 Good approximation by splines with variable knots II. Lecture Notes in Mathematics 363: Numerical Solution of Differential Equations, G.A. Watson, Ed., Berlin, Springer-Verlag, 12–20.Google Scholar
  15. DE BOOR, C. 1976 Total positivity of the spline collocation matrix. Indiana Univ. J. Math. 25, 541–551.MathSciNetCrossRefMATHGoogle Scholar
  16. DE BOOR, C. 1978 A Practical Guide to Splines. New York, Springer-Verlag.CrossRefMATHGoogle Scholar
  17. DE BOOR, C. and PINKUS, A. 1977 Backward error analysis for totally positive linear systems. Numer. Math. 27, 485–490.MathSciNetCrossRefMATHGoogle Scholar
  18. DE BOOR, C. and RICE, J.R. 1968 Least squares cubic spline approximation II — variable knots. Purdue University Report No. CSD TR 21.Google Scholar
  19. FORD, B., BENTLEY, J., DU CROZ, J.J. and HAGUE, S.J. 1979 The NAG Library "machine". Software-Practice and Experience 9, 56–72.CrossRefGoogle Scholar
  20. GENTLEMAN, W.M. 1974 Basic procedures for large, sparse or weighted linear least squares problems. Appl. Statist. 23, 448–454.CrossRefGoogle Scholar
  21. HAYES, J.G. 1978 Data-fitting algorithms available, in preparation, and in prospect, for the NAG Library. Numerical Software-Needs and Availability, D.A.H. Jacobs, Ed., London, Academic Press.Google Scholar
  22. HAYES, J.G. 1982 Curved knot lines and surfaces with ruled segments. To appear in proceedings of Dundee conference on Numerical Analysis, June 23–26, 1981, G.A. Watson, Ed., Berlin, Springer-Verlag.CrossRefGoogle Scholar
  23. HAYES, J.G. and HALLIDAY, J. 1974 The least-squares fitting of cubic spline surfaces to general data sets. J. Inst. Math. Appl. 14, 89–103.MathSciNetCrossRefMATHGoogle Scholar
  24. KARLIN, S. 1968 Total Positivity, Vol. 1. Stanford, California, Stanford University Press.MATHGoogle Scholar
  25. KOZAK, J. 1980 On the choice of the exterior knots in the B-spline basis for a spline space. University of Wisconsin Report 2148.Google Scholar
  26. POWELL, M.J.D. 1970 Curve fitting by splines in one variable. Numerical Approximation to Functions and Data, J.G. Hayes, Ed., London, Athlone Press, 65–83.Google Scholar
  27. RICE, J.R. 1969 The Approximation of Functions, Vol. II: Advanced Topics, Reading, Mass., Addison-Wesley.Google Scholar
  28. SCHOENBERG, I.J. and WHITNEY, Anne 1953 On Pólya frequency functions III. Trans. Am. Math. Soc. 74, 246–259.Google Scholar

Copyright information

© Spring-Verlag 1982

Authors and Affiliations

  • M. G. Cox
    • 1
  1. 1.National Physical LaboratoryTeddingtonUK

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