The stable evaluation of multivariate simplex splines

Author:
Thomas A. Grandine

Journal:
Math. Comp. **50** (1988), 197-205

MSC:
Primary 65D07; Secondary 41A15, 41A63

DOI:
https://doi.org/10.1090/S0025-5718-1988-0917827-2

MathSciNet review:
917827

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper gives a general method for the stable evaluation of multivariate simplex splines, based on the well-known recurrence relation of Micchelli [12]. This paper deals with two problems which arise in the implementation of the recurrence relation. First, the coefficients in the recurrence are shown to be efficiently computable via the dual simplex method of linear programminig. Secondly, the problem of evaluation along mesh boundaries is discussed in detail.

**[1]**Robert G. Bland, "New finite pivoting rules for the simplex method,"*Math. Oper. Res.*, v. 2, 1977, pp. 103-107. MR**0459599 (56:17791)****[2]**Carl de Boor, "Splines as linear combinations of*B*-splines," in*Approximation Theory II*(G.G. Lorentz, C. K. Chui, and L. L. Schumaker, eds.), Academic Press, New York, 1976, pp. 1-47.**[3]**Carl de Boor,*A Practical Guide to Splines*, Springer-Verlag, Berlin and New York, 1978. MR**507062 (80a:65027)****[4]**Carl de Boor, "Topics in multivariate approximation theory,"*in Topics in Numerical Analysis*(P. Turner, ed.), Lecture Notes in Math., vol. 965, Springer-Verlag, Berlin and New York, 1982, pp. 39-78. MR**690430 (84i:41047)****[5]**Carl de Boor & Klaus Höllig, "*B*-splines from parallelepipeds,"*J. Analyse Math.*, v. 42, 1982/83, pp. 99-115.**[6]**W. Dahmen & C. A. Micchelli, "Numerical algorithms for least squares approximation by multivariate*B*-splines," in*Numerical Methods of Approximation Theory*, vol. 6 (Collatz, Meinardus, and Werner, eds.), Birkhäuser, Basel, 1981.**[7]**G. B. Dantzig,*Linear Programming and Extensions*, Princeton Univ. Press, Princeton, N. J., 1963. MR**0201189 (34:1073)****[8]**H. Hakopian, "Multivariate spline functions,*B*-spline basis and polynomial interpolations,"*SIAM J. Numer. Anal.*, v. 19, 1982, pp. 510-517. MR**656466 (83h:41009)****[9]**Klaus Höllig, "A remark on multivariate*B*-splines,"*J. Approx. Theory*, v. 33, 1981, pp. 119-125. MR**643907 (84h:41015)****[10]**Olvi Mangasarian,*Linear Programming Lecture Notes*, Manuscript, 1978.**[11]**R. H. J. Gmelig Meyling,*An Algorithm for Constructing Configurations of Knots for Bivariate B-splines*, Mathematisch Instituut, Universiteit van Amsterdam, Report 85-06, 1985.**[12]**C. A. Micchelli, "On a numerically efficient method for computing multivariate*B*-splines," in*Multivariate Approximation Theory*(W. Schempp and K. Zeller, eds.), ISNM**51**, Birkhäuser, Basel, 1979.**[13]**C. A. Micchelli, "A constructive approach to Kergin interpolation in : Multivariate*B*-splines and Lagrange interpolation,"*Rocky Mountain J. Math.*, v. 10, 1980, pp. 485-497. MR**590212 (84i:41002)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1988-0917827-2

Keywords:
*B*-spline,
simplex spline,
multivariate,
recurrence relation,
linear programming,
simplex method

Article copyright:
© Copyright 1988
American Mathematical Society