The stable evaluation of multivariate simplex splines
Author:
Thomas A. Grandine
Journal:
Math. Comp. 50 (1988), 197205
MSC:
Primary 65D07; Secondary 41A15, 41A63
MathSciNet review:
917827
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Abstract: This paper gives a general method for the stable evaluation of multivariate simplex splines, based on the wellknown 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.
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 [1]
 Robert G. Bland, "New finite pivoting rules for the simplex method," Math. Oper. Res., v. 2, 1977, pp. 103107. MR 0459599 (56:17791)
 [2]
 Carl de Boor, "Splines as linear combinations of Bsplines," in Approximation Theory II (G.G. Lorentz, C. K. Chui, and L. L. Schumaker, eds.), Academic Press, New York, 1976, pp. 147.
 [3]
 Carl de Boor, A Practical Guide to Splines, SpringerVerlag, 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, SpringerVerlag, Berlin and New York, 1982, pp. 3978. MR 690430 (84i:41047)
 [5]
 Carl de Boor & Klaus Höllig, "Bsplines from parallelepipeds," J. Analyse Math., v. 42, 1982/83, pp. 99115.
 [6]
 W. Dahmen & C. A. Micchelli, "Numerical algorithms for least squares approximation by multivariate Bsplines," in Numerical Methods of Approximation Theory, vol. 6 (Collatz, Meinardus, and Werner, eds.), Birkhäuser, Basel, 1981.
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 G. B. Dantzig, Linear Programming and Extensions, Princeton Univ. Press, Princeton, N. J., 1963. MR 0201189 (34:1073)
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 H. Hakopian, "Multivariate spline functions, Bspline basis and polynomial interpolations," SIAM J. Numer. Anal., v. 19, 1982, pp. 510517. MR 656466 (83h:41009)
 [9]
 Klaus Höllig, "A remark on multivariate Bsplines," J. Approx. Theory, v. 33, 1981, pp. 119125. 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 Bsplines, Mathematisch Instituut, Universiteit van Amsterdam, Report 8506, 1985.
 [12]
 C. A. Micchelli, "On a numerically efficient method for computing multivariate Bsplines," 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 Bsplines and Lagrange interpolation," Rocky Mountain J. Math., v. 10, 1980, pp. 485497. MR 590212 (84i:41002)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198809178272
PII:
S 00255718(1988)09178272
Keywords:
Bspline,
simplex spline,
multivariate,
recurrence relation,
linear programming,
simplex method
Article copyright:
© Copyright 1988
American Mathematical Society
