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Formulas for bivariate hyperosculatory interpolation


Author: Herbert E. Salzer
Journal: Math. Comp. 25 (1971), 119-133
MSC: Primary 65.20
DOI: https://doi.org/10.1090/S0025-5718-1971-0287671-8
MathSciNet review: 0287671
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Abstract: For a given function $ f(x,y)$, bivariate hyperosculatory interpolation formulas are obtained by employing a suitably constructed binary nic $ {p_n}(x,y)$ that is fitted to the values of $ f(x,y)$ and its first and second partial derivatives at the m points $ ({x_i},{y_i})$ of a rectangular $ h \times k$ Cartesian grid, where ( $ ({x_i},{y_i}) = ({x_0} + {p_i}h,{y_0} + {q_i}k),{p_i}$ and $ {q_i}$ are small integers $ \geqq 0,i = 0(1)m - 1,m \geqq 2$. In terms of the variables (p, q), where $ x = {x_0} + ph,y = {y_0} + qk$ (and $ f(x,y) = F(p,q)$), we have $ {p_n}(x,y) = {P_n}(p,q)$. Often, for $ {P_n}(p,q)$ having a specified desirable form, this problem turns out to be insoluble for every configuration of the points $ ({x_i},{y_i})$. When this is not the case, it generally requires considerable investigation to find a practical configuration of points $ ({x_i},{y_i})$ for which there is a solution of the form $ {P_n}(p,q)$. Formulas are found for choices of $ {P_n}(p,q)$, and soluble configurations of points $ ({x_i},{y_i})$, that have dominant remainder terms in

$\displaystyle {h^r}{k^s}{f_{x \ldots x(r\;{\text{times}})y \ldots y(s\;{\text{times}})}}({x_0},{y_0})$

whose orders $ r + s$ are as high as possible. Three two-point formulas, two three-point formulas and one four-point formula, including all remainder terms through the order

$\displaystyle r + s = \left( {\begin{array}{*{20}{c}} {n,} \hfill & {{\text{for... ...l \\ {n + 1,} \hfill & {{\text{for}}\;m = 3,4} \hfill \\ \end{array} } \right),$

are given here in convenient matrix form.

References [Enhancements On Off] (What's this?)

  • [1] H. E. Salzer & G. M. Kimbro, Tables for Bivariate Osculatory Interpolation over a Cartesian Grid, Convair Astronautics, 1958, 40 pp. Note especially the introductory text, by the present writer, on pp. 1-23. MR 0099106 (20:5550)
  • [2] H. E. Salzer, Some New Divided Difference Algorithms for Two Variables, Proc. Sympos. Numerical Approximation (Madison, Wis., 1958), Univ. of Wisconsin Press, Madison, Wis., 1959, pp. 61-98. MR 21 #960. MR 0102166 (21:960)
  • [3] H. E. Salzer, "Divided differences for functions of two variables for irregularly spaced arguments," Numer. Math., v. 6, 1964, pp. 68-77. MR 29 #2939. MR 0165659 (29:2939)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1971-0287671-8
Keywords: Bivariate hyperosculatory interpolation, bivariate interpolation, hyperosculatory interpolation, Cartesian interpolation, interpolation, remainder formulas
Article copyright: © Copyright 1971 American Mathematical Society

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