   ISSN 1088-6842(online) ISSN 0025-5718(print)

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 ${h^r}{k^s}{f_{x \ldots x(r\;{\text {times}})y \ldots y(s\;{\text {times}})}}({x_0},{y_0})$ 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 $r + s = \left ( {\begin {array}{*{20}{c}} {n,} \hfill & {{\text {for}}\;m = 2} \hfill \\ {n + 1,} \hfill & {{\text {for}}\;m = 3,4} \hfill \\ \end {array} } \right ),$ are given here in convenient matrix form.

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• Herbert E. Salzer and Genevieve M. Kimbro, Tables for bivariate osculatory interpolation over a Cartesian grid., Convair Division of General Dynamics Corporation, San Diego, Calif., 1958. MR 0099106
• Herbert E. Salzer, Some new divided difference algorithms for two variables, On numerical approximation. Proceedings of a Symposium, Madison, April 21-23, 1958, Edited by R. E. Langer. Publication no. 1 of the Mathematics Research Center, U.S. Army, the University of Wisconsin, The University of Wisconsin Press, Madison, 1959, pp. 61–98. MR 0102166
• Herbert E. Salzer, Divided differences for functions of two variables for irregularly spaced arguments, Numer. Math. 6 (1964), 68–77. MR 165659, DOI https://doi.org/10.1007/BF01386056

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