Formulas for bivariate hyperosculatory interpolation

Abstract

For a given function f ( x , y ) f(x,y) , bivariate hyperosculatory interpolation formulas are obtained by employing a suitably constructed binary nic p n ( x , y ) {p_n}(x,y) that is fitted to the values of f ( x , y ) f(x,y) and its first and second partial derivatives at the m points ( x i , y i ) ({x_i},{y_i}) of a rectangular h × k h \times k Cartesian grid, where ( ( x i , y i ) = ( x 0 + p i h , y 0 + q i k ) , p i ({x_i},{y_i}) = ({x_0} + {p_i}h,{y_0} + {q_i}k),{p_i} and q i {q_i} are small integers ≧ 0 , i = 0 ( 1 ) m − 1 , m ≧ 2 \geqq 0,i = 0(1)m - 1,m \geqq 2 . In terms of the variables (p, q), where x = x 0 + p h , y = y 0 + q k x = {x_0} + ph,y = {y_0} + qk (and f ( x , y ) = F ( p , q ) f(x,y) = F(p,q) ), we have p n ( x , y ) = P n ( p , q ) {p_n}(x,y) = {P_n}(p,q) . Often, for P n ( p , q ) {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 ) ({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 ) ({x_i},{y_i}) for which there is a solution of the form P n ( p , q ) {P_n}(p,q) . Formulas are found for choices of P n ( p , q ) {P_n}(p,q) , and soluble configurations of points ( x i , y i ) ({x_i},{y_i}) , that have dominant remainder terms in \[ h r k s f x … x ( r times ) y … y ( s times ) ( x 0 , y 0 ) {h^r}{k^s}{f_{x \ldots x(r\;{\text {times}})y \ldots y(s\;{\text {times}})}}({x_0},{y_0}) \] whose orders r + s 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 \[ r + s = ( n , a m p ; for m = 2 n + 1 , a m p ; for m = 3 , 4 ) , r + s = \left ( {\begin {array}{*{20}{c}} {n,} \hfill & {{\text {for}}\;m = 2} \hfill \\ {n + 1,} \hfill & {{\text {for}}\;m = 3,4} \hfill \\ \end {array} } \right ), \] are given here in convenient matrix form.