Algorithm 684: C1- and C2-interplation on triangles with quintic and nonic bivariate polynomials

TLDR: The innovative part of the algorithm is the formulation of a bivariate nonic Hermite polynomial with C2-property contained in modules CBSIDE, CSINSD, and CSHORN, which describes a twice differentiable polynometric representation on a set of triangles.

Abstract: The innovative part of the algorithm is the formulation of a bivariate nonic Hermite polynomial with C2-property [7] contained in modules CBSIDE, CSINSD, and CSHORN. These formulas are of principal value. They describe a twice differentiable polynomial representation on a set of triangles. To be of immediate use, some interfaces were designed, and the new modules were integrated into Renka’s triangle interpolation package [9]. Especially all procedures related to the generation and the handling of triangles are taken or may be taken from that algorithm. Also, the method of derivative estimation is identical with Renka’s global method [S]. The higher derivatives of order n are generated by taking the derivatives of order n 1 as input (n = 2, 3, 4). For the sake of completeness, equivalent routines for Cl-interpolation (once differentiable) are supplied, based on the well known quintic triangular element [l, 3, 41. In general, the Cl-routines are approximately 3 times faster than their C ‘-equivalents and need less memory. The main user interfaces to the package are the routines C2GRID and ClGRID. They interpolate values of a rectangular grid to a given set of irregularly distributed points (scattered data interpolation in two dimensions). Because the array of grid points is scanned for every triangie, the coefficients for each polynomial are computed only once, and the fast evaluation phase of the Taylor representation [7] can be exploited most efficiently. If the mesh lines of the rectangular grid are relatively wide with respect to the irregularly distributed