Block-based Thiele-like blending rational interpolation

Thiele-type continued fraction interpolation may be the favoured nonlinear interpolation in the sense that it is constructed by means of the inverse differences which can be calculated recursively and produce useful intermediate results. However, Thiele's interpolation is in fact a point-based interpolation by which we mean that a new interpolating continued fraction with one more degree of its numerator or denominator polynomial is obtained by adding a tail to the current one, or in other words, this can be reshaped simply by adding a new support point to the current set of support points once at a time. In this paper we introduce so-called block-based inverse differences to extend the point-based Thiele-type interpolation to the block-based Thiele-like blending rational interpolation. The construction process may be outlined as follows: first of all, divide the original set of support points into some subsets (blocks), then construct each block by using whatever interpolation means, linear or rational, and finally assemble these blocks by Thiele's method to shape the whole interpolation scheme. Clearly, many flexible rational interpolation schemes can be obtained in this case including the classical Thiele's continued fraction interpolation as its special case. As an extension, a bivariate analogy is also discussed and numerical examples are given to show the effectiveness of our method.