Newton-Thiele's rational interpolants (Q1587029)

For the interpolation of a bivariate function \(f(x,y)\), one can construct a Newton interpolating polynomial in \(x\) whose coefficients are Newton interpolating polynomials in \(y\). This is the bivariate Newton interpolating polynomial. To construct a rational interpolant one can construct a Thiele continued fraction in \(x\) whose coefficients are Thiele continued fractions in \(y\). This is a branched continued fraction interpolant. Thiele-Newton rational interpolants use a Thiele continued fraction in \(x\) with coefficients that are Newton polynomials in \(y\). Newton-Thiele interpolants are like Thiele-Newton interpolants with the role of Thiele continued fraction and Newton polynomial interchanged. The latter are the subject of this paper. They are constructed using ``blending differences'' which is a new type of differences being a mixture of divided differences and inverse differences. This paper gives a constructive algorithm to compute these interpolants, discusses an expression for the interpolation error and considers the confluent case when the grid of interpolations collapses to one point.