How to compute polynomial interpolation of a RATIONAL functi

How to compute polynomial interpolation of a RATIONAL function, for example, 1/(1+x^2)?

Solution

The rational interpolation is one of the most difficult methods of interpolation. Its advantages are the high accuracy and absence of the problems which are typical for polynomial interpolation.

The given function has no poles on real axis d defines the interpolating scheme power and, consequently, its accuracy. It seems that the bigger d we choose, the less interpolation error we will get, but this effect has an upper bound

This function is the classical example of a function which cannot be interpolated by a polynomial on an equidistant grid.