Practical, Reliable, Rational Interpolation

An algorithm incorporating features essential for practical,reliable, rational interpolation is explained. This algorithmgenerates a Thiele-Werner continued fraction representationof the interpolant. A backward error analysis is presented forthe algorithm, as well as for its special cases of Newton polynomialinterpolation and Thiele rational interpolation. This is madepossible by introducing into the Newton method, Thiele methodand Werner method a strategy for selecting the interpolationpoints in an optimal order.