Fast and stable computation of the barycentric representation of rational interpolants

Abstract

We present numerical algorithms for solving the rational interpolation problem (r.i.p.) when both the numerators and the denominators are expressed with respect to the Newton basis. These algorithms generalize some classical procedures which were developed for dealing with polynomials given in power form. Our approach is based on the reduction of the r.i.p. to computing a block triangular decomposition of certain structured matrices. These matrices generalize the Hankel structure in the sense that they belong to the kernel of a suitable displacement operator which reduces to that defining the Hankel structure when the interpolation nodes collapse to zero. The computational cost and the stability properties of the resulting algorithms depend on the criterion we choose for the step size regulation. Jumping from a nonsingular leading principal submatrix to the next one leads to an algorithm for r.i.p. which reaches the optimal sequential complexity. Conversely, jumping from a well-conditioned leading principal submatrix to a next one allows us to control the accuracy of the computed approximations.