A Weakly Stable Algorithm for Padé Approximants and the Inversion of Hankel Matrices

Abstract

A new algorithm, Algorithm NPADE, is presented for numerically computing Padé approximants in a weakly stable fashion. By this, it is meant that if the problem is well conditioned, then Algorithm NPADE produces a good solution. No restrictions are imposed on the problem being solved. Except in certain pathological cases, the cost of the algorithm is $O( n^2 )$, where n is the maximum degree of the polynomials comprising the Padé approximant. The operation of Algorithm NPADE is controlled by a single parameter. Bounds are obtained for the computed solution and it is seen that they are a function of this parameter. Experimental results show that the bounds, while crude, reflect the actual behavior of the error. In addition, it is shown how better bounds can easily be obtained a posteriors. As another application of Algorithm NPADE, it is shown that it can be used to compute stably, in a weak sense, the inverse of a Hankel or Toeplitz matrix.