The purpose of this routine is to reconstruct a rational function  in x from its image  where u and m are polynomials in , and  is a field of characteristic 0. Given positive integers N and D, ratrecon returns the unique rational function  if it exists satisfying , , , and . Otherwise ratrecon returns FAIL, indicating that no such polynomials n and d exist.  The rational function r exists and is unique up to multiplication by a constant in  provided the following conditions hold:

If the integers N and D are not specified, they both default to be the integer .

Note, in order to use this routine to reconstruct a rational function  from u satisfying , the modulus m being used must be chosen to be relatively prime to d. Otherwise the reconstruction returns FAIL.

The special case of  corresponds to computing the N,D Pade approximate to the series u of order .

For the special case of , the polynomial  is the inverse of u in  provided u and m are relatively prime.

Error, (in ratrecon) degree bounds too big