rational function reconstruction
ratrecon(u, m, x, N, D)
polynomials in x
(optional) non-negative integers
The purpose of this routine is to reconstruct a rational function nd in x from its image umodm where u and m are polynomials in Fx, and F is a field of characteristic 0. Given positive integers N and D, ratrecon returns the unique rational function r=nd if it exists satisfying r=umodm, degree⁡n,x≤N, degree⁡d,x≤D, and lcoeff⁡d,x=1. 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 F provided the following conditions hold:
If the integers N and D are not specified, they both default to be the integer floordegree⁡m,x−12).
Note, in order to use this routine to reconstruct a rational function r=nd from u satisfying r=umodm, the modulus m being used must be chosen to be relatively prime to d. Otherwise the reconstruction returns FAIL.
The special case of m=xk corresponds to computing the N,D Pade approximate to the series u of order O⁡xk.
For the special case of N=0, the polynomial dn is the inverse of u in Fxm provided u and m are relatively prime.
s ≔ convert⁡series⁡ⅇx,x,polynom
Error, (in ratrecon) degree bounds too big
r ≔ ratrecon⁡x−1,x3−2,x,0,2
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