ratrecon
rational function reconstruction
Calling Sequence
Parameters
Description
Examples
ratrecon(u, m, x, N, D)
u, m
-
polynomials in x
x
name
N, D
(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, degreen,x≤N, degreed,x≤D, and lcoeffd,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:
N+D<degreem,x
degxGCDd,m=0
If the integers N and D are not specified, they both default to be the integer floordegreem,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 Oxk.
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≔convertseriesexpx,x,polynom
s≔1+x+12x2+16x3+124x4+1120x5
ratrecons,x6,x,3,2
20+13x3+3x2+12xx2−8x+20
ratrecons,x6,x,2,3
−3x2−24x−60x3−9x2+36x−60
ratrecons,x6,x,3,3
Error, (in ratrecon) degree bounds too big
ratreconx2+1,x3,x,1,1
FAIL
r≔ratreconx−1,x3−2,x,0,2
r≔1x2+x+1
remx−1r,x3−2,x
1
See Also
convert/ratpoly
gcdex
iratrecon
Ratrecon
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