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ratrecon

rational function reconstruction

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

ratrecon(u, m, x, N, D)

Parameters

u, m

-

polynomials in x

x

-

name

N, D

-

(optional) non-negative integers

Description

• 

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,xN, degreed,xD, 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&plus;D<degreem&comma;x

deg&lsqb;x&rsqb;GCDd&comma;m&equals;0

• 

If the integers N and D are not specified, they both default to be the integer floordegreem,x12).

• 

Note, in order to use this routine to reconstruct a rational function r&equals;nd from u satisfying r&equals;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&equals;xk corresponds to computing the N,D Pade approximate to the series u of order Oxk.

• 

For the special case of N&equals;0, the polynomial dn is the inverse of u in Fxm provided u and m are relatively prime.

Examples

sconvertseries&ExponentialE;x&comma;x&comma;polynom

s:=1&plus;x&plus;12x2&plus;16x3&plus;124x4&plus;1120x5

(1)

ratrecons&comma;x6&comma;x&comma;3&comma;2

20&plus;13x3&plus;3x2&plus;12xx28x&plus;20

(2)

ratrecons&comma;x6&comma;x&comma;2&comma;3

3x224x60x39x2&plus;36x60

(3)

ratrecons&comma;x6&comma;x&comma;3&comma;3

Error, (in ratrecon) degree bounds too big

ratreconx2&plus;1&comma;x3&comma;x&comma;1&comma;1

FAIL

(4)

rratreconx1&comma;x32&comma;x&comma;0&comma;2

r:=1x2&plus;x&plus;1

(5)

remx1r&comma;x32&comma;x

1

(6)

See Also

convert/ratpoly

gcdex

iratrecon

Ratrecon

 


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