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EuclideanReduction

  

compute the smallest degree pair of univariate polynomials by Euclidean-like unimodular reduction

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

EuclideanReduction(a, b, z, tau = eps, out)

Parameters

a, b

-

univariate numeric polynomials

z

-

name; indeterminate for a and b

tau = eps

-

(optional) equation where eps is of type numeric and non-negative; stability parameter

out

-

(optional) equation of the form output = obj where obj is 'UR' or a list containing one or more instances of this name; select result objects to compute

Description

• 

The EuclideanReduction(a, b, z) command returns the last numerically well-conditioned basis accepted by the Coprime algorithm [2].  This corresponds to the smallest degree pair of polynomials in the sequence of numerically well-behaved polynomial remainders that can be obtained from (a,b) by unimodular reduction.

• 

It thus provides the user with a pair of polynomials that generates the same ideal generated by (a,b) but with degrees that are, in general, much smaller. Furthermore, the highest degree component of such a reduced pair is a good candidate for an epsilon-GCD of (a,b).

• 

The optional stability parameter tau can be set to any non-negative value eps to control the quality of the output. Decreasing eps yields a more reliable solution. Increasing eps reduces the degrees of the returned basis.

  

As specified by the out option, Maple returns an expression sequence containing the following:

  

* UR contains a 2 by 2 unimodular matrix polynomial U in z such that a,b.U=a',b' where (a', b') is the last basis accepted by the algorithm of [2].

Examples

withSNAP:

az612.4z5+50.18112+62.53z4163.542z3+232.9776z2170.69184z

a:=z612.4z5+50.18112+62.53z4163.542z3+232.9776z2170.69184z

(1)

bz517.6z4+118.26z3372.992z2274.09272+538.3333z

b:=z517.6z4+118.26z3372.992z2274.09272+538.3333z

(2)

EuclideanReductiona,b,z

4.z445.3201452919811z3+182.643498183851z2301.305647387539z+164.902292707461,3.48999306545113z325.4412393835563z2+55.5055044834614z34.9173360478200

(3)

EuclideanReductiona,b,z,τ=1.10-8

0.250000000000000z20.875000000000067z+0.660000000000049,1.1657341758564110-14z+3.4972025275692410-15

(4)

References

  

Beckermann, B., and Labahn, G. "A fast and numerically stable Euclidean-like algorithm for detecting relatively prime numerical polynomials." Journal of Symbolic Computation. Vol. 26, (1998): 691-714.

  

Beckermann, B., and Labahn, G. "When are two numerical polynomials relatively prime?" Journal of Symbolic Computation. Vol. 26, (1998): 677-689.

See Also

SNAP[DistanceToCommonDivisors]

SNAP[EpsilonGCD]

 


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