compute the smallest degree pair of univariate polynomials by Euclidean-like unimodular reduction
EuclideanReduction(a, b, z, tau = eps, out)
univariate numeric polynomials
name; indeterminate for a and b
tau = eps
(optional) equation where eps is of type numeric and non-negative; stability parameter
(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
The EuclideanReduction(a, b, z) command returns the last numerically well-conditioned basis accepted by the Coprime algorithm . 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 .
a ≔ z6−12.4⁢z5+50.18112+62.53⁢z4−163.542⁢z3+232.9776⁢z2−170.69184⁢z
b ≔ z5−17.6⁢z4+118.26⁢z3−372.992⁢z2−274.09272+538.3333⁢z
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.
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