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.
Download Help Document
What kind of issue would you like to report? (Optional)
Thank you for submitting feedback on this help document. Your feedback will be used
to improve Maple's help in the future.