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 $\left(a,b\right)\.U=\left(a\',b\'\right)$ where (a', b') is the last basis accepted by the algorithm of [2].

$\mathrm{with}\left(\mathrm{SNAP}\right)\:$

$a≔z^6-12.4z^5+50.18112+62.53z^4-163.542z^3+232.9776z^2-170.69184z$

$a≔z^6-12.4z^5+50.18112+62.53z^4-163.542z^3+232.9776z^2-170.69184z$

$b≔z^5-17.6z^4+118.26z^3-372.992z^2-274.09272+538.3333z$

$b≔z^5-17.6z^4+118.26z^3-372.992z^2-274.09272+538.3333z$

$\mathrm{EuclideanReduction}\left(a\,b\,z\right)$

$\left[4.z^4-45.3201452919812z^3+182.643498183852z^2-301.305647387541z+164.902292707462\,3.48999306545107z^3-25.4412393835560z^2+55.5055044834605z-34.9173360478195\right]$

$\mathrm{EuclideanReduction}\left(a\,b\,z\,\mathrm{\tau }=1.\times {10}^{\mathrm{-8}}\right)$

$\left[0.250000000000000z^2-0.875000000000161z+0.659999999999995\,-1.07691633388640\times {10}^{\mathrm{-14}}z-5.54556400800266\times {10}^{\mathrm{-14}}\right]$

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.