EuclideanReduction - Maple Help

SNAP

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

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

Examples

 > $\mathrm{with}\left(\mathrm{SNAP}\right):$
 > $a≔{z}^{6}-12.4{z}^{5}+50.18112+62.53{z}^{4}-163.542{z}^{3}+232.9776{z}^{2}-170.69184z$
 ${a}{≔}{{z}}^{{6}}{-}{12.4}{}{{z}}^{{5}}{+}{50.18112}{+}{62.53}{}{{z}}^{{4}}{-}{163.542}{}{{z}}^{{3}}{+}{232.9776}{}{{z}}^{{2}}{-}{170.69184}{}{z}$ (1)
 > $b≔{z}^{5}-17.6{z}^{4}+118.26{z}^{3}-372.992{z}^{2}-274.09272+538.3333z$
 ${b}{≔}{{z}}^{{5}}{-}{17.6}{}{{z}}^{{4}}{+}{118.26}{}{{z}}^{{3}}{-}{372.992}{}{{z}}^{{2}}{-}{274.09272}{+}{538.3333}{}{z}$ (2)
 > $\mathrm{EuclideanReduction}\left(a,b,z\right)$
 $\left[{4.}{}{{z}}^{{4}}{-}{45.3201452919812}{}{{z}}^{{3}}{+}{182.643498183852}{}{{z}}^{{2}}{-}{301.305647387541}{}{z}{+}{164.902292707462}{,}{3.48999306545107}{}{{z}}^{{3}}{-}{25.4412393835561}{}{{z}}^{{2}}{+}{55.5055044834610}{}{z}{-}{34.9173360478198}\right]$ (3)
 > $\mathrm{EuclideanReduction}\left(a,b,z,\mathrm{\tau }=1.×{10}^{-8}\right)$
 $\left[{0.250000000000000}{}{{z}}^{{2}}{-}{0.875000000000147}{}{z}{+}{0.659999999999992}{,}{-}{9.32587340685131}{×}{{10}}^{{-15}}{}{z}{-}{5.26245713672324}{×}{{10}}^{{-14}}\right]$ (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.