The EpsilonGCD(a, b, z) command returns a univariate numeric polynomial g with a positive float epsilon such that g is an epsilon-GCD for the input polynomials (a,b). (See [2] for a definition of an epsilon-GCD.)

This epsilon-GCD g is derived from the stable algorithm of [2] as follows. The algorithm of [2] computes a numerical pseudo remainder sequence (ai,bi) for (a,b) in a weakly stable way, accepting only the pairs that are well-conditioned (because the others produce instability). The maximum index i for which (ai,bi) is accepted yields the epsilon-GCD g=ai provided the norm of bi is small enough in a sense given in [2]. The value of eta depends in particular on the value of bi and on the 1-norm of the residual error at the last accepted step.

If the problem is poorly conditioned, the EpsilonGCD(a, b, z) command may return FAIL (rather than a meaningless answer). Here, ill-conditioning is a function of the parameter tau. Its default value is the cubic root of the current value of the Digits variable. Decreasing the value of tau yields a more reliable solution. Increasing the value of tau reduces the risk of failure.

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

$a≔-0.2313432836z^4+0.003500000000z^3-0.1753694030z^2-0.3397276119z-0.0003395522388$

$a≔-0.2313432836z^4+0.003500000000z^3-0.1753694030z^2-0.3397276119z-0.0003395522388$

$b≔-0.2313432836z^3+0.003731343284z^2-0.1753731343z-0.3395522388$

$b≔-0.2313432836z^3+0.003731343284z^2-0.1753731343z-0.3395522388$

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

$0.125000000000000z^3-0.00201612903232778z^2+0.0947580644934703z+0.183467741918054,2.90497387244780\times {10}^{\mathrm{-10}}$

$a≔z^2+3.1z-2$

$a≔z^2+3.1z-2$

$b≔2z^3+1.5$

$b≔2z^3+1.5$

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

$\mathrm{FAIL}$

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

$0.876183368229130$

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.

Corless, R.M.; Gianni, P.M.; Trager, B.M.; and Watt, S.M. "The singular value decomposition for polynomial systems." ISSAC'95, pp. 195-207. ACM Press, 1995.

Karmarkar, N., and Lakshman, Y.N. "Approximate polynomial greatest common divisors and nearest singular polynomials." ISSAC'96, pp. 35-39. ACM Press, 1996.