EpsilonGCD - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

Online Help

All Products    Maple    MapleSim


SNAP

  

EpsilonGCD

  

compute an epsilon-GCD for a pair of univariate numeric polynomials

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

EpsilonGCD(a, b, z, tau = eps)

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

Description

• 

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.

Examples

(1)

(2)

(3)

(4)

(5)

(6)

(7)

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.

  

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.

See Also

SNAP[DistanceToCommonDivisors]

SNAP[QuasiGCD]

 


Download Help Document