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

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Numerical Computations : SNAP Package : SNAP/EpsilonGCD

SNAP[EpsilonGCD] - compute an epsilon-GCD for a pair of univariate numeric polynomials

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

withSNAP:

a:=0.2313432836z4+0.003500000000z30.1753694030z20.3397276119z0.0003395522388

a:=0.2313432836z4+0.003500000000z30.1753694030z20.3397276119z0.0003395522388

(1)

b:=0.2313432836z3+0.003731343284z20.1753731343z0.3395522388

b:=0.2313432836z3+0.003731343284z20.1753731343z0.3395522388

(2)

EpsilonGCDa,b,z

0.125000000000000z30.00201612903232778z2+0.0947580644934703z+0.183467741918054,2.9049738724478010-10

(3)

a:=z2+3.1z2

a:=z2+3.1z2

(4)

b:=2z3+1.5

b:=2z3+1.5

(5)

EpsilonGCDa,b,z

FAIL

(6)

DistanceToCommonDivisorsa,b,z

0.876183368229130

(7)

See Also

SNAP[DistanceToCommonDivisors], SNAP[QuasiGCD]

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.


Download Help Document

Was this information helpful?



Please add your Comment (Optional)
E-mail Address (Optional)
What is ? This question helps us to combat spam