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QuasiGCD

  

compute Schoenhage's quasi-GCD for a pair of univariate numeric polynomials

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

QuasiGCD(a, b, z, tau = eta)

Parameters

a, b

-

univariate numeric polynomials

z

-

name; indeterminate for a and b

tau = eta

-

(optional) equation where eta is of type numeric and non-negative; stability parameter

Description

• 

The QuasiGCD(a, b, z) command returns a univariate numeric polynomial g with a positive float eta such that g is a quasi-GCD with precision eta for the input polynomials (a,b). (See [3,2] for a definition of a quasi-GCD in the sense of Schoenhage.)

  

This quasi-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 quasi-GCD g=ai provided the 1-norm of bi is small enough in a sense precised 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 QuasiGCD(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:

a0.2313432836z4+0.003500000000z30.1753694030z20.3397276119z0.0003395522388

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

(1)

b0.2313432836z3+0.003731343284z20.1753731343z0.3395522388

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

(2)

QuasiGCDa,b,z

0.125000000000000z30.00201612903232778z2+0.0947580644934703z+0.183467741918054,1.8429775360680010-9

(3)

az2+3.1z2

a:=z2+3.1z2

(4)

b2z3+1.5

b:=2z3+1.5

(5)

QuasiGCDa,b,z

FAIL

(6)

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.

  

Schoenhage, A. "Quasi-GCD computations" Journal of Complexity. Vol. 1, (1985): 118-137.

See Also

SNAP[DistanceToCommonDivisors]

SNAP[EpsilonGCD]

 


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