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SNAP

  

AreCoprime

  

determine if two numeric polynomials are relatively prime up to a given error bound

  

IsSingular

  

determine if a numeric polynomial has a double root up to a given error bound

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

AreCoprime(a, b, z, eps, out)

IsSingular(a, z, eps, out)

Parameters

a, b

-

univariate numeric polynomials

z

-

name; indeterminate for a and b

eps

-

non-negative numeric; error bound

out

-

(optional) equation of the form output = obj where obj is 'BC' or a list containing one or more instances of this name; select result objects to compute

Description

• 

The AreCoprime(a, b, z, eps) command checks whether univariate numeric polynomials a, b in z remain coprime after perturbations of order eps. This is done by computing reliable lower and upper bounds on the distance between the pair (a, b) and the set of the univariate complex polynomial pairs in z with degrees that do not exceed those of a and b, and that have at least one common root. (See SNAP[DistanceToCommonDivisors].)

  

The lower bound LB is obtained using the SNAP[DistanceToCommonDivisors] function. The upper bound UP is the minimum between the 1-norm of a and the 1-norm of b.

  

If eps > UP, false is returned;

  

if eps < LB, true is returned;

  

if LB <= eps <= UP, FAIL is returned because it is impossible to provide a reliable answer.

• 

The IsSingular(a, z, eps) command checks whether the univariate numeric polynomial a in z has common roots up to perturbations of order eps. It essentially calls AreCoprime(a, b, z) with b = diff(a, z).

• 

The output option (out) determines the content of the returned expression sequence.

  

As specified by the out option, Maple returns an expression sequence containing one or more BC, which is the list [v, u] of the numeric polynomials in z that satisfy av+bu=1 and degreeu&comma;z<degreea&comma;z and degreev&comma;z<degreeb&comma;z (bezout coefficients for coprime polynomials a and b). This list is empty if the routine returns false or FAIL.

Examples

withSNAP&colon;

a0.1z2&plus;1.5z0.2

a:=0.1z2&plus;1.5z0.2

(1)

b0.2z3&plus;0.15

b:=0.2z3&plus;0.15

(2)

AreCoprimea&comma;b&comma;z&comma;0.5

false

(3)

AreCoprimea&comma;b&comma;z&comma;0.1

true

(4)

cAreCoprimea&comma;b&comma;z&comma;0.1&comma;output&equals;&apos;BC&apos;

c:=true&comma;0.871004100817766z20.112232907263963z0.0585145926760834&comma;0.435502050408883z&plus;6.58864720976522

(5)

fnormalexpandac21&plus;bc22

1.

(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.

See Also

expand

SNAP[DistanceToCommonDivisors]

SNAP[DistanceToSingularPolynomials]

 


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