Package edu.jas.ufd

Class FactorAbstract<C extends GcdRingElem<C>>

java.lang.Object

edu.jas.ufd.FactorAbstract<C>

Type Parameters:

C - coefficient type

All Implemented Interfaces:

Factorization<C>, java.io.Serializable

Direct Known Subclasses:

FactorAbsolute, FactorInteger, FactorQuotient, FactorRealAlgebraic, FactorRealReal

public abstract class FactorAbstract<C extends GcdRingElem<C>>
extends java.lang.Object
implements Factorization<C>

Abstract factorization algorithms class. This class contains implementations of all methods of the Factorization interface, except the method for factorization of a squarefree polynomial. The methods to obtain squarefree polynomials delegate the computation to the GreatestCommonDivisor classes and are included for convenience.

Author:

Heinz Kredel

See Also:

FactorFactory, Serialized Form

Field Summary

Fields

Modifier and Type	Field	Description
protected GreatestCommonDivisorAbstract<C>	engine	Gcd engine for base coefficients.
protected SquarefreeAbstract<C>	sengine	Squarefree decompositon engine for base coefficients.

Constructor Summary

Constructors

Modifier	Constructor	Description
protected	FactorAbstract()	No argument constructor.
	FactorAbstract(RingFactory<C> cfac)	Constructor.

Method Summary

Modifier and Type	Method	Description
java.util.SortedMap<GenPolynomial<C>,java.lang.Long>	baseFactors(GenPolynomial<C> P)	Univariate GenPolynomial factorization.
java.util.List<GenPolynomial<C>>	baseFactorsRadical(GenPolynomial<C> P)	Univariate GenPolynomial factorization ignoring multiplicities.
abstract java.util.List<GenPolynomial<C>>	baseFactorsSquarefree(GenPolynomial<C> P)	Univariate GenPolynomial factorization of a squarefree polynomial.
GenPolynomial<C>	basePrimitivePart(GenPolynomial<C> P)	GenPolynomial base primitive part.
java.util.SortedMap<GenPolynomial<C>,java.lang.Long>	factors(GenPolynomial<C> P)	GenPolynomial factorization.
long	factorsDegree(java.util.SortedMap<GenPolynomial<C>,java.lang.Long> F)	Degree of a factorization.
java.util.List<GenPolynomial<C>>	factorsRadical(GenPolynomial<C> P)	GenPolynomial factorization ignoring multiplicities.
java.util.List<GenPolynomial<C>>	factorsRadical(java.util.List<GenPolynomial<C>> L)	GenPolynomial list factorization ignoring multiplicities.
java.util.List<GenPolynomial<C>>	factorsSquarefree(GenPolynomial<C> P)	GenPolynomial factorization of a squarefree polynomial, using Kronecker substitution.
java.util.List<GenPolynomial<C>>	factorsSquarefreeKronecker(GenPolynomial<C> P)	GenPolynomial factorization of a squarefree polynomial, using Kronecker substitution.
java.util.List<GenPolynomial<C>>	factorsSquarefreeOptimize(GenPolynomial<C> P)	GenPolynomial factorization of a multivariate squarefree polynomial, using Kronecker substitution and variable order optimization.
boolean	isFactorization(GenPolynomial<C> P, java.util.List<GenPolynomial<C>> F)	GenPolynomial is factorization.
boolean	isFactorization(GenPolynomial<C> P, java.util.SortedMap<GenPolynomial<C>,java.lang.Long> F)	GenPolynomial is factorization.
boolean	isIrreducible(GenPolynomial<C> P)	GenPolynomial test if is irreducible.
boolean	isRecursiveFactorization(GenPolynomial<GenPolynomial<C>> P, java.util.SortedMap<GenPolynomial<GenPolynomial<C>>,java.lang.Long> F)	GenPolynomial is factorization.
boolean	isReducible(GenPolynomial<C> P)	GenPolynomial test if a non trivial factorization exists.
boolean	isSquarefree(GenPolynomial<C> P)	GenPolynomial test if is squarefree.
java.util.List<GenPolynomial<C>>	normalizeFactorization(java.util.List<GenPolynomial<C>> F)	Normalize factorization. p'_i > 0 for i > 1 and p'_1 !
GenPolynomial<C>	primitivePart(GenPolynomial<C> P)	GenPolynomial primitive part.
java.util.SortedMap<GenPolynomial<GenPolynomial<C>>,java.lang.Long>	recursiveFactors(GenPolynomial<GenPolynomial<C>> P)	Recursive GenPolynomial factorization.
java.util.List<GenPolynomial<GenPolynomial<C>>>	recursiveFactorsSquarefree(GenPolynomial<GenPolynomial<C>> P)	Recursive GenPolynomial factorization of a squarefree polynomial.
java.util.SortedMap<GenPolynomial<C>,java.lang.Long>	squarefreeFactors(GenPolynomial<C> P)	GenPolynomial squarefree factorization.
GenPolynomial<C>	squarefreePart(GenPolynomial<C> P)	GenPolynomial greatest squarefree divisor.
java.lang.String	toString()	Get the String representation.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, wait, wait, wait

Field Detail

engine

protected final GreatestCommonDivisorAbstract<C extends GcdRingElem<C>> engine

Gcd engine for base coefficients.

sengine

protected final SquarefreeAbstract<C extends GcdRingElem<C>> sengine

Squarefree decompositon engine for base coefficients.

Constructor Detail

FactorAbstract

protected FactorAbstract()

No argument constructor.

FactorAbstract

public FactorAbstract(RingFactory<C> cfac)

Constructor.

Parameters:

cfac - coefficient ring factory.

Method Detail

toString

public java.lang.String toString()

Get the String representation.

Overrides:

toString in class java.lang.Object

See Also:

Object.toString()

isIrreducible

public boolean isIrreducible(GenPolynomial<C> P)

GenPolynomial test if is irreducible.

Specified by:

isIrreducible in interface Factorization<C extends GcdRingElem<C>>

Parameters:

P - GenPolynomial.

Returns:

true if P is irreducible, else false.

isReducible

public boolean isReducible(GenPolynomial<C> P)

GenPolynomial test if a non trivial factorization exists.

Specified by:

isReducible in interface Factorization<C extends GcdRingElem<C>>

Parameters:

P - GenPolynomial.

Returns:

true if P is reducible, else false.

isSquarefree

public boolean isSquarefree(GenPolynomial<C> P)

GenPolynomial test if is squarefree.

Specified by:

isSquarefree in interface Factorization<C extends GcdRingElem<C>>

Parameters:

P - GenPolynomial.

Returns:

true if P is squarefree, else false.

factorsSquarefreeOptimize

public java.util.List<GenPolynomial<C>> factorsSquarefreeOptimize(GenPolynomial<C> P)

GenPolynomial factorization of a multivariate squarefree polynomial, using Kronecker substitution and variable order optimization.

Parameters:

P - squarefree and primitive! (respectively monic) multivariate GenPolynomial over the ring C.

Returns:

[p_1,...,p_k] with P = prod_{i=1,...,r} p_i.

factorsSquarefree

public java.util.List<GenPolynomial<C>> factorsSquarefree(GenPolynomial<C> P)

GenPolynomial factorization of a squarefree polynomial, using Kronecker substitution.

Specified by:

factorsSquarefree in interface Factorization<C extends GcdRingElem<C>>

Parameters:

P - squarefree and primitive! (respectively monic) GenPolynomial.

Returns:

[p_1,...,p_k] with P = prod_{i=1,...,r} p_i.

factorsSquarefreeKronecker

public java.util.List<GenPolynomial<C>> factorsSquarefreeKronecker(GenPolynomial<C> P)

GenPolynomial factorization of a squarefree polynomial, using Kronecker substitution.

Parameters:

P - squarefree and primitive! (respectively monic) GenPolynomial.

Returns:

[p_1,...,p_k] with P = prod_{i=1,...,r} p_i.

baseFactorsRadical

public java.util.List<GenPolynomial<C>> baseFactorsRadical(GenPolynomial<C> P)

Univariate GenPolynomial factorization ignoring multiplicities.

Parameters:

P - GenPolynomial in one variable.

Returns:

[p_1, ..., p_k] with P = prod_{i=1,...,k} p_i**{e_i} for some e_i.

baseFactors

public java.util.SortedMap<GenPolynomial<C>,java.lang.Long> baseFactors(GenPolynomial<C> P)

Univariate GenPolynomial factorization.

Parameters:

P - GenPolynomial in one variable.

Returns:

[p_1 -> e_1, ..., p_k -> e_k] with P = prod_{i=1,...,k} p_i**e_i.

baseFactorsSquarefree

public abstract java.util.List<GenPolynomial<C>> baseFactorsSquarefree(GenPolynomial<C> P)

Univariate GenPolynomial factorization of a squarefree polynomial.

Parameters:

P - squarefree and primitive! GenPolynomial in one variable.

Returns:

[p_1, ..., p_k] with P = prod_{i=1,...,k} p_i.

factorsRadical

public java.util.List<GenPolynomial<C>> factorsRadical(GenPolynomial<C> P)

GenPolynomial factorization ignoring multiplicities.

Specified by:

factorsRadical in interface Factorization<C extends GcdRingElem<C>>

Parameters:

P - GenPolynomial.

Returns:

[p_1, ..., p_k] with P = prod_{i=1,...,k} p_i**{e_i} for some e_i.

factorsRadical

public java.util.List<GenPolynomial<C>> factorsRadical(java.util.List<GenPolynomial<C>> L)

GenPolynomial list factorization ignoring multiplicities.

Parameters:

L - list of GenPolynomials.

Returns:

[p_1, ..., p_k] with p = prod_{i=1,...,k} p_i**{e_i} for some e_i for all p in L.

factors

public java.util.SortedMap<GenPolynomial<C>,java.lang.Long> factors(GenPolynomial<C> P)

GenPolynomial factorization.

Specified by:

factors in interface Factorization<C extends GcdRingElem<C>>

Parameters:

P - GenPolynomial.

Returns:

[p_1 -> e_1, ..., p_k -> e_k] with P = prod_{i=1,...,k} p_i**e_i.

squarefreePart

public GenPolynomial<C> squarefreePart(GenPolynomial<C> P)

GenPolynomial greatest squarefree divisor. Delegates computation to a GreatestCommonDivisor class.

Specified by:

squarefreePart in interface Factorization<C extends GcdRingElem<C>>

Parameters:

P - GenPolynomial.

Returns:

squarefree(P).

primitivePart

public GenPolynomial<C> primitivePart(GenPolynomial<C> P)

GenPolynomial primitive part. Delegates computation to a GreatestCommonDivisor class.

Parameters:

P - GenPolynomial.

Returns:

primitivePart(P).

basePrimitivePart

public GenPolynomial<C> basePrimitivePart(GenPolynomial<C> P)

GenPolynomial base primitive part. Delegates computation to a GreatestCommonDivisor class.

Parameters:

P - GenPolynomial.

Returns:

basePrimitivePart(P).

squarefreeFactors

public java.util.SortedMap<GenPolynomial<C>,java.lang.Long> squarefreeFactors(GenPolynomial<C> P)

GenPolynomial squarefree factorization. Delegates computation to a GreatestCommonDivisor class.

Specified by:

squarefreeFactors in interface Factorization<C extends GcdRingElem<C>>

Parameters:

P - GenPolynomial.

Returns:

[p_1 -> e_1, ..., p_k -> e_k] with P = prod_{i=1,...,k} p_i**e_i.

isFactorization

public boolean isFactorization(GenPolynomial<C> P,
                               java.util.List<GenPolynomial<C>> F)

GenPolynomial is factorization.

Specified by:

isFactorization in interface Factorization<C extends GcdRingElem<C>>

Parameters:

P - GenPolynomial.

F - = [p_1,...,p_k].

Returns:

true if P = prod_{i=1,...,r} p_i, else false.

isFactorization

public boolean isFactorization(GenPolynomial<C> P,
                               java.util.SortedMap<GenPolynomial<C>,java.lang.Long> F)

GenPolynomial is factorization.

Specified by:

isFactorization in interface Factorization<C extends GcdRingElem<C>>

Parameters:

P - GenPolynomial.

F - = [p_1 -> e_1, ..., p_k -> e_k].

Returns:

true if P = prod_{i=1,...,k} p_i**e_i , else false.

factorsDegree

public long factorsDegree(java.util.SortedMap<GenPolynomial<C>,java.lang.Long> F)

Degree of a factorization.

Parameters:

F - a factors map [p_1 -> e_1, ..., p_k -> e_k].

Returns:

sum_{i=1,...,k} degree(p_i)*e_i.

isRecursiveFactorization

public boolean isRecursiveFactorization(GenPolynomial<GenPolynomial<C>> P,
                                        java.util.SortedMap<GenPolynomial<GenPolynomial<C>>,java.lang.Long> F)

GenPolynomial is factorization.

Parameters:

P - GenPolynomial.

F - = [p_1 -> e_1, ..., p_k -> e_k].

Returns:

true if P = prod_{i=1,...,k} p_i**e_i , else false.

recursiveFactorsSquarefree

public java.util.List<GenPolynomial<GenPolynomial<C>>> recursiveFactorsSquarefree(GenPolynomial<GenPolynomial<C>> P)

Recursive GenPolynomial factorization of a squarefree polynomial.

Parameters:

P - squarefree recursive GenPolynomial.

Returns:

[p_1,...,p_k] with P = prod_{i=1, ..., k} p_i.

recursiveFactors

public java.util.SortedMap<GenPolynomial<GenPolynomial<C>>,java.lang.Long> recursiveFactors(GenPolynomial<GenPolynomial<C>> P)

Recursive GenPolynomial factorization.

Parameters:

P - recursive GenPolynomial.

Returns:

[p_1 -> e_1, ..., p_k -> e_k] with P = prod_{i=1,...,k} p_i**e_i.

normalizeFactorization

public java.util.List<GenPolynomial<C>> normalizeFactorization(java.util.List<GenPolynomial<C>> F)

Normalize factorization. p'_i > 0 for i > 1 and p'_1 != 1 if k > 1.

Parameters:

F - = [p_1,...,p_k].

Returns:

F' = [p'_1,...,p'_k].