edu.jas.ufd

Class FactorAbsolute<C extends GcdRingElem<C>>

java.lang.Object

edu.jas.ufd.FactorAbstract<C>

edu.jas.ufd.FactorAbsolute<C>

Type Parameters:

C - coefficient type

All Implemented Interfaces:

Factorization<C>, java.io.Serializable

Direct Known Subclasses:

FactorAlgebraic, FactorAlgebraicPrim, FactorComplex, FactorModular, FactorRational

public abstract class FactorAbsolute<C extends GcdRingElem<C>>
extends FactorAbstract<C>

Absolute factorization algorithms class. This class contains implementations of methods for factorization over algebraically closed fields. The required field extension is computed along with the factors. The methods have been tested for prime fields of characteristic zero, that is for BigRational. It might eventually also be used for prime fields of non-zero characteristic, that is with ModInteger. The field extension may yet not be minimal.

Author:

Heinz Kredel

See Also:

Serialized Form

Field Summary

Fields inherited from class edu.jas.ufd.FactorAbstract

engine, sengine

Constructor Summary

Constructors

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

Method Summary

Modifier and Type	Method and Description
PartialFraction<C>	baseAlgebraicPartialFraction(GenPolynomial<C> A, GenPolynomial<C> P) Univariate GenPolynomial algebraic partial fraction decomposition, Absolute factorization for elementary integration algorithm to linear factors.
PartialFraction<C>	baseAlgebraicPartialFractionIrreducibleAbsolute(GenPolynomial<C> A, GenPolynomial<C> P) Univariate GenPolynomial algebraic partial fraction decomposition, via absolute factorization to linear factors.
FactorsMap<C>	baseFactorsAbsolute(GenPolynomial<C> P) GenPolynomial absolute base factorization of a polynomial.
Factors<C>	baseFactorsAbsoluteIrreducible(GenPolynomial<C> P) GenPolynomial base absolute factorization of a irreducible polynomial.
FactorsList<C>	baseFactorsAbsoluteSquarefree(GenPolynomial<C> P) GenPolynomial absolute base factorization of a squarefree polynomial.
FactorsMap<C>	factorsAbsolute(GenPolynomial<C> P) GenPolynomial absolute factorization of a polynomial.
Factors<C>	factorsAbsoluteIrreducible(GenPolynomial<C> P) GenPolynomial absolute factorization of a irreducible polynomial.
FactorsList<C>	factorsAbsoluteSquarefree(GenPolynomial<C> P) GenPolynomial absolute factorization of a squarefree polynomial.
boolean	isAbsoluteFactorization(Factors<C> facs) GenPolynomial is absolute factorization.
boolean	isAbsoluteFactorization(FactorsList<C> facs) GenPolynomial is absolute factorization.
boolean	isAbsoluteFactorization(FactorsMap<C> facs) GenPolynomial is absolute factorization.
boolean	isAbsoluteIrreducible(GenPolynomial<C> P) GenPolynomial test if is absolute irreducible.
java.lang.String	toString() Get the String representation.

Methods inherited from class edu.jas.ufd.FactorAbstract

baseFactors, baseFactorsRadical, baseFactorsSquarefree, basePrimitivePart, factors, factorsDegree, factorsRadical, factorsRadical, factorsSquarefree, factorsSquarefreeKronecker, factorsSquarefreeOptimize, isFactorization, isFactorization, isIrreducible, isRecursiveFactorization, isReducible, isSquarefree, normalizeFactorization, primitivePart, recursiveFactors, recursiveFactorsSquarefree, squarefreeFactors, squarefreePart

Methods inherited from class java.lang.Object

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

Constructor Detail

FactorAbsolute

protected FactorAbsolute()

No argument constructor. Note: can't use this constructor.

FactorAbsolute

public FactorAbsolute(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 FactorAbstract<C extends GcdRingElem<C>>

See Also:

Object.toString()

isAbsoluteIrreducible

public boolean isAbsoluteIrreducible(GenPolynomial<C> P)

GenPolynomial test if is absolute irreducible.

Parameters:

P - GenPolynomial.

Returns:

true if P is absolute irreducible, else false.

baseFactorsAbsolute

public FactorsMap<C> baseFactorsAbsolute(GenPolynomial<C> P)

GenPolynomial absolute base factorization of a polynomial.

Parameters:

P - univariate GenPolynomial.

Returns:

factors map container: [p_1 -> e_1, ..., p_k -> e_k] with P = prod_{i=1,...,k} p_i**e_i. Note: K(alpha) not yet minimal.

baseFactorsAbsoluteSquarefree

public FactorsList<C> baseFactorsAbsoluteSquarefree(GenPolynomial<C> P)

GenPolynomial absolute base factorization of a squarefree polynomial.

Parameters:

P - squarefree and primitive univariate GenPolynomial.

Returns:

factors list container: [p_1,...,p_k] with P = prod_{i=1, ..., k} p_i. Note: K(alpha) not yet minimal.

baseFactorsAbsoluteIrreducible

public Factors<C> baseFactorsAbsoluteIrreducible(GenPolynomial<C> P)

GenPolynomial base absolute factorization of a irreducible polynomial.

Parameters:

P - irreducible! univariate GenPolynomial.

Returns:

factors container: [p_1,...,p_k] with P = prod_{i=1, ..., k} p_i in K(alpha)[x] for suitable alpha and p_i irreducible over L[x], where K \subset K(alpha) \subset L is an algebraically closed field over K. Note: K(alpha) not yet minimal.

baseAlgebraicPartialFraction

public PartialFraction<C> baseAlgebraicPartialFraction(GenPolynomial<C> A,
                                                       GenPolynomial<C> P)

Univariate GenPolynomial algebraic partial fraction decomposition, Absolute factorization for elementary integration algorithm to linear factors.

Parameters:

A - univariate GenPolynomial, deg(A) < deg(P).

P - univariate squarefree GenPolynomial, gcd(A,P) == 1.

Returns:

partial fraction container.

baseAlgebraicPartialFractionIrreducibleAbsolute

public PartialFraction<C> baseAlgebraicPartialFractionIrreducibleAbsolute(GenPolynomial<C> A,
                                                                          GenPolynomial<C> P)

Univariate GenPolynomial algebraic partial fraction decomposition, via absolute factorization to linear factors.

Parameters:

A - univariate GenPolynomial, deg(A) < deg(P).

P - univariate irreducible GenPolynomial, gcd(A,P) == 1.

Returns:

partial fraction container.

factorsAbsolute

public FactorsMap<C> factorsAbsolute(GenPolynomial<C> P)

GenPolynomial absolute factorization of a polynomial.

Parameters:

P - GenPolynomial.

Returns:

factors map container: [p_1 -> e_1, ..., p_k -> e_k] with P = prod_{i=1,...,k} p_i**e_i. Note: K(alpha) not yet minimal.

factorsAbsoluteSquarefree

public FactorsList<C> factorsAbsoluteSquarefree(GenPolynomial<C> P)

GenPolynomial absolute factorization of a squarefree polynomial.

Parameters:

P - squarefree and primitive GenPolynomial.

Returns:

factors list container: [p_1,...,p_k] with P = prod_{i=1, ..., k} p_i. Note: K(alpha) not yet minimal.

factorsAbsoluteIrreducible

public Factors<C> factorsAbsoluteIrreducible(GenPolynomial<C> P)

GenPolynomial absolute factorization of a irreducible polynomial.

Parameters:

P - irreducible! GenPolynomial.

Returns:

factors container: [p_1,...,p_k] with P = prod_{i=1, ..., k} p_i in K(alpha)[x] for suitable alpha and p_i irreducible over L[x], where K \subset K(alpha) \subset L is an algebraically closed field over K. Note: K(alpha) not yet minimal.

isAbsoluteFactorization

public boolean isAbsoluteFactorization(Factors<C> facs)

GenPolynomial is absolute factorization.

Parameters:

facs - factors container.

Returns:

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

isAbsoluteFactorization

public boolean isAbsoluteFactorization(FactorsList<C> facs)

GenPolynomial is absolute factorization.

Parameters:

facs - factors list container.

Returns:

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

isAbsoluteFactorization

public boolean isAbsoluteFactorization(FactorsMap<C> facs)

GenPolynomial is absolute factorization.

Parameters:

facs - factors map container.

Returns:

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