Package edu.jas.fd

Class FDUtil

java.lang.Object

edu.jas.fd.FDUtil

public class FDUtil
extends java.lang.Object

Solvable polynomials factorization domain utilities, for example recursive pseudo remainder.

Author:

Heinz Kredel

Constructor Summary

Constructors

Constructor	Description
FDUtil()

Method Summary

Modifier and Type	Method	Description
static <C extends GcdRingElem<C>> GenSolvablePolynomial<C>	basePseudoLeftDivide(GenSolvablePolynomial<C> P, GenSolvablePolynomial<C> S)	GenSolvablePolynomial sparse pseudo divide.
static <C extends GcdRingElem<C>> GenSolvablePolynomial<GenPolynomial<C>>	integralFromQuotientCoefficients(GenSolvablePolynomialRing<GenPolynomial<C>> fac, GenSolvablePolynomial<SolvableQuotient<C>> A)	Integral solvable polynomial from solvable rational function coefficients.
static <C extends GcdRingElem<C>> java.util.List<GenSolvablePolynomial<GenPolynomial<C>>>	integralFromQuotientCoefficients(GenSolvablePolynomialRing<GenPolynomial<C>> fac, java.util.Collection<GenSolvablePolynomial<SolvableQuotient<C>>> L)	Integral solvable polynomial from solvable rational function coefficients.
static <C extends GcdRingElem<C>> boolean	isLeftBasePseudoQuotientRemainder(GenPolynomial<C> P, GenPolynomial<C> S, GenPolynomial<C> q, GenPolynomial<C> r)	Is GenSolvablePolynomial left base pseudo quotient and remainder.
static <C extends GcdRingElem<C>> boolean	isRecursivePseudoQuotientRemainder(GenSolvablePolynomial<GenPolynomial<C>> P, GenSolvablePolynomial<GenPolynomial<C>> S, GenSolvablePolynomial<GenPolynomial<C>> q, GenSolvablePolynomial<GenPolynomial<C>> r)	Is recursive GenSolvablePolynomial pseudo quotient and remainder.
static <C extends GcdRingElem<C>> boolean	isRecursiveRightPseudoQuotientRemainder(GenSolvablePolynomial<GenPolynomial<C>> P, GenSolvablePolynomial<GenPolynomial<C>> S, GenSolvablePolynomial<GenPolynomial<C>> q, GenSolvablePolynomial<GenPolynomial<C>> r)	Is recursive GenSolvablePolynomial right pseudo quotient and remainder.
static <C extends GcdRingElem<C>> boolean	isRightBasePseudoQuotientRemainder(GenPolynomial<C> P, GenPolynomial<C> S, GenPolynomial<C> q, GenPolynomial<C> r)	Is GenSolvablePolynomial right base pseudo quotient and remainder.
static <C extends GcdRingElem<C>> GenSolvablePolynomial<C>	leftBasePseudoQuotient(GenSolvablePolynomial<C> P, GenSolvablePolynomial<C> S)	GenSolvablePolynomial sparse pseudo quotient for univariate polynomials or exact division.
static <C extends GcdRingElem<C>> GenSolvablePolynomial<C>[]	leftBasePseudoQuotientRemainder(GenSolvablePolynomial<C> P, GenSolvablePolynomial<C> S)	GenSolvablePolynomial sparse pseudo quotient and remainder for univariate polynomials or exact division.
static <C extends GcdRingElem<C>> GenSolvablePolynomial<C>	leftBaseSparsePseudoRemainder(GenSolvablePolynomial<C> P, GenSolvablePolynomial<C> S)	GenSolvablePolynomial sparse pseudo remainder for univariate polynomials.
static <C extends GcdRingElem<C>> GenSolvablePolynomial<SolvableQuotient<C>>	quotientFromIntegralCoefficients(GenSolvablePolynomialRing<SolvableQuotient<C>> fac, GenSolvablePolynomial<GenPolynomial<C>> A)	Solvable rational function from integral solvable polynomial coefficients.
static <C extends GcdRingElem<C>> java.util.List<GenSolvablePolynomial<SolvableQuotient<C>>>	quotientFromIntegralCoefficients(GenSolvablePolynomialRing<SolvableQuotient<C>> fac, java.util.Collection<GenSolvablePolynomial<GenPolynomial<C>>> L)	Solvable rational function from integral solvable polynomial coefficients.
static <C extends GcdRingElem<C>> GenSolvablePolynomial<GenPolynomial<C>>	recursiveDivide(GenSolvablePolynomial<GenPolynomial<C>> P, GenSolvablePolynomial<C> s)	GenSolvablePolynomial left recursive quotient for recursive polynomials and exact division by coefficient ring element.
static <C extends GcdRingElem<C>> GenSolvablePolynomial<GenPolynomial<C>>	recursiveDivideRightEval(GenSolvablePolynomial<GenPolynomial<C>> P, GenSolvablePolynomial<C> s)	GenSolvablePolynomial recursive quotient for recursive polynomials and exact division by coefficient ring element.
static <C extends GcdRingElem<C>> GenSolvablePolynomial<GenPolynomial<C>>	recursiveLeftDivide(GenSolvablePolynomial<GenPolynomial<C>> P, GenSolvablePolynomial<C> s)	GenSolvablePolynomial recursive quotient for recursive polynomials and partial left exact division by coefficient ring element.
static <C extends GcdRingElem<C>> GenSolvablePolynomial<GenPolynomial<C>>	recursivePseudoQuotient(GenSolvablePolynomial<GenPolynomial<C>> P, GenSolvablePolynomial<GenPolynomial<C>> S)	GenSolvablePolynomial recursive pseudo quotient for recursive polynomials.
static <C extends GcdRingElem<C>> GenSolvablePolynomial<GenPolynomial<C>>[]	recursivePseudoQuotientRemainder(GenSolvablePolynomial<GenPolynomial<C>> P, GenSolvablePolynomial<GenPolynomial<C>> S)	GenSolvablePolynomial recursive pseudo quotient and remainder for recursive polynomials.
static <C extends GcdRingElem<C>> GenSolvablePolynomial<GenPolynomial<C>>	recursiveRightDivide(GenSolvablePolynomial<GenPolynomial<C>> P, GenSolvablePolynomial<C> s)	GenSolvablePolynomial recursive quotient for recursive polynomials and partial right exact division by coefficient ring element.
static <C extends GcdRingElem<C>> GenSolvablePolynomial<GenPolynomial<C>>	recursiveRightPseudoQuotient(GenSolvablePolynomial<GenPolynomial<C>> P, GenSolvablePolynomial<GenPolynomial<C>> S)	GenSolvablePolynomial recursive right pseudo quotient for recursive polynomials.
static <C extends GcdRingElem<C>> GenSolvablePolynomial<GenPolynomial<C>>[]	recursiveRightPseudoQuotientRemainder(GenSolvablePolynomial<GenPolynomial<C>> P, GenSolvablePolynomial<GenPolynomial<C>> S)	GenSolvablePolynomial right sparse pseudo quotient and remainder for recursive solvable polynomials.
static <C extends GcdRingElem<C>> GenSolvablePolynomial<GenPolynomial<C>>	recursiveRightSparsePseudoRemainder(GenSolvablePolynomial<GenPolynomial<C>> P, GenSolvablePolynomial<GenPolynomial<C>> S)	GenSolvablePolynomial right sparse pseudo remainder for recursive solvable polynomials.
static <C extends GcdRingElem<C>> GenSolvablePolynomial<GenPolynomial<C>>	recursiveSparsePseudoRemainder(GenSolvablePolynomial<GenPolynomial<C>> P, GenSolvablePolynomial<GenPolynomial<C>> S)	GenSolvablePolynomial sparse pseudo remainder for recursive solvable polynomials.
static <C extends GcdRingElem<C>> GenSolvablePolynomial<C>	rightBasePseudoQuotient(GenSolvablePolynomial<C> P, GenSolvablePolynomial<C> S)	GenSolvablePolynomial right sparse pseudo quotient for univariate polynomials or exact division.
static <C extends GcdRingElem<C>> GenSolvablePolynomial<C>[]	rightBasePseudoQuotientRemainder(GenSolvablePolynomial<C> P, GenSolvablePolynomial<C> S)	GenSolvablePolynomial right sparse pseudo quotient and remainder for univariate polynomials or exact division.
static <C extends GcdRingElem<C>> GenSolvablePolynomial<C>	rightBaseSparsePseudoRemainder(GenSolvablePolynomial<C> P, GenSolvablePolynomial<C> S)	GenSolvablePolynomial sparse right pseudo remainder for univariate polynomials.

Methods inherited from class java.lang.Object

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

Constructor Detail

FDUtil

public FDUtil()

Method Detail

leftBaseSparsePseudoRemainder

public static <C extends GcdRingElem<C>> GenSolvablePolynomial<C> leftBaseSparsePseudoRemainder(GenSolvablePolynomial<C> P,
                                                                                                GenSolvablePolynomial<C> S)

GenSolvablePolynomial sparse pseudo remainder for univariate polynomials.

Type Parameters:

C - coefficient type.

Parameters:

P - GenSolvablePolynomial.

S - nonzero GenSolvablePolynomial.

Returns:

remainder with ore(ldcf(S)^m') P = quotient * S + remainder. m' ≤ deg(P)-deg(S)

rightBaseSparsePseudoRemainder

public static <C extends GcdRingElem<C>> GenSolvablePolynomial<C> rightBaseSparsePseudoRemainder(GenSolvablePolynomial<C> P,
                                                                                                 GenSolvablePolynomial<C> S)

GenSolvablePolynomial sparse right pseudo remainder for univariate polynomials.

Type Parameters:

C - coefficient type.

Parameters:

P - GenSolvablePolynomial.

S - nonzero GenSolvablePolynomial.

Returns:

remainder with P ore(ldcf(S)^m') = S * quotient + remainder. m' ≤ deg(P)-deg(S)

leftBasePseudoQuotient

public static <C extends GcdRingElem<C>> GenSolvablePolynomial<C> leftBasePseudoQuotient(GenSolvablePolynomial<C> P,
                                                                                         GenSolvablePolynomial<C> S)

GenSolvablePolynomial sparse pseudo quotient for univariate polynomials or exact division.

Type Parameters:

C - coefficient type.

Parameters:

P - GenSolvablePolynomial.

S - nonzero GenSolvablePolynomial.

Returns:

quotient with ore(ldcf(S)^m') P = quotient * S + remainder. m' ≤ deg(P)-deg(S)

rightBasePseudoQuotient

public static <C extends GcdRingElem<C>> GenSolvablePolynomial<C> rightBasePseudoQuotient(GenSolvablePolynomial<C> P,
                                                                                          GenSolvablePolynomial<C> S)

GenSolvablePolynomial right sparse pseudo quotient for univariate polynomials or exact division.

Type Parameters:

C - coefficient type.

Parameters:

P - GenSolvablePolynomial.

S - nonzero GenSolvablePolynomial.

Returns:

quotient with P ore(ldcf(S)^m') = S * quotient + remainder. m' ≤ deg(P)-deg(S)

leftBasePseudoQuotientRemainder

public static <C extends GcdRingElem<C>> GenSolvablePolynomial<C>[] leftBasePseudoQuotientRemainder(GenSolvablePolynomial<C> P,
                                                                                                    GenSolvablePolynomial<C> S)

GenSolvablePolynomial sparse pseudo quotient and remainder for univariate polynomials or exact division.

Type Parameters:

C - coefficient type.

Parameters:

P - GenSolvablePolynomial.

S - nonzero GenSolvablePolynomial.

Returns:

[ quotient, remainder ] with ore(ldcf(S)^m') P = quotient * S + remainder. m' ≤ deg(P)-deg(S)

rightBasePseudoQuotientRemainder

public static <C extends GcdRingElem<C>> GenSolvablePolynomial<C>[] rightBasePseudoQuotientRemainder(GenSolvablePolynomial<C> P,
                                                                                                     GenSolvablePolynomial<C> S)

GenSolvablePolynomial right sparse pseudo quotient and remainder for univariate polynomials or exact division.

Type Parameters:

C - coefficient type.

Parameters:

P - GenSolvablePolynomial.

S - nonzero GenSolvablePolynomial.

Returns:

[ quotient, remainder ] with P ore(ldcf(S)^m') = S * quotient + remainder. m' ≤ deg(P)-deg(S)

isLeftBasePseudoQuotientRemainder

public static <C extends GcdRingElem<C>> boolean isLeftBasePseudoQuotientRemainder(GenPolynomial<C> P,
                                                                                   GenPolynomial<C> S,
                                                                                   GenPolynomial<C> q,
                                                                                   GenPolynomial<C> r)

Is GenSolvablePolynomial left base pseudo quotient and remainder. For univariate polynomials. todo: Ore condition

Type Parameters:

C - coefficient type.

Parameters:

P - base GenSolvablePolynomial.

S - nonzero base GenSolvablePolynomial.

Returns:

true, if P = q * S + r, else false.

isRightBasePseudoQuotientRemainder

public static <C extends GcdRingElem<C>> boolean isRightBasePseudoQuotientRemainder(GenPolynomial<C> P,
                                                                                    GenPolynomial<C> S,
                                                                                    GenPolynomial<C> q,
                                                                                    GenPolynomial<C> r)

Is GenSolvablePolynomial right base pseudo quotient and remainder. For univariate polynomials. todo: Ore condition

Type Parameters:

C - coefficient type.

Parameters:

P - base GenSolvablePolynomial.

S - nonzero base GenSolvablePolynomial.

Returns:

true, if P = S * q + r, else false.

isRecursivePseudoQuotientRemainder

public static <C extends GcdRingElem<C>> boolean isRecursivePseudoQuotientRemainder(GenSolvablePolynomial<GenPolynomial<C>> P,
                                                                                    GenSolvablePolynomial<GenPolynomial<C>> S,
                                                                                    GenSolvablePolynomial<GenPolynomial<C>> q,
                                                                                    GenSolvablePolynomial<GenPolynomial<C>> r)

Is recursive GenSolvablePolynomial pseudo quotient and remainder. For recursive polynomials.

Type Parameters:

C - coefficient type.

Parameters:

P - recursive GenSolvablePolynomial.

S - nonzero recursive GenSolvablePolynomial.

Returns:

true, if P ~= q * S + r, else false.

recursiveSparsePseudoRemainder

public static <C extends GcdRingElem<C>> GenSolvablePolynomial<GenPolynomial<C>> recursiveSparsePseudoRemainder(GenSolvablePolynomial<GenPolynomial<C>> P,
                                                                                                                GenSolvablePolynomial<GenPolynomial<C>> S)

GenSolvablePolynomial sparse pseudo remainder for recursive solvable polynomials.

Type Parameters:

C - coefficient type.

Parameters:

P - recursive GenSolvablePolynomial.

S - nonzero recursive GenSolvablePolynomial.

Returns:

remainder with ore(ldcf(S)^m') P = quotient * S + remainder.

See Also:

.

recursivePseudoQuotient

public static <C extends GcdRingElem<C>> GenSolvablePolynomial<GenPolynomial<C>> recursivePseudoQuotient(GenSolvablePolynomial<GenPolynomial<C>> P,
                                                                                                         GenSolvablePolynomial<GenPolynomial<C>> S)

GenSolvablePolynomial recursive pseudo quotient for recursive polynomials.

Type Parameters:

C - coefficient type.

Parameters:

P - recursive GenSolvablePolynomial.

S - nonzero recursive GenSolvablePolynomial.

Returns:

quotient with ore(ldcf(S)^m') P = quotient * S + remainder.

recursivePseudoQuotientRemainder

public static <C extends GcdRingElem<C>> GenSolvablePolynomial<GenPolynomial<C>>[] recursivePseudoQuotientRemainder(GenSolvablePolynomial<GenPolynomial<C>> P,
                                                                                                                    GenSolvablePolynomial<GenPolynomial<C>> S)

GenSolvablePolynomial recursive pseudo quotient and remainder for recursive polynomials.

Type Parameters:

C - coefficient type.

Parameters:

P - recursive GenSolvablePolynomial.

S - nonzero recursive GenSolvablePolynomial.

Returns:

[ quotient, remainder ] with ore(ldcf(S)^m') P = quotient * S + remainder.

isRecursiveRightPseudoQuotientRemainder

public static <C extends GcdRingElem<C>> boolean isRecursiveRightPseudoQuotientRemainder(GenSolvablePolynomial<GenPolynomial<C>> P,
                                                                                         GenSolvablePolynomial<GenPolynomial<C>> S,
                                                                                         GenSolvablePolynomial<GenPolynomial<C>> q,
                                                                                         GenSolvablePolynomial<GenPolynomial<C>> r)

Is recursive GenSolvablePolynomial right pseudo quotient and remainder. For recursive polynomials.

Type Parameters:

C - coefficient type.

Parameters:

P - recursive GenSolvablePolynomial.

S - nonzero recursive GenSolvablePolynomial.

Returns:

true, if P ~= S * q + r, else false.

recursiveRightSparsePseudoRemainder

public static <C extends GcdRingElem<C>> GenSolvablePolynomial<GenPolynomial<C>> recursiveRightSparsePseudoRemainder(GenSolvablePolynomial<GenPolynomial<C>> P,
                                                                                                                     GenSolvablePolynomial<GenPolynomial<C>> S)

GenSolvablePolynomial right sparse pseudo remainder for recursive solvable polynomials. Note: uses right multiplication of P by ldcf(S), not always applicable.

Type Parameters:

C - coefficient type.

Parameters:

P - recursive GenSolvablePolynomial.

S - nonzero recursive GenSolvablePolynomial.

Returns:

remainder with P ore(ldcf(S)^m') = quotient * S + remainder.

See Also:

.

recursiveRightPseudoQuotient

public static <C extends GcdRingElem<C>> GenSolvablePolynomial<GenPolynomial<C>> recursiveRightPseudoQuotient(GenSolvablePolynomial<GenPolynomial<C>> P,
                                                                                                              GenSolvablePolynomial<GenPolynomial<C>> S)

GenSolvablePolynomial recursive right pseudo quotient for recursive polynomials.

Type Parameters:

C - coefficient type.

Parameters:

P - recursive GenSolvablePolynomial.

S - nonzero recursive GenSolvablePolynomial.

Returns:

quotient with P ore(ldcf(S)^m') = S * quotient + remainder.

recursiveRightPseudoQuotientRemainder

public static <C extends GcdRingElem<C>> GenSolvablePolynomial<GenPolynomial<C>>[] recursiveRightPseudoQuotientRemainder(GenSolvablePolynomial<GenPolynomial<C>> P,
                                                                                                                         GenSolvablePolynomial<GenPolynomial<C>> S)

GenSolvablePolynomial right sparse pseudo quotient and remainder for recursive solvable polynomials. Note: uses right multiplication of P by ldcf(S), not always applicable.

Type Parameters:

C - coefficient type.

Parameters:

P - recursive GenSolvablePolynomial.

S - nonzero recursive GenSolvablePolynomial.

Returns:

remainder with P ore(ldcf(S)^m') = S * quotient + remainder.

See Also:

.

recursiveDivideRightEval

public static <C extends GcdRingElem<C>> GenSolvablePolynomial<GenPolynomial<C>> recursiveDivideRightEval(GenSolvablePolynomial<GenPolynomial<C>> P,
                                                                                                          GenSolvablePolynomial<C> s)

GenSolvablePolynomial recursive quotient for recursive polynomials and exact division by coefficient ring element.

Type Parameters:

C - coefficient type.

Parameters:

P - recursive GenSolvablePolynomial.

s - GenSolvablePolynomial.

Returns:

P/s.

recursiveDivide

public static <C extends GcdRingElem<C>> GenSolvablePolynomial<GenPolynomial<C>> recursiveDivide(GenSolvablePolynomial<GenPolynomial<C>> P,
                                                                                                 GenSolvablePolynomial<C> s)

GenSolvablePolynomial left recursive quotient for recursive polynomials and exact division by coefficient ring element.

Type Parameters:

C - coefficient type.

Parameters:

P - recursive GenSolvablePolynomial.

s - GenSolvablePolynomial.

Returns:

q = this/s, with q * s = P.

recursiveRightDivide

public static <C extends GcdRingElem<C>> GenSolvablePolynomial<GenPolynomial<C>> recursiveRightDivide(GenSolvablePolynomial<GenPolynomial<C>> P,
                                                                                                      GenSolvablePolynomial<C> s)

GenSolvablePolynomial recursive quotient for recursive polynomials and partial right exact division by coefficient ring element.

Type Parameters:

C - coefficient type.

Parameters:

P - recursive GenSolvablePolynomial.

s - GenSolvablePolynomial.

Returns:

Q with s * Q = P.

basePseudoLeftDivide

public static <C extends GcdRingElem<C>> GenSolvablePolynomial<C> basePseudoLeftDivide(GenSolvablePolynomial<C> P,
                                                                                       GenSolvablePolynomial<C> S)

GenSolvablePolynomial sparse pseudo divide. For univariate polynomials or exact division.

Type Parameters:

C - coefficient type.

Parameters:

P - GenSolvablePolynomial.

S - nonzero GenSolvablePolynomial.

Returns:

quotient with ldcf(S)^m' P = quotient * S + remainder. m' ≤ deg(P)-deg(S)

recursiveLeftDivide

public static <C extends GcdRingElem<C>> GenSolvablePolynomial<GenPolynomial<C>> recursiveLeftDivide(GenSolvablePolynomial<GenPolynomial<C>> P,
                                                                                                     GenSolvablePolynomial<C> s)

GenSolvablePolynomial recursive quotient for recursive polynomials and partial left exact division by coefficient ring element.

Type Parameters:

C - coefficient type.

Parameters:

P - recursive GenSolvablePolynomial.

s - GenSolvablePolynomial.

Returns:

Q with P = Q * s.

integralFromQuotientCoefficients

public static <C extends GcdRingElem<C>> GenSolvablePolynomial<GenPolynomial<C>> integralFromQuotientCoefficients(GenSolvablePolynomialRing<GenPolynomial<C>> fac,
                                                                                                                  GenSolvablePolynomial<SolvableQuotient<C>> A)

Integral solvable polynomial from solvable rational function coefficients. Represent as polynomial with integral solvable polynomial coefficients by multiplication with the lcm(??) of the numerators of the rational function coefficients.

Parameters:

fac - result polynomial factory.

A - polynomial with solvable rational function coefficients to be converted.

Returns:

polynomial with integral solvable polynomial coefficients.

integralFromQuotientCoefficients

public static <C extends GcdRingElem<C>> java.util.List<GenSolvablePolynomial<GenPolynomial<C>>> integralFromQuotientCoefficients(GenSolvablePolynomialRing<GenPolynomial<C>> fac,
                                                                                                                                  java.util.Collection<GenSolvablePolynomial<SolvableQuotient<C>>> L)

Integral solvable polynomial from solvable rational function coefficients. Represent as polynomial with integral solvable polynomial coefficients by multiplication with the lcm(??) of the numerators of the solvable rational function coefficients.

Parameters:

fac - result polynomial factory.

L - list of polynomials with solvable rational function coefficients to be converted.

Returns:

list of polynomials with integral solvable polynomial coefficients.

quotientFromIntegralCoefficients

public static <C extends GcdRingElem<C>> GenSolvablePolynomial<SolvableQuotient<C>> quotientFromIntegralCoefficients(GenSolvablePolynomialRing<SolvableQuotient<C>> fac,
                                                                                                                     GenSolvablePolynomial<GenPolynomial<C>> A)

Solvable rational function from integral solvable polynomial coefficients. Represent as polynomial with type SolvableQuotient coefficients.

Parameters:

fac - result polynomial factory.

A - polynomial with integral solvable polynomial coefficients to be converted.

Returns:

polynomial with type SolvableQuotient coefficients.

quotientFromIntegralCoefficients

public static <C extends GcdRingElem<C>> java.util.List<GenSolvablePolynomial<SolvableQuotient<C>>> quotientFromIntegralCoefficients(GenSolvablePolynomialRing<SolvableQuotient<C>> fac,
                                                                                                                                     java.util.Collection<GenSolvablePolynomial<GenPolynomial<C>>> L)

Solvable rational function from integral solvable polynomial coefficients. Represent as polynomial with type SolvableQuotient coefficients.

Parameters:

fac - result polynomial factory.

L - list of polynomials with integral solvable polynomial coefficients to be converted.

Returns:

list of polynomials with type SolvableQuotient coefficients.