edu.jas.fd

Class FDUtil

java.lang.Object

edu.jas.fd.FDUtil

public class FDUtil
extends java.lang.Object

Factorization domain utilities, for example recursive pseudo remainder.

Constructor Summary

Constructors

Constructor and Description
FDUtil()

Method Summary

Modifier and Type	Method and Description
static <C extends GcdRingElem<C>> GenSolvablePolynomial<C>	basePseudoQuotient(GenSolvablePolynomial<C> P, GenSolvablePolynomial<C> S) GenSolvablePolynomial sparse pseudo quotient for univariate polynomials or exact division.
static <C extends GcdRingElem<C>> GenSolvablePolynomial<C>[]	basePseudoQuotientRemainder(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>	baseSparsePseudoRemainder(GenSolvablePolynomial<C> P, GenSolvablePolynomial<C> S) GenSolvablePolynomial sparse pseudo remainder for univariate polynomials.
static <C extends RingElem<C>> GenSolvablePolynomial<GenPolynomial<C>>	evalAsRightRecursivePolynomial(GenSolvablePolynomial<GenPolynomial<C>> R) Evaluate RecSolvablePolynomial as right coefficients polynomial.
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>> 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>> 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 RingElem<C>> boolean	isRightRecursivePolynomial(GenSolvablePolynomial<GenPolynomial<C>> P, GenSolvablePolynomial<GenPolynomial<C>> R) Test RecSolvablePolynomial right coefficients polynomial.
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<SolvableQuotient<C>>	quotientFromIntegralCoefficients(GenSolvablePolynomialRing<SolvableQuotient<C>> fac, GenSolvablePolynomial<GenPolynomial<C>> A) 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 recursive quotient for recursive polynomials and 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>>	recursiveSparsePseudoRemainder(GenSolvablePolynomial<GenPolynomial<C>> P, GenSolvablePolynomial<GenPolynomial<C>> S) GenSolvablePolynomial sparse pseudo remainder for recursive solvable polynomials.
static <C extends RingElem<C>> GenSolvablePolynomial<GenPolynomial<C>>	recursiveSparsePseudoRemainderRight(GenSolvablePolynomial<GenPolynomial<C>> P, GenSolvablePolynomial<GenPolynomial<C>> S) GenSolvablePolynomial right sparse pseudo remainder for recursive solvable polynomials.
static <C extends RingElem<C>> GenSolvablePolynomial<GenPolynomial<C>>	rightRecursivePolynomial(GenSolvablePolynomial<GenPolynomial<C>> P) RecSolvablePolynomial right coefficients from left coefficients.

Methods inherited from class java.lang.Object

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

Constructor Detail

FDUtil

public FDUtil()

Method Detail

baseSparsePseudoRemainder

public static <C extends GcdRingElem<C>> GenSolvablePolynomial<C> baseSparsePseudoRemainder(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 ldcf(S)^m' P = quotient * S + remainder. m' ≤ deg(P)-deg(S)

See Also:

GenPolynomial.remainder(edu.jas.poly.GenPolynomial).

basePseudoQuotient

public static <C extends GcdRingElem<C>> GenSolvablePolynomial<C> basePseudoQuotient(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 ldcf(S)^m' P = quotient * S + remainder. m' ≤ deg(P)-deg(S)

See Also:

GenPolynomial.divide(edu.jas.poly.GenPolynomial).

basePseudoQuotientRemainder

public static <C extends GcdRingElem<C>> GenSolvablePolynomial<C>[] basePseudoQuotientRemainder(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 ldcf(S)^m' P = quotient * S + remainder. m' ≤ deg(P)-deg(S)

See Also:

GenPolynomial.divide(edu.jas.poly.GenPolynomial).

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:

PolyUtil.recursiveSparsePseudoRemainder(edu.jas.poly.GenPolynomial,edu.jas.poly.GenPolynomial).

recursiveSparsePseudoRemainderRight

public static <C extends RingElem<C>> GenSolvablePolynomial<GenPolynomial<C>> recursiveSparsePseudoRemainderRight(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 ldcf(S)^m' = quotient * S + remainder.

See Also:

PolyUtil.recursiveSparsePseudoRemainder(edu.jas.poly.GenPolynomial,edu.jas.poly.GenPolynomial).

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.

See Also:

Note: not always meaningful and working

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.

See Also:

GenPolynomial.remainder(edu.jas.poly.GenPolynomial).

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.

See Also:

GenPolynomial.remainder(edu.jas.poly.GenPolynomial).

recursiveDivide

public static <C extends GcdRingElem<C>> GenSolvablePolynomial<GenPolynomial<C>> recursiveDivide(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:

this/s.

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:

this/s.

rightRecursivePolynomial

public static <C extends RingElem<C>> GenSolvablePolynomial<GenPolynomial<C>> rightRecursivePolynomial(GenSolvablePolynomial<GenPolynomial<C>> P)

RecSolvablePolynomial right coefficients from left coefficients. Note: R is represented as a polynomial with left coefficients, the implementation can at the moment not distinguish between left and right coefficients.

Type Parameters:

C - coefficient type.

Parameters:

P - GenSolvablePolynomial.

Returns:

R = sum( Xⁱ b_i ), with P = sum(a_i Xⁱ ) and eval(sum(Xⁱ b_i)) == sum(a_i Xⁱ)

evalAsRightRecursivePolynomial

public static <C extends RingElem<C>> GenSolvablePolynomial<GenPolynomial<C>> evalAsRightRecursivePolynomial(GenSolvablePolynomial<GenPolynomial<C>> R)

Evaluate RecSolvablePolynomial as right coefficients polynomial. Note: R is represented as a polynomial with left coefficients, the implementation can at the moment not distinguish between left and right coefficients.

Type Parameters:

C - coefficient type.

Parameters:

R - GenSolvablePolynomial with right coefficients.

Returns:

P as evaluated polynomial R. R = sum( Xⁱ b_i ), P = sum(a_i Xⁱ ) = eval(sum(Xⁱ b_i))

isRightRecursivePolynomial

public static <C extends RingElem<C>> boolean isRightRecursivePolynomial(GenSolvablePolynomial<GenPolynomial<C>> P,
                                                                         GenSolvablePolynomial<GenPolynomial<C>> R)

Test RecSolvablePolynomial right coefficients polynomial. Note: R is represented as a polynomial with left coefficients, the implementation can at the moment not distinguish between left and right coefficients.

Type Parameters:

C - coefficient type.

Parameters:

P - GenSolvablePolynomial with left coefficients.

R - GenSolvablePolynomial with right coefficients.

Returns:

true, if R is polynomial with right coefficients of P. R = sum( Xⁱ b_i ), with P = sum(a_i Xⁱ ) and eval(sum(Xⁱ b_i)) == sum(a_i Xⁱ)

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.