Package edu.jas.poly

Class PolyUtil

java.lang.Object

edu.jas.poly.PolyUtil

public class PolyUtil
extends java.lang.Object

Polynomial utilities, for example conversion between different representations, evaluation and interpolation.

Author:

Heinz Kredel

Constructor Summary

Constructors

Constructor	Description
PolyUtil()

Method Summary

Modifier and Type	Method	Description
static <C extends RingElem<C>> C	absNorm(GenPolynomial<C> p)	Absoulte norm.
static <C extends GcdRingElem<C>> GenPolynomial<AlgebraicNumber<C>>	algebraicFromComplex(GenPolynomialRing<AlgebraicNumber<C>> fac, GenPolynomial<Complex<C>> A)	AlgebraicNumber from complex coefficients.
static <C extends RingElem<C>> GenPolynomial<C>	baseDensePseudoQuotient(GenPolynomial<C> P, GenPolynomial<C> S)	GenPolynomial dense pseudo quotient.
static <C extends RingElem<C>> GenPolynomial<C>	baseDensePseudoRemainder(GenPolynomial<C> P, GenPolynomial<C> S)	GenPolynomial dense pseudo remainder.
static <C extends RingElem<C>> GenPolynomial<C>	baseDeriviative(GenPolynomial<C> P)	GenPolynomial polynomial derivative main variable.
static <C extends RingElem<C>> GenPolynomial<C>	baseDeriviative(GenPolynomial<C> P, int r)	GenPolynomial polynomial partial derivative variable r.
static <C extends RingElem<C>> GenPolynomial<C>	baseIntegral(GenPolynomial<C> P)	GenPolynomial polynomial integral main variable.
static <C extends RingElem<C>> GenPolynomial<C>	basePseudoDivide(GenPolynomial<C> P, GenPolynomial<C> S)	GenPolynomial sparse pseudo divide.
static <C extends RingElem<C>> GenPolynomial<C>[]	basePseudoQuotientRemainder(GenPolynomial<C> P, GenPolynomial<C> S)	GenPolynomial sparse pseudo quotient and remainder.
static <C extends RingElem<C>> GenPolynomial<C>	basePseudoRemainder(GenPolynomial<C> P, GenPolynomial<C> S)	Deprecated. (forRemoval=true) Use baseSparsePseudoRemainder(edu.jas.poly.GenPolynomial,edu.jas.poly.GenPolynomial) instead
static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>	baseRecursiveDivide(GenPolynomial<GenPolynomial<C>> P, C s)	GenPolynomial base divide.
static <C extends RingElem<C>> GenPolynomial<C>	baseRemainderPoly(GenPolynomial<C> P, C s)	GenPolynomial coefficient wise remainder.
static <C extends RingElem<C>> GenPolynomial<C>	baseSparsePseudoRemainder(GenPolynomial<C> P, GenPolynomial<C> S)	GenPolynomial sparse pseudo remainder.
static <C extends RingElem<C> & Modular> GenPolynomial<C>	chineseRemainder(GenPolynomialRing<C> fac, GenPolynomial<C> A, C mi, GenPolynomial<C> B)	ModInteger chinese remainder algorithm on coefficients.
static <C extends RingElem<C>> GenPolynomial<C>	coefficientBasePseudoDivide(GenPolynomial<C> P, C s)	GenPolynomial pseudo divide.
static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>	coefficientPseudoDivide(GenPolynomial<GenPolynomial<C>> P, GenPolynomial<C> s)	GenPolynomial pseudo divide.
static <C extends RingElem<C>> long	coeffMaxDegree(GenPolynomial<GenPolynomial<C>> A)	Maximal degree in the coefficient polynomials.
static <C extends RingElem<C> & Rational> GenPolynomial<Complex<BigDecimal>>	complexDecimalFromRational(GenPolynomialRing<Complex<BigDecimal>> fac, GenPolynomial<Complex<C>> A)	Convert to complex decimal coefficients.
static <C extends GcdRingElem<C>> GenPolynomial<Complex<C>>	complexFromAlgebraic(GenPolynomialRing<Complex<C>> fac, GenPolynomial<AlgebraicNumber<C>> A)	Complex from algebraic coefficients.
static <C extends GcdRingElem<C>> GenPolynomial<Complex<C>>	complexFromAny(GenPolynomialRing<Complex<C>> fac, GenPolynomial<C> A)	Complex from ring element coefficients.
static GenPolynomial<BigComplex>	complexFromRational(GenPolynomialRing<BigComplex> fac, GenPolynomial<BigRational> A)	Complex from rational coefficients.
static <C extends GcdRingElem<C>> GenPolynomial<AlgebraicNumber<C>>	convertRecursiveToAlgebraicCoefficients(GenPolynomialRing<AlgebraicNumber<C>> pfac, GenPolynomial<GenPolynomial<C>> A)	Convert to AlgebraicNumber coefficients.
static <C extends GcdRingElem<C>> GenPolynomial<AlgebraicNumber<C>>	convertToAlgebraicCoefficients(GenPolynomialRing<AlgebraicNumber<C>> pfac, GenPolynomial<C> A)	Convert to AlgebraicNumber coefficients.
static <C extends GcdRingElem<C>> GenPolynomial<AlgebraicNumber<C>>	convertToRecAlgebraicCoefficients(int depth, GenPolynomialRing<AlgebraicNumber<C>> pfac, GenPolynomial<C> A)	Convert to recursive AlgebraicNumber coefficients.
static <C extends RingElem<C> & Rational> GenPolynomial<BigDecimal>	decimalFromRational(GenPolynomialRing<BigDecimal> fac, GenPolynomial<C> A)	Convert to decimal coefficients.
static <C extends RingElem<C>> GenPolynomial<C>	distribute(GenPolynomialRing<C> dfac, GenPolynomial<GenPolynomial<C>> B)	Distribute a recursive polynomial to a generic polynomial.
static <C extends RingElem<C>> java.util.List<GenPolynomial<C>>	distribute(GenPolynomialRing<C> dfac, java.util.List<GenPolynomial<GenPolynomial<C>>> L)	Distribute a recursive polynomial list to a generic polynomial list.
static <C extends RingElem<C>> GenPolynomial<C>	evaluate(GenPolynomialRing<C> cfac, GenPolynomialRing<GenPolynomial<C>> rfac, GenPolynomialRing<GenPolynomial<C>> nfac, GenPolynomialRing<C> dfac, GenPolynomial<C> A, C a)	Evaluate at k-th variable.
static <C extends RingElem<C>> C	evaluateAll(RingFactory<C> cfac, GenPolynomial<C> A, java.util.List<C> a)	Evaluate all variables.
static <C extends RingElem<C>> GenPolynomial<C>	evaluateFirst(GenPolynomialRing<C> cfac, GenPolynomialRing<C> dfac, GenPolynomial<C> A, C a)	Evaluate at first (lowest) variable.
static <C extends RingElem<C>> GenPolynomial<C>	evaluateFirstRec(GenPolynomialRing<C> cfac, GenPolynomialRing<C> dfac, GenPolynomial<GenPolynomial<C>> A, C a)	Evaluate at first (lowest) variable.
static <C extends RingElem<C>> GenPolynomial<C>	evaluateMain(GenPolynomialRing<C> cfac, GenPolynomial<C> A, C a)	Evaluate at main variable.
static <C extends RingElem<C>> java.util.List<GenPolynomial<C>>	evaluateMain(GenPolynomialRing<C> cfac, java.util.List<GenPolynomial<C>> L, C a)	Evaluate at main variable.
static <C extends RingElem<C>> C	evaluateMain(RingFactory<C> cfac, GenPolynomial<C> A, C a)	Evaluate at main variable.
static <C extends RingElem<C>> java.util.List<C>	evaluateMain(RingFactory<C> cfac, java.util.List<GenPolynomial<C>> L, C a)	Evaluate at main variable.
static <C extends RingElem<C>> GenPolynomial<C>	evaluateMainRecursive(GenPolynomialRing<C> cfac, GenPolynomial<GenPolynomial<C>> A, C a)	Evaluate at main variable.
static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>	extendCoefficients(GenPolynomialRing<GenPolynomial<C>> pfac, GenPolynomial<GenPolynomial<C>> A, int j, long k)	Extend coefficient variables.
static <C extends RingElem<C>> GenSolvablePolynomial<GenPolynomial<C>>	extendCoefficients(GenSolvablePolynomialRing<GenPolynomial<C>> pfac, GenSolvablePolynomial<GenPolynomial<C>> A, int j, long k)	Extend coefficient variables.
static BigInteger	factorBound(ExpVector e)	Factor coefficient bound.
static <C extends GcdRingElem<C>> GenPolynomial<GenPolynomial<C>>	fromAlgebraicCoefficients(GenPolynomialRing<GenPolynomial<C>> rfac, GenPolynomial<AlgebraicNumber<C>> A)	From AlgebraicNumber coefficients.
static <C extends RingElem<C>> GenPolynomial<C>	fromIntegerCoefficients(GenPolynomialRing<C> fac, GenPolynomial<BigInteger> A)	From BigInteger coefficients.
static <C extends RingElem<C>> java.util.List<GenPolynomial<C>>	fromIntegerCoefficients(GenPolynomialRing<C> fac, java.util.List<GenPolynomial<BigInteger>> L)	From BigInteger coefficients.
static GenPolynomial<BigRational>	imaginaryPart(GenPolynomialRing<BigRational> fac, GenPolynomial<BigComplex> A)	Imaginary part.
static <C extends RingElem<C>> GenPolynomial<C>	imaginaryPartFromComplex(GenPolynomialRing<C> fac, GenPolynomial<Complex<C>> A)	Imaginary part.
static <C extends RingElem<C> & Modular> GenPolynomial<BigInteger>	integerFromModularCoefficients(GenPolynomialRing<BigInteger> fac, GenPolynomial<C> A)	BigInteger from ModInteger coefficients, symmetric.
static <C extends RingElem<C> & Modular> java.util.List<GenPolynomial<BigInteger>>	integerFromModularCoefficients(GenPolynomialRing<BigInteger> fac, java.util.List<GenPolynomial<C>> L)	BigInteger from ModInteger coefficients, symmetric.
static <C extends RingElem<C> & Modular> GenPolynomial<BigInteger>	integerFromModularCoefficientsPositive(GenPolynomialRing<BigInteger> fac, GenPolynomial<C> A)	BigInteger from ModInteger coefficients, positive.
static GenPolynomial<BigInteger>	integerFromRationalCoefficients(GenPolynomialRing<BigInteger> fac, GenPolynomial<BigRational> A)	BigInteger from BigRational coefficients.
static GenPolynomial<BigInteger>	integerFromRationalCoefficients(GenPolynomialRing<BigInteger> fac, java.math.BigInteger gcd, java.math.BigInteger lcm, GenPolynomial<BigRational> A)	BigInteger from BigRational coefficients.
static java.util.List<GenPolynomial<BigInteger>>	integerFromRationalCoefficients(GenPolynomialRing<BigInteger> fac, java.util.List<GenPolynomial<BigRational>> L)	BigInteger from BigRational coefficients.
static java.lang.Object[]	integerFromRationalCoefficientsFactor(GenPolynomialRing<BigInteger> fac, GenPolynomial<BigRational> A)	BigInteger from BigRational coefficients.
static <C extends RingElem<C>> GenPolynomial<C>	interpolate(GenPolynomialRing<C> fac, GenPolynomial<C> A, GenPolynomial<C> M, C mi, C a, C am)	Univariate polynomial interpolation.
static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>	interpolate(GenPolynomialRing<GenPolynomial<C>> fac, GenPolynomial<GenPolynomial<C>> A, GenPolynomial<C> M, C mi, GenPolynomial<C> B, C am)	ModInteger interpolate on first variable.
static <C extends RingElem<C>> java.util.List<GenPolynomial<C>>	intersect(GenPolynomialRing<C> R, java.util.List<GenPolynomial<C>> F)	Intersection.
static <C extends RingElem<C>> java.util.List<GenSolvablePolynomial<C>>	intersect(GenSolvablePolynomialRing<C> R, java.util.List<GenSolvablePolynomial<C>> F)	Intersection.
static <C extends RingElem<C>> java.util.List<GenWordPolynomial<C>>	intersect(GenWordPolynomialRing<C> R, java.util.List<GenWordPolynomial<C>> F)	Intersection.
static <C extends RingElem<C>> boolean	isBasePseudoQuotientRemainder(GenPolynomial<C> P, GenPolynomial<C> S, GenPolynomial<C> q, GenPolynomial<C> r)	Is GenPolynomial pseudo quotient and remainder.
static <C extends RingElem<C>> boolean	isRecursivePseudoQuotientRemainder(GenPolynomial<GenPolynomial<C>> P, GenPolynomial<GenPolynomial<C>> S, GenPolynomial<GenPolynomial<C>> q, GenPolynomial<GenPolynomial<C>> r)	Is recursive GenPolynomial pseudo quotient and remainder.
static <C extends RingElem<C>> java.util.List<ExpVector>	leadingExpVector(java.util.List<GenPolynomial<C>> L)	Polynomial list leading exponent vectors.
static <C extends RingElem<C>,D extends RingElem<D>> GenPolynomial<D>	map(GenPolynomialRing<D> ring, GenPolynomial<C> p, UnaryFunctor<C,D> f)	Map a unary function to the coefficients.
static <C extends RingElem<C>> long	maxDegree(java.util.List<GenPolynomial<C>> P)	Maximal degree of polynomial list.
static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>	monic(GenPolynomial<GenPolynomial<C>> p)	GenPolynomial monic, i.e. leadingBaseCoefficient == 1.
static <C extends RingElem<C>> GenSolvablePolynomial<GenPolynomial<C>>	monic(GenSolvablePolynomial<GenPolynomial<C>> p)	GenSolvablePolynomial monic, i.e. leadingBaseCoefficient == 1.
static <C extends RingElem<C>> java.util.List<GenPolynomial<C>>	monic(java.util.List<GenPolynomial<C>> L)	Polynomial list monic.
static <C extends RingElem<C>> java.util.List<GenPolynomial<GenPolynomial<C>>>	monicRec(java.util.List<GenPolynomial<GenPolynomial<C>>> L)	Recursive polynomial list monic.
static GenPolynomial<BigRational>	realPart(GenPolynomialRing<BigRational> fac, GenPolynomial<BigComplex> A)	Real part.
static <C extends RingElem<C>> GenPolynomial<C>	realPartFromComplex(GenPolynomialRing<C> fac, GenPolynomial<Complex<C>> A)	Real part.
static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>	recursive(GenPolynomialRing<GenPolynomial<C>> rfac, GenPolynomial<C> A)	Recursive representation.
static <C extends RingElem<C>> java.util.List<GenPolynomial<GenPolynomial<C>>>	recursive(GenPolynomialRing<GenPolynomial<C>> rfac, java.util.List<GenPolynomial<C>> L)	Recursive representation.
static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>	recursiveDensePseudoRemainder(GenPolynomial<GenPolynomial<C>> P, GenPolynomial<GenPolynomial<C>> S)	GenPolynomial dense pseudo remainder.
static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>	recursiveDeriviative(GenPolynomial<GenPolynomial<C>> P)	GenPolynomial recursive polynomial derivative main variable.
static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>	recursiveDivide(GenPolynomial<GenPolynomial<C>> P, GenPolynomial<C> s)	GenPolynomial divide.
static <C extends RingElem<C>> GenWordPolynomial<GenPolynomial<C>>	recursiveDivide(GenWordPolynomial<GenPolynomial<C>> P, GenPolynomial<C> s)	GenPolynomial divide.
static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>	recursivePseudoDivide(GenPolynomial<GenPolynomial<C>> P, GenPolynomial<GenPolynomial<C>> S)	GenPolynomial recursive pseudo divide.
static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>	recursivePseudoRemainder(GenPolynomial<GenPolynomial<C>> P, GenPolynomial<GenPolynomial<C>> S)	Deprecated. (forRemoval=true) Use recursiveSparsePseudoRemainder(edu.jas.poly.GenPolynomial,edu.jas.poly.GenPolynomial) instead
static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>	recursiveSparsePseudoRemainder(GenPolynomial<GenPolynomial<C>> P, GenPolynomial<GenPolynomial<C>> S)	GenPolynomial sparse pseudo remainder.
static <C extends RingElem<C>> GenPolynomial<C>	removeUnusedLowerVariables(GenPolynomial<C> p)	Remove all lower variables which do not occur in polynomial.
static <C extends RingElem<C>> GenPolynomial<C>	removeUnusedMiddleVariables(GenPolynomial<C> p)	Remove upper block of middle variables which do not occur in polynomial.
static <C extends RingElem<C>> GenPolynomial<C>	removeUnusedUpperVariables(GenPolynomial<C> p)	Remove all upper variables which do not occur in polynomial.
static <C extends RingElem<C>> GenPolynomial<C>	selectWithVariable(java.util.List<GenPolynomial<C>> P, int i)	Select polynomial with univariate leading term in variable i.
static <C extends RingElem<C>> GenPolynomial<C>	seriesOfTaylor(GenPolynomial<C> f, C a)	Taylor series for polynomial.
static <C extends RingElem<C>> GenPolynomial<C>	substituteMain(GenPolynomial<C> A, GenPolynomial<C> s)	Substitute main variable.
static <C extends RingElem<C>> GenPolynomial<C>	substituteUnivariate(GenPolynomial<C> f, GenPolynomial<C> t)	Substitute univariate polynomial.
static <C extends RingElem<C>> GenPolynomial<C>	substituteUnivariateMult(GenPolynomial<C> f, GenPolynomial<C> t)	Substitute univariate polynomial with multivariate coefficients.
static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>	switchVariables(GenPolynomial<GenPolynomial<C>> P)	Recursive GenPolynomial switch varaible blocks.
static <C extends RingElem<C>> GenPolynomial<Complex<C>>	toComplex(GenPolynomialRing<Complex<C>> fac, GenPolynomial<C> A)	Complex from real polynomial.
static Product<ModInteger>	toProduct(ProductRing<ModInteger> pfac, BigInteger c)	Product representation.
static <C extends RingElem<C>> Product<GenPolynomial<C>>	toProduct(ProductRing<GenPolynomial<C>> pfac, C c, ExpVector e)	Product representation.
static <C extends RingElem<C>> Product<GenPolynomial<C>>	toProduct(ProductRing<GenPolynomial<C>> pfac, GenPolynomial<C> A)	Product representation.
static GenPolynomial<Product<ModInteger>>	toProduct(GenPolynomialRing<Product<ModInteger>> pfac, GenPolynomial<BigInteger> A)	Product representation.
static java.util.List<GenPolynomial<Product<ModInteger>>>	toProduct(GenPolynomialRing<Product<ModInteger>> pfac, java.util.List<GenPolynomial<BigInteger>> L)	Product representation.
static <C extends GcdRingElem<C>> Product<C>	toProductGen(ProductRing<C> pfac, C c)	Product representation.
static <C extends GcdRingElem<C>> GenPolynomial<Product<C>>	toProductGen(GenPolynomialRing<Product<C>> pfac, GenPolynomial<C> A)	Product representation.
static <C extends GcdRingElem<C>> java.util.List<GenPolynomial<Product<C>>>	toProductGen(GenPolynomialRing<Product<C>> pfac, java.util.List<GenPolynomial<C>> L)	Product representation.
static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>	toRecursive(GenPolynomialRing<GenPolynomial<C>> rfac, GenPolynomial<C> A)	To recursive representation.
static <C extends RingElem<C>> GenSolvablePolynomial<GenPolynomial<C>>	toRecursive(GenSolvablePolynomialRing<GenPolynomial<C>> rfac, GenSolvablePolynomial<C> A)	To recursive representation.
static <C extends RingElem<C>> long	totalDegree(java.util.List<GenPolynomial<C>> P)	Total degree of polynomial list.
static <C extends RingElem<C>> long	totalDegreeLeadingTerm(java.util.List<GenPolynomial<C>> P)	Maximal degree of leading terms of a polynomial list.
static <C extends RingElem<C>> java.util.List<GenWordPolynomial<C>>	wordMonic(java.util.List<GenWordPolynomial<C>> L)	Word polynomial list monic.

Methods inherited from class java.lang.Object

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

Constructor Detail

PolyUtil

public PolyUtil()

Method Detail

recursive

public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> recursive(GenPolynomialRing<GenPolynomial<C>> rfac,
                                                                                GenPolynomial<C> A)

Recursive representation. Represent as polynomial in i variables with coefficients in n-i variables. Works for arbitrary term orders.

Type Parameters:

C - coefficient type.

Parameters:

rfac - recursive polynomial ring factory.

A - polynomial to be converted.

Returns:

Recursive represenations of this in the ring rfac.

distribute

public static <C extends RingElem<C>> GenPolynomial<C> distribute(GenPolynomialRing<C> dfac,
                                                                  GenPolynomial<GenPolynomial<C>> B)

Distribute a recursive polynomial to a generic polynomial. Works for arbitrary term orders.

Type Parameters:

C - coefficient type.

Parameters:

dfac - combined polynomial ring factory of coefficients and this.

B - polynomial to be converted.

Returns:

distributed polynomial.

recursive

public static <C extends RingElem<C>> java.util.List<GenPolynomial<GenPolynomial<C>>> recursive(GenPolynomialRing<GenPolynomial<C>> rfac,
                                                                                                java.util.List<GenPolynomial<C>> L)

Recursive representation. Represent as polynomials in i variables with coefficients in n-i variables. Works for arbitrary term orders.

Type Parameters:

C - coefficient type.

Parameters:

rfac - recursive polynomial ring factory.

L - list of polynomials to be converted.

Returns:

Recursive represenations of the list in the ring rfac.

distribute

public static <C extends RingElem<C>> java.util.List<GenPolynomial<C>> distribute(GenPolynomialRing<C> dfac,
                                                                                  java.util.List<GenPolynomial<GenPolynomial<C>>> L)

Distribute a recursive polynomial list to a generic polynomial list. Works for arbitrary term orders.

Type Parameters:

C - coefficient type.

Parameters:

dfac - combined polynomial ring factory of coefficients and this.

L - list of polynomials to be converted.

Returns:

distributed polynomial list.

integerFromModularCoefficients

public static <C extends RingElem<C> & Modular> GenPolynomial<BigInteger> integerFromModularCoefficients(GenPolynomialRing<BigInteger> fac,
                                                                                                         GenPolynomial<C> A)

BigInteger from ModInteger coefficients, symmetric. Represent as polynomial with BigInteger coefficients by removing the modules and making coefficients symmetric to 0.

Parameters:

fac - result polynomial factory.

A - polynomial with ModInteger coefficients to be converted.

Returns:

polynomial with BigInteger coefficients.

integerFromModularCoefficients

public static <C extends RingElem<C> & Modular> java.util.List<GenPolynomial<BigInteger>> integerFromModularCoefficients(GenPolynomialRing<BigInteger> fac,
                                                                                                                         java.util.List<GenPolynomial<C>> L)

BigInteger from ModInteger coefficients, symmetric. Represent as polynomial with BigInteger coefficients by removing the modules and making coefficients symmetric to 0.

Parameters:

fac - result polynomial factory.

L - list of polynomials with ModInteger coefficients to be converted.

Returns:

list of polynomials with BigInteger coefficients.

integerFromModularCoefficientsPositive

public static <C extends RingElem<C> & Modular> GenPolynomial<BigInteger> integerFromModularCoefficientsPositive(GenPolynomialRing<BigInteger> fac,
                                                                                                                 GenPolynomial<C> A)

BigInteger from ModInteger coefficients, positive. Represent as polynomial with BigInteger coefficients by removing the modules.

Parameters:

fac - result polynomial factory.

A - polynomial with ModInteger coefficients to be converted.

Returns:

polynomial with BigInteger coefficients.

integerFromRationalCoefficients

public static GenPolynomial<BigInteger> integerFromRationalCoefficients(GenPolynomialRing<BigInteger> fac,
                                                                        GenPolynomial<BigRational> A)

BigInteger from BigRational coefficients. Represent as polynomial with BigInteger coefficients by multiplication with the lcm of the numerators of the BigRational coefficients.

Parameters:

fac - result polynomial factory.

A - polynomial with BigRational coefficients to be converted.

Returns:

polynomial with BigInteger coefficients.

integerFromRationalCoefficientsFactor

public static java.lang.Object[] integerFromRationalCoefficientsFactor(GenPolynomialRing<BigInteger> fac,
                                                                       GenPolynomial<BigRational> A)

BigInteger from BigRational coefficients. Represent as polynomial with BigInteger coefficients by multiplication with the gcd of the numerators and the lcm of the denominators of the BigRational coefficients.
Author: Axel Kramer

Parameters:

fac - result polynomial factory.

A - polynomial with BigRational coefficients to be converted.

Returns:

Object[] with 3 entries: [0]=gcd [1]=lcm and [2]=polynomial with BigInteger coefficients.

integerFromRationalCoefficients

public static GenPolynomial<BigInteger> integerFromRationalCoefficients(GenPolynomialRing<BigInteger> fac,
                                                                        java.math.BigInteger gcd,
                                                                        java.math.BigInteger lcm,
                                                                        GenPolynomial<BigRational> A)

BigInteger from BigRational coefficients. Represent as polynomial with BigInteger coefficients by multiplication with the gcd of the numerators and the lcm of the denominators of the BigRational coefficients.

Parameters:

fac - result polynomial factory.

gcd - of rational coefficient numerators.

lcm - of rational coefficient denominators.

A - polynomial with BigRational coefficients to be converted.

Returns:

polynomial with BigInteger coefficients.

integerFromRationalCoefficients

public static java.util.List<GenPolynomial<BigInteger>> integerFromRationalCoefficients(GenPolynomialRing<BigInteger> fac,
                                                                                        java.util.List<GenPolynomial<BigRational>> L)

BigInteger from BigRational coefficients. Represent as list of polynomials with BigInteger coefficients by multiplication with the lcm of the numerators of the BigRational coefficients of each polynomial.

Parameters:

fac - result polynomial factory.

L - list of polynomials with BigRational coefficients to be converted.

Returns:

polynomial list with BigInteger coefficients.

fromIntegerCoefficients

public static <C extends RingElem<C>> GenPolynomial<C> fromIntegerCoefficients(GenPolynomialRing<C> fac,
                                                                               GenPolynomial<BigInteger> A)

From BigInteger coefficients. Represent as polynomial with type C coefficients, e.g. ModInteger or BigRational.

Type Parameters:

C - coefficient type.

Parameters:

fac - result polynomial factory.

A - polynomial with BigInteger coefficients to be converted.

Returns:

polynomial with type C coefficients.

fromIntegerCoefficients

public static <C extends RingElem<C>> java.util.List<GenPolynomial<C>> fromIntegerCoefficients(GenPolynomialRing<C> fac,
                                                                                               java.util.List<GenPolynomial<BigInteger>> L)

From BigInteger coefficients. Represent as list of polynomials with type C coefficients, e.g. ModInteger or BigRational.

Type Parameters:

C - coefficient type.

Parameters:

fac - result polynomial factory.

L - list of polynomials with BigInteger coefficients to be converted.

Returns:

list of polynomials with type C coefficients.

decimalFromRational

public static <C extends RingElem<C> & Rational> GenPolynomial<BigDecimal> decimalFromRational(GenPolynomialRing<BigDecimal> fac,
                                                                                               GenPolynomial<C> A)

Convert to decimal coefficients.

Parameters:

fac - result polynomial factory.

A - polynomial with Rational coefficients to be converted.

Returns:

polynomial with BigDecimal coefficients.

complexDecimalFromRational

public static <C extends RingElem<C> & Rational> GenPolynomial<Complex<BigDecimal>> complexDecimalFromRational(GenPolynomialRing<Complex<BigDecimal>> fac,
                                                                                                               GenPolynomial<Complex<C>> A)

Convert to complex decimal coefficients.

Parameters:

fac - result polynomial factory.

A - polynomial with complex Rational coefficients to be converted.

Returns:

polynomial with Complex BigDecimal coefficients.

realPart

public static GenPolynomial<BigRational> realPart(GenPolynomialRing<BigRational> fac,
                                                  GenPolynomial<BigComplex> A)

Real part.

Parameters:

fac - result polynomial factory.

A - polynomial with BigComplex coefficients to be converted.

Returns:

polynomial with real part of the coefficients.

imaginaryPart

public static GenPolynomial<BigRational> imaginaryPart(GenPolynomialRing<BigRational> fac,
                                                       GenPolynomial<BigComplex> A)

Imaginary part.

Parameters:

fac - result polynomial factory.

A - polynomial with BigComplex coefficients to be converted.

Returns:

polynomial with imaginary part of coefficients.

realPartFromComplex

public static <C extends RingElem<C>> GenPolynomial<C> realPartFromComplex(GenPolynomialRing<C> fac,
                                                                           GenPolynomial<Complex<C>> A)

Real part.

Parameters:

fac - result polynomial factory.

A - polynomial with BigComplex coefficients to be converted.

Returns:

polynomial with real part of the coefficients.

imaginaryPartFromComplex

public static <C extends RingElem<C>> GenPolynomial<C> imaginaryPartFromComplex(GenPolynomialRing<C> fac,
                                                                                GenPolynomial<Complex<C>> A)

Imaginary part.

Parameters:

fac - result polynomial factory.

A - polynomial with BigComplex coefficients to be converted.

Returns:

polynomial with imaginary part of coefficients.

toComplex

public static <C extends RingElem<C>> GenPolynomial<Complex<C>> toComplex(GenPolynomialRing<Complex<C>> fac,
                                                                          GenPolynomial<C> A)

Complex from real polynomial.

Parameters:

fac - result polynomial factory.

A - polynomial with C coefficients to be converted.

Returns:

polynomial with Complex coefficients.

complexFromRational

public static GenPolynomial<BigComplex> complexFromRational(GenPolynomialRing<BigComplex> fac,
                                                            GenPolynomial<BigRational> A)

Complex from rational coefficients.

Parameters:

fac - result polynomial factory.

A - polynomial with BigRational coefficients to be converted.

Returns:

polynomial with BigComplex coefficients.

complexFromAny

public static <C extends GcdRingElem<C>> GenPolynomial<Complex<C>> complexFromAny(GenPolynomialRing<Complex<C>> fac,
                                                                                  GenPolynomial<C> A)

Complex from ring element coefficients.

Parameters:

fac - result polynomial factory.

A - polynomial with RingElem coefficients to be converted.

Returns:

polynomial with Complex coefficients.

fromAlgebraicCoefficients

public static <C extends GcdRingElem<C>> GenPolynomial<GenPolynomial<C>> fromAlgebraicCoefficients(GenPolynomialRing<GenPolynomial<C>> rfac,
                                                                                                   GenPolynomial<AlgebraicNumber<C>> A)

From AlgebraicNumber coefficients. Represent as polynomial with type GenPolynomial<C> coefficients, e.g. ModInteger or BigRational.

Parameters:

rfac - result polynomial factory.

A - polynomial with AlgebraicNumber coefficients to be converted.

Returns:

polynomial with type GenPolynomial<C> coefficients.

convertToAlgebraicCoefficients

public static <C extends GcdRingElem<C>> GenPolynomial<AlgebraicNumber<C>> convertToAlgebraicCoefficients(GenPolynomialRing<AlgebraicNumber<C>> pfac,
                                                                                                          GenPolynomial<C> A)

Convert to AlgebraicNumber coefficients. Represent as polynomial with AlgebraicNumber coefficients, C is e.g. ModInteger or BigRational.

Parameters:

pfac - result polynomial factory.

A - polynomial with C coefficients to be converted.

Returns:

polynomial with AlgebraicNumber<C> coefficients.

convertToRecAlgebraicCoefficients

public static <C extends GcdRingElem<C>> GenPolynomial<AlgebraicNumber<C>> convertToRecAlgebraicCoefficients(int depth,
                                                                                                             GenPolynomialRing<AlgebraicNumber<C>> pfac,
                                                                                                             GenPolynomial<C> A)

Convert to recursive AlgebraicNumber coefficients. Represent as polynomial with recursive AlgebraicNumber coefficients, C is e.g. ModInteger or BigRational.

Parameters:

depth - recursion depth of AlgebraicNumber coefficients.

pfac - result polynomial factory.

A - polynomial with C coefficients to be converted.

Returns:

polynomial with AlgebraicNumber<C> coefficients.

convertRecursiveToAlgebraicCoefficients

public static <C extends GcdRingElem<C>> GenPolynomial<AlgebraicNumber<C>> convertRecursiveToAlgebraicCoefficients(GenPolynomialRing<AlgebraicNumber<C>> pfac,
                                                                                                                   GenPolynomial<GenPolynomial<C>> A)

Convert to AlgebraicNumber coefficients. Represent as polynomial with AlgebraicNumber coefficients, C is e.g. ModInteger or BigRational.

Parameters:

pfac - result polynomial factory.

A - recursive polynomial with GenPolynomial<BigInteger> coefficients to be converted.

Returns:

polynomial with AlgebraicNumber<C> coefficients.

complexFromAlgebraic

public static <C extends GcdRingElem<C>> GenPolynomial<Complex<C>> complexFromAlgebraic(GenPolynomialRing<Complex<C>> fac,
                                                                                        GenPolynomial<AlgebraicNumber<C>> A)

Complex from algebraic coefficients.

Parameters:

fac - result polynomial factory.

A - polynomial with AlgebraicNumber coefficients Q(i) to be converted.

Returns:

polynomial with Complex coefficients.

algebraicFromComplex

public static <C extends GcdRingElem<C>> GenPolynomial<AlgebraicNumber<C>> algebraicFromComplex(GenPolynomialRing<AlgebraicNumber<C>> fac,
                                                                                                GenPolynomial<Complex<C>> A)

AlgebraicNumber from complex coefficients.

Parameters:

fac - result polynomial factory over Q(i).

A - polynomial with Complex coefficients to be converted.

Returns:

polynomial with AlgebraicNumber coefficients.

chineseRemainder

public static <C extends RingElem<C> & Modular> GenPolynomial<C> chineseRemainder(GenPolynomialRing<C> fac,
                                                                                  GenPolynomial<C> A,
                                                                                  C mi,
                                                                                  GenPolynomial<C> B)

ModInteger chinese remainder algorithm on coefficients.

Parameters:

fac - GenPolynomial<ModInteger> result factory with A.coFac.modul*B.coFac.modul = C.coFac.modul.

A - GenPolynomial<ModInteger>.

B - other GenPolynomial<ModInteger>.

mi - inverse of A.coFac.modul in ring B.coFac.

Returns:

S = cra(A,B), with S mod A.coFac.modul == A and S mod B.coFac.modul == B.

monic

public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> monic(GenPolynomial<GenPolynomial<C>> p)

GenPolynomial monic, i.e. leadingBaseCoefficient == 1. If leadingBaseCoefficient is not invertible returns this unmodified.

Type Parameters:

C - coefficient type.

Parameters:

p - recursive GenPolynomial<GenPolynomial>.

Returns:

monic(p).

monic

public static <C extends RingElem<C>> GenSolvablePolynomial<GenPolynomial<C>> monic(GenSolvablePolynomial<GenPolynomial<C>> p)

GenSolvablePolynomial monic, i.e. leadingBaseCoefficient == 1. If leadingBaseCoefficient is not invertible returns this unmodified.

Type Parameters:

C - coefficient type.

Parameters:

p - recursive GenSolvablePolynomial<GenPolynomial>.

Returns:

monic(p).

monic

public static <C extends RingElem<C>> java.util.List<GenPolynomial<C>> monic(java.util.List<GenPolynomial<C>> L)

Polynomial list monic.

Type Parameters:

C - coefficient type.

Parameters:

L - list of polynomials with field coefficients.

Returns:

list of polynomials with leading coefficient 1.

wordMonic

public static <C extends RingElem<C>> java.util.List<GenWordPolynomial<C>> wordMonic(java.util.List<GenWordPolynomial<C>> L)

Word polynomial list monic.

Type Parameters:

C - coefficient type.

Parameters:

L - list of word polynomials with field coefficients.

Returns:

list of word polynomials with leading coefficient 1.

monicRec

public static <C extends RingElem<C>> java.util.List<GenPolynomial<GenPolynomial<C>>> monicRec(java.util.List<GenPolynomial<GenPolynomial<C>>> L)

Recursive polynomial list monic.

Type Parameters:

C - coefficient type.

Parameters:

L - list of recursive polynomials with field coefficients.

Returns:

list of polynomials with leading base coefficient 1.

leadingExpVector

public static <C extends RingElem<C>> java.util.List<ExpVector> leadingExpVector(java.util.List<GenPolynomial<C>> L)

Polynomial list leading exponent vectors.

Type Parameters:

C - coefficient type.

Parameters:

L - list of polynomials.

Returns:

list of leading exponent vectors.

extendCoefficients

public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> extendCoefficients(GenPolynomialRing<GenPolynomial<C>> pfac,
                                                                                         GenPolynomial<GenPolynomial<C>> A,
                                                                                         int j,
                                                                                         long k)

Extend coefficient variables. Extend all coefficient ExpVectors by i elements and multiply by x_j^k.

Parameters:

pfac - extended polynomial ring factory (by i variables in the coefficients).

j - index of variable to be used for multiplication.

k - exponent for x_j.

Returns:

extended polynomial.

extendCoefficients

public static <C extends RingElem<C>> GenSolvablePolynomial<GenPolynomial<C>> extendCoefficients(GenSolvablePolynomialRing<GenPolynomial<C>> pfac,
                                                                                                 GenSolvablePolynomial<GenPolynomial<C>> A,
                                                                                                 int j,
                                                                                                 long k)

Extend coefficient variables. Extend all coefficient ExpVectors by i elements and multiply by x_j^k.

Parameters:

pfac - extended polynomial ring factory (by i variables in the coefficients).

j - index of variable to be used for multiplication.

k - exponent for x_j.

Returns:

extended polynomial.

toRecursive

public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> toRecursive(GenPolynomialRing<GenPolynomial<C>> rfac,
                                                                                  GenPolynomial<C> A)

To recursive representation. Represent as polynomial in i+r variables with coefficients in i variables. Works for arbitrary term orders.

Type Parameters:

C - coefficient type.

Parameters:

rfac - recursive polynomial ring factory.

A - polynomial to be converted.

Returns:

Recursive represenations of A in the ring rfac.

toRecursive

public static <C extends RingElem<C>> GenSolvablePolynomial<GenPolynomial<C>> toRecursive(GenSolvablePolynomialRing<GenPolynomial<C>> rfac,
                                                                                          GenSolvablePolynomial<C> A)

To recursive representation. Represent as solvable polynomial in i+r variables with coefficients in i variables. Works for arbitrary term orders.

Type Parameters:

C - coefficient type.

Parameters:

rfac - recursive solvable polynomial ring factory.

A - solvable polynomial to be converted.

Returns:

Recursive represenations of A in the ring rfac.

baseRemainderPoly

public static <C extends RingElem<C>> GenPolynomial<C> baseRemainderPoly(GenPolynomial<C> P,
                                                                         C s)

GenPolynomial coefficient wise remainder.

Type Parameters:

C - coefficient type.

Parameters:

P - GenPolynomial.

s - nonzero coefficient.

Returns:

coefficient wise remainder.

basePseudoRemainder

@Deprecated
public static <C extends RingElem<C>> GenPolynomial<C> basePseudoRemainder(GenPolynomial<C> P,
                                                                           GenPolynomial<C> S)

Deprecated.

(forRemoval=true) Use baseSparsePseudoRemainder(edu.jas.poly.GenPolynomial,edu.jas.poly.GenPolynomial) instead

GenPolynomial sparse pseudo remainder. For univariate polynomials.

Type Parameters:

C - coefficient type.

Parameters:

P - GenPolynomial.

S - nonzero GenPolynomial.

Returns:

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

baseSparsePseudoRemainder

public static <C extends RingElem<C>> GenPolynomial<C> baseSparsePseudoRemainder(GenPolynomial<C> P,
                                                                                 GenPolynomial<C> S)

GenPolynomial sparse pseudo remainder. For univariate polynomials.

Type Parameters:

C - coefficient type.

Parameters:

P - GenPolynomial.

S - nonzero GenPolynomial.

Returns:

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

baseDensePseudoRemainder

public static <C extends RingElem<C>> GenPolynomial<C> baseDensePseudoRemainder(GenPolynomial<C> P,
                                                                                GenPolynomial<C> S)

GenPolynomial dense pseudo remainder. For univariate polynomials.

Parameters:

P - GenPolynomial.

S - nonzero GenPolynomial.

Returns:

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

baseDensePseudoQuotient

public static <C extends RingElem<C>> GenPolynomial<C> baseDensePseudoQuotient(GenPolynomial<C> P,
                                                                               GenPolynomial<C> S)

GenPolynomial dense pseudo quotient. For univariate polynomials.

Parameters:

P - GenPolynomial.

S - nonzero GenPolynomial.

Returns:

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

basePseudoDivide

public static <C extends RingElem<C>> GenPolynomial<C> basePseudoDivide(GenPolynomial<C> P,
                                                                        GenPolynomial<C> S)

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

Type Parameters:

C - coefficient type.

Parameters:

P - GenPolynomial.

S - nonzero GenPolynomial.

Returns:

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

basePseudoQuotientRemainder

public static <C extends RingElem<C>> GenPolynomial<C>[] basePseudoQuotientRemainder(GenPolynomial<C> P,
                                                                                     GenPolynomial<C> S)

GenPolynomial sparse pseudo quotient and remainder. For univariate polynomials or exact division.

Type Parameters:

C - coefficient type.

Parameters:

P - GenPolynomial.

S - nonzero GenPolynomial.

Returns:

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

isBasePseudoQuotientRemainder

public static <C extends RingElem<C>> boolean isBasePseudoQuotientRemainder(GenPolynomial<C> P,
                                                                            GenPolynomial<C> S,
                                                                            GenPolynomial<C> q,
                                                                            GenPolynomial<C> r)

Is GenPolynomial pseudo quotient and remainder. For univariate polynomials.

Type Parameters:

C - coefficient type.

Parameters:

P - base GenPolynomial.

S - nonzero base GenPolynomial.

Returns:

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

recursiveDivide

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

GenPolynomial divide. For recursive polynomials. Division by coefficient ring element.

Type Parameters:

C - coefficient type.

Parameters:

P - recursive GenPolynomial.

s - GenPolynomial.

Returns:

this/s.

recursiveDivide

public static <C extends RingElem<C>> GenWordPolynomial<GenPolynomial<C>> recursiveDivide(GenWordPolynomial<GenPolynomial<C>> P,
                                                                                          GenPolynomial<C> s)

GenPolynomial divide. For recursive polynomials. Division by coefficient ring element.

Type Parameters:

C - coefficient type.

Parameters:

P - recursive GenPolynomial.

s - GenPolynomial.

Returns:

this/s.

baseRecursiveDivide

public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> baseRecursiveDivide(GenPolynomial<GenPolynomial<C>> P,
                                                                                          C s)

GenPolynomial base divide. For recursive polynomials. Division by coefficient ring element.

Type Parameters:

C - coefficient type.

Parameters:

P - recursive GenPolynomial.

s - coefficient.

Returns:

this/s.

recursivePseudoRemainder

@Deprecated
public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> recursivePseudoRemainder(GenPolynomial<GenPolynomial<C>> P,
                                                                                               GenPolynomial<GenPolynomial<C>> S)

Deprecated.

(forRemoval=true) Use recursiveSparsePseudoRemainder(edu.jas.poly.GenPolynomial,edu.jas.poly.GenPolynomial) instead

GenPolynomial sparse pseudo remainder. For recursive polynomials.

Type Parameters:

C - coefficient type.

Parameters:

P - recursive GenPolynomial.

S - nonzero recursive GenPolynomial.

Returns:

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

recursiveSparsePseudoRemainder

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

GenPolynomial sparse pseudo remainder. For recursive polynomials.

Type Parameters:

C - coefficient type.

Parameters:

P - recursive GenPolynomial.

S - nonzero recursive GenPolynomial.

Returns:

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

recursiveDensePseudoRemainder

public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> recursiveDensePseudoRemainder(GenPolynomial<GenPolynomial<C>> P,
                                                                                                    GenPolynomial<GenPolynomial<C>> S)

GenPolynomial dense pseudo remainder. For recursive polynomials.

Parameters:

P - recursive GenPolynomial.

S - nonzero recursive GenPolynomial.

Returns:

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

recursivePseudoDivide

public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> recursivePseudoDivide(GenPolynomial<GenPolynomial<C>> P,
                                                                                            GenPolynomial<GenPolynomial<C>> S)

GenPolynomial recursive pseudo divide. For recursive polynomials.

Type Parameters:

C - coefficient type.

Parameters:

P - recursive GenPolynomial.

S - nonzero recursive GenPolynomial.

Returns:

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

isRecursivePseudoQuotientRemainder

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

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

Type Parameters:

C - coefficient type.

Parameters:

P - recursive GenPolynomial.

S - nonzero recursive GenPolynomial.

Returns:

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

coefficientPseudoDivide

public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> coefficientPseudoDivide(GenPolynomial<GenPolynomial<C>> P,
                                                                                              GenPolynomial<C> s)

GenPolynomial pseudo divide. For recursive polynomials.

Type Parameters:

C - coefficient type.

Parameters:

P - recursive GenPolynomial.

s - nonzero GenPolynomial.

Returns:

quotient with ldcf(s)^m P = quotient * s + remainder.

coefficientBasePseudoDivide

public static <C extends RingElem<C>> GenPolynomial<C> coefficientBasePseudoDivide(GenPolynomial<C> P,
                                                                                   C s)

GenPolynomial pseudo divide. For polynomials.

Type Parameters:

C - coefficient type.

Parameters:

P - GenPolynomial.

s - nonzero coefficient.

Returns:

quotient with ldcf(s)^m P = quotient * s + remainder.

baseDeriviative

public static <C extends RingElem<C>> GenPolynomial<C> baseDeriviative(GenPolynomial<C> P)

GenPolynomial polynomial derivative main variable.

Type Parameters:

C - coefficient type.

Parameters:

P - GenPolynomial.

Returns:

deriviative(P).

baseDeriviative

public static <C extends RingElem<C>> GenPolynomial<C> baseDeriviative(GenPolynomial<C> P,
                                                                       int r)

GenPolynomial polynomial partial derivative variable r.

Type Parameters:

C - coefficient type.

Parameters:

P - GenPolynomial.

r - variable for partial deriviate.

Returns:

deriviative(P,r).

baseIntegral

public static <C extends RingElem<C>> GenPolynomial<C> baseIntegral(GenPolynomial<C> P)

GenPolynomial polynomial integral main variable.

Type Parameters:

C - coefficient type.

Parameters:

P - GenPolynomial.

Returns:

integral(P).

recursiveDeriviative

public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> recursiveDeriviative(GenPolynomial<GenPolynomial<C>> P)

GenPolynomial recursive polynomial derivative main variable.

Type Parameters:

C - coefficient type.

Parameters:

P - recursive GenPolynomial.

Returns:

deriviative(P).

factorBound

public static BigInteger factorBound(ExpVector e)

Factor coefficient bound. See SACIPOL.IPFCB: the product of all maxNorms of potential factors is less than or equal to 2**b times the maxNorm of A.

Parameters:

e - degree vector of a GenPolynomial A.

Returns:

2**b.

absNorm

public static <C extends RingElem<C>> C absNorm(GenPolynomial<C> p)

Absoulte norm. Square root of the sum of the squared coefficients.

Parameters:

p - GenPolynomial

Returns:

sqrt( sum_i |c_i|² ).

evaluateMainRecursive

public static <C extends RingElem<C>> GenPolynomial<C> evaluateMainRecursive(GenPolynomialRing<C> cfac,
                                                                             GenPolynomial<GenPolynomial<C>> A,
                                                                             C a)

Evaluate at main variable.

Type Parameters:

C - coefficient type.

Parameters:

cfac - coefficent polynomial ring factory.

A - recursive polynomial to be evaluated.

a - value to evaluate at.

Returns:

A( x_1, ..., x_{n-1}, a ).

evaluateMain

public static <C extends RingElem<C>> GenPolynomial<C> evaluateMain(GenPolynomialRing<C> cfac,
                                                                    GenPolynomial<C> A,
                                                                    C a)

Evaluate at main variable.

Type Parameters:

C - coefficient type.

Parameters:

cfac - coefficent polynomial ring factory.

A - distributed polynomial to be evaluated.

a - value to evaluate at.

Returns:

A( x_1, ..., x_{n-1}, a ).

evaluateMain

public static <C extends RingElem<C>> java.util.List<GenPolynomial<C>> evaluateMain(GenPolynomialRing<C> cfac,
                                                                                    java.util.List<GenPolynomial<C>> L,
                                                                                    C a)

Evaluate at main variable.

Type Parameters:

C - coefficient type.

Parameters:

cfac - coefficent ring factory.

L - list of univariate polynomials to be evaluated.

a - value to evaluate at.

Returns:

list( A( x_1, ..., x_{n-1}, a ) ) for A in L.

evaluateMain

public static <C extends RingElem<C>> C evaluateMain(RingFactory<C> cfac,
                                                     GenPolynomial<C> A,
                                                     C a)

Evaluate at main variable.

Type Parameters:

C - coefficient type.

Parameters:

cfac - coefficent ring factory.

A - univariate polynomial to be evaluated.

a - value to evaluate at.

Returns:

A( a ).

evaluateMain

public static <C extends RingElem<C>> java.util.List<C> evaluateMain(RingFactory<C> cfac,
                                                                     java.util.List<GenPolynomial<C>> L,
                                                                     C a)

Evaluate at main variable.

Type Parameters:

C - coefficient type.

Parameters:

cfac - coefficent ring factory.

L - list of univariate polynomial to be evaluated.

a - value to evaluate at.

Returns:

list( A( a ) ) for A in L.

evaluate

public static <C extends RingElem<C>> GenPolynomial<C> evaluate(GenPolynomialRing<C> cfac,
                                                                GenPolynomialRing<GenPolynomial<C>> rfac,
                                                                GenPolynomialRing<GenPolynomial<C>> nfac,
                                                                GenPolynomialRing<C> dfac,
                                                                GenPolynomial<C> A,
                                                                C a)

Evaluate at k-th variable.

Type Parameters:

C - coefficient type.

Parameters:

cfac - coefficient polynomial ring in k variables C[x_1, ..., x_k] factory.

rfac - coefficient polynomial ring C[x_1, ..., x_{k-1}] [x_k] factory, a recursive polynomial ring in 1 variable with coefficients in k-1 variables.

nfac - polynomial ring in n-1 varaibles C[x_1, ..., x_{k-1}] [x_{k+1}, ..., x_n] factory, a recursive polynomial ring in n-k+1 variables with coefficients in k-1 variables.

dfac - polynomial ring in n-1 variables. C[x_1, ..., x_{k-1}, x_{k+1}, ..., x_n] factory.

A - polynomial to be evaluated.

a - value to evaluate at.

Returns:

A( x_1, ..., x_{k-1}, a, x_{k+1}, ..., x_n).

evaluateFirst

public static <C extends RingElem<C>> GenPolynomial<C> evaluateFirst(GenPolynomialRing<C> cfac,
                                                                     GenPolynomialRing<C> dfac,
                                                                     GenPolynomial<C> A,
                                                                     C a)

Evaluate at first (lowest) variable.

Type Parameters:

C - coefficient type.

Parameters:

cfac - coefficient polynomial ring in first variable C[x_1] factory.

dfac - polynomial ring in n-1 variables. C[x_2, ..., x_n] factory.

A - polynomial to be evaluated.

a - value to evaluate at.

Returns:

A( a, x_2, ..., x_n).

evaluateFirstRec

public static <C extends RingElem<C>> GenPolynomial<C> evaluateFirstRec(GenPolynomialRing<C> cfac,
                                                                        GenPolynomialRing<C> dfac,
                                                                        GenPolynomial<GenPolynomial<C>> A,
                                                                        C a)

Evaluate at first (lowest) variable. Could also be called evaluateFirst(), but type erasure of parameter A does not allow the same name.

Type Parameters:

C - coefficient type.

Parameters:

cfac - coefficient polynomial ring in first variable C[x_1] factory.

dfac - polynomial ring in n-1 variables. C[x_2, ..., x_n] factory.

A - recursive polynomial to be evaluated.

a - value to evaluate at.

Returns:

A( a, x_2, ..., x_n).

evaluateAll

public static <C extends RingElem<C>> C evaluateAll(RingFactory<C> cfac,
                                                    GenPolynomial<C> A,
                                                    java.util.List<C> a)

Evaluate all variables.

Type Parameters:

C - coefficient type.

Parameters:

cfac - coefficient ring factory.

A - polynomial to be evaluated.

a - = (a_1, a_2, ..., a_n) a tuple of values to evaluate at.

Returns:

A(a_1, a_2, ..., a_n).

substituteMain

public static <C extends RingElem<C>> GenPolynomial<C> substituteMain(GenPolynomial<C> A,
                                                                      GenPolynomial<C> s)

Substitute main variable.

Parameters:

A - univariate polynomial.

s - polynomial for substitution.

Returns:

polynomial A(x <- s).

substituteUnivariate

public static <C extends RingElem<C>> GenPolynomial<C> substituteUnivariate(GenPolynomial<C> f,
                                                                            GenPolynomial<C> t)

Substitute univariate polynomial.

Parameters:

f - univariate polynomial.

t - polynomial for substitution.

Returns:

polynomial f(x <- t).

substituteUnivariateMult

public static <C extends RingElem<C>> GenPolynomial<C> substituteUnivariateMult(GenPolynomial<C> f,
                                                                                GenPolynomial<C> t)

Substitute univariate polynomial with multivariate coefficients.

Parameters:

f - univariate polynomial with multivariate coefficients.

t - polynomial for substitution.

Returns:

polynomial f(x <- t).

seriesOfTaylor

public static <C extends RingElem<C>> GenPolynomial<C> seriesOfTaylor(GenPolynomial<C> f,
                                                                      C a)

Taylor series for polynomial.

Parameters:

f - univariate polynomial.

a - expansion point.

Returns:

Taylor series (a polynomial) of f at a.

interpolate

public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> interpolate(GenPolynomialRing<GenPolynomial<C>> fac,
                                                                                  GenPolynomial<GenPolynomial<C>> A,
                                                                                  GenPolynomial<C> M,
                                                                                  C mi,
                                                                                  GenPolynomial<C> B,
                                                                                  C am)

ModInteger interpolate on first variable.

Type Parameters:

C - coefficient type.

Parameters:

fac - GenPolynomial result factory.

A - GenPolynomial.

M - GenPolynomial interpolation modul of A.

mi - inverse of M(am) in ring fac.coFac.

B - evaluation of other GenPolynomial.

am - evaluation point (interpolation modul) of B, i.e. P(am) = B.

Returns:

S, with S mod M == A and S(am) == B.

interpolate

public static <C extends RingElem<C>> GenPolynomial<C> interpolate(GenPolynomialRing<C> fac,
                                                                   GenPolynomial<C> A,
                                                                   GenPolynomial<C> M,
                                                                   C mi,
                                                                   C a,
                                                                   C am)

Univariate polynomial interpolation.

Type Parameters:

C - coefficient type.

Parameters:

fac - GenPolynomial result factory.

A - GenPolynomial.

M - GenPolynomial interpolation modul of A.

mi - inverse of M(am) in ring fac.coFac.

a - evaluation of other GenPolynomial.

am - evaluation point (interpolation modul) of a, i.e. P(am) = a.

Returns:

S, with S mod M == A and S(am) == a.

switchVariables

public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> switchVariables(GenPolynomial<GenPolynomial<C>> P)

Recursive GenPolynomial switch varaible blocks.

Type Parameters:

C - coefficient type.

Parameters:

P - recursive GenPolynomial in R[X,Y].

Returns:

this in R[Y,X].

totalDegreeLeadingTerm

public static <C extends RingElem<C>> long totalDegreeLeadingTerm(java.util.List<GenPolynomial<C>> P)

Maximal degree of leading terms of a polynomial list.

Returns:

maximum degree of the leading terms of a polynomial list.

totalDegree

public static <C extends RingElem<C>> long totalDegree(java.util.List<GenPolynomial<C>> P)

Total degree of polynomial list.

Returns:

total degree of the polynomial list.

maxDegree

public static <C extends RingElem<C>> long maxDegree(java.util.List<GenPolynomial<C>> P)

Maximal degree of polynomial list.

Returns:

maximal degree of the polynomial list.

coeffMaxDegree

public static <C extends RingElem<C>> long coeffMaxDegree(GenPolynomial<GenPolynomial<C>> A)

Maximal degree in the coefficient polynomials.

Type Parameters:

C - coefficient type.

Returns:

maximal degree in the coefficients.

map

public static <C extends RingElem<C>,D extends RingElem<D>> GenPolynomial<D> map(GenPolynomialRing<D> ring,
                                                                                       GenPolynomial<C> p,
                                                                                       UnaryFunctor<C,D> f)

Map a unary function to the coefficients.

Parameters:

ring - result polynomial ring factory.

p - polynomial.

f - evaluation functor.

Returns:

new polynomial with coefficients f(p(e)).

toProductGen

public static <C extends GcdRingElem<C>> java.util.List<GenPolynomial<Product<C>>> toProductGen(GenPolynomialRing<Product<C>> pfac,
                                                                                                java.util.List<GenPolynomial<C>> L)

Product representation.

Type Parameters:

C - coefficient type.

Parameters:

pfac - polynomial ring factory.

L - list of polynomials to be represented.

Returns:

Product represenation of L in the polynomial ring pfac.

toProductGen

public static <C extends GcdRingElem<C>> GenPolynomial<Product<C>> toProductGen(GenPolynomialRing<Product<C>> pfac,
                                                                                GenPolynomial<C> A)

Product representation.

Type Parameters:

C - coefficient type.

Parameters:

pfac - polynomial ring factory.

A - polynomial to be represented.

Returns:

Product represenation of A in the polynomial ring pfac.

toProductGen

public static <C extends GcdRingElem<C>> Product<C> toProductGen(ProductRing<C> pfac,
                                                                 C c)

Product representation.

Type Parameters:

C - coefficient type.

Parameters:

pfac - product ring factory.

c - coefficient to be represented.

Returns:

Product represenation of c in the ring pfac.

toProduct

public static <C extends RingElem<C>> Product<GenPolynomial<C>> toProduct(ProductRing<GenPolynomial<C>> pfac,
                                                                          C c,
                                                                          ExpVector e)

Product representation.

Type Parameters:

C - coefficient type.

Parameters:

pfac - product polynomial ring factory.

c - coefficient to be used.

e - exponent vector.

Returns:

Product represenation of c X^e in the ring pfac.

toProduct

public static <C extends RingElem<C>> Product<GenPolynomial<C>> toProduct(ProductRing<GenPolynomial<C>> pfac,
                                                                          GenPolynomial<C> A)

Product representation.

Type Parameters:

C - coefficient type.

Parameters:

pfac - product polynomial ring factory.

A - polynomial.

Returns:

Product represenation of the terms of A in the ring pfac.

toProduct

public static Product<ModInteger> toProduct(ProductRing<ModInteger> pfac,
                                            BigInteger c)

Product representation.

Parameters:

pfac - product ring factory.

c - coefficient to be represented.

Returns:

Product represenation of c in the ring pfac.

toProduct

public static GenPolynomial<Product<ModInteger>> toProduct(GenPolynomialRing<Product<ModInteger>> pfac,
                                                           GenPolynomial<BigInteger> A)

Product representation.

Parameters:

pfac - polynomial ring factory.

A - polynomial to be represented.

Returns:

Product represenation of A in the polynomial ring pfac.

toProduct

public static java.util.List<GenPolynomial<Product<ModInteger>>> toProduct(GenPolynomialRing<Product<ModInteger>> pfac,
                                                                           java.util.List<GenPolynomial<BigInteger>> L)

Product representation.

Parameters:

pfac - polynomial ring factory.

L - list of polynomials to be represented.

Returns:

Product represenation of L in the polynomial ring pfac.

intersect

public static <C extends RingElem<C>> java.util.List<GenPolynomial<C>> intersect(GenPolynomialRing<C> R,
                                                                                 java.util.List<GenPolynomial<C>> F)

Intersection. Intersection of a list of polynomials with a polynomial ring. The polynomial ring must be a contraction of the polynomial ring of the list of polynomials and the TermOrder must be an elimination order.

Parameters:

R - polynomial ring

F - list of polynomials

Returns:

R \cap F

intersect

public static <C extends RingElem<C>> java.util.List<GenSolvablePolynomial<C>> intersect(GenSolvablePolynomialRing<C> R,
                                                                                         java.util.List<GenSolvablePolynomial<C>> F)

Intersection. Intersection of a list of solvable polynomials with a solvable polynomial ring. The solvable polynomial ring must be a contraction of the solvable polynomial ring of the list of polynomials and the TermOrder must be an elimination order.

Parameters:

R - solvable polynomial ring

F - list of solvable polynomials

Returns:

R \cap F

intersect

public static <C extends RingElem<C>> java.util.List<GenWordPolynomial<C>> intersect(GenWordPolynomialRing<C> R,
                                                                                     java.util.List<GenWordPolynomial<C>> F)

Intersection. Intersection of a list of word polynomials with a word polynomial ring. The polynomial ring must be a contraction of the polynomial ring of the list of polynomials,

Parameters:

R - word polynomial ring

F - list of word polynomials

Returns:

R \cap F

removeUnusedUpperVariables

public static <C extends RingElem<C>> GenPolynomial<C> removeUnusedUpperVariables(GenPolynomial<C> p)

Remove all upper variables which do not occur in polynomial.

Parameters:

p - polynomial.

Returns:

polynomial with removed variables

removeUnusedLowerVariables

public static <C extends RingElem<C>> GenPolynomial<C> removeUnusedLowerVariables(GenPolynomial<C> p)

Remove all lower variables which do not occur in polynomial.

Parameters:

p - polynomial.

Returns:

polynomial with removed variables

removeUnusedMiddleVariables

public static <C extends RingElem<C>> GenPolynomial<C> removeUnusedMiddleVariables(GenPolynomial<C> p)

Remove upper block of middle variables which do not occur in polynomial.

Parameters:

p - polynomial.

Returns:

polynomial with removed variables

selectWithVariable

public static <C extends RingElem<C>> GenPolynomial<C> selectWithVariable(java.util.List<GenPolynomial<C>> P,
                                                                          int i)

Select polynomial with univariate leading term in variable i.

Parameters:

i - variable index.

Returns:

polynomial with head term in variable i