Documentation of 'edu.jas.poly.GenPolynomial' Java class
GenPolynomial
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edu.jas.poly

Class GenPolynomial<C extends RingElem<C>>

  • Type Parameters:
    C - coefficient type
    All Implemented Interfaces:
    AbelianGroupElem<GenPolynomial<C>>, Element<GenPolynomial<C>>, MonoidElem<GenPolynomial<C>>, RingElem<GenPolynomial<C>>, java.io.Serializable, java.lang.Comparable<GenPolynomial<C>>, java.lang.Iterable<Monomial<C>>
    Direct Known Subclasses:
    GenSolvablePolynomial

    public class GenPolynomial<C extends RingElem<C>>
extends java.lang.Object
implements RingElem<GenPolynomial<C>>, java.lang.Iterable<Monomial<C>>
    GenPolynomial generic polynomials implementing RingElem. n-variate ordered polynomials over C. Objects of this class are intended to be immutable. The implementation is based on TreeMap respectively SortedMap from exponents to coefficients. Only the coefficients are modeled with generic types, the exponents are fixed to ExpVector with long entries (this will eventually be changed in the future). C can also be a non integral domain, e.g. a ModInteger, i.e. it may contain zero divisors, since multiply() does now check for zeros. Note: multiply() now checks for wrong method dispatch for GenSolvablePolynomial.
    See Also:
    Serialized Form

    • Constructor Detail

      • GenPolynomial

        public GenPolynomial(GenPolynomialRing<C> r)
        Constructor for zero GenPolynomial.
        Parameters:
        r - polynomial ring factory.

      • GenPolynomial

        public GenPolynomial(GenPolynomialRing<C> r,
                     C c,
                     ExpVector e)
        Constructor for GenPolynomial c * xe.
        Parameters:
        r - polynomial ring factory.
        c - coefficient.
        e - exponent.

      • GenPolynomial

        public GenPolynomial(GenPolynomialRing<C> r,
                     C c)
        Constructor for GenPolynomial c * x0.
        Parameters:
        r - polynomial ring factory.
        c - coefficient.

      • GenPolynomial

        public GenPolynomial(GenPolynomialRing<C> r,
                     ExpVector e)
        Constructor for GenPolynomial xe.
        Parameters:
        r - polynomial ring factory.
        e - exponent.

    • Method Detail

      • length

        public int length()
        Length of GenPolynomial.
        Returns:
        number of coefficients of this GenPolynomial.

      • getMap

        public java.util.SortedMap<ExpVector,C> getMap()
        ExpVector to coefficient map of GenPolynomial.
        Returns:
        val as unmodifiable SortedMap.

      • doPutToMap

        public void doPutToMap(ExpVector e,
                       C c)
        Put an ExpVector to coefficient entry into the internal map of this GenPolynomial. Note: Do not use this method unless you are constructing a new polynomial. this is modified and breaks the immutability promise of this class.
        Parameters:
        c - coefficient.
        e - exponent.

      • doRemoveFromMap

        public void doRemoveFromMap(ExpVector e,
                            C c)
        Remove an ExpVector to coefficient entry from the internal map of this GenPolynomial. Note: Do not use this method unless you are constructing a new polynomial. this is modified and breaks the immutability promise of this class.
        Parameters:
        e - exponent.
        c - expected coefficient, null for ignore.

      • doPutToMap

        public void doPutToMap(java.util.SortedMap<ExpVector,C> vals)
        Put an a sorted map of exponents to coefficients into the internal map of this GenPolynomial. Note: Do not use this method unless you are constructing a new polynomial. this is modified and breaks the immutability promise of this class.
        Parameters:
        vals - sorted map of exponents and coefficients.

      • toString

        public java.lang.String toString()
        String representation of GenPolynomial.
        Overrides:
        toString in class java.lang.Object
        See Also:
        Object.toString()

      • toString

        public java.lang.String toString(java.lang.String[] v)
        String representation of GenPolynomial.
        Parameters:
        v - names for variables.
        See Also:
        Object.toString()

      • isConstant

        public boolean isConstant()
        Is GenPolynomial<C> a constant.
        Returns:
        If this is a constant polynomial then true is returned, else false.

      • isHomogeneous

        public boolean isHomogeneous()
        Is GenPolynomial<C> homogeneous.
        Returns:
        true, if this is homogeneous, else false.

      • equals

        public boolean equals(java.lang.Object B)
        Comparison with any other object.
        Specified by:
        equals in interface Element<GenPolynomial<C extends RingElem<C>>>
        Overrides:
        equals in class java.lang.Object
        Returns:
        true if this is equal to b, else false.
        See Also:
        Object.equals(java.lang.Object)

      • hashCode

        public int hashCode()
        Hash code for this polynomial.
        Specified by:
        hashCode in interface Element<GenPolynomial<C extends RingElem<C>>>
        Overrides:
        hashCode in class java.lang.Object
        Returns:
        the hashCode.
        See Also:
        Object.hashCode()

      • numberOfVariables

        public int numberOfVariables()
        Number of variables.
        Returns:
        ring.nvar.

      • leadingMonomial

        public java.util.Map.Entry<ExpVector,C> leadingMonomial()
        Leading monomial.
        Returns:
        first map entry.

      • leadingExpVector

        public ExpVector leadingExpVector()
        Leading exponent vector.
        Returns:
        first exponent.

      • trailingExpVector

        public ExpVector trailingExpVector()
        Trailing exponent vector.
        Returns:
        last exponent.

      • leadingBaseCoefficient

        public C leadingBaseCoefficient()
        Leading base coefficient.
        Returns:
        first coefficient.

      • trailingBaseCoefficient

        public C trailingBaseCoefficient()
        Trailing base coefficient.
        Returns:
        coefficient of constant term.

      • coefficient

        public C coefficient(ExpVector e)
        Coefficient.
        Parameters:
        e - exponent.
        Returns:
        coefficient for given exponent.

      • reductum

        public GenPolynomial<C> reductum()
        Reductum.
        Returns:
        this - leading monomial.

      • degree

        public long degree(int i)
        Degree in variable i.
        Returns:
        maximal degree in the variable i.

      • degree

        public long degree()
        Maximal degree.
        Returns:
        maximal degree in any variables.

      • totalDegree

        public long totalDegree()
        Total degree.
        Returns:
        total degree in any variables.

      • degreeVector

        public ExpVector degreeVector()
        Maximal degree vector.
        Returns:
        maximal degree vector of all variables.

      • maxNorm

        public C maxNorm()
        GenPolynomial maximum norm.
        Returns:
        ||this||.

      • sumNorm

        public C sumNorm()
        GenPolynomial sum norm.
        Returns:
        sum of all absolute values of coefficients.

      • sum

        public GenPolynomial<C> sum(C a,
                            ExpVector e)
        GenPolynomial addition. This method is not very efficient, since this is copied.
        Parameters:
        a - coefficient.
        e - exponent.
        Returns:
        this + a xe.

      • sum

        public GenPolynomial<C> sum(C a)
        GenPolynomial addition. This method is not very efficient, since this is copied.
        Parameters:
        a - coefficient.
        Returns:
        this + a x0.

      • subtract

        public GenPolynomial<C> subtract(C a,
                                 ExpVector e)
        GenPolynomial subtraction. This method is not very efficient, since this is copied.
        Parameters:
        a - coefficient.
        e - exponent.
        Returns:
        this - a xe.

      • subtract

        public GenPolynomial<C> subtract(C a)
        GenPolynomial subtract. This method is not very efficient, since this is copied.
        Parameters:
        a - coefficient.
        Returns:
        this + a x0.

      • subtractMultiple

        public GenPolynomial<C> subtractMultiple(C a,
                                         GenPolynomial<C> S)
        GenPolynomial subtract a multiple.
        Parameters:
        a - coefficient.
        S - GenPolynomial.
        Returns:
        this - a xe S.

      • subtractMultiple

        public GenPolynomial<C> subtractMultiple(C a,
                                         ExpVector e,
                                         GenPolynomial<C> S)
        GenPolynomial subtract a multiple.
        Parameters:
        a - coefficient.
        e - exponent.
        S - GenPolynomial.
        Returns:
        this - a xe S.

      • scaleSubtractMultiple

        public GenPolynomial<C> scaleSubtractMultiple(C b,
                                              C a,
                                              ExpVector e,
                                              GenPolynomial<C> S)
        GenPolynomial scale and subtract a multiple.
        Parameters:
        b - scale factor.
        a - coefficient.
        e - exponent.
        S - GenPolynomial.
        Returns:
        this * b - a xe S.

      • scaleSubtractMultiple

        public GenPolynomial<C> scaleSubtractMultiple(C b,
                                              ExpVector g,
                                              C a,
                                              ExpVector e,
                                              GenPolynomial<C> S)
        GenPolynomial scale and subtract a multiple.
        Parameters:
        b - scale factor.
        g - scale exponent.
        a - coefficient.
        e - exponent.
        S - GenPolynomial.
        Returns:
        this * a xg - a xe S.

      • multiply

        public GenPolynomial<C> multiply(C s)
        GenPolynomial multiplication. Product with coefficient ring element.
        Parameters:
        s - coefficient.
        Returns:
        this*s.

      • monic

        public GenPolynomial<C> monic()
        GenPolynomial monic, i.e. leadingCoefficient == 1. If leadingCoefficient is not invertible returns this unmodified.
        Returns:
        monic(this).

      • multiply

        public GenPolynomial<C> multiply(C s,
                                 ExpVector e)
        GenPolynomial multiplication. Product with ring element and exponent vector.
        Parameters:
        s - coefficient.
        e - exponent.
        Returns:
        this * s xe.

      • multiply

        public GenPolynomial<C> multiply(ExpVector e)
        GenPolynomial multiplication. Product with exponent vector.
        Parameters:
        e - exponent (!= null).
        Returns:
        this * xe.

      • multiply

        public GenPolynomial<C> multiply(java.util.Map.Entry<ExpVector,C> m)
        GenPolynomial multiplication. Product with 'monomial'.
        Parameters:
        m - 'monomial'.
        Returns:
        this * m.

      • divide

        public GenPolynomial<C> divide(C s)
        GenPolynomial division. Division by coefficient ring element. Fails, if exact division is not possible.
        Parameters:
        s - coefficient.
        Returns:
        this/s.

      • quotientRemainder

        public GenPolynomial<C>[] quotientRemainder(GenPolynomial<C> S)
        GenPolynomial division with remainder. Fails, if exact division by leading base coefficient is not possible. Meaningful only for univariate polynomials over fields, but works in any case.
        Parameters:
        S - nonzero GenPolynomial with invertible leading coefficient.
        Returns:
        [ quotient , remainder ] with this = quotient * S + remainder and deg(remainder) < deg(S) or remiander = 0.
        See Also:
        .

      • divideAndRemainder

        @Deprecated
public GenPolynomial<C>[] divideAndRemainder(GenPolynomial<C> S)
        Deprecated. use quotientRemainder()
        GenPolynomial division with remainder. Fails, if exact division by leading base coefficient is not possible. Meaningful only for univariate polynomials over fields, but works in any case.
        Parameters:
        S - nonzero GenPolynomial with invertible leading coefficient.
        Returns:
        [ quotient , remainder ] with this = quotient * S + remainder and deg(remainder) < deg(S) or remiander = 0.
        See Also:
        .

      • divide

        public GenPolynomial<C> divide(GenPolynomial<C> S)
        GenPolynomial division. Fails, if exact division by leading base coefficient is not possible. Meaningful only for univariate polynomials over fields, but works in any case.
        Specified by:
        divide in interface MonoidElem<GenPolynomial<C extends RingElem<C>>>
        Parameters:
        S - nonzero GenPolynomial with invertible leading coefficient.
        Returns:
        quotient with this = quotient * S + remainder.
        See Also:
        .

      • remainder

        public GenPolynomial<C> remainder(GenPolynomial<C> S)
        GenPolynomial remainder. Fails, if exact division by leading base coefficient is not possible. Meaningful only for univariate polynomials over fields, but works in any case.
        Specified by:
        remainder in interface MonoidElem<GenPolynomial<C extends RingElem<C>>>
        Parameters:
        S - nonzero GenPolynomial with invertible leading coefficient.
        Returns:
        remainder with this = quotient * S + remainder.
        See Also:
        .

      • egcd

        public GenPolynomial<C>[] egcd(GenPolynomial<C> S)
        GenPolynomial extended greatest comon divisor. Only for univariate polynomials over fields.
        Specified by:
        egcd in interface RingElem<GenPolynomial<C extends RingElem<C>>>
        Parameters:
        S - GenPolynomial.
        Returns:
        [ gcd(this,S), a, b ] with a*this + b*S = gcd(this,S).

      • hegcd

        public GenPolynomial<C>[] hegcd(GenPolynomial<C> S)
        GenPolynomial half extended greatest comon divisor. Only for univariate polynomials over fields.
        Parameters:
        S - GenPolynomial.
        Returns:
        [ gcd(this,S), a ] with a*this + b*S = gcd(this,S).

      • modInverse

        public GenPolynomial<C> modInverse(GenPolynomial<C> m)
        GenPolynomial modular inverse. Only for univariate polynomials over fields.
        Parameters:
        m - GenPolynomial.
        Returns:
        a with with a*this = 1 mod m.

      • extend

        public GenPolynomial<C> extend(GenPolynomialRing<C> pfac,
                               int j,
                               long k)
        Extend variables. Used e.g. in module embedding. Extend all ExpVectors by i elements and multiply by x_j^k.
        Parameters:
        pfac - extended polynomial ring factory (by i variables).
        j - index of variable to be used for multiplication.
        k - exponent for x_j.
        Returns:
        extended polynomial.

      • extendLower

        public GenPolynomial<C> extendLower(GenPolynomialRing<C> pfac,
                                    int j,
                                    long k)
        Extend lower variables. Used e.g. in module embedding. Extend all ExpVectors by i lower elements and multiply by x_j^k.
        Parameters:
        pfac - extended polynomial ring factory (by i variables).
        j - index of variable to be used for multiplication.
        k - exponent for x_j.
        Returns:
        extended polynomial.

      • contract

        public java.util.Map<ExpVector,GenPolynomial<C>> contract(GenPolynomialRing<C> pfac)
        Contract variables. Used e.g. in module embedding. Remove i elements of each ExpVector.
        Parameters:
        pfac - contracted polynomial ring factory (by i variables).
        Returns:
        Map of exponents and contracted polynomials. Note: could return SortedMap

      • contractCoeff

        public GenPolynomial<C> contractCoeff(GenPolynomialRing<C> pfac)
        Contract variables to coefficient polynomial. Remove i elements of each ExpVector, removed elements must be zero.
        Parameters:
        pfac - contracted polynomial ring factory (by i variables).
        Returns:
        contracted coefficient polynomial.

      • extendUnivariate

        public GenPolynomial<C> extendUnivariate(GenPolynomialRing<C> pfac,
                                         int i)
        Extend univariate to multivariate polynomial. This is an univariate polynomial in variable i of the polynomial ring, it is extended to the given polynomial ring.
        Parameters:
        pfac - extended polynomial ring factory.
        i - index of the variable of this polynomial in pfac.
        Returns:
        extended multivariate polynomial.

      • homogenize

        public GenPolynomial<C> homogenize(GenPolynomialRing<C> pfac)
        Make homogeneous.
        Parameters:
        pfac - extended polynomial ring factory (by 1 variable).
        Returns:
        homogeneous polynomial.

      • deHomogenize

        public GenPolynomial<C> deHomogenize(GenPolynomialRing<C> pfac)
        Dehomogenize.
        Parameters:
        pfac - contracted polynomial ring factory (by 1 variable).
        Returns:
        in homogeneous polynomial.

      • reverse

        public GenPolynomial<C> reverse(GenPolynomialRing<C> oring)
        Reverse variables. Used e.g. in opposite rings.
        Returns:
        polynomial with reversed variables.

      • coefficientIterator

        public java.util.Iterator<C> coefficientIterator()
        Iterator over coefficients.
        Returns:
        val.values().iterator().

      • exponentIterator

        public java.util.Iterator<ExpVector> exponentIterator()
        Iterator over exponents.
        Returns:
        val.keySet().iterator().

      • iterator

        public java.util.Iterator<Monomial<C>> iterator()
        Iterator over monomials.
        Specified by:
        iterator in interface java.lang.Iterable<Monomial<C extends RingElem<C>>>
        Returns:
        a PolyIterator.

      • map

        public GenPolynomial<C> map(UnaryFunctor<? super C,C> f)
        Map a unary function to the coefficients.
        Parameters:
        f - evaluation functor.
        Returns:
        new polynomial with coefficients f(this(e)).
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