Package edu.jas.poly

Class GenSolvablePolynomial<C extends RingElem<C>>

java.lang.Object

edu.jas.poly.GenPolynomial<C>

edu.jas.poly.GenSolvablePolynomial<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:

LocalSolvablePolynomial, QLRSolvablePolynomial, QuotSolvablePolynomial, RecSolvablePolynomial, RecSolvableWordPolynomial, ResidueSolvablePolynomial, ResidueSolvableWordPolynomial

public class GenSolvablePolynomial<C extends RingElem<C>>
extends GenPolynomial<C>

GenSolvablePolynomial generic solvable polynomials implementing RingElem. n-variate ordered solvable polynomials over C. Objects of this class are intended to be immutable. The implementation is based on TreeMap respectively SortedMap from exponents to coefficients by extension of GenPolybomial. Only the coefficients are modeled with generic types, the exponents are fixed to ExpVector with long, int, short entries (@see edu.jas.poly.ExpVector StorUnit).

Author:

Heinz Kredel

See Also:

Serialized Form

Field Summary

Fields

Modifier and Type	Field	Description
GenSolvablePolynomialRing<C>	ring	The factory for the solvable polynomial ring.

Fields inherited from class edu.jas.poly.GenPolynomial

blen, hash, val

Constructor Summary

Constructors

Modifier	Constructor	Description
	GenSolvablePolynomial(GenSolvablePolynomialRing<C> r)	Constructor for zero GenSolvablePolynomial.
	GenSolvablePolynomial(GenSolvablePolynomialRing<C> r, C c)	Constructor for GenSolvablePolynomial.
	GenSolvablePolynomial(GenSolvablePolynomialRing<C> r, C c, ExpVector e)	Constructor for GenSolvablePolynomial.
protected	GenSolvablePolynomial(GenSolvablePolynomialRing<C> r, java.util.SortedMap<ExpVector,C> v)	Constructor for GenSolvablePolynomial.

Method Summary

Modifier and Type	Method	Description
GenSolvablePolynomial<C>	copy()	Clone this GenSolvablePolynomial.
GenSolvablePolynomial<C>	divide(GenSolvablePolynomial<C> S)	GenSolvablePolynomial left division.
boolean	equals(java.lang.Object B)	Comparison with any other object.
GenSolvablePolynomial<C>	evalAsRightRecursivePolynomial()	Evaluate RecSolvablePolynomial as right coefficients polynomial.
GenSolvablePolynomialRing<C>	factory()	Get the corresponding element factory.
int	hashCode()	Hash code for this polynomial.
boolean	isRightRecursivePolynomial(GenSolvablePolynomial<C> R)	Test RecSolvablePolynomial right coefficients polynomial.
GenSolvablePolynomial<C>	leftMonic()	GenSolvablePolynomial left monic, i.e. leadingCoefficient == 1.
GenSolvablePolynomial<C>	monic()	GenSolvablePolynomial left monic, i.e. leadingCoefficient == 1.
GenSolvablePolynomial<C>	multiply(C b)	GenSolvablePolynomial multiplication.
GenSolvablePolynomial<C>	multiply(C b, C c)	GenSolvablePolynomial left and right multiplication.
GenSolvablePolynomial<C>	multiply(C b, ExpVector e)	GenSolvablePolynomial multiplication.
GenSolvablePolynomial<C>	multiply(C b, ExpVector e, C c, ExpVector f)	GenSolvablePolynomial left and right multiplication.
GenSolvablePolynomial<C>	multiply(ExpVector e)	GenSolvablePolynomial multiplication.
GenSolvablePolynomial<C>	multiply(ExpVector e, ExpVector f)	GenSolvablePolynomial left and right multiplication.
GenSolvablePolynomial<C>	multiply(GenSolvablePolynomial<C> Bp)	GenSolvablePolynomial multiplication.
GenSolvablePolynomial<C>	multiply(GenSolvablePolynomial<C> S, GenSolvablePolynomial<C> T)	GenSolvablePolynomial left and right multiplication.
GenSolvablePolynomial<C>	multiply(java.util.Map.Entry<ExpVector,C> m)	GenSolvablePolynomial multiplication.
GenSolvablePolynomial<C>	multiplyLeft(C b)	GenSolvablePolynomial multiplication.
GenSolvablePolynomial<C>	multiplyLeft(C b, ExpVector e)	GenSolvablePolynomial multiplication.
GenSolvablePolynomial<C>	multiplyLeft(ExpVector e)	GenSolvablePolynomial multiplication.
GenSolvablePolynomial<C>	multiplyLeft(java.util.Map.Entry<ExpVector,C> m)	GenSolvablePolynomial multiplication.
GenSolvablePolynomial<C>[]	quotientRemainder(GenSolvablePolynomial<C> S)	GenSolvablePolynomial left division with remainder.
GenSolvablePolynomial<C>	remainder(GenSolvablePolynomial<C> S)	GenSolvablePolynomial remainder by left division.
GenSolvablePolynomial<C>	rightDivide(GenSolvablePolynomial<C> S)	GenSolvablePolynomial right division.
GenSolvablePolynomial<C>	rightMonic()	GenSolvablePolynomial right monic, i.e. leadingCoefficient == 1.
GenSolvablePolynomial<C>[]	rightQuotientRemainder(GenSolvablePolynomial<C> S)	GenSolvablePolynomial right division with remainder.
GenSolvablePolynomial<C>	rightRecursivePolynomial()	RecSolvablePolynomial right coefficients from left coefficients.
GenSolvablePolynomial<C>	rightRemainder(GenSolvablePolynomial<C> S)	GenSolvablePolynomial remainder by right division.
GenSolvablePolynomial<C>	scaleSubtractMultiple(C b, C a, ExpVector e, GenSolvablePolynomial<C> S)	GenSolvablePolynomial scale and subtract a multiple.
GenSolvablePolynomial<C>	scaleSubtractMultiple(C b, C a, GenSolvablePolynomial<C> S)	GenSolvablePolynomial scale and subtract a multiple.
GenSolvablePolynomial<C>	scaleSubtractMultiple(C b, ExpVector g, C a, ExpVector e, GenSolvablePolynomial<C> S)	GenSolvablePolynomial scale and subtract a multiple.
GenSolvablePolynomial<C>	subtractMultiple(C a, ExpVector e, GenSolvablePolynomial<C> S)	GenSolvablePolynomial subtract a multiple.
GenSolvablePolynomial<C>	subtractMultiple(C a, GenSolvablePolynomial<C> S)	GenSolvablePolynomial subtract a multiple.

Methods inherited from class edu.jas.poly.GenPolynomial

abs, bitLength, coefficient, coefficientIterator, coeffPrimitivePart, compareTo, contract, contractCoeff, degree, degree, degreeMin, degreeVector, deHomogenize, deltaExpVectors, deltaExpVectors, divide, divide, doAddTo, doAddTo, doAddTo, doPutToMap, doPutToMap, doRemoveFromMap, egcd, exponentIterator, extend, extendLower, extendUnivariate, gcd, getMap, hegcd, homogenize, inflate, inverse, isConstant, isHomogeneous, isONE, isUnit, isWeightHomogeneous, isZERO, iterator, leadingBaseCoefficient, leadingExpVector, leadingFacetPolynomial, leadingMonomial, leadingWeightPolynomial, leftDivideCoeff, length, map, mapOnStream, mapOnStream, maxNorm, modInverse, monicRight, multiply, negate, negateAlt, numberOfVariables, quotientRemainder, reductum, remainder, reverse, rightDivideCoeff, rightGcd, scaleSubtractMultiple, scaleSubtractMultiple, scaleSubtractMultiple, signum, spliterator, squareNorm, subtract, subtract, subtract, subtract, subtractMultiple, subtractMultiple, sum, sum, sum, sum, sumNorm, toScript, toScriptFactory, toString, toString, totalDegree, trailingBaseCoefficient, trailingExpVector, weightDegree

Methods inherited from class java.lang.Object

clone, finalize, getClass, notify, notifyAll, wait, wait, wait

Methods inherited from interface java.lang.Iterable

forEach

Methods inherited from interface edu.jas.structure.MonoidElem

leftDivide, leftRemainder, power, rightDivide, rightRemainder, twosidedDivide, twosidedRemainder

Methods inherited from interface edu.jas.structure.RingElem

leftGcd

Field Detail

ring

public final GenSolvablePolynomialRing<C extends RingElem<C>> ring

The factory for the solvable polynomial ring. Hides super.ring.

Constructor Detail

GenSolvablePolynomial

public GenSolvablePolynomial(GenSolvablePolynomialRing<C> r)

Constructor for zero GenSolvablePolynomial.

Parameters:

r - solvable polynomial ring factory.

GenSolvablePolynomial

public GenSolvablePolynomial(GenSolvablePolynomialRing<C> r,
                             C c,
                             ExpVector e)

Constructor for GenSolvablePolynomial.

Parameters:

r - solvable polynomial ring factory.

c - coefficient.

e - exponent.

GenSolvablePolynomial

public GenSolvablePolynomial(GenSolvablePolynomialRing<C> r,
                             C c)

Constructor for GenSolvablePolynomial.

Parameters:

r - solvable polynomial ring factory.

c - coefficient.

GenSolvablePolynomial

protected GenSolvablePolynomial(GenSolvablePolynomialRing<C> r,
                                java.util.SortedMap<ExpVector,C> v)

Constructor for GenSolvablePolynomial.

Parameters:

r - solvable polynomial ring factory.

v - the SortedMap of some other (solvable) polynomial.

Method Detail

factory

public GenSolvablePolynomialRing<C> factory()

Get the corresponding element factory.

Specified by:

factory in interface Element<C extends RingElem<C>>

Overrides:

factory in class GenPolynomial<C extends RingElem<C>>

Returns:

factory for this Element.

See Also:

Element.factory()

copy

public GenSolvablePolynomial<C> copy()

Clone this GenSolvablePolynomial.

Specified by:

copy in interface Element<C extends RingElem<C>>

Overrides:

copy in class GenPolynomial<C extends RingElem<C>>

Returns:

copy of this.

See Also:

Object.clone()

equals

public boolean equals(java.lang.Object B)

Comparison with any other object.

Specified by:

equals in interface Element<C extends RingElem<C>>

Overrides:

equals in class GenPolynomial<C extends RingElem<C>>

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<C extends RingElem<C>>

Overrides:

hashCode in class GenPolynomial<C extends RingElem<C>>

Returns:

the hashCode.

See Also:

Object.hashCode()

multiply

public GenSolvablePolynomial<C> multiply(GenSolvablePolynomial<C> Bp)

GenSolvablePolynomial multiplication.

Parameters:

Bp - GenSolvablePolynomial.

Returns:

this*Bp, where * denotes solvable multiplication.

multiply

public GenSolvablePolynomial<C> multiply(GenSolvablePolynomial<C> S,
                                         GenSolvablePolynomial<C> T)

GenSolvablePolynomial left and right multiplication. Product with two polynomials.

Parameters:

S - GenSolvablePolynomial.

T - GenSolvablePolynomial.

Returns:

S*this*T.

multiply

public GenSolvablePolynomial<C> multiply(C b)

GenSolvablePolynomial multiplication. Product with coefficient ring element.

Overrides:

multiply in class GenPolynomial<C extends RingElem<C>>

Parameters:

b - coefficient.

Returns:

this*b, where * is coefficient multiplication.

multiply

public GenSolvablePolynomial<C> multiply(C b,
                                         C c)

GenSolvablePolynomial left and right multiplication. Product with coefficient ring element.

Parameters:

b - coefficient.

c - coefficient.

Returns:

b*this*c, where * is coefficient multiplication.

multiply

public GenSolvablePolynomial<C> multiply(ExpVector e)

GenSolvablePolynomial multiplication. Product with exponent vector.

Overrides:

multiply in class GenPolynomial<C extends RingElem<C>>

Parameters:

e - exponent.

Returns:

this * x^e, where * denotes solvable multiplication.

multiply

public GenSolvablePolynomial<C> multiply(ExpVector e,
                                         ExpVector f)

GenSolvablePolynomial left and right multiplication. Product with exponent vector.

Parameters:

e - exponent.

f - exponent.

Returns:

x^e * this * x^f, where * denotes solvable multiplication.

multiply

public GenSolvablePolynomial<C> multiply(C b,
                                         ExpVector e)

GenSolvablePolynomial multiplication. Product with ring element and exponent vector.

Overrides:

multiply in class GenPolynomial<C extends RingElem<C>>

Parameters:

b - coefficient.

e - exponent.

Returns:

this * b x^e, where * denotes solvable multiplication.

multiply

public GenSolvablePolynomial<C> multiply(C b,
                                         ExpVector e,
                                         C c,
                                         ExpVector f)

GenSolvablePolynomial left and right multiplication. Product with ring element and exponent vector.

Parameters:

b - coefficient.

e - exponent.

c - coefficient.

f - exponent.

Returns:

b x^e * this * c x^f, where * denotes solvable multiplication.

multiplyLeft

public GenSolvablePolynomial<C> multiplyLeft(C b,
                                             ExpVector e)

GenSolvablePolynomial multiplication. Left product with ring element and exponent vector.

Parameters:

b - coefficient.

e - exponent.

Returns:

b x^e * this, where * denotes solvable multiplication.

multiplyLeft

public GenSolvablePolynomial<C> multiplyLeft(ExpVector e)

GenSolvablePolynomial multiplication. Left product with exponent vector.

Parameters:

e - exponent.

Returns:

x^e * this, where * denotes solvable multiplication.

multiplyLeft

public GenSolvablePolynomial<C> multiplyLeft(C b)

GenSolvablePolynomial multiplication. Left product with coefficient ring element.

Overrides:

multiplyLeft in class GenPolynomial<C extends RingElem<C>>

Parameters:

b - coefficient.

Returns:

b*this, where * is coefficient multiplication.

multiplyLeft

public GenSolvablePolynomial<C> multiplyLeft(java.util.Map.Entry<ExpVector,C> m)

GenSolvablePolynomial multiplication. Left product with 'monomial'.

Parameters:

m - 'monomial'.

Returns:

m * this, where * denotes solvable multiplication.

multiply

public GenSolvablePolynomial<C> multiply(java.util.Map.Entry<ExpVector,C> m)

GenSolvablePolynomial multiplication. Product with 'monomial'.

Overrides:

multiply in class GenPolynomial<C extends RingElem<C>>

Parameters:

m - 'monomial'.

Returns:

this * m, where * denotes solvable multiplication.

subtractMultiple

public GenSolvablePolynomial<C> subtractMultiple(C a,
                                                 GenSolvablePolynomial<C> S)

GenSolvablePolynomial subtract a multiple.

Parameters:

a - coefficient.

S - GenSolvablePolynomial.

Returns:

this - a * S.

subtractMultiple

public GenSolvablePolynomial<C> subtractMultiple(C a,
                                                 ExpVector e,
                                                 GenSolvablePolynomial<C> S)

GenSolvablePolynomial subtract a multiple.

Parameters:

a - coefficient.

e - exponent.

S - GenSolvablePolynomial.

Returns:

this - a * x^e * S.

scaleSubtractMultiple

public GenSolvablePolynomial<C> scaleSubtractMultiple(C b,
                                                      C a,
                                                      GenSolvablePolynomial<C> S)

GenSolvablePolynomial scale and subtract a multiple.

Parameters:

b - scale factor.

a - coefficient.

S - GenSolvablePolynomial.

Returns:

b * this - a * S.

scaleSubtractMultiple

public GenSolvablePolynomial<C> scaleSubtractMultiple(C b,
                                                      C a,
                                                      ExpVector e,
                                                      GenSolvablePolynomial<C> S)

GenSolvablePolynomial scale and subtract a multiple.

Parameters:

b - scale factor.

a - coefficient.

e - exponent.

S - GenSolvablePolynomial.

Returns:

b * this - a * x^e * S.

scaleSubtractMultiple

public GenSolvablePolynomial<C> scaleSubtractMultiple(C b,
                                                      ExpVector g,
                                                      C a,
                                                      ExpVector e,
                                                      GenSolvablePolynomial<C> S)

GenSolvablePolynomial scale and subtract a multiple.

Parameters:

b - scale factor.

g - scale exponent.

a - coefficient.

e - exponent.

S - GenSolvablePolynomial.

Returns:

a * x^g * this - a * x^e * S.

monic

public GenSolvablePolynomial<C> monic()

GenSolvablePolynomial left monic, i.e. leadingCoefficient == 1. If leadingCoefficient is not invertible returns this abs value.

Overrides:

monic in class GenPolynomial<C extends RingElem<C>>

Returns:

ldcf(this)**(-1) * this.

leftMonic

public GenSolvablePolynomial<C> leftMonic()

GenSolvablePolynomial left monic, i.e. leadingCoefficient == 1. If leadingCoefficient is not invertible returns this abs value.

Returns:

ldcf(this)**(-1) * this.

rightMonic

public GenSolvablePolynomial<C> rightMonic()

GenSolvablePolynomial right monic, i.e. leadingCoefficient == 1. If leadingCoefficient is not invertible returns this abs value.

Returns:

this * ldcf(this)**(-1).

divide

public GenSolvablePolynomial<C> divide(GenSolvablePolynomial<C> S)

GenSolvablePolynomial left division. 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 GenSolvablePolynomial with invertible leading coefficient.

Returns:

quotient with this = quotient * S + remainder and deg(remainder) < deg(S) or remainder = 0.

See Also:

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

remainder

public GenSolvablePolynomial<C> remainder(GenSolvablePolynomial<C> S)

GenSolvablePolynomial remainder by left division. 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 GenSolvablePolynomial with invertible leading coefficient.

Returns:

remainder with this = quotient * S + remainder and deg(remainder) < deg(S) or remainder = 0.

See Also:

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

quotientRemainder

public GenSolvablePolynomial<C>[] quotientRemainder(GenSolvablePolynomial<C> S)

GenSolvablePolynomial left 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 GenSolvablePolynomial with invertible leading coefficient.

Returns:

[ quotient , remainder ] with this = quotient * S + remainder and deg(remainder) < deg(S) or remainder = 0.

See Also:

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

rightDivide

public GenSolvablePolynomial<C> rightDivide(GenSolvablePolynomial<C> S)

GenSolvablePolynomial right division. 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 GenSolvablePolynomial with invertible leading coefficient.

Returns:

quotient with this = S * quotient + remainder and deg(remainder) < deg(S) or remainder = 0.

See Also:

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

rightRemainder

public GenSolvablePolynomial<C> rightRemainder(GenSolvablePolynomial<C> S)

GenSolvablePolynomial remainder by right division. 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 GenSolvablePolynomial with invertible leading coefficient.

Returns:

remainder with this = S * quotient + remainder and deg(remainder) < deg(S) or remainder = 0.

See Also:

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

rightQuotientRemainder

public GenSolvablePolynomial<C>[] rightQuotientRemainder(GenSolvablePolynomial<C> S)

GenSolvablePolynomial right 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 GenSolvablePolynomial with invertible leading coefficient.

Returns:

[ quotient , remainder ] with this = S * quotient + remainder and deg(remainder) < deg(S) or remainder = 0.

See Also:

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

rightRecursivePolynomial

public GenSolvablePolynomial<C> rightRecursivePolynomial()

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.

Returns:

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

evalAsRightRecursivePolynomial

public GenSolvablePolynomial<C> evalAsRightRecursivePolynomial()

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.

Returns:

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

isRightRecursivePolynomial

public boolean isRightRecursivePolynomial(GenSolvablePolynomial<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.

Parameters:

R - GenSolvablePolynomial with right coefficients.

Returns:

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