Package edu.jas.poly

Class GenWordPolynomial<C extends RingElem<C>>

java.lang.Object

edu.jas.poly.GenWordPolynomial<C>

Type Parameters:

C - coefficient type

All Implemented Interfaces:

AbelianGroupElem<GenWordPolynomial<C>>, Element<GenWordPolynomial<C>>, MonoidElem<GenWordPolynomial<C>>, RingElem<GenWordPolynomial<C>>, java.io.Serializable, java.lang.Comparable<GenWordPolynomial<C>>, java.lang.Iterable<WordMonomial<C>>

public final class GenWordPolynomial<C extends RingElem<C>>
extends java.lang.Object
implements RingElem<GenWordPolynomial<C>>, java.lang.Iterable<WordMonomial<C>>

GenWordPolynomial generic polynomials implementing RingElem. Non-commutative string polynomials over C. Objects of this class are intended to be immutable. The implementation is based on TreeMap respectively SortedMap from words to coefficients. Only the coefficients are modeled with generic types, the "exponents" are fixed to Word. C can also be a non integral domain, e.g. a ModInteger, i.e. it may contain zero divisors, since multiply() does check for zeros.

Author:

Heinz Kredel

See Also:

Serialized Form

Field Summary

Fields

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

Constructor Summary

Constructors

Modifier	Constructor	Description
	GenWordPolynomial(GenWordPolynomialRing<C> r)	Constructor for zero GenWordPolynomial.
	GenWordPolynomial(GenWordPolynomialRing<C> r, C c)	Constructor for GenWordPolynomial c * x0.
	GenWordPolynomial(GenWordPolynomialRing<C> r, C c, ExpVector e)	Constructor for GenWordPolynomial c * xe.
	GenWordPolynomial(GenWordPolynomialRing<C> r, C c, Word e)	Constructor for GenWordPolynomial c * xe.
	GenWordPolynomial(GenWordPolynomialRing<C> r, ExpVector e)	Constructor for GenWordPolynomial xe.
	GenWordPolynomial(GenWordPolynomialRing<C> r, Word e)	Constructor for GenWordPolynomial xe.
protected	GenWordPolynomial(GenWordPolynomialRing<C> r, java.util.SortedMap<Word,C> v)	Constructor for GenWordPolynomial.

Method Summary

Modifier and Type	Method	Description
GenWordPolynomial<C>	abs()	GenWordPolynomial absolute value, i.e. leadingCoefficient > 0.
C	coefficient(Word e)	Coefficient.
java.util.Iterator<C>	coefficientIterator()	Iterator over coefficients.
int	compareTo(GenWordPolynomial<C> b)	GenWordPolynomial comparison.
GenWordPolynomial<C>	contract(GenWordPolynomialRing<C> fac)	GenWordPolynomial contraction.
GenWordPolynomial<C>	copy()	Copy this GenWordPolynomial.
long	degree()	Maximal degree.
GenWordPolynomial<C>	divide(C s)	GenWordPolynomial division.
GenWordPolynomial<C>	divide(GenWordPolynomial<C> S)	GenWordPolynomial division.
void	doPutToMap(Word e, C c)	Put a Word to coefficient entry into the internal map of this GenWordPolynomial.
void	doPutToMap(java.util.SortedMap<Word,C> vals)	Put an a sorted map of words to coefficients into the internal map of this GenWordPolynomial.
void	doRemoveFromMap(Word e, C c)	Remove a Word to coefficient entry from the internal map of this GenWordPolynomial.
GenWordPolynomial<C>[]	egcd(GenWordPolynomial<C> S)	GenWordPolynomial extended greatest common divisor.
boolean	equals(java.lang.Object B)	Comparison with any other object.
GenWordPolynomialRing<C>	factory()	Get the corresponding element factory.
GenWordPolynomial<C>	gcd(GenWordPolynomial<C> S)	GenWordPolynomial greatest common divisor.
java.util.SortedMap<Word,C>	getMap()	Word to coefficient map of GenWordPolynomial.
int	hashCode()	Hash code for this polynomial.
GenWordPolynomial<C>[]	hegcd(GenWordPolynomial<C> S)	GenWordPolynomial half extended greatest common divisor.
GenWordPolynomial<C>	inverse()	GenWordPolynomial inverse.
boolean	isConstant()	Is GenWordPolynomial<C> a constant.
boolean	isHomogeneous()	Is GenWordPolynomial<C> homogeneous.
boolean	isONE()	Is GenWordPolynomial<C> one.
boolean	isUnit()	Is GenWordPolynomial<C> a unit.
boolean	isZERO()	Is GenWordPolynomial<C> zero.
java.util.Iterator<WordMonomial<C>>	iterator()	Iterator over monomials.
C	leadingBaseCoefficient()	Leading base coefficient.
java.util.Map.Entry<Word,C>	leadingMonomial()	Leading monomial.
Word	leadingWord()	Leading word.
int	length()	Length of GenWordPolynomial.
GenWordPolynomial<C>	map(UnaryFunctor<? super C,C> f)	Map a unary function to the coefficients.
C	maxNorm()	GenWordPolynomial maximum norm.
GenWordPolynomial<C>	modInverse(GenWordPolynomial<C> m)	GenWordPolynomial modular inverse.
GenWordPolynomial<C>	monic()	GenWordPolynomial monic, i.e. leadingCoefficient == 1.
GenWordPolynomial<C>	multiply(C s)	GenWordPolynomial multiplication.
GenWordPolynomial<C>	multiply(C s, C t)	GenWordPolynomial multiplication.
GenWordPolynomial<C>	multiply(C s, Word e)	GenWordPolynomial multiplication.
GenWordPolynomial<C>	multiply(C s, Word e, C t, Word f)	GenWordPolynomial left and right multiplication.
GenWordPolynomial<C>	multiply(C s, Word e, Word f)	GenWordPolynomial left and right multiplication.
GenWordPolynomial<C>	multiply(GenWordPolynomial<C> S)	GenWordPolynomial multiplication.
GenWordPolynomial<C>	multiply(GenWordPolynomial<C> S, GenWordPolynomial<C> T)	GenWordPolynomial left and right multiplication.
GenWordPolynomial<C>	multiply(Word e)	GenWordPolynomial multiplication.
GenWordPolynomial<C>	multiply(Word e, Word f)	GenWordPolynomial left and right multiplication.
GenWordPolynomial<C>	multiply(java.util.Map.Entry<Word,C> m)	GenWordPolynomial multiplication.
GenWordPolynomial<C>	negate()	GenWordPolynomial negation.
int	numberOfVariables()	Number of variables.
GenWordPolynomial<C>[]	quotientRemainder(GenWordPolynomial<C> S)	GenWordPolynomial division with remainder.
GenWordPolynomial<C>	reductum()	Reductum.
GenWordPolynomial<C>	remainder(GenWordPolynomial<C> S)	GenWordPolynomial remainder.
int	signum()	GenWordPolynomial signum.
GenWordPolynomial<C>	subtract(C a)	GenWordPolynomial subtract.
GenWordPolynomial<C>	subtract(C a, Word e)	GenWordPolynomial subtraction.
GenWordPolynomial<C>	subtract(GenWordPolynomial<C> S)	GenWordPolynomial subtraction.
GenWordPolynomial<C>	sum(C a)	GenWordPolynomial addition.
GenWordPolynomial<C>	sum(C a, Word e)	GenWordPolynomial addition.
GenWordPolynomial<C>	sum(GenWordPolynomial<C> S)	GenWordPolynomial summation.
C	sumNorm()	GenWordPolynomial sum norm.
java.lang.String	toScript()	Get a scripting compatible string representation.
java.lang.String	toScriptFactory()	Get a scripting compatible string representation of the factory.
java.lang.String	toString()	String representation of GenWordPolynomial.
C	trailingBaseCoefficient()	Trailing base coefficient.
Word	trailingWord()	Trailing word.
java.util.Iterator<Word>	wordIterator()	Iterator over words.

Methods inherited from class java.lang.Object

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

Methods inherited from interface java.lang.Iterable

forEach, spliterator

Methods inherited from interface edu.jas.structure.MonoidElem

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

Methods inherited from interface edu.jas.structure.RingElem

leftGcd, rightGcd

Field Detail

ring

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

The factory for the polynomial ring.

Constructor Detail

GenWordPolynomial

public GenWordPolynomial(GenWordPolynomialRing<C> r)

Constructor for zero GenWordPolynomial.

Parameters:

r - polynomial ring factory.

GenWordPolynomial

public GenWordPolynomial(GenWordPolynomialRing<C> r,
                         C c,
                         Word e)

Constructor for GenWordPolynomial c * x^e.

Parameters:

r - polynomial ring factory.

c - coefficient.

e - word.

GenWordPolynomial

public GenWordPolynomial(GenWordPolynomialRing<C> r,
                         C c)

Constructor for GenWordPolynomial c * x⁰.

Parameters:

r - polynomial ring factory.

c - coefficient.

GenWordPolynomial

public GenWordPolynomial(GenWordPolynomialRing<C> r,
                         Word e)

Constructor for GenWordPolynomial x^e.

Parameters:

r - polynomial ring factory.

e - word.

GenWordPolynomial

public GenWordPolynomial(GenWordPolynomialRing<C> r,
                         ExpVector e)

Constructor for GenWordPolynomial x^e.

Parameters:

r - polynomial ring factory.

e - exponent vector.

GenWordPolynomial

public GenWordPolynomial(GenWordPolynomialRing<C> r,
                         C c,
                         ExpVector e)

Constructor for GenWordPolynomial c * x^e.

Parameters:

r - polynomial ring factory.

c - coefficient.

e - exponent vector.

GenWordPolynomial

protected GenWordPolynomial(GenWordPolynomialRing<C> r,
                            java.util.SortedMap<Word,C> v)

Constructor for GenWordPolynomial.

Parameters:

r - polynomial ring factory.

v - the SortedMap of some other polynomial.

Method Detail

factory

public GenWordPolynomialRing<C> factory()

Get the corresponding element factory.

Specified by:

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

Returns:

factory for this Element.

See Also:

Element.factory()

copy

public GenWordPolynomial<C> copy()

Copy this GenWordPolynomial.

Specified by:

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

Returns:

copy of this.

length

public int length()

Length of GenWordPolynomial.

Returns:

number of coefficients of this GenWordPolynomial.

getMap

public java.util.SortedMap<Word,C> getMap()

Word to coefficient map of GenWordPolynomial.

Returns:

val as unmodifiable SortedMap.

doPutToMap

public void doPutToMap(Word e,
                       C c)

Put a Word to coefficient entry into the internal map of this GenWordPolynomial. 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 - word.

doRemoveFromMap

public void doRemoveFromMap(Word e,
                            C c)

Remove a Word to coefficient entry from the internal map of this GenWordPolynomial. 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 - Word.

c - expected coefficient, null for ignore.

doPutToMap

public void doPutToMap(java.util.SortedMap<Word,C> vals)

Put an a sorted map of words to coefficients into the internal map of this GenWordPolynomial. 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 wordss and coefficients.

toString

public java.lang.String toString()

String representation of GenWordPolynomial.

Overrides:

toString in class java.lang.Object

See Also:

Object.toString()

toScript

public java.lang.String toScript()

Get a scripting compatible string representation.

Specified by:

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

Returns:

script compatible representation for this Element.

See Also:

Element.toScript()

toScriptFactory

public java.lang.String toScriptFactory()

Get a scripting compatible string representation of the factory.

Specified by:

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

Returns:

script compatible representation for this ElemFactory.

See Also:

Element.toScriptFactory()

isZERO

public boolean isZERO()

Is GenWordPolynomial<C> zero.

Specified by:

isZERO in interface AbelianGroupElem<C extends RingElem<C>>

Returns:

If this is 0 then true is returned, else false.

See Also:

AbelianGroupElem.isZERO()

isONE

public boolean isONE()

Is GenWordPolynomial<C> one.

Specified by:

isONE in interface MonoidElem<C extends RingElem<C>>

Returns:

If this is 1 then true is returned, else false.

See Also:

MonoidElem.isONE()

isUnit

public boolean isUnit()

Is GenWordPolynomial<C> a unit.

Specified by:

isUnit in interface MonoidElem<C extends RingElem<C>>

Returns:

If this is a unit then true is returned, else false.

See Also:

MonoidElem.isUnit()

isConstant

public boolean isConstant()

Is GenWordPolynomial<C> a constant.

Returns:

If this is a constant polynomial then true is returned, else false.

isHomogeneous

public boolean isHomogeneous()

Is GenWordPolynomial<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<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<C extends RingElem<C>>

Overrides:

hashCode in class java.lang.Object

Returns:

the hashCode.

See Also:

Object.hashCode()

compareTo

public int compareTo(GenWordPolynomial<C> b)

GenWordPolynomial comparison.

Specified by:

compareTo in interface java.lang.Comparable<C extends RingElem<C>>

Specified by:

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

Parameters:

b - GenWordPolynomial.

Returns:

sign(this-b).

signum

public int signum()

GenWordPolynomial signum.

Specified by:

signum in interface AbelianGroupElem<C extends RingElem<C>>

Returns:

sign(ldcf(this)).

numberOfVariables

public int numberOfVariables()

Number of variables.

Returns:

ring.alphabet.length().

leadingMonomial

public java.util.Map.Entry<Word,C> leadingMonomial()

Leading monomial.

Returns:

first map entry.

leadingWord

public Word leadingWord()

Leading word.

Returns:

first highest word.

trailingWord

public Word trailingWord()

Trailing word.

Returns:

last lowest word.

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(Word e)

Coefficient.

Parameters:

e - word.

Returns:

coefficient for given word.

reductum

public GenWordPolynomial<C> reductum()

Reductum.

Returns:

this - leading monomial.

degree

public long degree()

Maximal degree.

Returns:

maximal degree in any variables.

maxNorm

public C maxNorm()

GenWordPolynomial maximum norm.

Returns:

||this||.

sumNorm

public C sumNorm()

GenWordPolynomial sum norm.

Returns:

sum of all absolute values of coefficients.

sum

public GenWordPolynomial<C> sum(GenWordPolynomial<C> S)

GenWordPolynomial summation.

Specified by:

sum in interface AbelianGroupElem<C extends RingElem<C>>

Parameters:

S - GenWordPolynomial.

Returns:

this+S.

sum

public GenWordPolynomial<C> sum(C a,
                                Word e)

GenWordPolynomial addition. This method is not very efficient, since this is copied.

Parameters:

a - coefficient.

e - word.

Returns:

this + a e.

sum

public GenWordPolynomial<C> sum(C a)

GenWordPolynomial addition. This method is not very efficient, since this is copied.

Parameters:

a - coefficient.

Returns:

this + a x⁰.

subtract

public GenWordPolynomial<C> subtract(GenWordPolynomial<C> S)

GenWordPolynomial subtraction.

Specified by:

subtract in interface AbelianGroupElem<C extends RingElem<C>>

Parameters:

S - GenWordPolynomial.

Returns:

this-S.

subtract

public GenWordPolynomial<C> subtract(C a,
                                     Word e)

GenWordPolynomial subtraction. This method is not very efficient, since this is copied.

Parameters:

a - coefficient.

e - word.

Returns:

this - a e.

subtract

public GenWordPolynomial<C> subtract(C a)

GenWordPolynomial subtract. This method is not very efficient, since this is copied.

Parameters:

a - coefficient.

Returns:

this + a x⁰.

negate

public GenWordPolynomial<C> negate()

GenWordPolynomial negation.

Specified by:

negate in interface AbelianGroupElem<C extends RingElem<C>>

Returns:

-this.

abs

public GenWordPolynomial<C> abs()

GenWordPolynomial absolute value, i.e. leadingCoefficient > 0.

Specified by:

abs in interface AbelianGroupElem<C extends RingElem<C>>

Returns:

abs(this).

multiply

public GenWordPolynomial<C> multiply(GenWordPolynomial<C> S)

GenWordPolynomial multiplication.

Specified by:

multiply in interface MonoidElem<C extends RingElem<C>>

Parameters:

S - GenWordPolynomial.

Returns:

this*S.

multiply

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

GenWordPolynomial left and right multiplication. Product with two polynomials.

Parameters:

S - GenWordPolynomial.

T - GenWordPolynomial.

Returns:

S*this*T.

multiply

public GenWordPolynomial<C> multiply(C s)

GenWordPolynomial multiplication. Product with coefficient ring element.

Parameters:

s - coefficient.

Returns:

this*s.

multiply

public GenWordPolynomial<C> multiply(C s,
                                     C t)

GenWordPolynomial multiplication. Product with coefficient ring element.

Parameters:

s - coefficient.

t - coefficient.

Returns:

s*this*t.

monic

public GenWordPolynomial<C> monic()

GenWordPolynomial monic, i.e. leadingCoefficient == 1. If leadingCoefficient is not invertible returns this unmodified.

Returns:

monic(this).

multiply

public GenWordPolynomial<C> multiply(C s,
                                     Word e)

GenWordPolynomial multiplication. Product with ring element and word.

Parameters:

s - coefficient.

e - left word.

Returns:

this * s e.

multiply

public GenWordPolynomial<C> multiply(Word e,
                                     Word f)

GenWordPolynomial left and right multiplication. Product with ring element and two words.

Parameters:

e - left word.

f - right word.

Returns:

e * this * f.

multiply

public GenWordPolynomial<C> multiply(C s,
                                     Word e,
                                     Word f)

GenWordPolynomial left and right multiplication. Product with ring element and two words.

Parameters:

s - coefficient.

e - left word.

f - right word.

Returns:

e * this * s * f.

multiply

public GenWordPolynomial<C> multiply(C s,
                                     Word e,
                                     C t,
                                     Word f)

GenWordPolynomial left and right multiplication. Product with ring element and two words.

Parameters:

s - coefficient.

e - left word.

t - coefficient.

f - right word.

Returns:

s * e * this * t * f.

multiply

public GenWordPolynomial<C> multiply(Word e)

GenWordPolynomial multiplication. Product with word.

Parameters:

e - word (!= null).

Returns:

this * e.

multiply

public GenWordPolynomial<C> multiply(java.util.Map.Entry<Word,C> m)

GenWordPolynomial multiplication. Product with 'monomial'.

Parameters:

m - 'monomial'.

Returns:

this * m.

divide

public GenWordPolynomial<C> divide(C s)

GenWordPolynomial division. Division by coefficient ring element. Fails, if exact division is not possible.

Parameters:

s - coefficient.

Returns:

this/s.

quotientRemainder

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

GenWordPolynomial 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.

Specified by:

quotientRemainder in interface MonoidElem<C extends RingElem<C>>

Parameters:

S - nonzero GenWordPolynomial with invertible leading coefficient.

Returns:

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

See Also:

.

divide

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

GenWordPolynomial 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<C extends RingElem<C>>

Parameters:

S - nonzero GenWordPolynomial with invertible leading coefficient.

Returns:

quotient with this = quotient * S + remainder.

See Also:

.

remainder

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

GenWordPolynomial 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<C extends RingElem<C>>

Parameters:

S - nonzero GenWordPolynomial with invertible leading coefficient.

Returns:

remainder with this = quotient * S + remainder.

See Also:

.

gcd

public GenWordPolynomial<C> gcd(GenWordPolynomial<C> S)

GenWordPolynomial greatest common divisor. Only for univariate polynomials over fields.

Specified by:

gcd in interface RingElem<C extends RingElem<C>>

Parameters:

S - GenWordPolynomial.

Returns:

gcd(this,S).

egcd

public GenWordPolynomial<C>[] egcd(GenWordPolynomial<C> S)

GenWordPolynomial extended greatest common divisor. Only for univariate polynomials over fields.

Specified by:

egcd in interface RingElem<C extends RingElem<C>>

Parameters:

S - GenWordPolynomial.

Returns:

[ gcd(this,S), a, b ] with a*this + b*S = gcd(this,S).

hegcd

public GenWordPolynomial<C>[] hegcd(GenWordPolynomial<C> S)

GenWordPolynomial half extended greatest common divisor. Only for univariate polynomials over fields.

Parameters:

S - GenWordPolynomial.

Returns:

[ gcd(this,S), a ] with a*this + b*S = gcd(this,S).

inverse

public GenWordPolynomial<C> inverse()

GenWordPolynomial inverse. Required by RingElem. Throws not invertible exception.

Specified by:

inverse in interface MonoidElem<C extends RingElem<C>>

Returns:

x with this * x = 1, if it exists.

modInverse

public GenWordPolynomial<C> modInverse(GenWordPolynomial<C> m)

GenWordPolynomial modular inverse. Only for univariate polynomials over fields.

Parameters:

m - GenWordPolynomial.

Returns:

a with with a*this = 1 mod m.

coefficientIterator

public java.util.Iterator<C> coefficientIterator()

Iterator over coefficients.

Returns:

val.values().iterator().

wordIterator

public java.util.Iterator<Word> wordIterator()

Iterator over words.

Returns:

val.keySet().iterator().

iterator

public java.util.Iterator<WordMonomial<C>> iterator()

Iterator over monomials.

Specified by:

iterator in interface java.lang.Iterable<C extends RingElem<C>>

Returns:

a PolyIterator.

map

public GenWordPolynomial<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)).

contract

public GenWordPolynomial<C> contract(GenWordPolynomialRing<C> fac)

GenWordPolynomial contraction.

Parameters:

fac - GenWordPolynomialRing.

Returns:

this contracted to fac ring, if this in fac ring, null else.