Package edu.jas.arith

Class BigQuaternion

java.lang.Object

edu.jas.arith.BigQuaternion

All Implemented Interfaces:

AbelianGroupElem<BigQuaternion>, Element<BigQuaternion>, GcdRingElem<BigQuaternion>, MonoidElem<BigQuaternion>, RingElem<BigQuaternion>, StarRingElem<BigQuaternion>, java.io.Serializable, java.lang.Comparable<BigQuaternion>

Direct Known Subclasses:

BigQuaternionInteger

public class BigQuaternion
extends java.lang.Object
implements StarRingElem<BigQuaternion>, GcdRingElem<BigQuaternion>

BigQuaternion class based on BigRational implementing the RingElem interface and with the familiar MAS static method names. Objects of this class are immutable. The integer quaternion methods are implemented after https://de.wikipedia.org/wiki/Hurwitzquaternion see also https://en.wikipedia.org/wiki/Hurwitz_quaternion

Author:

Heinz Kredel

See Also:

Serialized Form

Field Summary

Fields

Modifier and Type	Field	Description
BigRational	im	Imaginary part i of the data structure.
BigRational	jm	Imaginary part j of the data structure.
BigRational	km	Imaginary part k of the data structure.
protected static java.util.Random	random
BigRational	re	Real part of the data structure.
BigQuaternionRing	ring	Corresponding BigQuaternion ring.

Constructor Summary

Constructors

Constructor	Description
BigQuaternion(BigQuaternionRing fac)	Constructor for a BigQuaternion with no arguments.
BigQuaternion(BigQuaternionRing fac, long r)	Constructor for a BigQuaternion from long.
BigQuaternion(BigQuaternionRing fac, BigComplex r)	Constructor for a BigQuaternion from BigComplex.
BigQuaternion(BigQuaternionRing fac, BigRational r)	Constructor for a BigQuaternion from BigRationals.
BigQuaternion(BigQuaternionRing fac, BigRational r, BigRational i)	Constructor for a BigQuaternion from BigRationals.
BigQuaternion(BigQuaternionRing fac, BigRational r, BigRational i, BigRational j)	Constructor for a BigQuaternion from BigRationals.
BigQuaternion(BigQuaternionRing fac, BigRational r, BigRational i, BigRational j, BigRational k)	Constructor for a BigQuaternion from BigRationals.
BigQuaternion(BigQuaternionRing fac, java.lang.String s)	The BigQuaternion string constructor accepts the following formats: empty string, "rational", or "rat i rat j rat k rat" with no blanks around i, j or k if used as polynoial coefficient.

Method Summary

Modifier and Type	Method	Description
BigQuaternion	abs()	Quaternion number absolute value.
long	bitLength()	Returns the number of bits in the representation of this BigQuaternion, including a sign bit.
BigQuaternion	ceil()	BigQuaternion ceiling, component wise.
int	compareTo(BigQuaternion b)	Since quaternion numbers are unordered, we use lexicographical order of re, im, jm and km.
BigQuaternion	conjugate()	BigQuaternion conjugate.
BigQuaternion	copy()	Clone this.
BigQuaternion	divide(BigQuaternion b)	BigQuaternion right divide.
BigQuaternion	divide(BigRational b)	BigQuaternion divide.
BigQuaternion[]	egcd(BigQuaternion S)	BigQuaternion extended greatest common divisor.
boolean	equals(java.lang.Object b)	Comparison with any other object.
BigQuaternionRing	factory()	Get the corresponding element factory.
BigQuaternion	floor()	BigQuaternion floor, component wise.
BigQuaternion	gcd(BigQuaternion S)	Quaternion number greatest common divisor.
BigRational	getIm()	Get the imaginary part im.
BigRational	getJm()	Get the imaginary part jm.
BigRational	getKm()	Get the imaginary part km.
BigRational	getRe()	Get the real part.
int	hashCode()	Hash code for this BigQuaternion.
BigQuaternion	inverse()	BigQuaternion inverse.
boolean	isEntier()	Is BigQuaternion entier element.
boolean	isIMAG()	Is BigQuaternion imaginary one.
boolean	isONE()	Is BigQuaternion number one.
static boolean	isQONE(BigQuaternion A)	Is BigQuaternion number one.
static boolean	isQZERO(BigQuaternion A)	Is Quaternion number zero.
boolean	isUnit()	Is BigQuaternion unit element.
boolean	isZERO()	Is BigQuaternion number zero.
BigQuaternion	leftDivide(BigQuaternion b)	BigQuaternion left divide.
BigQuaternion	leftGcd(BigQuaternion S)	Quaternion number greatest common divisor.
BigQuaternion	multiply(BigQuaternion B)	BigQuaternion multiply.
BigQuaternion	multiply(BigRational b)	BigQuaternion multiply with BigRational.
BigQuaternion	negate()	BigQuaternion number negative.
BigQuaternion	norm()	Quaternion number norm.
static BigRational	QABS(BigQuaternion A)	Quaternion number absolute value.
static BigQuaternion	QCON(BigQuaternion A)	Quaternion number conjugate.
static BigQuaternion	QDIF(BigQuaternion A, BigQuaternion B)	Quaternion number difference.
static BigQuaternion	QINV(BigQuaternion A)	Quaternion number inverse.
static BigQuaternion	QNEG(BigQuaternion A)	Quaternion number negative.
static BigQuaternion	QPROD(BigQuaternion A, BigQuaternion B)	Quaternion number product.
static BigQuaternion	QQ(BigQuaternion A, BigQuaternion B)	Quaternion number quotient.
static BigQuaternion	QSUM(BigQuaternion A, BigQuaternion B)	Quaternion number sum.
BigQuaternion[]	quotientRemainder(BigQuaternion S)	Quotient and remainder by division of this by S.
BigQuaternion	remainder(BigQuaternion S)	BigQuaternion remainder.
BigQuaternion	rightDivide(BigQuaternion b)	BigQuaternion right divide.
BigQuaternion	rightGcd(BigQuaternion S)	Quaternion number greatest common divisor.
BigQuaternionInteger	roundToHurwitzian()	BigQuaternion round to next Hurwitz integer.
BigQuaternionInteger	roundToLipschitzian()	BigQuaternion round to next Lipschitz integer.
int	signum()	Since quaternion numbers are unordered, we use lexicographical order of re, im, jm and km.
BigQuaternion	subtract(BigQuaternion B)	BigQuaternion subtraction.
BigQuaternion	sum(BigQuaternion B)	BigQuaternion summation.
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()	Get the string representation.

Methods inherited from class java.lang.Object

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

Methods inherited from interface edu.jas.structure.MonoidElem

leftRemainder, power, rightRemainder, twosidedDivide, twosidedRemainder

Field Detail

re

public final BigRational re

Real part of the data structure.

im

public final BigRational im

Imaginary part i of the data structure.

jm

public final BigRational jm

Imaginary part j of the data structure.

km

public final BigRational km

Imaginary part k of the data structure.

ring

public final BigQuaternionRing ring

Corresponding BigQuaternion ring.

random

protected static final java.util.Random random

Constructor Detail

BigQuaternion

public BigQuaternion(BigQuaternionRing fac,
                     BigRational r,
                     BigRational i,
                     BigRational j,
                     BigRational k)

Constructor for a BigQuaternion from BigRationals.

Parameters:

fac - BigQuaternionRing.

r - BigRational.

i - BigRational.

j - BigRational.

k - BigRational.

BigQuaternion

public BigQuaternion(BigQuaternionRing fac,
                     BigRational r,
                     BigRational i,
                     BigRational j)

Constructor for a BigQuaternion from BigRationals.

Parameters:

fac - BigQuaternionRing.

r - BigRational.

i - BigRational.

j - BigRational.

BigQuaternion

public BigQuaternion(BigQuaternionRing fac,
                     BigRational r,
                     BigRational i)

Constructor for a BigQuaternion from BigRationals.

Parameters:

fac - BigQuaternionRing.

r - BigRational.

i - BigRational.

BigQuaternion

public BigQuaternion(BigQuaternionRing fac,
                     BigRational r)

Constructor for a BigQuaternion from BigRationals.

Parameters:

fac - BigQuaternionRing.

r - BigRational.

BigQuaternion

public BigQuaternion(BigQuaternionRing fac,
                     BigComplex r)

Constructor for a BigQuaternion from BigComplex.

Parameters:

fac - BigQuaternionRing.

r - BigComplex.

BigQuaternion

public BigQuaternion(BigQuaternionRing fac,
                     long r)

Constructor for a BigQuaternion from long.

Parameters:

fac - BigQuaternionRing.

r - long.

BigQuaternion

public BigQuaternion(BigQuaternionRing fac)

Constructor for a BigQuaternion with no arguments.

Parameters:

fac - BigQuaternionRing.

BigQuaternion

public BigQuaternion(BigQuaternionRing fac,
                     java.lang.String s)
              throws java.lang.NumberFormatException

The BigQuaternion string constructor accepts the following formats: empty string, "rational", or "rat i rat j rat k rat" with no blanks around i, j or k if used as polynoial coefficient.

Parameters:

fac - BigQuaternionRing.

s - String.

Throws:

java.lang.NumberFormatException

Method Detail

factory

public BigQuaternionRing factory()

Get the corresponding element factory.

Specified by:

factory in interface Element<BigQuaternion>

Returns:

factory for this Element.

See Also:

Element.factory()

copy

public BigQuaternion copy()

Clone this.

Specified by:

copy in interface Element<BigQuaternion>

Returns:

Creates and returns a copy of this Element.

See Also:

Object.clone()

getRe

public BigRational getRe()

Get the real part.

Returns:

re.

getIm

public BigRational getIm()

Get the imaginary part im.

Returns:

im.

getJm

public BigRational getJm()

Get the imaginary part jm.

Returns:

jm.

getKm

public BigRational getKm()

Get the imaginary part km.

Returns:

km.

toString

public java.lang.String toString()

Get the string representation. Is compatible with the string constructor.

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<BigQuaternion>

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<BigQuaternion>

Returns:

script compatible representation for this ElemFactory.

See Also:

Element.toScriptFactory()

isQZERO

public static boolean isQZERO(BigQuaternion A)

Is Quaternion number zero.

Parameters:

A - BigQuaternion.

Returns:

true if A is 0, else false.

isZERO

public boolean isZERO()

Is BigQuaternion number zero.

Specified by:

isZERO in interface AbelianGroupElem<BigQuaternion>

Returns:

true if this is 0, else false.

See Also:

AbelianGroupElem.isZERO()

isQONE

public static boolean isQONE(BigQuaternion A)

Is BigQuaternion number one.

Parameters:

A - is a quaternion number.

Returns:

true if A is 1, else false.

isONE

public boolean isONE()

Is BigQuaternion number one.

Specified by:

isONE in interface MonoidElem<BigQuaternion>

Returns:

true if this is 1, else false.

See Also:

MonoidElem.isONE()

isIMAG

public boolean isIMAG()

Is BigQuaternion imaginary one.

Returns:

true if this is i, else false.

isUnit

public boolean isUnit()

Is BigQuaternion unit element.

Specified by:

isUnit in interface MonoidElem<BigQuaternion>

Returns:

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

See Also:

MonoidElem.isUnit()

isEntier

public boolean isEntier()

Is BigQuaternion entier element.

Returns:

If this is an integer Hurwitz element then true is returned, else false.

equals

public boolean equals(java.lang.Object b)

Comparison with any other object.

Specified by:

equals in interface Element<BigQuaternion>

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 BigQuaternion.

Specified by:

hashCode in interface Element<BigQuaternion>

Overrides:

hashCode in class java.lang.Object

Returns:

the hashCode.

See Also:

Object.hashCode()

compareTo

public int compareTo(BigQuaternion b)

Since quaternion numbers are unordered, we use lexicographical order of re, im, jm and km.

Specified by:

compareTo in interface java.lang.Comparable<BigQuaternion>

Specified by:

compareTo in interface Element<BigQuaternion>

Parameters:

b - BigQuaternion.

Returns:

0 if b is equal to this, 1 if this is greater b and -1 else.

signum

public int signum()

Since quaternion numbers are unordered, we use lexicographical order of re, im, jm and km.

Specified by:

signum in interface AbelianGroupElem<BigQuaternion>

Returns:

0 if this is equal to 0; 1 if re > 0, or re == 0 and im > 0, or ...; -1 if re < 0, or re == 0 and im < 0, or ...

See Also:

AbelianGroupElem.signum()

sum

public BigQuaternion sum(BigQuaternion B)

BigQuaternion summation.

Specified by:

sum in interface AbelianGroupElem<BigQuaternion>

Parameters:

B - BigQuaternion.

Returns:

this+B.

QSUM

public static BigQuaternion QSUM(BigQuaternion A,
                                 BigQuaternion B)

Quaternion number sum.

Parameters:

A - BigQuaternion.

B - BigQuaternion.

Returns:

A+B.

QDIF

public static BigQuaternion QDIF(BigQuaternion A,
                                 BigQuaternion B)

Quaternion number difference.

Parameters:

A - BigQuaternion.

B - BigQuaternion.

Returns:

A-B.

subtract

public BigQuaternion subtract(BigQuaternion B)

BigQuaternion subtraction.

Specified by:

subtract in interface AbelianGroupElem<BigQuaternion>

Parameters:

B - BigQuaternion.

Returns:

this-B.

QNEG

public static BigQuaternion QNEG(BigQuaternion A)

Quaternion number negative.

Parameters:

A - is a quaternion number

Returns:

-A.

negate

public BigQuaternion negate()

BigQuaternion number negative.

Specified by:

negate in interface AbelianGroupElem<BigQuaternion>

Returns:

-this.

See Also:

AbelianGroupElem.negate()

QCON

public static BigQuaternion QCON(BigQuaternion A)

Quaternion number conjugate.

Parameters:

A - is a quaternion number.

Returns:

the quaternion conjugate of A.

conjugate

public BigQuaternion conjugate()

BigQuaternion conjugate.

Specified by:

conjugate in interface StarRingElem<BigQuaternion>

Returns:

conjugate(this).

norm

public BigQuaternion norm()

Quaternion number norm.

Specified by:

norm in interface StarRingElem<BigQuaternion>

Returns:

||this||.

See Also:

StarRingElem.norm()

abs

public BigQuaternion abs()

Quaternion number absolute value.

Specified by:

abs in interface AbelianGroupElem<BigQuaternion>

Returns:

|this|.

See Also:

AbelianGroupElem.abs()

QABS

public static BigRational QABS(BigQuaternion A)

Quaternion number absolute value.

Parameters:

A - is a quaternion number.

Returns:

the absolute value of A, a rational number. Note: The square root is not jet implemented.

QPROD

public static BigQuaternion QPROD(BigQuaternion A,
                                  BigQuaternion B)

Quaternion number product.

Parameters:

A - BigQuaternion.

B - BigQuaternion.

Returns:

A*B.

multiply

public BigQuaternion multiply(BigRational b)

BigQuaternion multiply with BigRational.

Parameters:

b - BigRational.

Returns:

this*b.

multiply

public BigQuaternion multiply(BigQuaternion B)

BigQuaternion multiply.

Specified by:

multiply in interface MonoidElem<BigQuaternion>

Parameters:

B - BigQuaternion.

Returns:

this*B.

QINV

public static BigQuaternion QINV(BigQuaternion A)

Quaternion number inverse.

Parameters:

A - is a non-zero quaternion number.

Returns:

S with S * A = A * S = 1.

inverse

public BigQuaternion inverse()

BigQuaternion inverse.

Specified by:

inverse in interface MonoidElem<BigQuaternion>

Returns:

S with S * this = this * S = 1.

See Also:

MonoidElem.inverse()

remainder

public BigQuaternion remainder(BigQuaternion S)

BigQuaternion remainder.

Specified by:

remainder in interface MonoidElem<BigQuaternion>

Parameters:

S - BigQuaternion.

Returns:

0.

QQ

public static BigQuaternion QQ(BigQuaternion A,
                               BigQuaternion B)

Quaternion number quotient.

Parameters:

A - BigQuaternion.

B - BigQuaternion.

Returns:

R/S.

divide

public BigQuaternion divide(BigQuaternion b)

BigQuaternion right divide.

Specified by:

divide in interface MonoidElem<BigQuaternion>

Parameters:

b - BigQuaternion.

Returns:

this * b**(-1).

rightDivide

public BigQuaternion rightDivide(BigQuaternion b)

BigQuaternion right divide.

Specified by:

rightDivide in interface MonoidElem<BigQuaternion>

Parameters:

b - BigQuaternion.

Returns:

this * b**(-1).

leftDivide

public BigQuaternion leftDivide(BigQuaternion b)

BigQuaternion left divide.

Specified by:

leftDivide in interface MonoidElem<BigQuaternion>

Parameters:

b - BigQuaternion.

Returns:

b**(-1) * this.

divide

public BigQuaternion divide(BigRational b)

BigQuaternion divide.

Parameters:

b - BigRational.

Returns:

this/b.

quotientRemainder

public BigQuaternion[] quotientRemainder(BigQuaternion S)

Quotient and remainder by division of this by S.

Specified by:

quotientRemainder in interface MonoidElem<BigQuaternion>

Parameters:

S - a quaternion number

Returns:

[this*S**(-1), this - (this*S**(-1))*S].

gcd

public BigQuaternion gcd(BigQuaternion S)

Quaternion number greatest common divisor.

Specified by:

gcd in interface RingElem<BigQuaternion>

Parameters:

S - BigQuaternion.

Returns:

gcd(this,S).

leftGcd

public BigQuaternion leftGcd(BigQuaternion S)

Quaternion number greatest common divisor.

Specified by:

leftGcd in interface RingElem<BigQuaternion>

Parameters:

S - BigQuaternion.

Returns:

leftCcd(this,S).

rightGcd

public BigQuaternion rightGcd(BigQuaternion S)

Quaternion number greatest common divisor.

Specified by:

rightGcd in interface RingElem<BigQuaternion>

Parameters:

S - BigQuaternion.

Returns:

rightCcd(this,S).

egcd

public BigQuaternion[] egcd(BigQuaternion S)

BigQuaternion extended greatest common divisor.

Specified by:

egcd in interface RingElem<BigQuaternion>

Parameters:

S - BigQuaternion.

Returns:

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

bitLength

public long bitLength()

Returns the number of bits in the representation of this BigQuaternion, including a sign bit. It is equivalent to re.bitLength()+im.bitLength()+jm.bitLength()+km.bitLength().)

Returns:

number of bits in the representation of this BigQuaternion, including a sign bit.

ceil

public BigQuaternion ceil()

BigQuaternion ceiling, component wise.

Returns:

ceiling of this.

floor

public BigQuaternion floor()

BigQuaternion floor, component wise.

Returns:

floor of this.

roundToLipschitzian

public BigQuaternionInteger roundToLipschitzian()

BigQuaternion round to next Lipschitz integer. BigQuaternion with all integer components.

Returns:

Lipschitz integer of this.

roundToHurwitzian

public BigQuaternionInteger roundToHurwitzian()

BigQuaternion round to next Hurwitz integer. BigQuaternion with all integer or all 1/2 times integer components.

Returns:

Hurwitz integer near this.