Package edu.jas.arith

Class BigQuaternionInteger

java.lang.Object

edu.jas.arith.BigQuaternion

edu.jas.arith.BigQuaternionInteger

All Implemented Interfaces:

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

public final class BigQuaternionInteger
extends BigQuaternion

Integer 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 inherited from class edu.jas.arith.BigQuaternion

im, jm, km, random, re, ring

Constructor Summary

Constructors

Constructor	Description
BigQuaternionInteger(BigQuaternionRing fac)	Constructor for a BigQuaternion with no arguments.
BigQuaternionInteger(BigQuaternionRing fac, long r)	Constructor for a BigQuaternion from long.
BigQuaternionInteger(BigQuaternionRing fac, BigComplex r)	Constructor for a BigQuaternion from BigComplex.
BigQuaternionInteger(BigQuaternionRing fac, BigQuaternion q)	Constructor for a BigQuaternionInteger from BigQuaternion.
BigQuaternionInteger(BigQuaternionRing fac, BigRational r)	Constructor for a BigQuaternion from BigRationals.
BigQuaternionInteger(BigQuaternionRing fac, BigRational r, BigRational i)	Constructor for a BigQuaternion from BigRationals.
BigQuaternionInteger(BigQuaternionRing fac, BigRational r, BigRational i, BigRational j)	Constructor for a BigQuaternion from BigRationals.
BigQuaternionInteger(BigQuaternionRing fac, BigRational r, BigRational i, BigRational j, BigRational k)	Constructor for a BigQuaternion from BigRationals.
BigQuaternionInteger(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.
BigQuaternionInteger	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.
BigQuaternionRing	factory()	Get the corresponding element factory.
BigQuaternion	gcd(BigQuaternion S)	Quaternion number greatest common divisor.
BigQuaternion	inverse()	BigQuaternion inverse.
boolean	isPrime()	Quaternion number test if it is a prime number.
BigQuaternion	leftDivide(BigQuaternion b)	BigQuaternion left divide.
BigQuaternion	leftGcd(BigQuaternion S)	Integer quaternion number left greatest common divisor.
BigQuaternion[]	leftQuotientAndRemainder(BigQuaternion b)	Integral quotient and remainder by left division of this by S.
BigQuaternion	leftRemainder(BigQuaternion a)	Left remainder.
static BigQuaternion	QINV(BigQuaternion A)	Quaternion number inverse.
static BigQuaternion	QQ(BigQuaternion A, BigQuaternion B)	Quaternion number quotient.
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)	Integer quaternion number right greatest common divisor.
BigQuaternion[]	rightQuotientAndRemainder(BigQuaternion b)	Integral quotient and remainder by right division of this by S.
BigQuaternion	rightRemainder(BigQuaternion a)	Right remainder.

Methods inherited from class edu.jas.arith.BigQuaternion

bitLength, ceil, compareTo, conjugate, equals, floor, getIm, getJm, getKm, getRe, hashCode, isEntier, isIMAG, isONE, isQONE, isQZERO, isUnit, isZERO, multiply, multiply, multiplyLeft, negate, norm, QABS, QCON, QDIF, QNEG, QPROD, QSUM, roundToHurwitzian, roundToLipschitzian, signum, subtract, sum, toScript, toScriptFactory, toString

Methods inherited from class java.lang.Object

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

Methods inherited from interface edu.jas.structure.MonoidElem

power, twosidedDivide, twosidedRemainder

Constructor Detail

BigQuaternionInteger

public BigQuaternionInteger(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.

BigQuaternionInteger

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

Constructor for a BigQuaternion from BigRationals.

Parameters:

fac - BigQuaternionRing.

r - BigRational.

i - BigRational.

j - BigRational.

BigQuaternionInteger

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

Constructor for a BigQuaternion from BigRationals.

Parameters:

fac - BigQuaternionRing.

r - BigRational.

i - BigRational.

BigQuaternionInteger

public BigQuaternionInteger(BigQuaternionRing fac,
                            BigRational r)

Constructor for a BigQuaternion from BigRationals.

Parameters:

fac - BigQuaternionRing.

r - BigRational.

BigQuaternionInteger

public BigQuaternionInteger(BigQuaternionRing fac,
                            BigComplex r)

Constructor for a BigQuaternion from BigComplex.

Parameters:

fac - BigQuaternionRing.

r - BigComplex.

BigQuaternionInteger

public BigQuaternionInteger(BigQuaternionRing fac,
                            BigQuaternion q)

Constructor for a BigQuaternionInteger from BigQuaternion.

Parameters:

fac - BigQuaternionRing.

q - BigQuaternion.

BigQuaternionInteger

public BigQuaternionInteger(BigQuaternionRing fac,
                            long r)

Constructor for a BigQuaternion from long.

Parameters:

fac - BigQuaternionRing.

r - long.

BigQuaternionInteger

public BigQuaternionInteger(BigQuaternionRing fac)

Constructor for a BigQuaternion with no arguments.

Parameters:

fac - BigQuaternionRing.

BigQuaternionInteger

public BigQuaternionInteger(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>

Overrides:

factory in class BigQuaternion

Returns:

factory for this Element.

See Also:

Element.factory()

copy

public BigQuaternionInteger copy()

Clone this.

Specified by:

copy in interface Element<BigQuaternion>

Overrides:

copy in class BigQuaternion

Returns:

Creates and returns a copy of this Element.

See Also:

Object.clone()

abs

public BigQuaternion abs()

Quaternion number absolute value.

Specified by:

abs in interface AbelianGroupElem<BigQuaternion>

Overrides:

abs in class BigQuaternion

Returns:

|this|**2. Note: returns the norm(this).

See Also:

AbelianGroupElem.abs()

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>

Overrides:

inverse in class 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>

Overrides:

remainder in class BigQuaternion

Parameters:

S - BigQuaternion.

Returns:

this - this * b**(-1).

QQ

public static BigQuaternion QQ(BigQuaternion A,
                               BigQuaternion B)

Quaternion number quotient.

Parameters:

A - BigQuaternion.

B - BigQuaternion.

Returns:

R * B**(-1).

divide

public BigQuaternion divide(BigQuaternion b)

BigQuaternion right divide.

Specified by:

divide in interface MonoidElem<BigQuaternion>

Overrides:

divide in class BigQuaternion

Parameters:

b - BigQuaternion.

Returns:

this * b**(-1).

rightDivide

public BigQuaternion rightDivide(BigQuaternion b)

BigQuaternion right divide.

Specified by:

rightDivide in interface MonoidElem<BigQuaternion>

Overrides:

rightDivide in class BigQuaternion

Parameters:

b - BigQuaternion.

Returns:

this * b**(-1).

leftDivide

public BigQuaternion leftDivide(BigQuaternion b)

BigQuaternion left divide.

Specified by:

leftDivide in interface MonoidElem<BigQuaternion>

Overrides:

leftDivide in class BigQuaternion

Parameters:

b - BigQuaternion.

Returns:

b**(-1) * this.

divide

public BigQuaternion divide(BigRational b)

BigQuaternion divide.

Overrides:

divide in class BigQuaternion

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>

Overrides:

quotientRemainder in class 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>

Overrides:

gcd in class BigQuaternion

Parameters:

S - BigQuaternion.

Returns:

gcd(this,S).

egcd

public BigQuaternion[] egcd(BigQuaternion S)

BigQuaternion extended greatest common divisor.

Specified by:

egcd in interface RingElem<BigQuaternion>

Overrides:

egcd in class BigQuaternion

Parameters:

S - BigQuaternion.

Returns:

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

leftQuotientAndRemainder

public BigQuaternion[] leftQuotientAndRemainder(BigQuaternion b)

Integral quotient and remainder by left division of this by S. This must be also an integral (Hurwitz) quaternion number.

Parameters:

b - an integral (Hurwitz) quaternion number

Returns:

[round(b**(-1)) this, this - b * (round(b**(-1)) this)].

rightQuotientAndRemainder

public BigQuaternion[] rightQuotientAndRemainder(BigQuaternion b)

Integral quotient and remainder by right division of this by S. This must be also an integral (Hurwitz) quaternion number.

Parameters:

b - an integral (Hurwitz) quaternion number

Returns:

[this round(b**(-1)), this - this (round(b**(-1)) b)].

leftRemainder

public BigQuaternion leftRemainder(BigQuaternion a)

Left remainder.

Specified by:

leftRemainder in interface MonoidElem<BigQuaternion>

Overrides:

leftRemainder in class BigQuaternion

Parameters:

a - element.

Returns:

r = this - (a/left) * a, where left * a = this.

rightRemainder

public BigQuaternion rightRemainder(BigQuaternion a)

Right remainder.

Specified by:

rightRemainder in interface MonoidElem<BigQuaternion>

Overrides:

rightRemainder in class BigQuaternion

Parameters:

a - element.

Returns:

r = this - a * (a/right), where a * right = this.

leftGcd

public BigQuaternion leftGcd(BigQuaternion S)

Integer quaternion number left greatest common divisor.

Specified by:

leftGcd in interface RingElem<BigQuaternion>

Overrides:

leftGcd in class BigQuaternion

Parameters:

S - integer BigQuaternion.

Returns:

leftGcd(this,S).

rightGcd

public BigQuaternion rightGcd(BigQuaternion S)

Integer quaternion number right greatest common divisor.

Specified by:

rightGcd in interface RingElem<BigQuaternion>

Overrides:

rightGcd in class BigQuaternion

Parameters:

S - integer BigQuaternion.

Returns:

rightGcd(this,S).

isPrime

public boolean isPrime()

Quaternion number test if it is a prime number.

Returns:

isPrime(norm(this))