edu.jas.arith

Class BigOctonion

java.lang.Object

edu.jas.arith.BigOctonion

All Implemented Interfaces:

AbelianGroupElem<BigOctonion>, AbelianGroupFactory<BigOctonion>, Element<BigOctonion>, ElemFactory<BigOctonion>, GcdRingElem<BigOctonion>, MonoidElem<BigOctonion>, MonoidFactory<BigOctonion>, RingElem<BigOctonion>, RingFactory<BigOctonion>, StarRingElem<BigOctonion>, java.io.Serializable, java.lang.Comparable<BigOctonion>

public final class BigOctonion
extends java.lang.Object
implements StarRingElem<BigOctonion>, GcdRingElem<BigOctonion>, RingFactory<BigOctonion>

BigOctonion class based on BigRational implementing the RingElem interface and with the familiar MAS static method names. Objects of this class are immutable.

Author:

Heinz Kredel

See Also:

Serialized Form

Field Summary

Fields

Modifier and Type	Field and Description
BigOctonion	I The constant i.
BigQuaternion	oi Second part of the data structure.
BigOctonion	ONE The constant 1.
BigQuaternion	or First part of the data structure.
BigOctonion	ZERO The constant 0.

Constructor Summary

Constructors

Constructor and Description
BigOctonion(BigQuaternion r) Constructor for a BigOctonion from BigQuaternion.
BigOctonion(BigQuaternion r, BigQuaternion i) Constructor for a BigOctonion from Quaternions.
BigOctonion(BigQuaternionRing fac) Constructor for a BigOctonion with no arguments.
BigOctonion(BigQuaternionRing fac, BigComplex r) Constructor for a BigOctonion from BigComplex.
BigOctonion(BigQuaternionRing fac, BigRational r) Constructor for a BigOctonion from BigRational.
BigOctonion(BigQuaternionRing fac, long r) Constructor for a BigOctonion from long.
BigOctonion(BigQuaternionRing fac, java.lang.String s) The BigOctonion string constructor accepts the following formats: empty string, "quaternion", or "quat o quat" with no blanks around o if used as polynoial coefficient.

Method Summary

Modifier and Type	Method and Description
BigOctonion	abs() Octonion number absolute value.
long	bitLength() Returns the number of bits in the representation of this BigOctonion, including a sign bit.
java.math.BigInteger	characteristic() Characteristic of this ring.
int	compareTo(BigOctonion b) Since quaternion numbers are unordered, we use lexicographical order of re, im, jm and km.
BigOctonion	conjugate() BigOctonion conjugate.
BigOctonion	copy() Clone this.
BigOctonion	copy(BigOctonion c) Copy BigOctonion element c.
BigOctonion	divide(BigOctonion b) BigOctonion divide.
BigOctonion	divide(BigRational b) BigOctonion divide.
BigOctonion[]	egcd(BigOctonion S) BigOctonion extended greatest common divisor.
boolean	equals(java.lang.Object b) Comparison with any other object.
BigOctonion	factory() Get the corresponding element factory.
BigOctonion	fromInteger(java.math.BigInteger a) Get a BigOctonion element from a BigInteger.
BigOctonion	fromInteger(long a) Get a BigOctonion element from a long.
BigOctonion	gcd(BigOctonion S) Octonion number greatest common divisor.
java.util.List<BigOctonion>	generators() Get a list of the generating elements.
BigQuaternion	getI() Get the oi part.
BigOctonion	getONE() Get the one element.
BigQuaternion	getR() Get the or part.
BigOctonion	getZERO() Get the zero element.
int	hashCode() Hash code for this BigOctonion.
BigOctonion	inverse() BigOctonion inverse.
boolean	isAssociative() Query if this ring is associative.
boolean	isCommutative() Query if this ring is commutative.
boolean	isField() Query if this ring is a field.
boolean	isFinite() Is this structure finite or infinite.
boolean	isIMAG() Is BigOctonion imaginary one.
boolean	isONE() Is BigOctonion number one.
static boolean	isOONE(BigOctonion A) Is BigOctonion number one.
static boolean	isOZERO(BigOctonion A) Is Octonion number zero.
boolean	isUnit() Is BigOctonion unit element.
boolean	isZERO() Is BigOctonion number zero.
BigOctonion	leftDivide(BigOctonion b) BigOctonion left divide.
BigOctonion	multiply(BigOctonion B) BigOctonion multiply.
BigOctonion	negate() BigOctonion number negative.
BigOctonion	norm() Octonion number norm.
static BigRational	OABS(BigOctonion A) Octonion number absolute value.
static BigOctonion	OCON(BigOctonion A) Octonion number conjugate.
static BigOctonion	ODIF(BigOctonion A, BigOctonion B) Octonion number difference.
static BigOctonion	OINV(BigOctonion A) Octonion number inverse.
static BigOctonion	ONEG(BigOctonion A) Octonion number negative.
static BigOctonion	OPROD(BigOctonion A, BigOctonion B) Octonion number product.
static BigOctonion	OQ(BigOctonion A, BigOctonion B) Octonion number quotient.
static BigOctonion	OSUM(BigOctonion A, BigOctonion B) Octonion number sum.
BigOctonion	parse(java.io.Reader r) Parse quaternion number from Reader.
BigOctonion	parse(java.lang.String s) Parse quaternion number from String.
BigOctonion[]	quotientRemainder(BigOctonion S) Quotient and remainder by division of this by S.
BigOctonion	random(int n) BigOctonion random.
BigOctonion	random(int n, java.util.Random rnd) BigOctonion random.
BigOctonion	remainder(BigOctonion S) BigOctonion remainder.
BigOctonion	rightDivide(BigOctonion b) BigOctonion right divide.
int	signum() Since quaternion numbers are unordered, we use lexicographical order of re, im, jm and km.
BigOctonion	subtract(BigOctonion B) BigOctonion subtraction.
BigOctonion	sum(BigOctonion B) BigOctonion 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.RingElem

leftGcd, rightGcd

Methods inherited from interface edu.jas.structure.MonoidElem

leftRemainder, power, rightRemainder, twosidedDivide, twosidedRemainder

Field Detail

or

public final BigQuaternion or

First part of the data structure.

oi

public final BigQuaternion oi

Second part of the data structure.

ZERO

public BigOctonion ZERO

The constant 0.

ONE

public BigOctonion ONE

The constant 1.

I

public BigOctonion I

The constant i.

Constructor Detail

BigOctonion

public BigOctonion(BigQuaternion r,
                   BigQuaternion i)

Constructor for a BigOctonion from Quaternions.

Parameters:

r - BigQuaternion.

i - BigQuaternion.

BigOctonion

public BigOctonion(BigQuaternion r)

Constructor for a BigOctonion from BigQuaternion.

Parameters:

r - BigQuaternion.

BigOctonion

public BigOctonion(BigQuaternionRing fac,
                   BigComplex r)

Constructor for a BigOctonion from BigComplex.

Parameters:

fac - BigQuaternionRing.

r - BigComplex.

BigOctonion

public BigOctonion(BigQuaternionRing fac,
                   BigRational r)

Constructor for a BigOctonion from BigRational.

Parameters:

fac - BigQuaternionRing.

r - BigRational.

BigOctonion

public BigOctonion(BigQuaternionRing fac,
                   long r)

Constructor for a BigOctonion from long.

Parameters:

fac - BigQuaternionRing.

r - long.

BigOctonion

public BigOctonion(BigQuaternionRing fac)

Constructor for a BigOctonion with no arguments.

Parameters:

fac - BigQuaternionRing.

BigOctonion

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

The BigOctonion string constructor accepts the following formats: empty string, "quaternion", or "quat o quat" with no blanks around o if used as polynoial coefficient.

Parameters:

fac - BigQuaternionRing.

s - String.

Throws:

java.lang.NumberFormatException

Method Detail

factory

public BigOctonion factory()

Get the corresponding element factory.

Specified by:

factory in interface Element<BigOctonion>

Returns:

factory for this Element.

See Also:

Element.factory()

generators

public java.util.List<BigOctonion> generators()

Get a list of the generating elements.

Specified by:

generators in interface ElemFactory<BigOctonion>

Returns:

list of generators for the algebraic structure.

See Also:

ElemFactory.generators()

isFinite

public boolean isFinite()

Is this structure finite or infinite.

Specified by:

isFinite in interface ElemFactory<BigOctonion>

Returns:

true if this structure is finite, else false.

See Also:

ElemFactory.isFinite()

copy

public BigOctonion copy()

Clone this.

Specified by:

copy in interface Element<BigOctonion>

Returns:

Creates and returns a copy of this Element.

See Also:

Object.clone()

copy

public BigOctonion copy(BigOctonion c)

Copy BigOctonion element c.

Specified by:

copy in interface ElemFactory<BigOctonion>

Parameters:

c - BigOctonion.

Returns:

a copy of c.

getZERO

public BigOctonion getZERO()

Get the zero element.

Specified by:

getZERO in interface AbelianGroupFactory<BigOctonion>

Returns:

0 as BigOctonion.

getONE

public BigOctonion getONE()

Get the one element.

Specified by:

getONE in interface MonoidFactory<BigOctonion>

Returns:

q as BigOctonion.

isCommutative

public boolean isCommutative()

Query if this ring is commutative.

Specified by:

isCommutative in interface MonoidFactory<BigOctonion>

Returns:

false.

isAssociative

public boolean isAssociative()

Query if this ring is associative.

Specified by:

isAssociative in interface MonoidFactory<BigOctonion>

Returns:

false.

isField

public boolean isField()

Query if this ring is a field.

Specified by:

isField in interface RingFactory<BigOctonion>

Returns:

true.

characteristic

public java.math.BigInteger characteristic()

Characteristic of this ring.

Specified by:

characteristic in interface RingFactory<BigOctonion>

Returns:

characteristic of this ring.

fromInteger

public BigOctonion fromInteger(java.math.BigInteger a)

Get a BigOctonion element from a BigInteger.

Specified by:

fromInteger in interface ElemFactory<BigOctonion>

Parameters:

a - BigInteger.

Returns:

a BigOctonion.

fromInteger

public BigOctonion fromInteger(long a)

Get a BigOctonion element from a long.

Specified by:

fromInteger in interface ElemFactory<BigOctonion>

Parameters:

a - long.

Returns:

a BigOctonion.

getR

public BigQuaternion getR()

Get the or part.

Returns:

or.

getI

public BigQuaternion getI()

Get the oi part.

Returns:

oi.

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<BigOctonion>

Specified by:

toScript in interface ElemFactory<BigOctonion>

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<BigOctonion>

Returns:

script compatible representation for this ElemFactory.

See Also:

Element.toScriptFactory()

isOZERO

public static boolean isOZERO(BigOctonion A)

Is Octonion number zero.

Parameters:

A - BigOctonion.

Returns:

true if A is 0, else false.

isZERO

public boolean isZERO()

Is BigOctonion number zero.

Specified by:

isZERO in interface AbelianGroupElem<BigOctonion>

Returns:

true if this is 0, else false.

See Also:

AbelianGroupElem.isZERO()

isOONE

public static boolean isOONE(BigOctonion A)

Is BigOctonion number one.

Parameters:

A - is a quaternion number.

Returns:

true if A is 1, else false.

isONE

public boolean isONE()

Is BigOctonion number one.

Specified by:

isONE in interface MonoidElem<BigOctonion>

Returns:

true if this is 1, else false.

See Also:

MonoidElem.isONE()

isIMAG

public boolean isIMAG()

Is BigOctonion imaginary one.

Returns:

true if this is i, else false.

isUnit

public boolean isUnit()

Is BigOctonion unit element.

Specified by:

isUnit in interface MonoidElem<BigOctonion>

Returns:

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

See Also:

MonoidElem.isUnit()

equals

public boolean equals(java.lang.Object b)

Comparison with any other object.

Specified by:

equals in interface Element<BigOctonion>

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 BigOctonion.

Specified by:

hashCode in interface Element<BigOctonion>

Overrides:

hashCode in class java.lang.Object

Returns:

the hashCode.

See Also:

Object.hashCode()

compareTo

public int compareTo(BigOctonion b)

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

Specified by:

compareTo in interface Element<BigOctonion>

Specified by:

compareTo in interface java.lang.Comparable<BigOctonion>

Parameters:

b - BigOctonion.

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<BigOctonion>

Returns:

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

See Also:

AbelianGroupElem.signum()

sum

public BigOctonion sum(BigOctonion B)

BigOctonion summation.

Specified by:

sum in interface AbelianGroupElem<BigOctonion>

Parameters:

B - BigOctonion.

Returns:

this+B.

OSUM

public static BigOctonion OSUM(BigOctonion A,
                               BigOctonion B)

Octonion number sum.

Parameters:

A - BigOctonion.

B - BigOctonion.

Returns:

A+B.

ODIF

public static BigOctonion ODIF(BigOctonion A,
                               BigOctonion B)

Octonion number difference.

Parameters:

A - BigOctonion.

B - BigOctonion.

Returns:

A-B.

subtract

public BigOctonion subtract(BigOctonion B)

BigOctonion subtraction.

Specified by:

subtract in interface AbelianGroupElem<BigOctonion>

Parameters:

B - BigOctonion.

Returns:

this-B.

ONEG

public static BigOctonion ONEG(BigOctonion A)

Octonion number negative.

Parameters:

A - is a octonion number

Returns:

-A.

negate

public BigOctonion negate()

BigOctonion number negative.

Specified by:

negate in interface AbelianGroupElem<BigOctonion>

Returns:

-this.

See Also:

AbelianGroupElem.negate()

OCON

public static BigOctonion OCON(BigOctonion A)

Octonion number conjugate.

Parameters:

A - is a quaternion number.

Returns:

the quaternion conjugate of A.

conjugate

public BigOctonion conjugate()

BigOctonion conjugate.

Specified by:

conjugate in interface StarRingElem<BigOctonion>

Returns:

conjugate(this).

norm

public BigOctonion norm()

Octonion number norm.

Specified by:

norm in interface StarRingElem<BigOctonion>

Returns:

||this||.

See Also:

StarRingElem.norm()

abs

public BigOctonion abs()

Octonion number absolute value.

Specified by:

abs in interface AbelianGroupElem<BigOctonion>

Returns:

|this|.

See Also:

AbelianGroupElem.abs()

OABS

public static BigRational OABS(BigOctonion A)

Octonion 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.

OPROD

public static BigOctonion OPROD(BigOctonion A,
                                BigOctonion B)

Octonion number product.

Parameters:

A - BigOctonion.

B - BigOctonion.

Returns:

A*B.

multiply

public BigOctonion multiply(BigOctonion B)

BigOctonion multiply.

Specified by:

multiply in interface MonoidElem<BigOctonion>

Parameters:

B - BigOctonion.

Returns:

this*B.

OINV

public static BigOctonion OINV(BigOctonion A)

Octonion number inverse.

Parameters:

A - is a non-zero quaternion number.

Returns:

S with S * A = 1.

inverse

public BigOctonion inverse()

BigOctonion inverse.

Specified by:

inverse in interface MonoidElem<BigOctonion>

Returns:

S with S * this = 1.

See Also:

MonoidElem.inverse()

remainder

public BigOctonion remainder(BigOctonion S)

BigOctonion remainder.

Specified by:

remainder in interface MonoidElem<BigOctonion>

Parameters:

S - BigOctonion.

Returns:

0.

OQ

public static BigOctonion OQ(BigOctonion A,
                             BigOctonion B)

Octonion number quotient.

Parameters:

A - BigOctonion.

B - BigOctonion.

Returns:

R/S.

divide

public BigOctonion divide(BigOctonion b)

BigOctonion divide.

Specified by:

divide in interface MonoidElem<BigOctonion>

Parameters:

b - BigOctonion.

Returns:

this * b**(-1).

rightDivide

public BigOctonion rightDivide(BigOctonion b)

BigOctonion right divide.

Specified by:

rightDivide in interface MonoidElem<BigOctonion>

Parameters:

b - BigOctonion.

Returns:

this * b**(-1).

leftDivide

public BigOctonion leftDivide(BigOctonion b)

BigOctonion left divide.

Specified by:

leftDivide in interface MonoidElem<BigOctonion>

Parameters:

b - BigOctonion.

Returns:

b**(-1) * this.

divide

public BigOctonion divide(BigRational b)

BigOctonion divide.

Parameters:

b - BigRational.

Returns:

this/b.

quotientRemainder

public BigOctonion[] quotientRemainder(BigOctonion S)

Quotient and remainder by division of this by S.

Specified by:

quotientRemainder in interface MonoidElem<BigOctonion>

Parameters:

S - a octonion number

Returns:

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

random

public BigOctonion random(int n)

BigOctonion random. Random rational numbers A, B, C and D are generated using random(n). Then R is the quaternion number with real part A and imaginary parts B, C and D.

Specified by:

random in interface ElemFactory<BigOctonion>

Parameters:

n - such that 0 ≤ A, B, C, D ≤ (2ⁿ-1).

Returns:

R, a random BigOctonion.

random

public BigOctonion random(int n,
                          java.util.Random rnd)

BigOctonion random. Random rational numbers A, B, C and D are generated using RNRAND(n). Then R is the quaternion number with real part A and imaginary parts B, C and D.

Specified by:

random in interface ElemFactory<BigOctonion>

Parameters:

n - such that 0 ≤ A, B, C, D ≤ (2ⁿ-1).

rnd - is a source for random bits.

Returns:

R, a random BigOctonion.

parse

public BigOctonion parse(java.lang.String s)

Parse quaternion number from String.

Specified by:

parse in interface ElemFactory<BigOctonion>

Parameters:

s - String.

Returns:

BigOctonion from s.

parse

public BigOctonion parse(java.io.Reader r)

Parse quaternion number from Reader.

Specified by:

parse in interface ElemFactory<BigOctonion>

Parameters:

r - Reader.

Returns:

next BigOctonion from r.

gcd

public BigOctonion gcd(BigOctonion S)

Octonion number greatest common divisor.

Specified by:

gcd in interface RingElem<BigOctonion>

Parameters:

S - BigOctonion.

Returns:

gcd(this,S).

egcd

public BigOctonion[] egcd(BigOctonion S)

BigOctonion extended greatest common divisor.

Specified by:

egcd in interface RingElem<BigOctonion>

Parameters:

S - BigOctonion.

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 BigOctonion, including a sign bit. It is equivalent to or.bitLength() + oi.bitLength().)

Returns:

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