jhypercomplex.algebras

Class BinaryAlgebra

java.lang.Object

jhypercomplex.BinaryProductStructure

jhypercomplex.algebras.BinaryAlgebra

Direct Known Subclasses:

CayleyDicksonAlgebra, CliffordAlgebra, OctonarySubAlgebra, TensoredAlgebra

public abstract class BinaryAlgebra
extends BinaryProductStructure

Copyright © 2005-2015 by Markus Maute. All rights reserved.

Version:

0.1

Author:

Markus Maute (http:///www.jalgebra.com)

Field Summary

Fields inherited from class jhypercomplex.BinaryProductStructure

ARG_NUM, CIRCULAR, COMP_NUM, components, HYPERBOLIC, ZERO_PRECISSION

Method Summary

Methods

Modifier and Type	Method and Description
BinaryAlgebra	add(BinaryProductStructure a)
BinaryAlgebra	add(BinaryProductStructure a, boolean collect_terms) Add an element from the algebra.
static BinaryAlgebra	applyLeibnizRule(BinaryAlgebra h) TODO: meanwhile see 'getLeftCovariantDerivative ()'
BinaryAlgebra	calculateBCHWith(BinaryAlgebra a, int from_order, int to_order) The Baker Campbell Hausdorff formula (BCH) is calculated for a specified range of orders.
BinaryAlgebra	getAntiCommutatorWith(BinaryAlgebra a) Calculates the anti-commutator of the element a1 of the algebra with another one a2 which is defined as: {a1, a2} = a1*a2 + a2*a1
abstract BinaryAlgebra	getClone()
BinaryAlgebra	getCommutatorWith(BinaryAlgebra a)
BinaryAlgebra	getCommutatorWith(BinaryAlgebra a, boolean collect_terms) Calculates the commutator of an element a1 of the algebra with another one a2.
java.lang.String[]	getComponentsOfGrade(int grade, java.util.TreeMap<java.lang.Integer,java.lang.Integer> grades)
static BinaryAlgebra	getDifference(BinaryAlgebra a1, BinaryAlgebra a2)
abstract BinaryAlgebra	getGrade(int grade)
static BinaryAlgebra	getGradedInnerProduct(BinaryAlgebra a1, BinaryAlgebra a2, boolean with_grade_0)
static java.util.HashSet<BinaryProductStructure>	getIntersection(BinaryProductStructure[] h1, BinaryProductStructure[] h2) Two sets of hypernumbers are compared.
BinaryAlgebra	getLeftCovariantDerivative(BinaryAlgebra covariant_deriv_op, BinaryAlgebra h) The covariant derivative of the hypernumber is calculated.
int	getNumberOfComponents(int grade, java.util.TreeMap<java.lang.Integer,java.lang.Integer> grades)
abstract int	getNumberOfGrades()
static BinaryAlgebra	getOuterProduct(BinaryAlgebra a1, BinaryAlgebra a2)
static BinaryAlgebra	getProduct(BinaryAlgebra a, BinaryAlgebra b)
static BinaryAlgebra	getScalarProduct(BinaryAlgebra a1, BinaryAlgebra a2) The scalar product (= real inner product) <.|.> of two algebra a1 and a2 is defined as = 1/2(a1a2*+a2a1*)
static BinaryAlgebra	getSum(BinaryAlgebra a1, BinaryAlgebra a2)
static BinaryAlgebra	getTernaryInnerProduct(BinaryAlgebra a1, BinaryAlgebra a2, BinaryAlgebra a3) The scalar product (= real inner product) <.|.|.> of three algebras a1 and a2 and a3 is defined as = 1/12((ab)^*c + a(bc^*) + 5 permutations)
abstract boolean	isCayleyDicksonAlgebra()
static boolean	isCommutative(BinaryAlgebra a1, BinaryAlgebra a2)
boolean	isEqual(BinaryProductStructure h) Compares the components of the hypernumber with those of another one.
static boolean	isOrthogonal(BinaryAlgebra a1, BinaryAlgebra a2)
boolean	isOrthogonalTo(BinaryAlgebra a)
boolean	isProjectionOperator()
BinaryAlgebra	rightMultiplyWith(BinaryAlgebra a)
BinaryAlgebra	subtract(BinaryAlgebra a) Subtracts an element from an algebra.

Methods inherited from class jhypercomplex.BinaryProductStructure

asString, asString, collectTerms, compressComponents, conjugate, dual, getBasis, getBasisElementAsString, getBasisElementsAsString, getClosedPairsAsString, getComponent, getComponents, getComponentsAsString, getComponentsAsString, getConjugate, getDiagonalProducts, getDual, getExpressionAsList, getFormattedMultTable, getHermitianConjugate, getImaginaryPart, getInstance, getMTab, getMultiplicationTableAsArray, getMultiplicativeOrder, getMultiplicativeOrder, getMultTableDiagonal, getNegated, getNewInstance, getNextBitmap, getNonzeroComponentsNumbered, getNonzeroComponentsNumbered, getNormedMultiplicationTable, getNormSquared, getNormSquaredAsValue, getNumberOfClosedElements, getNumberOfClosedPairs, getNumberOfComponents, getNumberOfNonzeroComponents, getNumericalInverse, getProduct, getProductWithScalar, getPseudoScalarComponent, getRandomClosedPair, getRandomNonClosedPair, getScalarComponent, getSubalgebras, getSubalgebrasAsString, getSubalgebrasSignatures, getTrace, getTraceRespDeterminant, getVectorDerivative, getVectorProduct, isAutomorphism, isCayleyDicksonLoop, isComponentZero, isLinearlyDependent, isNumerical, isNumericalityDetected, isProductClosed, isZero, multiplyWithScalar, negate, resetComponents, rightMultiplyWith, rightMultiplyWith, setBasis, setComponent, setComponentAsBasisNames, setComponentExclusively, setComponents, setComponents, setIndexedComponents, setMultiplicationTable, setMultiplicationTable, setNumerical, setNumerical, setPseudoScalarComponent, setRandomComponents, setRandomComponents, setRandomComponents, setRandomComponentsAll, setRepresentation, setScalarComponent, setSignTable, simplifyNumericalFactors, simplifyNumericalSummands, simplifyPlusMinus, square

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Method Detail

getClone

public abstract BinaryAlgebra getClone()

Specified by:

getClone in class BinaryProductStructure

getNumberOfGrades

public abstract int getNumberOfGrades()
                               throws java.lang.Exception

Throws:

java.lang.Exception

getGrade

public abstract BinaryAlgebra getGrade(int grade)
                                throws java.lang.Exception

Throws:

java.lang.Exception

isCayleyDicksonAlgebra

public abstract boolean isCayleyDicksonAlgebra()
                                        throws java.lang.Exception

Throws:

java.lang.Exception

isEqual

public boolean isEqual(BinaryProductStructure h)
                throws java.lang.Exception

Compares the components of the hypernumber with those of another one. Different ordering of summands and products is taken into account, i.e. x3+x1*x2 = x2*x1+x3

Returns:

'true' iff all components are the same.

Throws:

java.lang.Exception

getIntersection

public static java.util.HashSet<BinaryProductStructure> getIntersection(BinaryProductStructure[] h1,
                                                        BinaryProductStructure[] h2)
                                                                 throws java.lang.Exception

Two sets of hypernumbers are compared. The set of hypernumbers that occur in both sets is returned.

Parameters:

h1 - First set with hypernumbers.

h2 - Second set with hypernumbers.

Returns:

Intersection of h1 and h2.

Throws:

java.lang.Exception

getNumberOfComponents

public int getNumberOfComponents(int grade,
                        java.util.TreeMap<java.lang.Integer,java.lang.Integer> grades)

Parameters:

grade - Grade.

Returns:

Number of components for a given grade.

getComponentsOfGrade

public java.lang.String[] getComponentsOfGrade(int grade,
                                      java.util.TreeMap<java.lang.Integer,java.lang.Integer> grades)

Returns:

Vector components in the order defined by the basis.

add

public BinaryAlgebra add(BinaryProductStructure a)

add

public BinaryAlgebra add(BinaryProductStructure a,
                boolean collect_terms)

Add an element from the algebra.

Parameters:

a - Algebraic element to be added.

collect_terms - true: Terms are added or subtracted if possible. false: Terms are not added or subtracted. (This is interesting if they contain functions and differential operators for which the order is relevant. The simplification procedure ignores the order).

Returns:

Result of addition.

subtract

public BinaryAlgebra subtract(BinaryAlgebra a)
                       throws java.lang.Exception

Subtracts an element from an algebra.

Parameters:

a - Algebraic element to be subtracted.

Returns:

Result of subtraction.

Throws:

java.lang.Exception

getTernaryInnerProduct

public static BinaryAlgebra getTernaryInnerProduct(BinaryAlgebra a1,
                                   BinaryAlgebra a2,
                                   BinaryAlgebra a3)
                                            throws java.lang.Exception

The scalar product (= real inner product) <.|.|.> of three algebras a1 and a2 and a3 is defined as = 1/12((ab)^*c + a(bc^*) + 5 permutations)

Parameters:

a1 - Element 1 of the algebra.

a2 - Element 2 of the algebra.

a3 - Element 3 of the algebra.

Returns:

Scalar product of a1 and a2 and a3.

Throws:

java.lang.Exception

getScalarProduct

public static BinaryAlgebra getScalarProduct(BinaryAlgebra a1,
                             BinaryAlgebra a2)
                                      throws java.lang.Exception

The scalar product (= real inner product) <.|.> of two algebra a1 and a2 is defined as = 1/2(a1a2*+a2a1*)

Parameters:

a1 - Element 1 of the algebra.

a2 - Element 2 of the algebra.

Returns:

Scalar product of a1 and a2.

Throws:

java.lang.Exception

getCommutatorWith

public BinaryAlgebra getCommutatorWith(BinaryAlgebra a,
                              boolean collect_terms)
                                throws java.lang.Exception

Calculates the commutator of an element a1 of the algebra with another one a2. which is defined as: [a1, a2] = a1*a2 - a2*a1

Parameters:

a2 - Second algebra.

collect_terms - true: Terms are added or subtracted if possible. false: Terms are not added or subtracted. (This is interesting if they contain functions and differential operators for which the order is relevant. The simplification procedure ignores the order).

Returns:

The commutator of the element of the algebra with another one.

Throws:

java.lang.Exception

getAntiCommutatorWith

public BinaryAlgebra getAntiCommutatorWith(BinaryAlgebra a)
                                    throws java.lang.Exception

Calculates the anti-commutator of the element a1 of the algebra with another one a2 which is defined as: {a1, a2} = a1*a2 + a2*a1

Parameters:

a - Second element of the algebra.

Returns:

The anti-commutator of an element of the algebra with another one.

Throws:

java.lang.Exception

getLeftCovariantDerivative

public BinaryAlgebra getLeftCovariantDerivative(BinaryAlgebra covariant_deriv_op,
                                       BinaryAlgebra h)
                                         throws java.lang.Exception

The covariant derivative of the hypernumber is calculated. This is carried out in 2 steps: i) The covariant derivative operator (which is also a hypernumber) is left-multiplied with the hypernumber. ii) The Leibniz rule for the partial derivatives is applied component-wise. THE CHAIN RULE OF DIFFERENTIATION IS NOT IMPLEMENTED AT THE MOMENT ! To simplify things, only 2 rules apply: i) A partial derivative takes the form d[] (only one variable is allowed. It only acts on functions, i.e. d[x]*x = 0 as x is treated as constant. ii) A function takes the form [, , ...] 'd' is not allowed for the function name as it is reserved for the partial derivative. Example: Left covariant derivative = (d[x]+A[y],1,1,x) hypernumber = (A[x,y,z]*B[X], A[x,y,z], a*b,0) The result of the action of the 1-st component of the covariant derivative on the first component of the hypernumber is: (d[x]+A[y])(A[x,y,z]*B[x])= d[x]A[x,y,z]*B[x]+ A[x,y,z]*d[x]B[x]+ A[y]*A[x,y,z]*B[x] The partial derivatives in the result only acts on the subsequent function. The '*' is omitted such that when the function is invoked a second time no ambiguity arises.

Parameters:

covariant_deriv_op - Covariant derivative operator.

h - Hypernumber upon which the covariant derivative is to act upon.

Returns:

Left covariant derivative of hypernumber.

Throws:

java.lang.Exception

rightMultiplyWith

public BinaryAlgebra rightMultiplyWith(BinaryAlgebra a)
                                throws java.lang.Exception

Throws:

java.lang.Exception

getProduct

public static BinaryAlgebra getProduct(BinaryAlgebra a,
                       BinaryAlgebra b)
                                throws java.lang.Exception

Throws:

java.lang.Exception

getSum

public static BinaryAlgebra getSum(BinaryAlgebra a1,
                   BinaryAlgebra a2)
                            throws java.lang.Exception

Throws:

java.lang.Exception

getDifference

public static BinaryAlgebra getDifference(BinaryAlgebra a1,
                          BinaryAlgebra a2)
                                   throws java.lang.Exception

Throws:

java.lang.Exception

isProjectionOperator

public boolean isProjectionOperator()
                             throws java.lang.Exception

Throws:

java.lang.Exception

getCommutatorWith

public BinaryAlgebra getCommutatorWith(BinaryAlgebra a)
                                throws java.lang.Exception

Throws:

java.lang.Exception

isCommutative

public static boolean isCommutative(BinaryAlgebra a1,
                    BinaryAlgebra a2)
                             throws java.lang.Exception

Throws:

java.lang.Exception

isOrthogonalTo

public boolean isOrthogonalTo(BinaryAlgebra a)
                       throws java.lang.Exception

Throws:

java.lang.Exception

isOrthogonal

public static boolean isOrthogonal(BinaryAlgebra a1,
                   BinaryAlgebra a2)
                            throws java.lang.Exception

Throws:

java.lang.Exception

getOuterProduct

public static BinaryAlgebra getOuterProduct(BinaryAlgebra a1,
                            BinaryAlgebra a2)
                                     throws java.lang.Exception

Throws:

java.lang.Exception

getGradedInnerProduct

public static BinaryAlgebra getGradedInnerProduct(BinaryAlgebra a1,
                                  BinaryAlgebra a2,
                                  boolean with_grade_0)
                                           throws java.lang.Exception

Throws:

java.lang.Exception

calculateBCHWith

public BinaryAlgebra calculateBCHWith(BinaryAlgebra a,
                             int from_order,
                             int to_order)
                               throws java.lang.Exception

The Baker Campbell Hausdorff formula (BCH) is calculated for a specified range of orders. It is given by: [A,B] = h1 + h2 + h3 + ... with h1, h2, ... being the orders that are given by h1 = A + B h2 = 1/2 * [A,B] h3 = 1/12 * [A,[A,B]] - 1/12 * [B,[A,B]] h4 = -1/24 * [B,[A,[A,B]]] and [,] the commutator. The implementation is currently up to order 4.

Parameters:

a - Algebra.

from_order - lowest order of order range. Possible values: 1,2,3,4.

to_order - largest order of order range. Possible values: 1,2,3,4.

Returns:

The result of the BCH formula for the given range of orders.

Throws:

java.lang.Exception

applyLeibnizRule

public static BinaryAlgebra applyLeibnizRule(BinaryAlgebra h)

TODO: meanwhile see 'getLeftCovariantDerivative ()'

Parameters:

h - Hypernumber

Returns: