public class Sedenion extends CayleyDicksonAlgebra
CD_LIKE0, CD_LIKE1, CD_LIKE10, CD_LIKE11, CD_LIKE12, CD_LIKE13, CD_LIKE14, CD_LIKE15, CD_LIKE16, CD_LIKE17, CD_LIKE18, CD_LIKE19, CD_LIKE2, CD_LIKE20, CD_LIKE21, CD_LIKE22, CD_LIKE23, CD_LIKE24, CD_LIKE25, CD_LIKE26, CD_LIKE27, CD_LIKE28, CD_LIKE29, CD_LIKE3, CD_LIKE30, CD_LIKE31, CD_LIKE4, CD_LIKE5, CD_LIKE6, CD_LIKE7, CD_LIKE8, CD_LIKE9, CLASSICAL, CLASSICAL2, COMPLEX, MAX_TYPE, MIN_TYPE, OCTONION, QUATERNION, SEDENION, TRIGINTADUONION, TWISTED, TWISTED2
ARG_NUM, CIRCULAR, COMP_NUM, components, HYPERBOLIC, ZERO_PRECISSION
Constructor and Description |
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Sedenion()
Constructors and the like *
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Sedenion(Sedenion s) |
Sedenion(java.lang.String components) |
Modifier and Type | Method and Description |
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Sedenion |
add(Sedenion sedenion) |
Sedenion |
conjugate()
Arithmetics *
|
static void |
doCayleyDicksonDoubling(java.lang.String signature) |
static void |
doCayleyDicksonDoubling(java.lang.String signatures,
java.lang.String vector_basis) |
static void |
doCayleyDicksonDoubling(java.lang.String signatures,
java.lang.String vector_basis,
int type)
A sequence of 4 Cayley-Dickson doublings is carried out starting from the real numbers and resulting
in the sedenion algebra.
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static Sedenion |
getAntiAssociator(Sedenion s1,
Sedenion s2,
Sedenion s3) |
static Sedenion |
getAntiCommutator(Sedenion s1,
Sedenion s2) |
static Sedenion |
getAssociationType(int nr,
Sedenion s1,
Sedenion s2,
Sedenion s3,
Sedenion s4) |
static Sedenion |
getAssociationType(int nr,
Sedenion s1,
Sedenion s2,
Sedenion s3,
Sedenion s4,
Sedenion s5) |
static Sedenion |
getAssociationTypeCommutator(int nr,
Sedenion s1,
Sedenion s2,
Sedenion s3,
Sedenion s4) |
static Sedenion |
getAssociationTypeCommutator(int nr,
Sedenion s1,
Sedenion s2,
Sedenion s3,
Sedenion s4,
Sedenion s5) |
static Sedenion |
getAssociator(Sedenion s1,
Sedenion s2,
Sedenion s3) |
java.lang.String[] |
getBasis()
Multiplication table and basis *
|
java.lang.String[] |
getBivectorComponents() |
java.lang.String |
getBivectorPartAsString() |
java.lang.String |
getBivectorPartAsString(boolean with_blanks) |
Sedenion |
getClone() |
static Sedenion |
getCommutator(Sedenion s1,
Sedenion s2) |
static Sedenion |
getCommutator(Sedenion s1,
Sedenion s2,
boolean collect_terms) |
static Sedenion |
getCommutatorJacobian(Sedenion s1,
Sedenion s2,
Sedenion s3) |
static Sedenion |
getCommutatorSaglian(Sedenion s1,
Sedenion s2,
Sedenion s3,
Sedenion s4) |
static Sedenion |
getConjugate(Sedenion s) |
java.lang.String |
getDiagonalVectorProducts() |
static Sedenion |
getDifference(Sedenion s1,
Sedenion s2) |
java.lang.String[] |
getEvenGradeComponents() |
java.lang.String |
getEvenGradePartAsString() |
java.lang.String |
getEvenGradePartAsString(boolean with_blanks) |
static Sedenion |
getFirstHetztelPeresian(Sedenion s1,
Sedenion s2,
Sedenion s3,
Sedenion s4,
Sedenion s5) |
Sedenion |
getGrade(int grade) |
static java.util.TreeMap<java.lang.Integer,java.lang.Integer> |
getGrades() |
java.lang.String |
getGrading() |
static Sedenion |
getImaginaryPart(Sedenion s) |
static Sedenion |
getInverse(Sedenion s) |
static Sedenion |
getJacobian(Sedenion s1,
Sedenion s2,
Sedenion s3) |
static Sedenion |
getJordanAssociator(Sedenion s1,
Sedenion s2,
Sedenion s3) |
java.lang.String[][] |
getMTab()
Multiplication table and basis *
|
static MultiplicationTable |
getMultiplicationTable() |
static java.lang.String[][] |
getMultiplicationTableAsArray() |
static java.lang.String |
getMultTableAsHtml() |
java.util.Collection<java.lang.String> |
getNCycleGrades(int ncycle)
Intended for usage in context with multiplication tables generated by Cayley Dickson
doubling.
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static Sedenion |
getNegated(Sedenion s) |
BinaryProductStructure |
getNewInstance() |
static java.lang.String[][] |
getNormedMultiplicationTable() |
static java.lang.String |
getNormedMultiplicationtableAsHtml() |
int |
getNumberOfComponents()
Components *
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int |
getNumberOfGrades() |
static Sedenion |
getNumericalDifference(Sedenion s,
Sedenion t) |
static Sedenion |
getNumericalProduct(Sedenion s,
Sedenion t)
Multiplies two sedenions numerically.
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static Sedenion |
getNumericalSum(Sedenion s,
Sedenion t) |
static java.util.HashSet<OctonarySubAlgebra> |
getOctonarySubalgebras() |
java.util.LinkedList<MultiplicationTable> |
getOrder4Subalgebras() |
java.util.LinkedList<MultiplicationTable> |
getOrder8Subalgebras() |
static Sedenion |
getProduct(Sedenion s1,
Sedenion s2) |
static Sedenion |
getScalarProduct(Sedenion s1,
Sedenion s2) |
static Sedenion |
getSecondHetztelPeresian(Sedenion s1,
Sedenion s2,
Sedenion s3,
Sedenion s4) |
static java.lang.String |
getSignature() |
static java.util.HashSet<java.lang.String> |
getSubalgebras(int order) |
static java.util.HashSet<int[]> |
getSubalgebrasUnsigned(int order) |
java.lang.String[] |
getSubBasis()
CayleyDicksonAlgebra, based on one of the following doubling formulas:
Doubling_type:
CLASSICAL e = (a,b)(c,d) = (ac-ld*b, da+bc*)
CLASSICAL2 e = (a,b)(c,d) = (ac-ldb*, a*d+cb)
TWISTED e = (a,b)(c,d) = (ac-lb*d, da*+bc)
TWISTED2 e = (a,b)(c,d) = (ac-lbd*, ad+c*b)
WARREN_SMITH_16_ON
CD_LIKE0 e = (a,b)(c,d) = (ca-lb*d, da*+bc)
CD_LIKE1 e = (a,b)(c,d) = (ca-ldb*, da*+bc)
CD_LIKE2 e = (a,b)(c,d) = (ca-lb*d, a*d+cb)
CD_LIKE3 e = (a,b)(c,d) = (ca-ldb*, a*d+cb)
CD_LIKE5 e = (a,b)(c,d) = (ac-ldb*, da*+bc)
CD_LIKE6 e = (a,b)(c,d) = (ac-lb*d, a*d+cb)
CD_LIKE7 (= CLASSICAL2) e = (a,b)(c,d) = (ac-ldb*, a*d+cb)
CD_LIKE8 e = (a,b)(c,d) = (ca-lbd*, da*+bc)
CD_LIKE9 e = (a,b)(c,d) = (ca-ld*b, da*+bc)
CD_LIKE10 e = (a,b)(c,d) = (ca-lbd*, a*d+cb)
CD_LIKE11 e = (a,b)(c,d) = (ca-ld*b, a*d+cb)
CD_LIKE12 e = (a,b)(c,d) = (ac-lbd*, da*+bc)
CD_LIKE13 e = (a,b)(c,d) = (ac-ld*b, da*+bc)
CD_LIKE14 e = (a,b)(c,d) = (ac-lbd*, a*d+cb)
CD_LIKE15 e = (a,b)(c,d) = (ac-ld*b, a*d+cb)
CD_LIKE16 e = (a,b)(c,d) = (ca-lb*d, ad+c*b)
CD_LIKE17 e = (a,b)(c,d) = (ca-ldb*, ad+c*b)
CD_LIKE18 e = (a,b)(c,d) = (ca-lb*d, da+bc*)
CD_LIKE19 e = (a,b)(c,d) = (ca-ldb*, da+bc*)
CD_LIKE20 e = (a,b)(c,d) = (ac-lb*d, ad+c*b)
CD_LIKE21 e = (a,b)(c,d) = (ac-ldb*, ad+c*b)
CD_LIKE22 e = (a,b)(c,d) = (ac-lb*d, da+bc*)
CD_LIKE23 e = (a,b)(c,d) = (ac-ldb*, da+bc*)
CD_LIKE24 e = (a,b)(c,d) = (ca-lbd*, ad+c*b)
CD_LIKE25 e = (a,b)(c,d) = (ca-ld*b, ad+c*b)
CD_LIKE26 e = (a,b)(c,d) = (ca-lbd*, da+bc*)
CD_LIKE27 e = (a,b)(c,d) = (ca-ld*b, da+bc*)
CD_LIKE28 (= TWISTED_CAYLEY_DICKSON2) e = (a,b)(c,d) = (ac-lbd*, ad+c*b)
CD_LIKE29 e = (a,b)(c,d) = (ac-ld*b, ad+c*b)
CD_LIKE30 e = (a,b)(c,d) = (ac-lbd*, da+bc*)
CD_LIKE31 (= CLASSICAL) e = (a,b)(c,d) = (ac-ld*b, da+bc*)
with
'e' defining the set of basis elements of the new algebra.
'*' being conjugation.
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static Sedenion |
getSum(Sedenion s1,
Sedenion s2) |
static Sedenion |
getTernaryInnerProduct(Sedenion s1,
Sedenion s2,
Sedenion s3) |
java.lang.String[] |
getTrivectorComponents() |
java.lang.String |
getTrivectorPartAsString() |
java.lang.String |
getTrivectorPartAsString(boolean with_blanks) |
java.lang.String[] |
getVectorComponents() |
java.lang.String |
getVectorPartAsString() |
java.lang.String |
getVectorPartAsString(boolean with_blanks) |
java.lang.String |
getVectorSignature() |
boolean |
isCayleyDicksonAlgebra() |
Sedenion |
multiplyWithScalar(java.lang.String scalar)
Multiplication with a scalar.
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Sedenion |
negate()
Negates the hypernumber: h --> -h.
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Sedenion |
rightMultiplyWith(BinaryProductStructure s) |
void |
setBiVectorComponents(java.lang.String bivector)
Sets the 6 bivector components.
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void |
setBiVectorComponentsExclusively(java.lang.String bivector)
Sets the 6 bivector components.
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void |
setDefaultTable() |
void |
setEvenGradeComponents(java.lang.String comps)
Sets the 8 even grade components.
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void |
setEvenGradeComponentsExclusively(java.lang.String comps)
Sets the 8 even grade components.
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static void |
setMultiplicationTable(java.lang.String[][] mult_table) |
void |
setTriVectorComponents(java.lang.String trivector)
Sets the 4 trivector components.
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void |
setTriVectorComponentsExclusively(java.lang.String trivector)
Sets the 4 trivector components.
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void |
setVectorComponents(java.lang.String vector)
Sets the 4 vector components.
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void |
setVectorComponents(java.lang.String[] vector_comps) |
void |
setVectorComponentsExclusively(java.lang.String vector)
Sets the 4 vector components.
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Sedenion |
square() |
Sedenion |
subtract(Sedenion sedenion) |
getCDTypeAsString
add, add, applyLeibnizRule, calculateBCHWith, getAntiCommutatorWith, getCommutatorWith, getCommutatorWith, getComponentsOfGrade, getDifference, getGradedInnerProduct, getIntersection, getLeftCovariantDerivative, getNumberOfComponents, getOuterProduct, getProduct, getScalarProduct, getSum, getTernaryInnerProduct, isCommutative, isEqual, isOrthogonal, isOrthogonalTo, isProjectionOperator, rightMultiplyWith, subtract
asString, asString, collectTerms, compressComponents, dual, getBasisElementAsString, getBasisElementsAsString, getClosedPairsAsString, getComponent, getComponents, getComponentsAsString, getComponentsAsString, getConjugate, getDiagonalProducts, getDual, getExpressionAsList, getFormattedMultTable, getHermitianConjugate, getImaginaryPart, getInstance, getMultiplicativeOrder, getMultiplicativeOrder, getMultTableDiagonal, getNegated, getNextBitmap, getNonzeroComponentsNumbered, getNonzeroComponentsNumbered, getNormedMultiplicationTable, getNormSquared, getNormSquaredAsValue, getNumberOfClosedElements, getNumberOfClosedPairs, getNumberOfNonzeroComponents, getNumericalInverse, getProduct, getProductWithScalar, getPseudoScalarComponent, getRandomClosedPair, getRandomNonClosedPair, getScalarComponent, getSubalgebras, getSubalgebrasAsString, getSubalgebrasSignatures, getTrace, getTraceRespDeterminant, getVectorDerivative, getVectorProduct, isAutomorphism, isCayleyDicksonLoop, isComponentZero, isLinearlyDependent, isNumerical, isNumericalityDetected, isProductClosed, isZero, resetComponents, rightMultiplyWith, setBasis, setComponent, setComponentAsBasisNames, setComponentExclusively, setComponents, setComponents, setIndexedComponents, setMultiplicationTable, setNumerical, setNumerical, setPseudoScalarComponent, setRandomComponents, setRandomComponents, setRandomComponents, setRandomComponentsAll, setRepresentation, setScalarComponent, setSignTable, simplifyNumericalFactors, simplifyNumericalSummands, simplifyPlusMinus
public Sedenion()
public Sedenion(Sedenion s)
public Sedenion(java.lang.String components) throws java.lang.Exception
java.lang.Exception
public BinaryProductStructure getNewInstance()
getNewInstance
in class BinaryProductStructure
public Sedenion getClone()
getClone
in class BinaryAlgebra
public java.lang.String[] getBasis()
getBasis
in class BinaryProductStructure
public static java.lang.String[][] getMultiplicationTableAsArray()
public static MultiplicationTable getMultiplicationTable()
public static void setMultiplicationTable(java.lang.String[][] mult_table)
public java.lang.String[][] getMTab()
BinaryProductStructure
getMTab
in class BinaryProductStructure
public void setDefaultTable()
public static java.lang.String getMultTableAsHtml()
public static java.lang.String[][] getNormedMultiplicationTable()
public static java.lang.String getNormedMultiplicationtableAsHtml()
public static void doCayleyDicksonDoubling(java.lang.String signatures, java.lang.String vector_basis, int type) throws java.lang.Exception
signatures
- Comma delimited string of 4 signatures with "+" specifying a non-split and "-" a
split doubling. (e.g. "+,-,+,+").vector_basis
- java.lang.Exception
public static void doCayleyDicksonDoubling(java.lang.String signature) throws java.lang.Exception
java.lang.Exception
public static void doCayleyDicksonDoubling(java.lang.String signatures, java.lang.String vector_basis) throws java.lang.Exception
java.lang.Exception
public int getNumberOfComponents()
getNumberOfComponents
in class BinaryProductStructure
public static Sedenion getImaginaryPart(Sedenion s) throws java.lang.Exception
java.lang.Exception
public void setVectorComponents(java.lang.String vector) throws java.lang.Exception
vector
- 4 vector components, formatting: comma seperated, e.g. "x1,x2,x3,t".java.lang.Exception
public void setVectorComponents(java.lang.String[] vector_comps) throws java.lang.Exception
java.lang.Exception
public void setVectorComponentsExclusively(java.lang.String vector) throws java.lang.Exception
vector
- 4 vector components, formatting: comma seperated, e.g. "x1,x2,x3,t".java.lang.Exception
public java.lang.String[] getVectorComponents()
public java.lang.String getVectorPartAsString(boolean with_blanks)
no_blanks
- 'true' No blanks between summands.
'false' A blank between summands.public java.lang.String getVectorPartAsString()
public void setBiVectorComponents(java.lang.String bivector) throws java.lang.Exception
vector
- 6 bivector components, formatting: comma seperated, e.g. "a,b,c,d".java.lang.Exception
public void setBiVectorComponentsExclusively(java.lang.String bivector) throws java.lang.Exception
vector
- 4 vector components, formatting: comma seperated, e.g. "a,b,c,d".java.lang.Exception
public java.lang.String[] getBivectorComponents()
public java.lang.String getBivectorPartAsString(boolean with_blanks)
no_blanks
- 'true' No blanks between summands.
'false' A blank between summands.public void setTriVectorComponents(java.lang.String trivector) throws java.lang.Exception
vector
- 4 trivector components, formatting: comma seperated, e.g. "a,b,c,d".java.lang.Exception
public void setTriVectorComponentsExclusively(java.lang.String trivector) throws java.lang.Exception
vector
- 4 vector components, formatting: comma seperated, e.g. "a,b,c,d".java.lang.Exception
public java.lang.String getBivectorPartAsString()
public java.lang.String[] getTrivectorComponents()
public java.lang.String getTrivectorPartAsString(boolean with_blanks)
no_blanks
- 'true' No blanks between summands.
'false' A blank between summands.public java.lang.String getTrivectorPartAsString()
public void setEvenGradeComponents(java.lang.String comps) throws java.lang.Exception
spinor
- 8 even grade components, formatting: comma seperated,
e.g. "s,p1,p2,p3,p4,p5,p6,w".java.lang.Exception
public void setEvenGradeComponentsExclusively(java.lang.String comps) throws java.lang.Exception
spinor
- 8 even grade components, formatting: comma seperated,
e.g. "s,p1,p2,p3,p4,p5,p6,w".java.lang.Exception
public java.lang.String[] getEvenGradeComponents()
public java.lang.String getEvenGradePartAsString(boolean with_blanks)
no_blanks
- 'true' No blanks between summands.
'false' A blank between summands.public java.lang.String getEvenGradePartAsString()
public java.util.LinkedList<MultiplicationTable> getOrder4Subalgebras() throws java.lang.Exception
java.lang.Exception
public java.util.LinkedList<MultiplicationTable> getOrder8Subalgebras() throws java.lang.Exception
java.lang.Exception
public Sedenion conjugate() throws java.lang.Exception
conjugate
in class BinaryProductStructure
java.lang.Exception
public Sedenion square() throws java.lang.Exception
square
in class BinaryProductStructure
java.lang.Exception
public Sedenion rightMultiplyWith(BinaryProductStructure s) throws java.lang.Exception
rightMultiplyWith
in class BinaryProductStructure
java.lang.Exception
public Sedenion negate() throws java.lang.Exception
BinaryProductStructure
negate
in class BinaryProductStructure
java.lang.Exception
public static Sedenion getNegated(Sedenion s) throws java.lang.Exception
java.lang.Exception
public static Sedenion getSum(Sedenion s1, Sedenion s2) throws java.lang.Exception
java.lang.Exception
public static Sedenion getDifference(Sedenion s1, Sedenion s2) throws java.lang.Exception
java.lang.Exception
public Sedenion subtract(Sedenion sedenion) throws java.lang.Exception
java.lang.Exception
public Sedenion multiplyWithScalar(java.lang.String scalar) throws java.lang.Exception
BinaryProductStructure
multiplyWithScalar
in class BinaryProductStructure
scalar
- Scalar.java.lang.Exception
public static Sedenion getInverse(Sedenion s) throws java.lang.Exception
java.lang.Exception
public static Sedenion getConjugate(Sedenion s) throws java.lang.Exception
java.lang.Exception
public static Sedenion getProduct(Sedenion s1, Sedenion s2) throws java.lang.Exception
java.lang.Exception
public static Sedenion getNumericalProduct(Sedenion s, Sedenion t) throws java.lang.Exception
s1
- Sedenion 1 with numerical components.s2
- Sedenion 2 with numerical components.java.lang.Exception
public static Sedenion getNumericalSum(Sedenion s, Sedenion t) throws java.lang.Exception
java.lang.Exception
public static Sedenion getNumericalDifference(Sedenion s, Sedenion t) throws java.lang.Exception
java.lang.Exception
public static Sedenion getCommutator(Sedenion s1, Sedenion s2) throws java.lang.Exception
java.lang.Exception
public static Sedenion getCommutator(Sedenion s1, Sedenion s2, boolean collect_terms) throws java.lang.Exception
java.lang.Exception
public static Sedenion getAntiCommutator(Sedenion s1, Sedenion s2) throws java.lang.Exception
java.lang.Exception
public static Sedenion getAssociator(Sedenion s1, Sedenion s2, Sedenion s3) throws java.lang.Exception
java.lang.Exception
public static Sedenion getJordanAssociator(Sedenion s1, Sedenion s2, Sedenion s3) throws java.lang.Exception
java.lang.Exception
public static Sedenion getAntiAssociator(Sedenion s1, Sedenion s2, Sedenion s3) throws java.lang.Exception
java.lang.Exception
public static Sedenion getScalarProduct(Sedenion s1, Sedenion s2) throws java.lang.Exception
java.lang.Exception
public static Sedenion getTernaryInnerProduct(Sedenion s1, Sedenion s2, Sedenion s3) throws java.lang.Exception
java.lang.Exception
public static Sedenion getJacobian(Sedenion s1, Sedenion s2, Sedenion s3) throws java.lang.Exception
java.lang.Exception
public static Sedenion getCommutatorJacobian(Sedenion s1, Sedenion s2, Sedenion s3) throws java.lang.Exception
java.lang.Exception
public static Sedenion getCommutatorSaglian(Sedenion s1, Sedenion s2, Sedenion s3, Sedenion s4) throws java.lang.Exception
java.lang.Exception
public static Sedenion getSecondHetztelPeresian(Sedenion s1, Sedenion s2, Sedenion s3, Sedenion s4) throws java.lang.Exception
java.lang.Exception
public static Sedenion getFirstHetztelPeresian(Sedenion s1, Sedenion s2, Sedenion s3, Sedenion s4, Sedenion s5) throws java.lang.Exception
java.lang.Exception
public static Sedenion getAssociationType(int nr, Sedenion s1, Sedenion s2, Sedenion s3, Sedenion s4) throws java.lang.Exception
java.lang.Exception
public static Sedenion getAssociationTypeCommutator(int nr, Sedenion s1, Sedenion s2, Sedenion s3, Sedenion s4) throws java.lang.Exception
java.lang.Exception
public static Sedenion getAssociationType(int nr, Sedenion s1, Sedenion s2, Sedenion s3, Sedenion s4, Sedenion s5) throws java.lang.Exception
java.lang.Exception
public static Sedenion getAssociationTypeCommutator(int nr, Sedenion s1, Sedenion s2, Sedenion s3, Sedenion s4, Sedenion s5) throws java.lang.Exception
java.lang.Exception
public java.util.Collection<java.lang.String> getNCycleGrades(int ncycle) throws java.lang.Exception
ncycle
- N cycle.java.lang.Exception
public java.lang.String getDiagonalVectorProducts()
public java.lang.String getVectorSignature()
public int getNumberOfGrades() throws java.lang.Exception
getNumberOfGrades
in class BinaryAlgebra
java.lang.Exception
public Sedenion getGrade(int grade) throws java.lang.Exception
getGrade
in class BinaryAlgebra
java.lang.Exception
public java.lang.String getGrading() throws java.lang.Exception
java.lang.Exception
public static java.util.TreeMap<java.lang.Integer,java.lang.Integer> getGrades() throws java.lang.Exception
java.lang.Exception
public static java.util.HashSet<java.lang.String> getSubalgebras(int order) throws java.lang.Exception
java.lang.Exception
public static java.util.HashSet<OctonarySubAlgebra> getOctonarySubalgebras() throws java.lang.Exception
java.lang.Exception
public static java.util.HashSet<int[]> getSubalgebrasUnsigned(int order) throws java.lang.Exception
java.lang.Exception
public static java.lang.String getSignature()
public boolean isCayleyDicksonAlgebra() throws java.lang.Exception
isCayleyDicksonAlgebra
in class BinaryAlgebra
java.lang.Exception
public java.lang.String[] getSubBasis() throws java.lang.Exception
CayleyDicksonAlgebra
java.lang.Exception